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. It seems to be very difficult, especially in
19721
ON CHARACTERISTIC CLASSES OF T-FOL1ATIONS
1043
the real case, to find foliations which distinguish between the different classes defined by
, g _ 1 = p _ 1 + r~l andf is continuously differentiable and has compact support, we have \\(ABBA)^f\\, , we can pass to the limit under the integral signs above. Thus, denoting by H\{z) and H2(z) the analytic functions in 0\ and O z , respectively, such that H i ' = F / G i , H 2 ' = F2G2', and tending to zero at infinity, we find that L{f,g) 0 conformally onto 0\ and such that <£(z) ^ z a s z - ^ °°. Because the boundary of Oi is a C™ curve, 0{z) and <£'(z) extend continuously to R(z) > Oand 0'(z) does not vanish in R(z) > 0 and is bounded there. Furthermore, the vector iMcM-1 = or = <*>. Our method, however, cannot possibly yield this result. Indeed, the inequality ||B(0)|| >cM,c> 0. as in the last part of the proof of Theorem 1. For additional references see refs. 2-4. 2. Related integral operators (actually a trigonometric polynomial) whose mean value over (0, 2 n) is zero. Several examples of kernels of this type could be considered, but we shall now state the problem in a more general form. (£)|. Then we have the formula ip(t) = i/3t+ f (eitx - 1 - /x(E') < /i(£) (i.e. if E' is a "nonexpanded" image of E, then its measure is not greater than that of E). Then we say that p is a /c-dimensional measure. At the time when the paper referred to appeared, many different A;-dimensional measures had been constructed (lengths of curves, Hausdorff measures, Caratheodory measures, and so on), satisfying the axioms (l)-(4). In this paper Kolmogorov obtained the following fundamental result: For any natural number k < n there are two special k-dimensional measures, the maximal measure JIk and the minimal measure [ik such that any k-dimensional measure /i of any Suslin set E lies between ~pk(E) and fik: £fc (E) < fi(E) < ~p,k (E). Both measures /xfc and ~pk are defined in a completely natural way. Let us turn to descriptive set theory. We shall not discuss it in detail. This field used to be at the epicenter of the interests of the Moscow Mathematical School, but relatively few are interested in it now. In 1916 P. S. Alexsandroff in Russia and F. Hausdorff in Germany independently solved the continuum problem for Borel sets, proving that any Borel set is either countable or has the power of the continuum. For this Alexsandroff and Hausdorff used the property that any Borel set can be obtained by means of some constructive procedure. A clear description of this procedure was given by Suslin: he called it A-operation. Given a sequence of sets £ = {ESl„,Sn}, numbered "by processions" { s i , . . . ,sn},Si £ N, we construct the set A{E) according to the following rule: 21. Since, in view of (5), (6) 0) is the number of negative eigenvalues of system (5). REFERENCES 1. 2. 3. 4. 5. 6. 0 (0 r\x, i}„Raumes, namlich einem solchen, in dem | 9 = 0 gilt. Dies Gebiet enthalt das Ebenenstiick der Ebene f3 = »?9 = 0 im Inneren. Nunmehr geht man von den samtlichen Geraden £x = konst., rj2 = konst. des soeben betrachteten dreidimensionalen Gebietes | ? = 0 des f1} f4, i]1, rj^Raumes aus und setzt die dort bekannten Funktionen (In diesen Gleichungen wird in der komplexen Differenzierbarkeit der aik zum einzigen wesentlichen Male die analytische Abhangigkeit der Differentialgleichung F = 0 von ihren Argumenten benutzt.) Weiter ist beispielsweise 'Al 0, continuous in y ^ 0. If we take the integral J* \ d\ | + \dp\ + | dv \, extended over the rc-axis, a parallel to the .-c-axis, or over the y-axis, it remains by Lemmas 1, 2, 3 bounded above by 6L (0). We now define the continuous function a(x, y) = y Id\I + Idy.I -+- \dv\ by extending the integral from (0, 0) to (0, y) to (x, y) over straight segments. For sufficiently small h > 0, the square q of the corners — h, +h, h + 2ih, —h + 2ih has the property that within q,
Furthermore, (AB — BA)f has first-order derivatives
(AB -
BA)f
\aXj
< cf\P,
(b)
where, again, c is independent off. THEOREM 2. Let h(x) be homogeneous of degree —n — 1 and locally integrable in\x\ > 0. Let b(x) have first-order derivatives in U, 1
Hx-
y)[b(x) -
b(y)]f(y)dy.
Ct maps L" continuously into L" and \\C(J)\\t < c|jgrad &||r||/||j, S \h(x)\dv, where the integral is extended over \x\ = I, dv denotes the surface area of \x\ = 1, and c depends on p and r but not on e. Furthermore, as e tends to zero Cf(f) 'converges in norm in L". A similar result holds if h(x) is odd provided that it belongs locally to L log + L in \x\ > 0 and that the functions Xj h(x), j = 1,2, ..., n, have mean value zero on \x\ = 1. This, however, will not be proved in the present note. THEOREM 3. Let F(t + is) be analytic in s > 0 and belong to Hv, 0 < p < ». Let S(F)(t) = {f x(t — u,s)\F'(u + is) \ 2du ds ]v', where x(t,s) is the characteristic 1092
103 VOL. 53, 1965
MATHEMATICS: A. P. CALDER6N
1093
function of the set s > 0, \t\ < s. Then there exist two positive constants C\ and c2 depending on p only, such that ci\\F(t)\\v < \\S(F)\\„ < cz\\F(t)\\p, where F(t) = Km «-M)
F(t + is). The novelty in the preceding statement is the first inequality for p < 1. A similar result for the function g of Littlewood and Paley when F has no zeros was proved by T. M. Flett (ref. 3), whose method we borrow partially. Actually, only the case p ^ 1 will be needed in this note, but its proof is no less laborious than that of the general case. Proof of Theorem 8: We will assume first that F(t + is) is analytic in s ^ 0 and that \ F\ (i2 + s2)k-* 0 a,s (t2 + s2) — » for every k > 0. Then, of course, F belongs to H" for every p > 0. We introduce now some notation. For a function G defined on the real line we write
M„(G) = J
G" dt\ ,
p > 0.
If G is also denned in the upper half-plane, we write m(G) = sup x(t - u,s)\G(u,s)\,
S(G) = [fx(t
- M,s)|grad G\ Hu ds]''*,
where x(t,$) is the characteristic function of the set s > 0, 11\ < s. By integration we obtain M22[S(G) ] = 2 J" s\ grad G\ 2dt ds. Now if 5 is any positive number, we set G = 1^1*, then a simple calculation gives A(G2) = 4|grad(?j 2 and an application of Green's formula yields M22(G) = 4 f
(0)
4
s\ grad Q\ 2dt ds = 2 M22[S(G)].
(1)
On account of the definition of G and the analyticity of F, we have the following well-known inequality Mv[m(G)} < c MP(G),
0 < p < «>.
(2)
Now let p J? 1, then S(G")2 = f
x(t
-
u , s ) | p G p - ' grad G\2du ds <
p2m{G)2p-2S{G)2,
that is, S(G") < pm(Gy~lS{G),
1 < p < c°.
Now let ccfi > 0, 0 < cr < 1, a<j + /9(1 -
Then
S(G)2 = f
f
x(t
~ u,s)|grad G\ 2du ds = c-fp-w-*)
(3)
( x |grad Ga\ 2)° {x\^&AG^\2Y-'duds,
whence from Holder's inequality we obtain
S(G) < [ i 5 ( G - ) ] ^ SiG^Y".
(4)
104 1094
MATHEMATICS:
A. P. CALDER6N
PROC. N. A. S.
Let us assume now that we have the inequality cMT(G)>MT[S(G)] for some r, r > 0. Let 0 < q < r and p = r/q.
(5) ilP
Then (3) applied to G
S(G) < pro(G1'1,),,~1<S(G!1/,,) =
gives
pm(G)i'-wS(&l»),
whence, applying Holder's inequality, we get < p*MilmWv-vi'SiG1")*]
Mt'[S(G)]
< p*M r/f [5(G l ")»]M r /( f ^,)[m(G)«»-»'»] = p« Mr<'[S(G1'p)]Mq[m(G)]
and from the last expression, (2), and (5) applied to GllP it follows that Ma<[S(G)] < cpq Mr«[G1»']Mg<'ip-»'p(G) = cp* M9"p(G)M
M,[S(G)] < cgMQ(G).
(6)
On account of (1), (5) holds with r = 2. Hence the preceding inequality holds for 0 < q < 2. Now we will show that (6) holds for 0 < q < a>. Since (5) implies (6) with q < r, it is enough to show that (6) holds for q > 4. Let h(t) > 0 be any bounded function with compact support. Then S(G)2hdt = |
f J
— at
h(t) f
x(t
~ M,s)jgrad G\Hu ds dt
%J — co
/
+ "> h(t)x(t - u,s) dt du ds.
Now we observe that if P(t,s) denotes the Poisson kernel for the half-plane, then x(M) — c sP(.t,$) a n d consequently I m/
h(t)x(t - u,s) dt < c\ — CD
«/
h(t)sP(t - u,s)dt
where H(t,s) is the Poisson integral of h(t).
r.
H(u,s),
— CD
Thus,
•+0O
S(G)2h dt < c f
| grad G\ 2s H(t,s)dt ds.
Now, from (0) we have A(G*H) = HAG2 + 2(grad G2) • (grad H) = 4H\ grad G| 2 + 2G(grad G) • (grad H) ^ 4H| grad G\2 - 2G\ grad G\ | grad H\ and
r. •+
S(G)2h dt<-f
sA(G*H)dt ds + '-\f sG\ grad G\ | grad H\ dt ds
105 VOL. 53, 1965
MATHEMATICS:
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1095
and applying Green's formula to the first term on the right 4 G2h
S(G)*hdt< — oo
rr U
— oo
+ - I 4
<
- f
dt S x(t — u,s) G|grad G\ (grad H\du ds
%/ — oo
G2/i d/ + - f
4 J-m
m(G) S(G) S(H)dt.
4 J -co
Now we set p = q/(q — 1) and apply the three-term Holder inequality with exponents 2q, 2q, p to the preceding integrals and get 4 f
S(G)*hdt < cM2q*(G)Mp(h) + cM2q[m(G)]M2q[S(G)]Mp[S(H)].
(7)
• / — oo
Since H is harmonic and 1 < p < m, we have MP[S(H)] < cpMP(h), and since 4 < g < o>; we also have M2q[m(G)] < cqM2q(G). Substituting in the preceding inequality, setting Mp{h) = 1, and taking the supremum of the left-hand side over all such h, we find that Mq[S(Gy] = M2q2[5(G)] < c M2q(G)[M2q(G) + M2qS(G)], and this implies that M2q[S(G)] < c' M2q(G) provided that M2q[S(G)] < » . To see that this is the case we observe that since m(G) is bounded, (7) holds with M„[m(G)] replacing M2q[m(G)] and Mq[S(G)] replacing M2q[S(G)] and from this, arguing as above, we obtain M2tnS(G)]
< cM2q\G)
+
cM„[m(G)]Mq[S(G)].
Since the right-hand side is finite for q = 2, it follows by induction that the lefthand side is finite for arbitrarily large q and hence for all q > 2. Thus (6) is established for 0 < q < °°. Now we prove the converse inequality. Let q > 0. Then (1) and (4) give Mq«(G) = ilf,*(G»/*) = 2M2iS(G"2)
< cM,^^")1'^^")'"-')],
where a = 2q/(q + 2), 0 = 2/q,
Mq<(G) < But
= M(5+2>/s2"[S((?""2)]
M^+1)lq[S{G"^y} MQ+1[S(G)W-^]
= M t 2 <'-" [5(G)].
Applying (6) to the right-hand side of the first of the preceding identities, and observing that M(q+2)iq[Ga"n} = Mqaqn(G), substitution in the preceding inequality yields Mq"{G) <
cqMqa°<(G)Mq™-°)[S(G)].
Since q — aaq = 2(1 —
(8)
106 1096
MATHEMATICS:
A. P. CALDER6N
PKOC. N. A. S.
To obtain (6) and (8) for F we set G = \F\ and observe that |grad G| = | F ' | . Finally, we must remove the conditions we imposed on F at the beginning of the proof. If F(z), z = t + is, is analytic in the upper half-plane and belongs to H", then F{z + i/n) = F„(z) is bounded there. Let now em(z) = exp( — z"m), where 0 < a < 1/4 and arg(z") is between 0 and ir/4. Then a simple calculation shows that
1
\em'(t + is)\2dtds
< c2a,
where c is independent of m. Consequently, S(em)2 < c2a. inequalities can be readily verified: S(Fnemy
< 2{S(Fn)2 +
< 2{S{Fny + m(FnyS(emy] S(Fn emY < 2"[S(Fny
Now, the following c*m(Fnya]
2
+
c"m{Fnya"i ].
Integrating we get M/[S(Fnen)] Since M/(Fn)
= lim M/(Fn
< 2>[M,'[S(Fn)]
+
em) and by (8), M/(Fn
Ca"i2M/[m{Fn)]}. em) < c / M/[S(Fn
em)\ from
m
the inequality above we obtain M/{Fn)
+ c p f t "/W/[m(F B )]],
< c / 2"[M/[S(Fri)}
and letting a tend to zero Mp{Fn) <
cp2Mp[S(Fn)].
Finally, as n tends to infinity, Mp(Fn) converges to MP(F) and S(Fn) increases and converges to S(F). Thus we can pass to the limit in the preceding inequality and obtain half of the desired result. To obtain the other half we observe that, since (^n em)' converges to Fn', we have S(Fn) = limm inf S(Fn em). Thus from (6) applied to Fn em and Fatou's lemma we get M,[S(Fn)]
and a passage to the limit completes the proof of the theorem. Proof of Theorem 2: We begin with the one-dimensional case. becomes simply x~2, and the proof reduces to estimate f
Ct(J)gdx = f
(x-
y)-2[b{x) - b(y)]g(x)f(y)dx
Here h{x)
dy
in terms of the norms of/, g, and b'. For this purpose there is no loss of generality in assuming that these functions are infinitely differentiable and have compact support. Let e(x) be the characteristic function of x > 0 and x(x) that of | x\ > e. Then b(x) = 1
e(x -
t)b'{t)dt,
and substituting, the integral above becomes
X
b'(t) f
(x - y)-2x(\x
- y\)[e(x - t) - e(y - t)]g(x)f(y)dx dy dt
107 VOL. 53, 1965
MATHEMATICS:
A. P. CALDER6N
1097
and the problem reduces to studying the class of the function represented by the inner integral. For this purpose we let z be a complex variable and set /*(*) = 1T- (
—'—/(*)<&,
3 = 1 if Im(a) > 0,
j = 2 if Im(z) < 0,
and define similarly gj(z). Then we have/(a:) = fi{x) — f2(x) and similarly for g. Furthermore, the f, belong to H", 1 < p < °°, in the corresponding half-planes and, with the notation of the preceding proof, we have Mp(fj)
l
(9)
Corresponding relations hold also for g and g,. We will study the contribution of /i to the integral in question, an analogous argument being applicable to f2. Let us introduce the following kernels K0(x,y,t) = (x - y)-*x(\x
- y\)[e(x - t) - e(y - t)] 2
Ky(x,y,t) = (x - y - ic)- [e(:r - f) - e(y - t)] K2{x,y,t) = l(x - t)* + {y - 0* + e 2 r' A
KJ(x,y,t)g(x)f1(y)dxdy
k2(t) = / K2{x,y,t)\g(x) fi(y)\dx
dy.
We are interested in estimating ko. On account of the inequality between the Kj stated above, we have \k0\ < | fci| + ck2 and thus it will suffice to estimate fci and k2. On account of the analyticity of fx(y) if x > t we have + •»
/
rt
(x - y - ie)-2fi(y) dy
-r
Ki(x,y,t)fx(y)
dy = I
— no
« / — oo
[(Z + is) -
(x -
*'e)]-2/i(< + is) d(is).
J s= 0
As readily seen, for x < t the integral on the left above is also given by this last expression. Thus, fa(0 = -
f
(*) |
J —a>
J
[(* + is) -
(x -
H>]-2/I(<
+ is) rf(is),
S=0
and interchanging the order of integration we get hit) = - f
/i(< + is) I
J s =0
[(< + is) - (a; - ie)]~2 g(x)
dxd(is).
J —a>
Since gr(x) = gri(a;) — gr2(aO and 92(2) is analytic in Im(z) < 0, its contribution to the inner integral above is zero and the value of this reduces to 2irig\'(t + is + it). Thus we have fci(0 = - 2xt I /x(f + is) £d'(f + is + te) d(is). J«=o Let us introduce now
108 1098
MATHEMATICS:
F{z) = - 2T* f
A. P. CALDERdN
PHOC. N. A. S.
hiz + is)gt'(z + is + u)d(is).
Js = 0
Then we have fci(t) = F(i). Furthermore, since / i and 3 / are bounded and 0(z~l) and 0{z~2), respectively, F(z) belongs to Hp, p > 1, and with the notation of the preceding proof we have (2T)- 1 S(F) < mVdSigAz and if q~l = p ~ l + r - - 1 ,1 < p, q< °=>, r < Mr,^{h)
= MTh^{F)
< cMHr^[S{F)}
+ U)) < ro(/i)<S(ffi) ro
, then by Theorem 3 and (9) we have
<
cM,[m(fd]M^[Sfa)]
< c Mp(fi)Ml/q-i(g{)
< c MP(f)Mg/^(g).
(10)
Now we estimate k2. We have f
K,(x,y,t) |/(2/) I dy< t[(x - t)* + e 2 ]" 1 sup <S2 f
"[(y - r)2
+ 5 2 r V ' | / ( 2 / ) | * / < c e[(x - t)* +
**]-%),
where / is the maximal function of Hardy and Littlewood associated with | / | . Consequently, \Mt)\
< ef(t) sup ef 6
Mr/^ih)
~[(x-
t)* + a 2 ]- 1 \g(x)\ dx < cf(t) g(«).
• / — 00
< cMv(])Mqli^{g)
<
cMv(l)MQlt^(g).
This combined with (10) shows that Mr/^&o) < c MV(J) M„it-i(g) where c depends on p and r but not on e. As readily seen, this implies that Mt[Ct(j)\ < cMTQ>')
M,(f). We now pass to discuss the n-dimensional case. As before, we assume that / and the partial derivatives bj of b are infinitely differentiable and have compact support. We denote by v a unit vector in Rn and by E its orthogonal complement and fix e, e > 0. Let s be a real variable and k(x, v) = I
h(v) s-2[b(x)
- b(x + vs)]f(x + vs) ds.
Jls\>e
Then setting y = x + vs, integration in polar coordinates shows that C.(/) = 'A f
k(x,y) dv,
(11)
where dv denotes the surface area element of the unit sphere in Rn. We now fix v and set a; = z + vt, where zeE. Then from the inequality for the one-dimensional case established above we get f
k(z + vt,vydt < c\ I
jgrad b (z +
vt,v)\TdtX''
X
u:>
\f(z + vt,v)\* dt
iip,
\h{v)\.
Integrating with respect to 2 over E and applying Holder's inequality to the righthand side, we obtain
109 VOL. 53, 1965
MATHEMATICS:
A. P. CALDERON
c [f\&adb\r
[f\k(x,v)\°dxyi*<
dxY"[f\f(x)\'
1099
dx]v> \h(v)\.
From this and Minkowski's integral inequality applied to (11) we obtain I | a ( / ) | | 5 < C | | g r a d b | | r | | / | | p / | f t W | dv, where c depends on p, q, and r but not on «. Concerning the convergence of Ct(J) as e tends to zero we merely observe that our assertion obviously holds if/and the bj are assumed to be infinitely differentiable and have compact support, whence the general case follows from the inequality above by approximation. Proof of Theorem 1: Since (b) can readily be obtained from (a) by duality, we shall only prove the latter. Let us consider first the case when k(x) is an odd function. There will be no loss in generality in assuming that k(x) is infinitely differentiable in | x\ > 0 and that / and the b, are infinitely differentiable and have compact support. Let fh bit and kj denote the jth. partial derivatives of /, b, and k, respectively. Then integration by parts yields f
k(x - y) [b(x) - b(y) ]fj(y) dy = +
f J
f
k{x - y) b,(y) f(y)
d-y
k,(x - y)[b(x) - b(y)]f(y) dy \x-y\>t
- f
k(ve)[b(x) - b{x + ve)]f{x + v^t"-1
dv,
where v5 denotes the j\h component of the unit vector v and dv denotes the surface area element of the unit sphere in Rn. Now, the first term on the right represents an ordinary truncated singular integral and its norm in Lq can be estimated in terms of the norms of bj and / . To estimate the norm of the second term we use Theorem 2, and in the last term we replace b(x)— b(x + ee) by »i
Jo
2ibj(x + tve)vj€ dt
and apply Minkowski's integral inequality to the resulting integral. Collecting results and letting e tend to zero, (a) follows. In the case when k(x) is even, the operator A can be represented as a finite sum of operators of the form AiA2 where A\ and A% have odd kernels and satisfy the hypothesis of the theorem (see ref. 2). Since d/dXj commutes with A-i, we have {AxAtB - BArAt) ~ = Ar(A2B - BA2) ~ + (A1B - BAJ — At, ax,ax, ax, since Ai and A2 are bounded in L" for every p, 1 < p < oo} the desired result follows. * This research was partly supported by the NSF grant GP-3984. 1 Calderon, A. P., and A. Zygmund, "Singular integral operators and differential equations," Am. J. Math., 79, 901-921 (1957). *Ibid., "On singular integrals," 78, 289-309 (1956). 3 Flett, T. M., "On some theorems of Littlewood and Paley," / . London Math. Soc, 31, 336-344 (1956). 4 To obtain Green's formula for the half-plane under our assumptions we apply it to n cos(n_1i) sin(ra-1s) and the function Gl or G2H over the square — » | ^ t fs »f, 0 < s < rnr, and let n tend to infinity.
110 Proc. Nail. Acad. Sci. USA Vol. 74, No. 4, pp. 1324-1327, April 1977
Cauchy integrals on Lipschitz curves and related operators (commutators/singular integrals/weighted inequalities) A. P. C A L D E R O N Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Contributed by A. P. Calderon, January 12, 1977 ABSTRACT In this note, we establish certain properties of the Cauchy integral on Lipschitz curves and prove the Lp~ boundedness of some related operators. In particular, we obtain the recent results of R. R. Coifman and Y. Meyer [(1976) "Commutateurs d'integrales singulieres:" Analyse harmonique d'Orsay n° 211, Universite Paris XI] on the continuity of the so-called commutator operators.
uniformly in A. Expressing k in terms of h, and using the uniform boundedness in L2 of the truncated Hilbert transform and Minkowski's integral inequality, we obtain the desired result. Our goal is to estimate the norm of B(\) in terms of A(A) and M, that is, a bound for v'{t). This in conjunction with Eq. 1 will give us an estimate for the norm of A(A) in terms of M alone. Let 0\ and Oi be the open subsets of the complex plane consisting of the points lying above and below the curve V, respectively. With a f u n c t i o n / ( / ) in Co™, we associate the functions Fi(io) and F2{w), analytic in 0\ and O2, respectively, given by
1. T h e C a u c h y integral T H E O R E M 1. Let T be a curve in the complex plane given by the equation z(t) = t + itp(t), where tp(t) is a real-valued function on the real line with a bounded derivative, and let
AVJ:
M
-dz(s), (>0 z(s)-z(t)" Then there exists a positive number a such that || ip' || „ < a implies that the operator sup(\ A^J\ is of weak type (1,1) and bounded in LP, 1 < p < °°, and that lim t —oA^f exists pointwise almost everywhere for f in L p , 1 < p < °°. Proof: We shall first consider the case in which
2xi Jr\*-t\>< \,-
Ax«/ = A(X)/ = - M i m
/(»)
f
/|.-(|>.z*(s)-zx(0 2x1 ,—0 J|s-!|>< where z x (() = 1 + iX^((), 0 < X < 1, and it), 0 < X < I)-*')] ;~6 J |fi - s | > . L [[z -» (^s ) - z * ( ( ) F B(Xtf = 2xi -i-lim
F)(w)
I
B(s)fds,
f e
Co-
[I +
/<*)
•dz(s), z(s) — w
wGOj.
[2]
[3]
/ ( ( ) = F,(z(t))
+
FMt))
!|F/(z(«))ll2s(i+l|A(l)ll)l|/||2.
(4)
We now introduce the operator
Jhrtt)- ds)} 2xt
J-~
{z(s)-z{t)-ibfJ
-JMdz(s), & > 0,
/ G C0"
As 8 —- 0, C 5 / converges in the mean of order 2 to
i -lim ,. rf
fMQ-Ws)],fMdz(s)
>,_0 J|«-i|>< \z{S)-z(t)f
2xj
[s)dzx(s).
-^'(Oz'tO-'/W so that B ( l ) / = l\m CJ + j-o 2
-^(t)t'(tr'f(t) + A{\W(t)z'(t)-\j(t).
[5]
To estimate the norm of the operator Cf = lim^ „oC?>f, we consider the bilinear form
il]
On the other hand, A(A) is uniformly bounded in L . To see this, we write the kernel of A (A) as 1
J-™
Consequently we have
2
s- r
r
F/(z(0) = | / ( t ) ± A ( l ) / .
They are well-defined, at least f o r / E Co™, and in this case A(X)/ and B(\)f are continuous functions of t. The operator B(A) is obtained as the formal derivative of A(A) with respect to A, but since for ( > 0 the integral in the definition of 6(A) is the derivative of the one in the definition of A (A) and they converge uniformly as « —*• 0, we have indeed A(A)/ = A ( 0 ) / +
2-rri
It is not difficult to see that these functions extend as C * functions to the curve V and that
d*M,
(zx(s) - zx( t))zx>(s)\n'
. (-D'" +1
Lij,g)= C + ~gCfdz(t) = Um C+"gCJd>'z(t), g £
k(\,s,t))
Setting
where, as is readily verified, Jr(\,s,() is infinitely differentiate and has a double Fourier transform h(\u,v) which is integrable
•At)1324
x:
e(t — u)(ff(u)du
Co".
Ill Proc. Natl, Acad. Sci. USA 74 (1977)
Mathematics: Calderon where e(t) is the characteristic function of t =i 0, substituting in the expression for C&f above and interchanging the order of integration we obtain
[z(s) - z(t) - ib}2 To calculate the inner double integral, we consider the functions Fj(w) and Gy(w) associated with / and g as in Eq. 2 and replace / and g by
1325
Evidently, the same inequality holds for 4>'{z) l, and multiplying we find that
[ § |0'(OIA][ f; |0'(OI-'^] < *2\i\m + M2) that is, the function | 0 ' ( O | belongs to the class A 2 of Muckenhoupt with constant TT2(M2 + 1). Returning to the function Hi, let us consider the functions Cl(0(3)), Fi((j>(z)), H\{(t>{z)) in the upper halfplane a n d the associated maximal and Lusin functions m(t) = s u p |Gi(<Mi + « + to))| |ll|
/ ( s ) = F 1 (z(*)) + F 2 (z(5)) g(t) = G1(z(t))
+
/-•
S(')
G2(z(t)).
Observing that
X
—
I W
= I J\u\
12
h - F i ( < M ( + « + 10)) rfudo \dz I
S(02 = f [-r J\u\
1 [2(s)-2(()-t«r
W
i<^( + " +
iv))\dudv. I
Because _ J2iriF,'(z(() + ii) 10
ifj = l if j = 2
and similarly for C ; , and denoting by T u t h e p a t h z = * + i ^ ( 0 , f > u, we obtain
^-H1(0(«)) = G1(«(z))^-F,W2)) dz dz we have S(r) < m(t)S(t)
and
f_*"s(0k'(*)l
L(/,g) = lim
-[xr m
X £ * " •
-ib))dzdu.
Because
2
= -i £ *
But the function | 0 ' ( O | is in the class A2 and consequently (see ref. 5)
f*J
\HMt))\\*'(*)\dt = fl~ \Hd
J*_*" m(t)*\<M)\dt < CM J I" \GdMt))\2Wt)\dt
V(«)[FI(2(M))C2(Z(U))
= CM f
+
"' \GMtW\z'(t)\dt
+ #1(2(11)) 4- H 2 (a(u))]rfu. [6] We proceed now to estimate the functions H,. For this purpose we let
f
+
~ S(O2|0'(O|A i cM f
+
~ |F,M0)| 2 k'(0|d«
=cM
Xr |Fi(z(,))|2|z ' (f)| ' i '
in which CM is a constant depending on M and, as above, z(t) = t + \
|arg0'(f)| ^ arctanM.
\\HMt))\\&cM\\FMt))U\GMt))h,
Taking the logarithm of 4>'{z) and using the m a x i m u m principle, we find that the preceding inequality is satisfied at all points of R(z) > 0. As is readily seen, this implies that
where c « is another constant depending only on M. Clearly, a similar inequality is valid for H 2 . Thus Eqs. 4 and 6 yield
0 < R(*'(z)) < \4>'{z)\ < R(
R((f>'(t))dt < 7rfl(0'( s + ia)\
and this combined with the preceding inequality gives |yy
f( \*'{t)\dt
< n\4,'(s
+ ia)\(l
+ M2)'/2.
|i.(/,g)|sc„(U|Aa)«)2«/i!2flg«2, which implies that IIC||
112 1326
Mathematics: Ca\6erbn
Proc. Natl. Acad. Sci. USA 74 (1977)
Evidently, the preceding argument and this last inequality are valid for A(\) and B(\), 0 < X < 1. Thus, Eq. 1 implies that
ce(t)<
|5(t)| < c - } t ( 0
l|Ati)/||p
[10]
Kp<<»
[11]
|A< 2 >/-A< 3 >/|
U(X)I < \\A(0)l + cM J ^ (^ + |Uwr)
[7]
where, as above, the norms are norms of operators in L 2 . But A(0) is just half the ordinary Hilbert transform so that ||.A(0)J| = lk- Consequently, the function | A ( X ) | , 0 < X < 1, is majorized by the solution of the differential equation /l
\2
[12]
where the constants c in Eqs. 9, 10, and 12 can be taken so as to depend only on the function TJ, and the one in Eq. 11 so as to depend on this function and on ll^'IU, and where m(J) is the ordinary Hardy-Little wood maximal function of / . Furthermore, we have
UMf - A<»flp < cWl.
1
1
1131
with c depending only on p and the function rj(t). To see this, we write
that is ||A(X)| ^ ( l - C i i X ) - 1 - ^ ,
0<X
(A">/-A<3'/)(«)=
1.
Let now a = supMcM ~', where the s u p r e m u m is taken over all M ^ 0 and the corresponding constants c ^ , CM > 1, for which Eq. 7 holds with M„ < M. Then if M j „ < a, and JI^'JL < M c M ~ 1 , setting X = | | ^ | U M - 1 < C M _ 1 < l a n d ^ = X<^, the preceding inequality becomes llAyll < (1 - c „ X ) - ' - i = (1 - U ^ I L M - ' C M ) -
i,
and letting M c M ~ J tend to a we finally obtain llA^U S ( l - l l ^ l U a - ' ) " 1 - | ,
ll?ll-
[81
To complete the proof of our theorem, we shall henceforth consider only functions tp with j||j« < a. Because the techniques we shall employ are standard we will merely outline our argument. First, we show that A^ is continuous in L p for 1 < p < °°. For this purpose, we let / be a function with vanishing integral and supported in the interval | J — £o| < lj% S. Then a simple calculation shows that
X,-,o|>S |A ' (/A ' )(,)l '" Scll/ " 1 Hence, from Eq. 8 and Theorem that IU^(//z')llp
1 in ref. 1 we conclude K p < 2 ,
A " » / = J"8(f)-'!)(« - s M l ) - ' ) ( A j A ' ) ( s ) d 2 ( S ) A'2)f= S{1 - 7((r - s)r\t))\(z(s) z(t))-
,
, (z(s) -
z(t)rlf(s)ds,
where JJ(/) is an even non-negative function in Co00 which equals 1 near the origin, e(t) is an arbitrary positive measurable function of t and
s«)= J\(«-sMf)-')
X-.o|>J A(2,/|di - dl/L provided that J / d ( = 0 and / ( ( ) = 0 for 11 - 10| > - p
k(t,u) = SHt)-'v((t =
-
s)t(t)-i)(z(u) -z(S))-'dz(s)-(z(u)-z(t))-» j5(()-',(((-sW()-i)((Z(U) - z(s))-< - (z(u) z(t))-i)dz(s).
Because \z'\ < 1 + a and a is finite (see the remark below), this last integral is readily seen to be majorized by cc(t){t — u)~2 with c depending only on the function rj. If on the other hand, t(t) <\t-u\< 2e(r), then
k(t,u)= f
XtrlM(t - sW)-*)
,
%J \s—t\
<2i{t)
-vdt-uMtr^izi^-zit^dzis) + 6(t)~Wt-uHt)-i)\im
f 6—0
X (z(u)~
|91
J&<\s-t\<2t{t)
zis^dzis)
+ (z(u) -
z(t))~l
and estimating the first integral in terms of rf and taking Eq. 10 into account, we find that in this case k(t,u) is majorized by c t ( ( ) _ 1 - Finally, if \t — u\ < t(t) we have the same preceding expression for k(t,u) with the last term omitted so that the same estimate as in the preceding case holds. From all this there follows |fc
with c depending only on p and li^LU. Now, the standard duality argument shows that the same result holds for 2 < p < °°. Consequently, A^ is bounded in LP, 1 < p < °°. Next consider the following operators with ip still in CQ°:
A™/ =
Sk(t,u)f(u)du
and for simplicity assume that TJ(() = 0 for J(j > 1 and TJ(/) = l f o r \t\ < % Then if |( - u\ > 2e((),
((f) 2 )- 1 ,
which implies that |AU>/-A<3>/|
where m(f) is the Hardy-Littlewood maximal function of/, and this in turn clearly implies Eq. 13. We are now near the completion of our proof. From Eqs. 11, 12, and 13 there follows that A ( 2 ) is bounded in LP, 1 < p < » with a norm that can be estimated in terms of p and ||i/IL only, and this combined with Eq. 9 and Theorem 1 in ref. 1 implies that A ! 2 ' is also of weak type (1,1). But then Eq. 12 allows us to conclude that the same result holds for A131. So far we have assumed that ^r is a function in Co™, but since all preceding estimates depend on JI^'IU only, as far as their dependence on \p is concerned, a passage to the limit shows that the same results and estimates hold for operators involving general functions tp with I ^ I U < «• Furthermore, the estimates for A ( 3 ' are independent of the function t(t), which is positive measurable but otherwise arbitrary, and this implies that sup | A ^ , / | is of weak type (1,1) and strong type (p,p), 1 < p < CD, whenever
113 Proc. Natl. Acad. Sci. USA 74 (1977)
Mathematics: Calderon
1327
Finally, to prove the pointwise existence of \im(^*oA^J, we observe that this limit clearly exists at (Q if / is in Co™ and /(*o) = 0, or if / is in Co™, / is constant near to and <^'(/o) exists. From this we conclude that if / is in Co™ then lim(—oA^J" exists almost everywhere, whence the general result follows from the fact that sup \A^J\ is of weak type (1,1) and strong type (p,p), 1
where p is sufficiently close to fl, we have
THEOREM 2. Let F(z) be analytic in the disc | z| < R and ^ a real Lipschitz function on the real line such that \\
This research was supported by National Science Foundation Grant MCS75-05567.
(W)(0-
f
F(2)(A z , ( /)(0-.
[15]
1
If z" — u + it), setting 1 = t ~ utp{t), s ~ s — u
L
J=S,
, -V(*^)/<.)*-
*J\s-t\>tS
—t
\
S —t
/
Then the operator supt\Lti\ is of weak type (1,1) and strong type (p,p), 1 < p < <», and limt^Q (L
AZJ = f
,
[s-t-z-Ws)
J\s-t\>* - ^m-'fWs,
\z\=p
[14]
1.
Benedek, A., Calderon, A. P. & Panzone, R. (1962) "Convolution operators on Banach space valued functions," Proc. Natl. Acad. Sci. USA 48, 356-365. 2. Calderon, A. P. (1965) "Commutators of singular integral operators," Proc, Natl. Acad. Sci. USA 53,1092-1099. 3. Coif man, R. R. & Meyer, Y. (1975) "On commutators of singular integrals and bilinear singular integrals," Trans. Am. Math. Soc. 212.315-331. 4. Coifman, R. R. & Meyer, Y. (1976) "Commutateurs d'integrales singulieres:" Analyse harmonique d'Orsay, n° 211., University Paris XL 5. Gundy, R. P. & Wheeden, R. L. (1975) "Weighted integral inequalities for ihe non-tangential maximal Junction, Lusin area integral, and Walsh-Paiey series," Studia Math. 49,107-124.
ON THE EXISTENCE OF CERTAIN SINGULAR INTEGRALS. By
A. P. CALDERON and A. ZYGMUND Dedicated to Professor MARCEL RIESZ, on the occasion of his 65th birthday
Introduction. Let / (x) and K (x) be two functions integrable over the interval (—oo, — oo). It is very well known that their composition + 00
f
j(t)K(x-t)dl
-00
exists, as an absolutely convergent integral, for almost every x. The integral can, however, exist almost everywhere even if K is not absolutely integrable. The most interesting special case is that of K (x) = 1/x. Let us set
31 J X — t -oo
The function / is called the conjugate of / (or the Hilbert transform of /). It exists for almost every value of x in the Priacipal Value sense:
-oc
x+e
Moreover it is known (See [9] or [7], p. 317) to satisfy the M. Riesz inequality
a)
[J\f)rdzY'zA,[f\frd*r, - oc
where Av The limit by dF(t), (— oo, + graphical
I
- oo
depends on p only. There are substitute result for p = 1 and p = oo. / exists almost everywhere also in the case when / (t)dt is replaced there wheie F (t) is any function of bounded variation over the whole interval oo). (For all this, see e.g. [7], Chapters VII and XI, where also biblioreferences can be found).
115 86
A. P. Calderon and A. Zygmund.
The corresponding problems for functions of several variables have been little investigated, and it is the purpose of this paper to obtain some results in this direction. To indicate the problems we are going to discuss let us consider two classical examples. Let / (s, t) be a function integrable over the whole plane, and let us consider in the half-space z > 0 the Newtonian potential u (x, y, z) of the masses with density / (g, t). Thus u (x, y, z) = fff
(s, t) ^ - ,
R* = (x - sf -r (y - tf + z\
the integration being extended over the whole plane. Let us also consider the partial derivatives
zj l / ( « > 0 ^ 3 - ,
ux = — \
\f(s,t)^^-dsdt.
Here — (4 TI)'1 M2 is the Poisson integral of /, and it is a classical fact that it tends to / (x, y) as 2 -> 0, at every point (x, y) at which / is the derivative of its indefinite integral. On the other hand, by formally replacing 2 by 0 in the formula for uz we obtain the singular integral
/ / It can be written in the form (2)
jjf(s,t)K(x
— s,y —
t)dsdt,
with (3)
K(x,y)
=
{x + V.
It is a simple matter to show that at every point (x0, y0) at which / is the derivative of its indefinite integral the existence of the integral is equivalent to the existence of lim MX, as the point (xr y, z) approaches (x0, y0, 0) non-tangentially (and that both expressions have the same value), but neither fact seems to have been established unconditionally. Here again the integral (2) is taken in the principal value sense, which in two dimensions means that first the integral is taken over the exterior of the circle with center (x„, y0) and radius e, and then s is made to tend to 0. Another example, of a somewhat similar nature, arises from considering in the plane the logarithmic potential « of masses with density / (s, t). Hence
116 On the Existence of Certain Singular Integrals. u(x,y)
= \ I f(s,t) log- dsdt;
87
r2 = (x — sf + {y — tf.
If in order to avoid unnecessary complications we assume that / vanishes in a neighborhood of infinity, then in any finite circle u is the convolution of two integrable functions, and so the integral converges absolutely almost everywhere. The integral obtained by formal differentiation, say with respect to x, is
and so, as a convolution of two integrable functions, again converges absolutely almost everywhere and represents a function integrable over any finite portion of the plane. Using this fact one proves without difficulty (see [1]) that the integral actually represents ux. Thus ux and w„ exist almost everywhere. Let us, however, differentiate the integral (4) formally once more, with respect to x and with respect to y. We get the integrals of type (2) with (5)
K ( X t y )
. ^ ^
t
K{Xty)__*M_,
respectively. These two kernels are not essentially different, since one is obtained from the other through a rotation of the axes by 45°. It may also be of interest to observe that they appear respectively as the real and imaginary parts of
1 _ 22
1 (x + i yf
The existence almost everywhere of the integrals (2) in the cases (5) has been established by Lichtenstein for functions / which are continuous (or, slightly more generally, Riemann integrable). This result seems not to have been superseded so far, though the existence almost everywhere of uxi, uvv, uly together with the relation uzz — uyy = — 2 T I / was established by Lichtenstein [6] (see also [2], [8]) in the much more general case of / quadratically integrable. The kernels (3) and (5) have one feature in common: they are of the form QMS'2'
x = qcos(p,
y = Q sin
where g (
117 88
A. P. Calderon and A. Zygmund.
Suppose we have a function / (xx, x2, . . . xn) integrable over the whole w-dimensional space, and a kernel K (xl, x2, . . . xn) = o~n Q (aj, a 2 , . . ., a„), where Xj = o cos a7 for all j , and a x , x 2 , . . ., an are the direction angles. What can be said about the existence and the properties of the integral (6)
f(xl,xi,...,
x„) = / /fo , s 2 , . . . , s„) K (xl — sx, . . ., xn — sn) ds! . . . dsn ?
An answer to this problem is our main object here. This is the plan of the paper. In Chapter I it will be shown that, if / 6 £ " , 1 < p < o o , then the integral (6) converges, in the metric L", to a function f 6 L", provided a) the mean value of Q over the unit sphere is zero, b) the function Q(a.l, a 2 , . . ., a„) satisfies a smoothness condition (See Chapter II). (In the case p = 2 condition b) can be considerably relaxed). The function / satisfies the condition analogous to (1). The cases p = 1 and p = oo are also investigated. The main result of Chapter II is that under conditions a) and b) the integral (6) exists almost everywhere not only for p> 1, but also for p= 1. The result holds, if f ds-L ... dsn is replaced by d[i, where (u is an arbitrary mass distribution with finite total mass. If / € i p , p> 1, the partial integrals of the integral (6), that is the integrals over the exterior of the sphere of radius e and center (xlt xit ... xn) are majorized by a function of L", independent of e. Chapter III is devoted to some applications of the results previously obtained to the problem of the differentiability of the potential. Other problems connected with our main topic will be considered in an another paper.
W o l f P r i z e in M a t h e m a t i c s , V o l . 2 ( p p . 1 19—1 6 4 )
eds. S. S. Chern and F. Hirzebruch © 2001 World Scientific Publishing Co.
A. N. KOLMOGOROV V. M. Tikhomirov Moscow State University Life Story Andrei Nikolaevich Kolmogorov, one of the greatest mathematicians of the twentieth century and one of the most outstanding scholars in the history of Russian science, was born on April 25, 1903. His mother Maria Yakovlevna Kolmogorova died in childbirth. The responsibility for his upbringing was taken on by his mother's sister, his aunt Vera Yakovlevna Kolmogorova (1863-1950). Maria Yakovlevna and Vera Yakovlevna were the daughters of Yakov Stepanovich Kolmogorov, a nobleman and rich landowner. Kolmogorov's first years were spent in the Tunoshna estate owned by his grandfather. The estate was situated near the old Russian town of Yaroslavl and the greatest Russian river, the Volga. Kolmogorov always said that his childhood was happy. Andrei was surrounded with love, kindness, attention and care. His relative's goal was to cultivate the child's curiosity and an interest in books and nature. Vera Kolmogorova, together with a good friend, organized a small school for children living near the estate. At this school the teaching was carried out "according to the latest educationals theories". a School children, with the help of their teachers, published the journal "The Swallow of Spring". The five-year-old boy "edited" the mathematical section of the journal. Kolmogorov wrote sixty years later: "The first impression of mathematical 'discovery' I experienced at the age of five or six, noticing the regularity in the identities 1 = l 2 , 1 + 3 = 2 2 , 1 + 3 + 5 = 3 2 , 1 + 3 + 5 + 7 = 4 2 , and so on." In the journal he had published this "discovery", and it could be considered his first "scientific publication". In 1910, when Kolmogorov reached the age of seven, he and his aunt moved to Moscow. There he entered one of the most progressive grammar schools (gymnasiums) of the time. It was organized by a group of radically inclined intelligentsia. Unlike the majority of grammar schools of that time, it was co-educational (this took place only at two grammar schools in Moscow) and practised many interesting experimental teaching methods. In the school, the spirit of freedom reigned and the teachers tried to support every sign of talent. Kolmogorov's range of interests at the age 10-14 was extremely broad. He took a serious interest in biology and physics, and by the age of 14 he was already familiar with higher mathematics. He was fond of chess, and fascinated by social problems. In particular, he thought about a constitution for a community, where the principles of higher justice were to be put into practice. He actually drafted the text of such a constitution. He took part in the elections for the Constituent Assembly in autumn 1917. Equally, his interest in history was especially profound. "The first scientific report which I made at the age of 17 at the University of Moscow," he recalled a
If words in quotation marks are not followed by the name of the author, they are from A. N. Kolmogorov.
120 later, "was a report to the seminar of Professor S. V. Bakhrushin about the history of Novgorod in the fifteenth century." Kolmogorov then made a small discovery, which was recognized by S. V. Bakhrushin, one of the leading Russian historians of that time. Kolmogorov asked the professor whether or not he should publish his result. The reply Bakhrushin gave him was: "Certainly not! You have found only one proof. That is very little for a historian. You need at least five proofs." The disappointment he felt at that moment might have influenced his future destiny, and he began to study mathematics, where just one proof suffices.b Nevertheless, he did not immediately decide to devote his life to mathematics. While he was studying at the University of Moscow in 1920, he enrolled in the Faculty of Metallurgy at the Institute for Chemical Technology, because he had intended to study practical engineering. But very soon his interest in mathematics outweighed "his doubts about the relevance of the profession of mathematician". He continued his studies only at the University, and from then on his whole life was linked to the University of Moscow. He immediately succumbed to the University's atmosphere of creative enthusiasm that had then existed there. The outstanding figure among the mathematicians at that time was Nikolai Nikolaevich Luzin. Kolmogorov came into lively scientific contact with some of Luzin's students, Pavel Sergeevich Alexsandroff and Pavel Samuilovich Uryson, who later played an important part in the development of topology. Once, in 1920, at Luzin's lecture, the 18-year-old freshman disproved one of the lecturer's hypothetical assertions. He presented his argument with confidence to the group of mathematicians present, and for the first time he attracted some attention. This resulted in P. S. Uryson inviting Kolmogorov to become his student. But it was the problems put forward by Luzin that drew the young man's attention. He began to construct a general theory of operations on sets, wishing to advance beyond the boundaries at which P. S. Alexsandroff and M. Ya. Suslin had stopped. Kolmogorov also attended V. V. Stepanov's seminar on trigonometrical series. There he solved one of the problems posed by Luzin, and after that Luzin "with some solemnity" (as Kolmogorov wrote) suggested that he be his student. "In relation to his pupils," Kolmogorov wrote not long before his death, "Luzin had a definite idea of which of them was destined for working on 'metrical' or 'descriptive' problems. My predestination was to deal with 'metrical' one(s)..." In the summer of 1922 (the work was dated 2 June 1922), Kolmogorov obtained an outstanding result: He constructed a Fourier series divergent almost everywhere, and this immediately brought him international repute. This was the beginning of his incomparable creative career. But earlier, in 1920-21, Kolmogorov realized his ideas on descriptive set theory. He wrote a long paper on this subject, dated February 1922. But this paper was devoted "to a direction that was not at all envisaged by Luzin." It was the reason that the first part of the paper was published in 1928 and the second part of it wasn't printed until 1987. Kolmogorov's manuscript devoted to history of Novgorod land-ownership in the 15th century was found in his papers after his death. It was published in 1994 and was highly appreciated by specialists. In the manuscript, Kolmogorov used some statistical methods for his historical conclusions.
121 In 1925, Kolmogorov graduated from the University of Moscow and became Luzin's research student. The same year saw the beginning of his work on probability theory (in collaboration with A. Ya. Khinchin). In 1929 he set out on a long boat journey along the Volga with Pavel Sergeevich Alexsandroff, and this marked the beginning of a friendship which lasted until Alexsandroff died on 16 November 1982. In 1935, Kolmogorov and Alexsandroff acquired a house not far from Moscow on the bank of the river Klyaz'ma, and it was in that house that most of their creative life continued. For more than 50 years, this house was the center of mathematical activity for several generations of Soviet mathematicians as well as many guests from abroad. Kolmogorov had accumulated many accomplishments in his field in his lifetime. By 1931, Kolmogorov became a Professor at Moscow University, and founded several departments. In 1939, he was elected full Member of the Academy of Sciences of the USSR. From 1954 to 1958 he was the Dean of the Faculty of Mechanics and Mathematics at the University of Moscow. From 1964 to 1966 and from 1973 to 1985 he was the President of the Moscow Mathematical Society. Between June 1930 and March 1931, Kolmogorov went on academic journeys abroad (Gottingen, Munich, Paris). There he met Hilbert, Caratheodory, Landau, Levy, Frechet, Lebesgue, Borel, Weyl, and other outstanding mathematicians. Kolmogorov often spoke of the exceptional significance of being able to do his research work in such a scholarly atmosphere. He felt privileged to have spent this time abroad, and to have established personal contacts with these scholars from different countries. In 1954, he spent two months as Professor at the Humboldt University in Berlin. In 1958, he was Professor at the University of Paris, for the spring term. Kolmogorov attended the International Congresses of Mathematicians in Amsterdam (1954), Stockholm (1962), Moscow (1966) and Nice (1970). At the Amsterdam Congress, Kolmogorov was invited to conclude the scientific program of the Congress with a lecture on dynamical systems (a lecture by J. von Neumann had opened the scientific program). More than twenty scientific organizations have elected Kolmogorov to be a Honorary Member. He was elected Member of the Royal Netherlands Academy of Sciences (1953), of the Royal Society of London (1964), of the National Academy of Sciences of the USA (1967), of the Academie des Sciences of Paris (1968), Honorary Member of the Roumanian Academy of Sciences (1965) (he had been a Corresponding Member since 1957), Foreign Member of the Polish Academy of Sciences (1956), Honorary Member of the Hungarian Academy of Sciences (1965). He was a Doctor Honoris Causa of the Universities of Paris, Stockholm, Warsaw, Budapest, of the Calcutta Mathematical Society, of the Royal Statistical Society of London, etc. Kolmogorov was awarded the State Prize of the USSR (jointly with A. Ya. Khinchin, 1941), the P. L. Chebyshev Prize of the Academy of Sciences of the USSR (jointly with B. V. Gnedenko, 1949), the highest Prize of Soviet Government (jointly with V. I. Arnol'd, 1965), the N. I. Lobachevskii Prize (1986), and the following international prizes: The Balzan Prize (1963), the Prize was awarded simultaneously but in other nominations to Pope John XXIII, the historian S. Morrison, the biologist K. Frisch, the composer P. Hindemith), the Wolf prize, the Helmholtz Gold Medal, and many others.
122 Kolmogorov devoted the last two and a half decades of his life mainly to the problem of instruction and teaching in secondary schools. Scientific Legacy There are different ways of speaking about the scientific achievements of a great scholar: We can give a chronological account, we can consider separately each scientific direction, or we can classify the papers according to their importance, and so on. I prefer to follow a different way. In the original mathematical work of an outstanding scientist, it is possible to differentiate several components: results (solutions to difficult problems, finding new formulas, etc.), ideas (introduction of new definitions, concepts and interpretation of old ones, enunciation of problems, working out the beginning of new scientific directions), theories (where the aim is to explain a group of phenomena), methods and conceptions. All these options are present in Kolmogorov's work. And we shall investigate what makes up each of these components. In his article "The Architecture of Mathematics" that describes his program, Nicolas Bourbaki notes with regret that there is no mathematician, even among those with the broadest erudition, who would not feel to be a stranger in certain fields of mathematics. The exceptions to this rule are the geniuses like Poincare and Hilbert, who left their mark in almost every mathematical area. In my opinion, there is complete justification to add the name of Andrei Nikolaevich Kolmogorov to this short list. The breadth of Kolmogorov's scientific range of interests is unique. Trigonometrical and orthogonal series, descriptive set theory, mathematical logic, the theory of measure and integral, topology, the theory of approximations, geometry, celestial mechanics, cohomology theory, turbulence, ergodic theory, superposition of functions, functional analysis, ... . And in addition to all this, the vast field of probability (Markov chains and processes, statistics, stationary processes, etc. — Kolmogorov transformed probability theory completely, starting from its foundations); information theory; the theory of automata; and still more — numerous works on applications to biology, geophysics, production control, the theory of marksmanship, the theory of versification, etc. And in almost every one of these fields, we owe to Kolmogorov fundamental theories, ideas, methods and outstanding results. First and foremost, it is necessary to mention the main conceptual line of his legacy. When the Academy of Sciences of the USSR decided to publish his collected papers, it was assumed that this publication would consist of two volumes. The question of how to divide the material into two parts was, naturally, passed on to the author himself to solve. Kolmogorov suggested dividing them into one volume of the papers on mathematics and mechanics and, in the other, papers on probability theory and information theory. This division reflected the fact that enormous world of mathematics is in fact itself divided into two parts, as it were into two kingdoms. In one, deterministic phenomena are studied, and in the other, random ones. A century ago the Kingdom of Chance occupied a very modest territory in our world of mathematics. Until the end of the century there was still no fundamental role for random phenomena in the natural sciences (although the foundations of
123 thermodynamics were already laid). Then the Kingdom of Chance began to colonize greater areas and became comparable to the rest of the mathematical world. (And as Kolmogorov once remarked, the Congress of the Bernoulli Society, are comparable to the International Congress of Mathematicians.) Kolmogorov was a leader of progress in both kingdoms, and he discovered many previously uncharted regions. At the end of his life he put forward a grandiose program of simultaneous and parallel study on the complexity of deterministic phenomena and the statistical determination of random phenomena. The basic idea is that the Kingdom of Order and the Kingdom of Chance have no real boundaries, that our mathematical world is one, and in principle, indivisible. The attempt to reveal the essence of the concepts of "order" and "chaos" crowns Kolmogorov's original work. In this concept, all the ideas of probability theory, the foundations of mathematical logic and the theory of algorithms, the ideas and methods of information theory, the results of ergodic theory and of the theory of dynamical systems, and intense speculations about the natural sciences are encompassed. But many of Kolmogorov's first works (on the theory of functions, on set theory, geometry, etc.) can be interpreted as a prologue to the realization of this immense program. And now let us turn to the concrete results Kolmogorov obtained. It is impossible to describe in one article all of his contributions to science with any totality, but whilst inevitably restricting myself, I would like to present Kolmogorov's original work as a whole, picking out the basic components and linking them together. The clarity and wide importance of the goals which Kolmogorov set for himself are striking. It is difficult to deny oneself the pleasure of trying to explain the significance of the results he obtained to the average reader, and not be disrespectful to a certain degree to his or her field of interest. We have already spoken of the main concept of his life. Now we shall speak of theories. At the very outset of his creative work, Kolmogorov was obsessed by "a vague desire to study mathematics, which has links with physics and the natural sciences". This led him to perceive the deep and mysterious secrets of nature. Here we must first focus on his papers on classical mechanics. Can the Solar system last forever? This is probably the central problem of astronomy. Is perpetual motion possible in simpler planetary systems, consisting, say, of only three bodies? Or, will perhaps the evolution of a planetary system always (on a set of full measure) end in catastrophe? It was natural to try to consider first the motion of systems close to integrable ones. Problems of this kind date back to Newton and Laplace, and Poincare, who called the problem of describing the evolution of Hamiltonian systems for small perturbations "the fundamental problem of dynamics". In his papers Kolmogorov solved the fundamental problem of dynamics for the three-body problem for the majority of initial conditions in the case of general position. As a direct consequence of Kolmogorov's theorems, it was found that a satellite of Jupiter moving in the plane of the circular orbit of Jupiter along an elliptical orbit, perturbed by Jupiter but not perturbing it, always remains in an elliptical orbit. Kolmogorov's theory proved to be applicable to a large number of problems in mechanics and physics (the problem of the stability of rapid rotation of an asymmetric rigid body around a fixed point; the problem of magnetic surfaces in systems of tokamak type, and so on). Kolmogorov gave a survey lecture on his results, to which we have referred, at the International Congress of Mathematicians
124 in Amsterdam. The ideas of Kolmogorov, developed in the works of V. I. Arnold and J. Moser, were called KAM theory (the theory of Kolmogorov, Arnold and Moser), the name now used by almost every mathematician. Kolmogorov made a significant contribution to the creation of the theory of stochastic processes. In 1931, he published his paper "Analytic methods in probability theory". The initial methodological position is remarkable for its simplicity and depth. Deterministic processes, in which initial position determines their further evolution, are replaced by processes where "the state x of the system at some moment of time t stipulates only a known probability for the occurrence of a possible state y at some later moment t > 0." This consideration led Kolmogorov to a definition of a Markov processes. He wrote it out in the general form of an integral equation, which had been found for special cases by Smoluhovskii. From this integral equation one can deduce direct equations (already seen in the work of such powerful physicists as Planck, Einstein, Fokker, Smoluhovskii) and inverse equations, which were not known to the physicists. In his work Kolmogorov combined the theory of heat, due to Fourier, the theory of Brownian motion, due to Einstein and Smoluhovskii, and a description of probabilistic random walks, on which Markov and his followers had worked, and also the ideas of Bachelier and Wiener, who constructed the first examples of stochastic processes. In their article, written on the occasion of Kolmogorov's fiftieth birthday, P. S. Alexsandroff and A. Ya. Khinchin wrote: "In the whole of probability theory in the twentieth century, it is difficult to name any other piece of research that has turned out to be so fundamental for the further development of science and its applications than this paper by Kolmogorov." In the course of his work on the theory of Markov processes, Kolmogorov began a fruitful collaboration with known physicists (S. I. Vavilov, M. A. Leontovich and others). Kolmogorov delighted in affirming that in his joint paper with Leontovich, the "physical" part was due to him and the mathematical part to Leontovich. Physical intuition often lead Kolmogorov, even in his research along purely abstract lines. According to him, cohomology theory occurred to him from visualizing descriptions of a flow of a liquid over a manifold. A clear understanding of the physical picture of undamped oscillations with a continuous spectrum served as a lodestar for the creation of the theory of stationary processes. Wiener was very proud of his construction of the theory of interpolation and filtration of stationary stochastic processes. It was with some bitterness that he acknowledged Kolmogorov's priority in this area. Stationary processes, being processes whose probability characteristics remain constant in time, are the idealization of a great number of stochastic natural phenomena (in the atmosphere, in the ocean, and so on); they occur constantly in technical applications, for example in radio engineering. One of the most important problems is the prediction of the future by observing the process over a period of time. It was this theory that Kolmogorov and Wiener worked out. It discovered innumerable applications in different fields of science and technology. The theory of stationary processes led Kolmogorov to consideration of turbulence problems. In 1941, he became interested in this field, which was absolutely new to him, and which belongs, strictly speaking, to the domain of physics, or, rather, hydro- and aerodynamics, rather than mathematics. There was no mathematical theory of turbulence before 1941, but there were several outstanding schol-
125 ars who tried to give phenomenological explanations of turbulence. Among these distinguished people were Jeffrey Taylor and von Karman. Kolmogorov had been thinking about turbulence for about half a year when War World II started, he then had to switch to other problems (he was engaged in correcting the artillery fire and bombing and radically reformed this science as well). Kolmogorov published only three articles on the theory of turbulence in "Doklady" at the Russian Academy. Kolmogorov's papers on turbulence had an enormous influence on the whole further development of this important field of the natural sciences. ... Fifty-eight years passed, and one of Kolmogorov's students, Akiva Jaglom, won a grant for a series of books on Kolmogorov's theory of turbulence. The science has been extended drastically. "How long are you going to write this book", I asked him. "You have probably heard about the recent sentence in an American court of law. The criminal was sentenced to three life terms and thirty more years. So, three lives and yet another thirty years might be enough," he replied. The motion of fluids and gases is subject to deterministic laws; however the very character of the motion turns out to be so complex that it calls to mind a stochastic process. Among Kolmogorov's concrete results one can point to the widely known "two-thirds law", which has the characteristic of a law of nature. In a turbulent flow (under certain conditions) the mean square of the difference of the velocities at two points, at a distance r (of mean sizes), is proportional tO r 2 / 3 . Kolmogorov made a very significant contribution to the natural sciences in his research inspired by biological problems. His work in biology led him to the solution of a series of problems in mathematical biology and to remarkable results in pure mathematics. In a joint paper with I. G. Petrovskii and N. S. Piskunov, which was stimulated by their work on biological problems, they were the first to construct a mathematical theory of the stability of solutions of travelling wave type for the diffusion equation with a nonlinear right-hand side. (The list of papers on this subject extends at present to some thousand items.) Exemplified on the physical side, the formulisation and description of the qualitative picture of the phenomenon is due to Kolmogorov. ("I had noticed how a Bickford fuse burns," he said of this, although the work arose from attempts to describe the propagation of a gene.) Biological problems led Kolmogorov to formulate a theory of branching processes. Had Kolmogorov done nothing more than create the theory of stochastic processes, the theory of turbulence and laid the foundations of KAM theory, he would still have been one of the greatest stars in the world of science. In his monograph "Limit distributions for sums of independent random variables" , Kolmogorov presented the results of the initial stage of the summation of random variables, which had begun in the work of Jacques Bernoulli, de Moivre and Laplace, and had been continued by Poisson, Tchebyshev, Markov, Bernstein and others. During the war Kolmogorov made an important contribution to the theory of ballistics, and according to the experts, his results completely transformed this field. Reduced to tables, graphs and nomograms, they helped the Allies to achieve victory in the World War II.
126 And finally we must mention his contribution to information theory, where (with Khinchin and others) he laid the mathematical foundations of the theory and obtained some interesting formulae and theorems. What are the fundamental results obtained by Kolmogorov? Let us begin with the metric theory of functions. The best known work from the first period of his scientific activity was his example of a Fourier series divergent everywhere. If x(-) is integrable on [-7r,7r] (i.e. x(-) E Li([-7r,7r])), then from it one can construct a Fourier series: J2kezxi*eik', where xk = (27r) _1 JTx(t)e~ihtdt. Fourier series date back to the eighteenth century, but the theory of their summation began to develop in the middle of the nineteenth century. The following result, discovered by the 19-year-old Kolmogorov, caused a sensation. Theorem 1. There is a function integrable on [—7r,7r] whose Fourier series diverges almost everywhere. In 1926, Kolmogorov constructed an example of a series that diverges everywhere. The question of whether for every square integrable function (i.e. a;(-) € I<2([7r,7r])) the Fourier series converges almost everywhere was raised by N. N. Luzin in 1915, was solved by Carleson in 1966, and was completed by Hunt (for any function x(-) in Lp([—n, n]) for p > 1, its Fourier series converges to x(-) almost everywhere). In particular, it was not known until Carleson's theory whether it is possible to replace £i([—rr,7r]) in Theorem 1 by C{[—7r,7r]), the set of continuous functions on [—7r,7r]. In the field of trigonometric series, in which Weyl, Hardy, Luzin, Men'shov, Bari and many other outstanding mathematicians worked, two results stand out by reason of their power and completeness: Kolmogorov's example and the Carleson-Hunt theorem. In connection with Luzin's problem it is appropriate to mention another remarkable result which Kolmogorov found jointly with D. E. Men'shov. Theorem 2. There is an orthogonal system on the unit interval consisting of functions of unit modulus and a square integrable function on the same interval such that the Fourier series of this function, with respect to the given system diverges everywhere. In the same note Kolmogorov stated the following result: There is a function in L2([—7r,7r]) such that the Fourier series with respect to a rearranged trigonometric system diverges almost everywhere (so that Carleson's theorem reflects a very deep fact concerning harmonics in their natural order). Kolmogorov did not publish his proof of this theorem, and this result was proved in 1960 by the Polish mathematician Zahorski. The next theorem, like the two previous ones, is very simple in its formulation, but its significance is not immediately obvious to anyone who is not connected with harmonic analysis. One of the most actively developing branches of harmonic analysis studies are problems on the convergence of Fourier sums and of other important integral operators in various function spaces. One of the most important operators is the Hilbert transform H, associating to a function x(-) the function z(-) that is conjugate to x(-) in the sense of complex analysis. We
127 have the formula Hx(t) = x(t) = - ( 2 7 r ) - 1 J x(t + T) cot(r/2)dr. Various generalizations of the Hilbert transform (singular integral operators) play a very important part in classical analysis. The Hilbert transform maps Lp([—7r,7r]) into LP([-TT, n]) for p > 1 (M. Riesz), or, as we say, this is a transformation of strong type (p,p) for p > 1. For p = 1, it is not a transformation of strong type (1,1). However, we have the following result. Theorem 3. Let x(-) be a function integrable on [—TT,TT]. Then the measure of those points where the Hilbert transform Hx(-) is greater than a in absolute value satisfies the inequality meas{t\ \Hx(t)\ > a} < Ca _1 ||o;(-)|| z , l([ _ 7ri7r]) . In modern language, this inequality means that the operator H is of weak type (1,1). Theorem 3 proved to be one of the starting points for a very broad field of mathematics, with which the names of Riesz, Marcinkiewicz, Zygmund, Calderon, Hardy, Littlewood, Paley, and others are associated. The very concept of weak type was very important in the theory of singular operators. It is very difficult to know which result to choose when talking of Kolmogorov's work on probability. Kolmogorov deemed this field to be his "narrow speciality", and in it he was unquestionably a world leader. His input here was enormous. At the beginning of his work in this field he found some fundamental theorems crowning the research of Chebyshev, Markov, Bernstein and others. In classical probability theory there were two fundamental themes: The law of large numbers and limit theorems. In this century, a third one has been added: The law of the iterated logarithm discovered in the simplest situations by Khinchin. In particular, Kolmogorov found a necessary and sufficient condition for the law of large numbers, and very significantly extended the limits of applicability of the law of the iterated logarithm. He made other fundamental discoveries. We shall mention only his definitive results related to the strong law of large numbers. We owe the concept of strong law to Borel. A sequence of independent random variables satisfies the strong law of large numbers if its sequence of arithmetic means converges almost certainly to some number (unlike the law of large numbers, where it converges in a weaker sense — in the sense of measure). Theorem 4. Let {{„} n gN be a sequence of independent equally distributed variables. Then the condition of finiteness of the expectation of |£i| (and hence of all |£ n |) is necessary and sufficient for it to satisfy the strong law of large numbers. One of the basic and natural questions is: What distributions are limits of sums of mutually independent normalized terms? It turned out that under natural assumptions the answer was only the infinitely divisible distributions introduced by Bruno de Finetti. A random variable subordinate to such a law can be represented as sum of n independent equally distributed terms, for any n. The problem then was how to describe infinitely divisible laws.
128 Theorem 5. Let $(•) be a characteristic function of an infinitely divisible distribution with finite dispersion,
itx)d\/x2,
JR
where (3 £ R, and dX is the Borel measure on R. Later, using a completely different method, P. Levy removed the requirement of finite dispersion (replacing the last term by {eitx - 1 - itx) i i ^ L - dX{x))
J£'
and then Khinchin showed that Levy's general result could be easily obtained by Kolmogorov's method. "In this way," Khinchin wrote, "this method gives the simplest and clearest substantiation to-date of the canonical form of infinitely divisible laws, making their totality easy to visualize." We spoke earlier of the theory of stochastic processes, and here we shall mention only two of Kolmogorov's remarkable results, concerning stationary random sequences: A criterion for the regularity of a sequence (i.e. a criterion that observation of a sequence implies the influx of new information) and a formula for the error of prediction. The question of regularity is primary in the theory of stationary sequences and processes. M. G. Krein solved it for processes. In mathematical statistics, Kolmogorov proved one of the most fundamental results in this discipline. Given a random variable £, whose distribution function is F(-), let {xj}™=1 be the result of n independent observations of this variable arranged in increasing order. The function 0
Fn(x) = { k/n 1
if x < x\, if Xk < x < Xk+i, if x > xn
is called the empirical distribution function. One of the fundamental questions in statistics then arises: To what extent does the empirical distribution function Fn(-) "resemble" the function F(-) itself? We then ask: Is it true that P(sup \F(x) — Fn(x)\ < e) —¥ 1 as n —> oo? Glivenko gave the answer to this question, and in the same issue of the journal that he did so, Kolmogorov not only answered this question, but he also gave the limit distribution of the variable Dn = -y^nsup \Fn{x) — F(x)\. More precisely, the following theorem holds: Theorem 6. Let F(-) be a continuous distribution function. $(A) = P{Dn(x) < A} as n —• oo uniformly in A tends to $(A) = ^ ( - l ) f c e x p ( - 2 A ; 2 A 2 ) fcez [Kolmogorov's
distribution).
Then the probability
129 The question on how to measure the degree of approximation of Fn(-) to F(-) occurred to such great scholars as von Mises and Cramer, but they were far from answering it. It is only natural that Kolmogorov's test is found in all textbooks on mathematical statistics. There is one result in topology of which Kolmogorov spoke with satisfaction and of which he was, I think, proud. A map / : X —> Y of a topological space X onto a topological space Y is called open if the image of any open set under this map is open. "The problem of whether the dimension can increase under an open map was of very great interest to P. S. Alexsandroff. For some time we worked together on proving the nonexistence of an open dimension-raising map. During these attempts the reasons for our failure were gradually revealed. This analysis of our failures led in the end to a counterexample." Thus they proved the following result. Theorem 7. There exists an open map of a one-dimensional compactum onto a two-dimensional one. This work acted as a stimulus in other areas of research, among which we must first mention the work of L. V. Keldysh. In particular, by a very complicated procedure, she constructed an open map of a one-dimensional compactum onto a square. Many years later, I. M. Kozlovskii constructed an open map from a onedimensional compactum onto a square by a very transparent and simple method, using Kolmogorov's construction as a fundamental element. In 1956, Kolmogorov obtained a result on the summation of random variables, of which he had been dreaming since the thirties. He was clearly satisfied when he reported on this work at a seminar. We have already spoken of infinitely divisible distributions. We denote the totality of such distributions by D. Let F(-) be the distribution function of the random variable £ and <&„(•, .F) be the distribution function of the sum of n independent random variables that have the distribution F(-). We recall that only infinitely divisible distributions can be limits of sums of independent equally distributed random variables. Is it true that the distributions of sums of equal independent terms are uniformly close to D? The answer is given by the following theorem. Theorem 8. There is a constant C such that for any distribution F(-) and for any n G N there is a \t(-) £ D such that
I^F)-^)!^^-1/5. At once the question arose about the exponent 1/5: Can it be improved? Among those working on this were Prokhorov, Meshalkin, and Kolmogorov himself. The definitive solution was found by T. V. Arak in 1981-1982; the optimal exponent turned out to be 2/3. Do functions of several variables exist? That would seem to be a pointless question; of course they exist. Let us phrase this question more precisely. Some functions of three variables can be expressed in terms of functions of two variables, say, fi(x,y,z) = (
130 The conviction that functions of three variables are not reducible to functions of two variables (in the sense that not all are representable as superpositions of functions of two variables) was so great that Hilbert (Hilbert again) indicated a concrete analytic function of three variables which (as he thought) could not be represented as a superposition of continuous functions of two variables (Hilbert's thirteenth problem). Alas! this expectation was not to be fulfilled. It turned out that any continuous function of any number of variables can be represented as a superposition of continuous functions by using only one function of two variables, namely s(x, y) = x + y, and the remaining functions are of one variable. Here is the story of this discovery. In 1956, Kolmogorov proved in an article that any continuous function of four variables can be represented as a superposition of continuous functions of three variables. Refuting Hilbert's conjecture was reduced to a concrete problem about the representation of functions defined on universal trees in R3. In Kolmogorov's opinion, this paper of 1956 was his most complicated technical achievement, requiring the longest period of uninterrupted thought in his life. Once again he left the final step to his successors. In the spring of the following year a third-year undergraduate student named V. I. Arnold obtained the necessary result about functions on universal trees, and thus Hilbert's thirteenth problem was completely solved (i.e. Hilbert's conjecture was disproved). Soon after that Kolmogorov devised a comparatively simple, natural and very beautiful construction, which led to the following remarkable theorem, which we have already mentioned. Theorem 9. For any integer n > 2 there are continuous functions ipij(-) defined on I — [0,1] such that any function f continuous on In can be represented in the form 2n+l
f(xu...
,xn) = ^2 Xil
/ n
^2^ij(xj)
where the functions x(-) ore continuous on R. We conclude this section devoted to results with a theorem relating to the ergodic theory of dynamical systems, in which a solution was given to a problem raised by von Neumann and dated back more than 25 years. A dynamical system is a pair (T,A), where T = (T, E, fx) is a measure space, and A : T —> T is a measure-preserving map. In the space L2(T) the map A generates a natural unitary operator Ux(t) = x(At). The spectrum of this operator is called the spectrum of the dynamical system. The problem of the classification of dynamical systems up to an isomorphism was studied for many years as the basic problem in ergodic theory. At the beginning of 1930, von Neumann proved that two dynamical systems with a common point spectrum are isomorphic. It was assumed to be plausible that the spectrum always uniquely determines the dynamical system. "Bernouilli schemes" provide an example of dynamical systems with a special kind of spectrum. We suppose that independent tests have been carried out, for which 1 occurs with probability p, and 0 with probability 1— p, 0 < p < 1. Such a procedure gives a measure on the set of all sequences of 0's and l's, and a shift by one step in such sequences generates a dynamical system, known as a Bernoulli scheme. Theorem 10. Bernoulli schemes for different p are nonisomorphic.
131 This led to the solution of von Neumann's problem and signaled the beginning of a period of rapid growth in this theory. We have cited only ten results from the rich collection of Kolmogorov's theorems. Again we must express our astonishment at the number of fundamental original problems (therefore so simple in their formulation) that he managed to solve. Now it is time to turn to the ideological aspects (ideas) of Kolmogorov's creativity. In his early period, Kolmogorov spent a lot of time on the interpretation (within the framework of the axiomatic method) of the limits of the fundamental concepts of mathematics. We give his answers to questions about the most primary structures. What is an integral? Recall the evolution of this concept: Archimedes, Newton, Leibnitz, Cauchy, Riemann; then the Lebesgue integral, then the integrals of Stieltjes, Hellinger and others; later the integration of vector-valued functions (in infinite-dimensional Frechet space). Kolmogorov proposed the following simple definition of integral, which covers all the above. Let T be some set and M. a distinguished system of subsets of T such that A £ M, B £ M. ^> An B £ A4. Further, assume given a generally speaking multivalued function with values in a vector space, defined on any A £ M.. We define the integral of / on the set T for a given system M. as the limit (in the sense of Shatunovskii-Moore-Smith) of the Riemann sums
(Rf)r(T) = £
f(Tn),
where T = {Tn} is a partition of T, i.e. T = UnTn, % l~l Tj = $, i ^ j . We denote this integral by M. = J fdT. This construction subsumes all the concepts of integration we spoke of earlier (Riemann, Lebesgue, Stieltjes, and so on). It implies, in particular, the following most unusual fact: The Lebesgue integral is a limit of Riemann (not Lebesgue!) sums. Kolmogorov had a high opinion of his paper "Study of the concept of the integral" and often expressed regret that its definition had not yet taken root. Is it possible to extend substantially the concepts of derivative, integral and summation of divergent series? This was the problem considered by Kolmogorov in a paper of 1925. It turned out that the search for effective methods of differentiation, integration or summation of series came up against the same difficulty as the effective construction of nonmeasurable functions. For example, there is a method of summing divergent series that satisfies the conditions
(a) ^ Xn = Xi + Y^ Xn+1 , n€N
n€N
(b) ^2 «6N
aX
n = a XI
Xnj
ngN
and if by using this it is possible to define the sum of the series of the form ^ n g N s i n 3 n x , then it is possible to construct an effective example of a function that is nonmeasurable in the Lebesgue sense. What is the ^-dimensional measure of a set situated in n-dimensional Euclidean space? In his paper of 1933, Kolmogorov approached the problem from an axiomatic point of view. The following result holds. Definition. Let fi be a set function defined on all Suslin sets E £ R " (<^> E £ Sus(R™)) and satisfying the following axioms:
132 (1) E C U £ n , (E,En e Sus(R")) => n{E) < £ n e N / i ( £ „ ) , (2) Et G Sus(R"), ^ n £,- = $, u? = 1 Ei C £ =*• n = i M(#0 < M ( # ) , (3) M (J") = 1, where Jfc = [0, l] fc cube, denotes the k-dimensional Euclidean cube, (4) E' :=
A{E) = u(£ S l n ESlS2 n ESlS2S3 n • • • n £„...,„), where the union is carried out over all the sequences {s„} n 6 N. Then we say that A{£) is obtained from £ by A-operation. In 1916, the same year, Suslin obtained a result that made a great impression on his contemporaries. He proved that there are sets obtained from closed subsets of an interval by means of an A-operator that are not Borel sets. Sets obtained from closed ones by means of an ^-operator were called Suslin sets. Thus the class of Suslin sets (or A-sets) proved to be broader than that of Borel sets (B-sets) and it was soon discovered that many natural procedures of analysis lead to A-sets and j4-measurable functions that are not 5-measurable. In the autumn of 1921, Alexsandroff gave a course of lectures at the University of Moscow on descriptive set theory. Apparently he spoke in this course of the reoperation he had introduced — an operation "complementary" to the A-operation. Kolmogorov attended this course. He immediately tried to work out a maximal general scheme of set-theoretic operations (tf5-operations). Every map / : N ->• N generates a sequence {/(n)}„ 6 N- Let T = {/} be some family of such maps and £ = { E „ } „ 6 N some sequence of sets (for definitenesssubsets of the interval I — [0,1]). Definition. If the set $?(£) is formed by the rule $jr(5) = U/ejr ( n „ 6 N E / ( „ ) ) ,
133 then we say that it is obtained from £ by means of the Ss-operation generated byT. An A-operation is obviously a ^-operation. Kolmogorov defined in a general form a complementary ^-operation, of which AlexsandrofF's T-operation is a special case. Kolmogorov's first journey into mathematical research was devoted to these questions. The work was completed in January 1922, while he was a second-year undergraduate student. The paper of that particular research contains the germ of the general theory of operations on sets, which was later developed by Hausdorff, Kantorovich, Lyapunov and others. In the paper he proved the following fundamental result: There is a set of the form &?(£) the complement of which is not a set of the form $^r(5') for any sequence £' = {E'n}neN of (closed) sets E'n. Two of Kolmogorov's papers, from 1925 and 1932, are on mathematical logic. The 1925 paper is, in fact, the first paper on mathematical logic by a Russian mathematician. It was of great significance; witness its inclusion in Heijenoort's famous book on mathematical logic From Frege to Godel. From the beginning of the century there was no end to arguments about the essence of mathematics or the applicability of logical laws or paradoxes in set theory. All this led to a new line, "intuitionism". This recognized the invalidity of the application of "tertium non datur" (the law of the excluded middle) in the field of transfinite induction. In his paper "Tertium non datur", Kolmogorov proposed an interpretation of classical mathematics in which all of its propositions are converted into intuitionistic propositions. Thus, Kolmogorov proved that all finite conclusions obtained by means of transfinite application of the "tertium non datur" principle are true and can be proved without its help. In the 1932 paper, Kolmogorov proposed investigating "the logic of the solution of problems" (say, problems on construction) side-by-side with "the logic of proof. To the principle of syllogism: "if b follows from a, and c follows from b, then c follows from a", there corresponds "if the solution of problem a reduces to a solution of problem b, and the solution of problem b reduces to a solution of problem c, then the solution of problem a reduces to a solution of c". Kolmogorov introduced the corresponding symbolism and constructed a calculus of problems. The form of this calculus coincided with the system that was formulated slightly earlier by Heyting, of the axioms of intuitionistic logic. This gave another interpretation of intuitionistic logic. Kolmogorov published two papers on geometry. One of them ("On the foundations of projective geometry", 1932) reflects his youthful love of projective geometry "which was old-fashioned, but on which Aleksei Konstantinovich Vlasovc lectured in a truly talented way" (Kolmogorov always listed Vlasov among his teachers). In the early 1930s "synuhetic structures" were being developed extensively, when algebra united with topology (recall the representation theory of compact groups). Kolmogorov was at the beginning of "topological geometry". How beautiful and unexpected the links between geometry and topology can be is shown by the following result of Kolmogorov.
Professor of Moscow University; Kolmogorov heard his lectures devoted to projective geometry, when he was s student.
134
Let three systems of elements (called points, straight lines and planes) satisfy the projective axioms of combination, where each of the systems is a connected compactum and the relations of combination are continuous.01 Then a geometry that has the three given systems as objects is isomorphic to either R P 3 , C P 3 or H P , i.e. either real, or complex, or quaternion projective geometry. Here the most important thing is the idea of combining geometry and topology, along with the aesthetic element included in the formulation. The main point of the proof is the reference to Pontryagin's theorem (proved on Kolmogorov's suggestion) giving a description of all the connected locally compact topological skew-fields with countable base (there are only three: R, C and H). These facts undoubtedly belong to the golden treasure-house of topological algebra. This fully justifies the thought repeatedly expressed by Kolmogorov: Aesthetic motives not infrequently have interesting consequences. Four papers by Kolmogorov are concerned with functional analysis. In each of them there are important ideological elements. In a paper of 1931, Kolmogorov gave the first criterion for compactness in Lp. Later M. Riesz gave a similar criterion to the Arzela criterion — more usual and natural, but Kolmogorov's result remains significant, since it can easily be extended to any measure space, whereas Riesz's cannot. The best-known of these papers on functional analysis is undoubtedly a paper from 1934. This paper contains the definitions of a linear topological space and a bounded set in it (in his historical survey, Bourbaki singles out the importance of these concepts for the further development of functional analysis) and in which the following classical criterion of normability is proved: for a linear topological space to admit a norm it is necessary and sufficient that it has a bounded convex neighborhood of zero. At the source of the theory of normed rings created later by I. M. Gel'fand, is their joint paper of 1935 which contains the following beautiful result: Let T\ and T2 be two topological spaces (satisfying the first axiom of countability). For Ti and T2 to be homeomorphic it is necessary and sufficient that the rings C(T\) and CiT?) (of continuous functions on T\ and T2, respectively) or the rings Cb(T\) and Cb(T2) (of continuous bounded functions) be algebraically isomorphic; that is, an algebraic isomorphism of the algebras of continuous functions determines a topological homeomorphism of the compacta themselves. In Kolmogorov's papers on approximation theory, the formulation of the problem of revealing the connection between the concepts of approximation and smoothness is central. Kolmogorov owned the origination for the very important concepts of n-diameter and £-entropy. Kolmogorov made a fundamental contribution to algebraic topology. To him we owe the introduction of what is perhaps the central concept in the whole of this field — that of cohomology, one of the concepts fundamental to the whole of modern mathematics. Kolmogorov also defined the most significant operation in cohomology groups: the product operation, giving a ring structure. The American topologist Alexander had the same idea at the same time. Kolmogorov completed his own theory with the remarkable discovery of the duality We say that a straight line passing through two points depends continuously on these two points, and so on.
135 for closed sets in compact spaces. In the mid-1950s, Kolmogorov turned to the theory of algorithms and the theory of automata and set himself a task close in spirit to the one we discussed in connection with the theory of operations on sets. It is very well expressed in the notes to his 1958 joint paper with V. A. Uspenski: "Authors striving to interpret the concepts of 'a computable function' and 'an algorithm' have tried to survey the existing variants of the definitions in the literature and finally convinced themselves that there is no hidden possibility of extending the range of the concept of computable function'. The result of this work is the broadest (apparently, of all possible) definitions of the basic objects of the theory-algorithms and automata." In ergodic theory, Kolmogorov introduced two fundamental concepts — metric entropy (Kolmogorov's idea was definitively formulated by Ya. G. Sinai) and a quasiregular dynamical system. In almost every field to which he turned his attention, Kolmogorov initiated one or several principal scientific lines. From his research, tens or hundreds of research workers began to work, survey articles, monographs began to appear and scientific conferences were arranged. For example, in 1930s in the field of probability alone he laid the foundations of the theory of probability and of stochastic processes; he began to develop Markov processes, the theory of Markov chains with a countable number of states; in statistics, the theory of statistical criteria and (jointly with Khinchin) the theory of stationary processes. These directions appear as separate chapters in modern courses on probability theory, mathematical statistics and stochastic processes, and if we look at the content of these courses, we get the impression that the sum total of these new directions created by one man in the course of less than ten years can be compared with the sum total of the new directions that have emerged in these sciences over the last fifty years. Kolmogorov's ideas always had a remarkable vitality. They were quickly taken up, and eventually overtaken by successors, and were interwoven with others, forming scientific directions and schools. There is a multitude of methods, constructions and lemmas that Kolmogorov devised to solve individual problems, and they are constantly being used in further research. He developed particularly many methods in the theory of probability and stochastic processes (starting, say, with the well-known Kolmogorov inequality); he introduced many constructive ideas into the theory of functions, topology and other fields. For example, the number of uses of the comparison theory — an auxiliary result, which he obtained when proving the theorem on the intermediate derivative — is exceptionally large. The same applies to the principle of introducing a new invariant of entropy type to prove the nonisomorphism of various aggregates. But what had the greatest effect was his method of solving functional equations with small denominators — a special kind of implicit function theorem — which is the basis of KAM-theory. Peculiarity of the Creative Manner The creative power of a scholar can be seen in his achievements in the arts, whether in his originality as a writer, as a painter, or as a poet. The legacy of a great scholar, as any cultural phenomenon, has its own history, its own architecture, and possibly its own undeciphered plan. It is linked to profound conceptions, with consequences in culture in general.
136
At a superficial glance, Kolmogorov's genius may seen to be a pile of isolated and unrelated parts. Kolmogorov himself, it would seem, sometimes confirmed this verdict. "For me, prone to lack singleness of purpose ..." he wrote once to his teacher N. N. Luzin, and he repeated something similar many times. But if we consider his creativity more intently, we see that his efforts were all subject to one general plan and the diversity of what was planned and achieved was the result of the infinite number of goals he set for himself. Observing the creativity of great men brings us closer to discovering the nature of man. There are two opinions about the nature of genius. Thomas Mann — Kolmogorov's favorite author — dedicated his last work "Doktor Faustus" to one of them. For Mann, a genius is a person who, in order to achieve his goal which requires surmounting impossible difficulties, is compelled to make a pact with dark, demonic forces. In the novel the hero, Faust, consciously infects himself with a fatal disease, so that his inflamed brain can reflect the transcendental images and ideas. But there is another point of view. In Russian culture it is most clearly expressed in Pushkin's creative work (indeed in his own personality). In Pushkin's short tragedy 'Mozart and Salieri', Mozart was the shining chosen one who was given the gift of hearing the divine voice. It was most likely that Pushkin did not know the words written by Mozart in one of his letters, but he had a profound understanding of the essence of the very phenomenon of Mozart's genius. Mozart wrote, "Ideas come to me in quantities and with unusual facility. Where do they come from? I know nothing at all of this ... The work grows, I hear it more and more distinctly ... Then I comprehend it in one glance." To a large extent, the great discoveries made by Kolmogorov saw the light of day in a similar way. Kolmogorov was undoubtedly one of the "shining" geniuses. Over and over again one and the same motif crops up in his publications and discussions. Once, the remarkable Soviet geometer Boris Delone, appeared before participants of the School of the Mathematical Olympiad, and had expressed the idea that the only difference between a great scientific discovery and a good Olympiad problem is that solving the Olympiad problem requires five hours, but finding a powerful scientific result requires an investment of 5,000 hours. (Apparently this was the way Delone himself worked, and this was the manner in which many others such as Hilbert worked — we shall compare further the creativities of Hilbert and Kolmogorov.) Kolmogorov often recalled those words of Delone. But each time the talk turned to these famous 5,000 hours, he'd say with a feeling of some awkwardness and even of irritation, that he himself was unable to concentrate his thoughts for such a long time on one and the same problem. In an interview, Kolmogorov was asked the question "How do you work?" He said, "You read some books, you prepare your own lectures with some new variations or other, and suddenly, from the soil of this everyday work, some unexpected idea emerges, and, vaguely as yet, some completely different approach can be seen. Then having worked it out, almost everything else is neglected — and one thinks, thinks endlessly along the lines which have just appeared. Fortunately, I usually had the opportunity to do this, but in the whole history of my scientific discoveries such complete oblivion, cut off from everything else, might last for a week, sometimes possibly for two — not more." And then the sudden inspiration would come. Thus it was with the discovery of the trigonometrical series divergent almost everywhere, which was preceded by
137 three days of uninterrupted reflection and complete concentration; thus it was the final result, connected with Hilbert's thirteenth problem, and with the entropy invariant for dynamical systems and many others. After concentrating enormous energy there suddenly came the moment when the whole picture of the phenomenon appeared in its complete form. And then what? Wilhelm Ostwald's classification of scholars is well known. He divides them into classicists and romantics. "The first," Ostwald wrote, "can be compared to a mill, carefully grinding with logical millstones the initial ideas, in order to extract from them the most far-reaching possible consequences and to develop the theory to the limits of possible completion and perfection. The latter one must rather equate with generators of new ideas. Having stated the idea, they quickly lose interest in it, however brilliant it may have been, and take no part in the further development of it." This opinion was often quoted. I am not completely satisfied with Ostwald's classification, but the label "romantic" suits Kolmogorov very well. This accumulation of energy over a short span of time, which we spoke of earlier, led to a powerful explosion that caused gaping holes in what one would think of as impregnable bastions. Suddenly, there rushed in tens and hundreds of followers. Usually, Kolmogorov did not pursue the matter further; it was as though a creative tiredness would set in and he already had in his mind another goal in view. Something like that happened to him almost all the time. That was the case with the solution of Hilbert's thirteenth problem. When he had almost reached his goal, he declared that he would leave the completion of the work to his successors; and it was the same with problems in classical mechanics, ergodic theory and many other subjects. Kolmogorov was very much interested in the essence of creative talent. He distinguished algorithmic, purely geometric and logical faculties in man. Now, a division into two groups is more common (analytic and descriptive geometric), corresponding to the different functions of the two hemispheres of the brain. I find it difficult to say which of the qualities mentioned above were the most prominent in Kolmogorov's case. Analytic papers with an abundance of calculations and transformations exist side-by-side with his papers where "the art of a consistent, correctly partitioned logical argument" plays a fundamental role. And besides, in his research in function theory and topology, for example, there are geometric constructions which stand out by their beauty. By the way, Kolmogorov was left-handed from birth, and as a child he taught himself to use his right hand, which he did well. We cannot rule out that he may be one of that rare breed of men, each of whose hemispheres fulfills both functions, so that asymmetry is really nonexistent. An essential part of an intellect is its quickness of thought. P. S. Alexsandroff once spoke jokingly of "quick" and "slow" geniuses, and he included Hilbert among the latter. Kolmogorov was undoubtedly one of the "quick" geniuses, but what stuck me personally most of all about him was not so much the quickness of his thinking as the quickness of his grasp and his perception. I was often a witness of the way scarcely perceptible emanations or very vague rough drafts which came to his attention, were immediately incorporated into an ordered and complete system. (Sometimes one had the impression that Kolmogorov was one of those people who
138 have as it were "a priori" knowledge — it seemed he knew everything, although it was not quite clear when he had learned all of it.) And it was unusually interesting to observe how suddenly, at the right moment, things surfaced in his conciousness such as the universal tree of Menger and Kronrod's constructions, or the ideas of Poincare—Bogoliubov-Kryolv and the NewtonKantorovich method, the Pontryagin-Shnirei'man designs and Shannon's theory. It is natural to raise the questions: Who had the greatest influence on Kolmogorov, on "whose shoulders" did he stand, on what is his creativity mainly based? This subject is still awaiting a more careful study, and we shall only touch upon it. In the first volume of Kolmogorov's selected works there are 60 articles. In none of them does he express any thanks to his teachers or colleagues (for help, for useful advice, for suggesting the problems, or anything else). In 15 of these articles there is no bibliography at all, and in the others the work of 93 mathematicians is mentioned. Of scholars of former ages, only Aristotle and Leibniz are cited, of modern times (and in a not very important context) only Grothendieck is mentioned. All others are famous scientists from a preceding generation and from his own generation. Of his predecessors the most significant references are to Hadamard, Birkhoff, Borel, Brouwer, Hilbert, Caratheodory, Lebesgue, Luzin, Taylor, von Karman, Hardy and Hausdorff; in other volumes to Chebyshev, Bernstein, von Mises and Fisher. I think that is precisely these individuals who must be included among those of his predecessors that he was most indebted to. Somewhat surprising is the absence of references to Poincare. This is largely because Kolmogorov learned of Poincare's ideas by reading the works of Chazy and Chariier. The other mathematicians to whom Kolmogorov refers to were part of the current scientific scene. Here we must mention the great influence the works of Krylov-Bogoliubov and de Rham had on him. More than once in this article there has appeared the name of one of the greatest mathematicians of all time, David Hilbert. Here I would like to draw some comparisons between the genius of Hilbert and that of Kolmogorov. Intellectual achievements cannot be completely classified (they make up "a partially ordered set"), and geniuses cannot be compared. My comparison of them arises not from a wish to contrast one with the other or to elevate the one at the expense of the other, but from an attempt to demonstrate how far creative styles can differ and by what different paths they can achieve their great aims. Hilbert had a determination not matched I dare say by anyone else's. His scientific biography is sharply divided into periods, dedicated to work in one field. Hilbert's first creative period was devoted to algebra and arithmetic (the theory of invariants 1885-1893, algebraic number theory 1893-1898); in the first decade of the century in the course of two summers (1908-1909) he solved Waring's problem. Kolmogorov did not write any papers on algebra or arithmetic, just as Hilbert did not work in the field of random phenomena. The next stage of Hilbert's interests was geometry (1989-1902). It ended with his classic book Grundlagen der Geometrie. Kolmogorov did work in the field of geometry, in topology and algebraic topology in the 1930s, while working at the same time in the field of analysis, probability theory, function theory and many others. Then came Hilbert's "analytic" period (1900-1910), when he developed the calculus of variations and infinite-dimensional analysis (in particular, the theory
139 of integral equations). Kolmogorov turned to problems of mathematical analysis and the theory of functions in two main periods of his life (trigonometrical and orthogonal series, the theory of approximation, functional analysis in the 1930s, dynamical systems, superpositions, ergodic theory in the 1950s). Then Hilbert went through the period when he solved mathematical problems in the natural sciences, mainly in mathematical physics (1910-1922). Kolmogorov studied many problems in natural sciences: Biology and physics in the 1930s, turbulence, geology and meteorology in the 1940s, celestial mechanics in the 1950s, and oceangraphy in the 1970s. Hilbert's last period was devoted to the logical foundations of mathematics. Kolmogorov was interested in logic both in the 1930s and 1950s, which was in the very last period of his creativity. He, like Hilbert, concluded his creative life with work on logic. As we said, Hilbert did no work on random phenomena. The theory of probability, on the contrary, was a constant concern of Kolmogorov's. He was involved in it in from 1920s to 1950s, and moreover, he was not just involved in classical problems, but also in random processes, statistics, information theory and numerous applications. Hilbert could only work on one subject at a time. Kolmogorov, at the height of his creativity, could work down to the foundations, the roots, the very essence of the subject; Kolmogorov was striving to go forward, new goals beckoned him, he tried hard to conquer unattainable heights, leaving it to others to incorporate the new territories. But both achieved brilliant results, they introduced fundamental ideas, worked out deep theories, created fruitful methods and basic concepts. Furthermore, they were both great teachers. The number of mathematicians who they influenced cannot be counted. But, the number of their actual students can be compared. The list of Hilbert's disciples is given in his collected works. The specialities of the students correspond exactly to the various interests of their teacher at the given time. And now I would like to present a list of Kolmogorov's pupils with whom he had worked during the decade 1953-1963: V. M. Alekseev (classical mechanics), V. I. Arnol'd (superpositions, classical mechanics), G. I. Barenblatt (hydrodynamics), L. A. Bassalygo (information theory), Yu. K. Belyaev (stochastic processes), L. N. Bol'shev (mathematical statistics), A. A. Borovkov (probability theory), R. L. Dobrushin (probability theory), E. B. Dynkin (stochastic processes), V. D. Erokhin (approximation theory), S. V. Fomin (ergodic theory), I. M. Gel'fand, B. V. Gnedenko (probability theory), L. A. Levin (complexity), A. I. Mal'tsev (mathematical logic), P. Martin-Lef (complexity), Yu. T. Medvedev (mathematical logic), L. D. Meshalkin (probability theory, ergodic theory), V. S. Mikhalevich (probability theory), A. S. Monin (oceanology, turbulence), S. M. Nikol'skii (approximation theory), A. M. Obukhov (atmospheric physics, turbulence), M. S. Pinsker (information theory), Yu. V. Prokhorov (probability theory), Yu. A. Rozanov (stochastic processes), B. A. Sevast'yanov (stochastic processes), A. N. Shiryaev (stochastic processes), Ya. G. Sinai (ergodic theory), S. Kh. Sirazhdinov (probability theory), V. M. Tikhomirov (approximation theory), V. A. Uspenskii (mathematical logic), A. M. Yaglom (turbulence) and V. M. Zolotarev (probability theory). Kolmogorov did most of his research work in his house in Komarovka, outside Moscow. He loved this house very much and there he played the role of gardener. As a result of his fifty years of work in the Komarovka garden, it had huge birches,
140
larches, maples, crab-apples, and the whole garden was full of flowers-lilacs, roses, and peonies. Whatever Kolmogorov planted flourished. Once, many years ago, he gave a colleague a bush of sweet briar. This colleague was no longer alive and now little remains of the once rich and well-tended garden. Only the sweet briar, given by Kolmogorov, to this day does not cease to flower and bear fruit. This luck of a gardener illuminated both his original research and his work as a teacher; everything he touched flowered and bore fruit. Let me go back to where I started. The heritage of a great man is rich beyond price; the comprehension of which confirms our faith in mankind, and the personality of a genius sets a moral example and let us discover directions in life. What is conducive to a genius? What sets him in motion? How splendidly and happily were combined in the fate of Andrei Nikolaevich Kolmogorov a cheerful childhood, a remarkable secondary school, an unusual time resulting in an atmosphere of creative elan during his youth, and the outstanding School of Mathematics at the University of Moscow, early recognition, and an unusual gift of originality ... But let us not forget that without which even the combination of all these qualities would not lead to such a beneficent outcome: the presence of inner stimuli in the final analysis on the fundamental moral perseverance of the individual. Kolmogorov himself once said on this subject: "It is absolutely essential in science (as in poetry, music, etc.) that the man who has the right moral qualities should accept his work as a crucial duty." And here, it seems to me, the great commandment: the greatest stimulus in life is the voluntary acceptance of one's duty. In answer to the question: "By whom were you guided in life?" Kolmogorov replied thus: "I always considered that the most important thing is truth." Striving for truth is one of the basic aims in the life of a genius. Kolmogorov is one of those incomparable people who embellish life by the simple fact of their existence. The mere awareness of the fact that somewhere on earth, beats the heart of a man who is endowed with such perfect wisdom and such a pure soul gave wings, gave joy, gave the strength to live, and protecting us from evil and inspiring us to good deeds.
141 On Normability of a General Topological Linear Space 1
Our discussion is based on the following definition of topological linear space. A set E is a topological linear space if (a) for the elements of E the operations of addition and multiplication by real numbers are defined which satisfy the axioms for a linear space 2 ; (b) for any subset A of E its closure A C E is defined, which satisfies three axioms for a topological space 3 ; (c) the operations of addition and multiplication are continuous. Open sets of a topological space in the sense of footnote 3, regarded as neighborhoods of the points of these sets, satisfy in general the first three Hausdorff axioms 4 and the first axiom of separation. However, in our case of topological linear spaces as well as in the general case of topological groups, the second and third axioms of separation are necessarily fulfilled, i.e. the space is regular. This fact will be proved in §1. In this connection it seems quite natural that the general theory of linear functionals and operators should be developed in particular for topological linear spaces. A considerable part of this theory, however, has been developed only for the case of normed spaces, i.e. spaces where each element is associated with a non-negative number \x\ satisfying the following conditions: \ax\ = \a\ \x\, \x + y\<\x\
(1)
+ \y\;
(2)
and the topology of the space is determined by the distance between two elements: p{x,y) = \x-y\.
(3)
The question arises: which topological linear spaces can be normed? In other words, what conditions should be imposed so that a topological linear space can be endowed with a norm satisfying conditions (1) and (2) and determining the a priori topological relations in this space? In order to answer this question we will use the following definition.5 1 "Zur Normierbarkeit eines allgemeinen topologischen linearen Raumes", Stud. Math. 5 (1934), 29-33. 2 See S. Banach, Theorie des operations lineaires, Warszawa, 1931, Chap. 2. 3 See, for example, P. Alexandroff, "Uber stetige Abbildungen kompakter Raume", Math. Ann. 96 (1926), 555. These axioms read as follows:
1. A subset A C E consisting of at most one element coincides with its closure. 2. A = A for any set Ac E. 3. MUN = MUN. 4
See F. Hausdorff, Grundziige der Mengenlehre, Berlin, 1927, 227-229. S . Mazur and W. Orlicz, "Uber Folgen linearer operationen", Stud. Math. particularly p. 152. 5
4 (1933), 152-157,
142 Definition. A set A C E is said to be bounded if for any sequence {an} of real numbers and for any sequence {xn} of elements of A the condition an —>• 0 implies a-n^n —• 0 (where 0 in the latter relation denotes the origin of the space E). This definition enables us to state the following theorem: T h e o r e m . For a space E to be normable it is necessary and sufficient that in E there exist at least one bounded convex neighborhood of zero. Here the set A C E is said to be convex if x £ A, y £ A, A > 0 and fi>0
^±™£A.
imply
(4)
This theorem will be proved in §2. §1. In this section, E is an arbitrary topological group, i.e. an arbitrary topological space on which an operation of addition is defined satisfying all the group axioms, and the addition and subtraction are continuous. We will prove that in this case E is regular or, in other words, that for any neighborhood U(XQ) of an element XQ £ E there exists a neighborhood V(xo) which belongs to U(XQ) together with its closure. Using the transformation x' = x — XQ we can reduce this problem to the case XQ = 0. Thus, let a neighborhood U of zero be given. Since 0 + 0 = 0, we can, by the continuity of addition, find a neighborhood of zero V such that x' £ V, x" £ V implies x' + x" £ U. Now let x £ V. Since x — x — 0 and subtraction is continuous, there exists a neighborhood W(x) of x such that x' £ W(x), x" G W(x) implies x' — x" £ V. We now find a point x* that belongs simultaneously to W(x) and V (x* exists since x £V). Since x and x* belong to W(x), it follows that x — x* £V. Further, since x — x* and x* belong to V, it follows that x = (x — x*) + x* belongs to U, whence VCU, (5) and so on. §2. The necessity of the conditions of the theorem is obvious since the unit ball, consisting of all points with \x\ < 1, is a convex bounded neighborhood of zero. It remains to show that the conditions of the theorem are sufficient. Let U be a convex bounded neighborhood of zero. We denote by aU the set of points x = ax' such that x' £ U. For any a ^ 0 the set aU is a convex bounded neighborhood of zero. We define the norm by |a;| = sup \a\,
x £ E\ aU ,
(6)
where the supremum extends over all a satisfying the condition x £ E\ aU. It can be seen immediately that the norm defined by (6) satisfies condition (1). We now show that condition (2) also holds. To this end we first note that x £ aU and y £ aU imply Xx
+ ™£aU X+H for all A > 0 and /x > 0 and therefore it follows from |x| < a and \y\ < a that Xx + fiy —;
S ct.
143 Putting |a:| = A, \y\ = p., X + fx = a, a j\x\ \x + y\
y' = -y we obtain i 'i
= a,
a
\
i
\y\ = - W = a >
Xx' + fiy'
+ \y\
It remains to show that the distance p(x,y) = \x — y\ determines the same limiting relations as in the space E. Let us consider the point 0 and prove that the sets all, a > 0 form a complete system of neighborhoods of zero. Let W be an arbitrary neighborhood of zero. We have to find a > 0 such that all is contained in W. If this were impossible, there would exist sequences an —• 0 and xn G U such that the points anxn would lie outside W for all n and therefore the sequence anxn would not tend to zero, which contradicts the boundedness of U. Therefore the sets all form a complete system of neighborhoods of zero. It can be seen immediately that the ball |a;| < a is contained in the set aU. Therefore, if we prove that 0 is an interior point of every ball \x\ < a, a > 0, this will imply that the system of balls |ai| < a, a > 0, is equivalent to the complete system of neighborhoods all of zero. But each ball \x\ < a, a > 0 contains a neighborhood of zero, coinciding with the intersection 6 of ^alf and —^all. Indeed, if x e \aU and x £ — \aU, then x lies in every neighborhood bU with |6| > a/2, whence it follows that |a;| < ka < a. Using the transformation x' = x — Xo we can see that the system of balls with center at XQ is equivalent to the complete system of neighborhoods at this point. Thus, the theorem is proved completely. 26 May 1934
144
LIST OF W O R K S B Y A. N. KOLMOGOROV 1923 1. Une serie de Fourier-Lebesgue divergente presque partout, Fund. Math. 4, 324-328. 2. Sur l'ordre de grandeur des coefficients de la serie de Fourier-Lebesgue, Bull. Acad. Pol. Sci. A, 83-86. 1924 3. Une contribution a l'etude de la convergence des series de Fourier, Fund. Math. 5, 96-97. 4. (with G. A. Sehverstov) Sur la convergence des series de Fourier, C.R. Acad. Sci. Paris 178, 303-306. 1925 5. La definition axiomatique de 1'integrale, C.R. Acad. Sci. Paris 180, 110-111. 6. Sur les bornes de la generalisation de l'integrale. (Paper No. 6 in the Selected Works of A. N. Kohnogorov, Vol. 1, Kluwer, 1991.) 7. Sur la possibility de la definition generale de la derivee, de l'integrale et de la sommation des series divergentes, C.R. Acad. Sci. Paris 180, 362-364. 8. Sur les fonctions harmoniques conjuguees et les series de Fourier, Fund. Math. 7, 24-29. 9. On the tertium non datur principle, Mat. Sb. 32:4, 646-667 (in Russian). 10. (with A. Ya. Khinchin) Uber Konvergenz von Reihen, deren Glieder durch den Zufall bestimmt werden, Mat. Sb. 32:4, 668-677. 1926 11. (with G. A. Sehverstov) Sur la convergence des series de Fourier, Atii Accad. Naz. Lincei. Rend. 3, 307-310. 12. Une serie de Fourier-Lebesgue divergente partout, C.R. Acad. Sci. Paris 183, 1327-1329. 1927 13. Sur la loi des grands nombres, C.R. Acad. Sci. Paris 185, 917-919. 14. (with D. E. Men'shov) Sur la convergence des series de fonctions orthogonales, Math. Z. 26:2/3, 421-441. 1928 15. On operations on sets, Mat. Sb. 35:3/4, 414-422 (in Russian). 16. Sur une formule limite de M. A. Khinchine, C.R. Acad. Sci. Paris 186, 824-825. 17. Sur un procede d'integration de M. Denjoy, Fund. Math. 11, 27-28. 18. Uber die Summen durch den Zufall bestimmer unabhangiger Grossen, Math. Ann. 99, 309-319.
145 1929 19. Bemerkungen zu meiner Arbeit "Uber die Summen zufalliger Grossen", Math. Ann. 102, 484-488. 20. General measure theory and calculus of probabilities, in Proc. Communist Academy Math. Sect. Vol. I, 8-21 (in Russian). 21. Contemporary dispute on the nature of mathematics, Nauch. Slo. No. 6, 41-54 (in Russian). 22. Uber das Gesetz des iterierten Logarithmus, Math. Ann. 101, 126-135. 23. Sur la loi des grand nombres, Atti Accad. Naz. Lincei. Rend. 9:6, 470-474. 1930 24. Sur la loi forte des grands nombre, C.R. Acad. Sci. Paris 191, 910-912. 25. Zur topologish-gruppentheoretishen Begriindung der Geometrie, Nachr. Ges. Wiss. Gottingen. Fachgr. 1 (Mathematik) 8, 208-210. 26. Untersuchungen iiber den Integralbegriff, Math. Ann. 103, 654-696. 27. Sur la notion de la moyenne, Atti Accad. Naz. Lincei. Rend. 12:9, 388-391. 1931 28. Uber die analitishen Wahrscheinlichkeitsrechnung, Math. Ann. 104, 415-458. 29. Sur le probleme d'attente, Mat. Sb. 38:1/2, 101-106. 30. The method of medians in the theory of errors, Mat. Sb. 38:3/4, 47-50 (in Russian). 31. Eine Verallgemeinerung des Laplace-Liapounoffshen Satzes, Izv. Akad. Nauk SSSR OMEN, 959-962. 32. Uber Kompaktheit der Funktionenmengen bei der Konvergenz im Mittel, Nachr. Ges. Wiss. Gottingen 9, 60-63. 1932 33. Theory of functions of a real variable, in Mathematics in the USSR During 15 Years, GTTI, Moscow, Leningrad, pp. 37-48 (in Russian). 34. Sulla forma generale di un processo stocastico omogeneo. (Un problema di Bruno de Finetti), Atti Accad. Naz. Lincei. Rend. 15, 805-808. 35. Ancora sull forma generale di un processo stocastico omogeneo, Atti Accad. Naz. Lincei. Rend. 15, 866-869. 36. Zur Deutung der intuitionistischen Logik, Math. Z. 35, 58-65. 37. Zur Begriindung der projektiven Geometrie, Ann. Math. 33, 175-176. 38. (with P. S. Aleksandrov) Introduction to the Theory of Functions of a Real Variable (GTTI, Moscow, Leningrad) (in Russian).
39. 40. 41. 42. 43.
1933 (with P. S. Aleksandrov) Introduction to the Theory of Functions of a Real Variable (2nd ed.) (GTTI, Moscow, Leningrad). Grundbegriffe der Wahrscheinlichkeitsrechnung (Springer-Verlag). Beitrage zur Masstheorie, Math. Ann. 107, 351-366. (with M. Leontovich) Zur Berechnung der mittleren Brounschen Flache, Phys. Z. Sow. 4 : 1 , 1-13. Sulla determinazione empirica di una legge di distrebuzione, G. 1st. Ital. Attuar. 4, 83-91.
44. Uber die Grenzwersatze der WahrscheinUchkeitsrechnung, Izv. Akad. Nauk SSSR OMEN, 366-372. 45. Zur Theorie der stetigen zufalligen Prozesse, Math. Ann. 108, 149—160. 46. Sur la determination empirique d'une loi de distribution, Uch. Zap. MGU 1, 9-10. 47. On the question of the applicability of prognosis formulas found in a statistical manner, Zh. Geofiz. 3:1, 78-82 (in Russian). 1934 48. (with I. Ya. Verchenko) On points of discontinuity of functions of two variables, Dokl. Akad. Nauk SSSR 1:3, 105-106. 49. Zur Normierbarkeit eines allgemeinen topologischen linearen Raumes, Stud. Math. 5, 29-33. 50. (with I. Ya. Verchenko) Continuation of the study of points of discontinuity of functions of two variables, Dokl. Akad. Nauk SSSR 4:7, 361-362 (in Russian). 51. On convergence of series in orthogonal polynomials, Dokl. Akad. Nauk SSSR 1:6, 291-294 (in Russian). 52. Quelques remarques sur l'approximation des fonctions continues, Mat. Sb. 41:1, 99-103. 53. On certain new trends in probability theory, in Bull. USSR Second Congress of Mathematicians in Leningrad, June 24-30, 1934 (USSR Academy of Sciences Press) 8 (in Russian). 54. Modern mathematics, Front Nauki Tekh. No. 5/6, 25-28 (in Russian). 55. The Institute of Math, and Mech. at MGU, Front Nauki Tekh., No. 5/6, 75-78 (in Russian). 56. Zufallige Bewegungen (Zur Theorie der Brownschen Bewegung), Ann. Math. 35, 116-117.
57.
58. 59. 60.
61. 62.
63. 64.
1935 On certain modern trends in probability theory, in Proc. USSR Second Congress of Mathematicians in Leningrad, June 24-30, 1934, Vol. 1 (USSR Academy of Sciences Press), 349-358 (in Russian). Deviation from a formula of Hardy for partial isolation, Dokl. Akad. Nauk SSSR 3:3, 129-132 (in Russian). La transformation de Laplace dans les espaces lineaires, C.R. Acad. Set. Paris 200, 1717-1718. Zur Grossenordnung des restgliedes Fourierschen Reihen differenzzierbarer Funktionen, Ann. Math. 36, 521-526. 1936 Uber die beste Annaherung von Funktionen einer gegebenen Funktionenklasse, Ann. Math. 37, 107-110. Basic Notions of Probability Theory (ONTI, Moscow) (in Russian). Translated from German: Grundbegriffe der WahrscheinUchkeitsrechnung (SpringerVerlag, 1933). Equations, in BSE 56, 163-165 (in Russian). Modern mathematics, in Collection of Papers on the Philosophy of Mathematics (ONTI, Moscow), pp. 7-13 (in Russian).
147 65. Theory and practice in mathematics, Front Nauki Tekh. No. 5, 39-42 (in Russian). 66. Uber die Dualitat im Aufbau der kombinatorischen Topologie, Mat. Sb. 1, 97-102. 67. Anfangs griinde der Theorie der Markoffschen Ketten mit unendlich vielen moglichen Zustanden, Mat. Sb. 1, 607-610. 68. Homologierung des Komplexes mid des lokalbikompakten Raumes, Mat. Sb. 1, 701-706. 69. Zur Theorie der Markoffschen Ketten, Math. Ann. 112, 155-160. 70. On Plessner's condition for the law of large numbers, Mat. Sb. 1, 847-849 (in Russian). 71. (with P. S. Aleksandrov) Endliche Uberdeckungen topologischer Raumes, Fund. Math. 26, 267-271. 72. Les groupes de Betti des espaces localement bicompacts, C.R. Acad. Sci. Paris 202,1144-1147. 73. Proprietes des groupes de Betti des espaces localement bicompacts, C.R. Acad. Sci. Paris 202, 1325-1327. 74. Les groupes de Betti des espaces metriques, C.R. Acad. Sci. Paris 202, 1558-1560. 75. Cycles relatifs. Theoreme de dualite de M. Alexander, C.R. Acad. Sci. Paris 202,1641-1643. 76. Foreword to the Russian translation of the book by A. Heyting: Review of Studies on Foundations of Mathematics. Introduction to Proof Theory (ONTI, Moscow) (in Russian). 77. Sulla teoria di Volterra della lotta per l'esistenca, G. 1st. Ital. Attuar. 7, 74-80. 1937 78. Uber offene Abbildungen, Ann. Math. 38, 36-38. 79. Skew-symmetric quantities and topological invariants, in Proc. Seminar Vector and Tensor Anal, and Appl. to Geometry, Mechanics, and Physics, 4, (GONTI, Moscow), pp. 342-347 (in Russian). 80. Markov chains with countable set of states, Bull. MGU Mat. Mekh. 1:3, 1-16 (in Russian). 81. (with I. G. Petrovskii and N. S. Piskunov) Studies on the diffusion equation, combined with increase in the amount of matter and its application to a problem in biology, Bull. MGU Mat. Mekh. 1:6, 1-26 (in Russian). 82. On the statistical theory of crystallization of metals, Izv. Akad. Nauk SSSR Ser. Mat. No. 3, 355-360 (in Russian). 83. Ein vereinfachter Beweis der Birkhoff-Khintchineschen Ergodensatzes, Mat. Sb. 2, 367-368. 84. Zur Umkehrbarkeit der statistischen Naturgesetze, Math. Ann. 113, 766-772. 85. (with P. S. Aleksandrov) Editing and supplementing the translation of the book by F. Hausdorff: Set Theory (GONTI, Moscow) (in Russian). 86. The continuum, in BSE 34, 139-140. 1938 87. (with P. S. Aleksandrov) Introduction to the Theory of Functions of a Real Variable (3rded. (revised)), (GONTI, Moscow) (in Russian) (lsted.: Moscow, 1932).
148 88. 89. 90. 91. 92. 93. 94. 95. 96. 97.
98. 99. 100. 101.
102.
A. A. Markov, in BSE 38, 152-153 (in Russian). Mathematics, in BSE 38, 359-402 (in Russian). Mathematical induction, in BSE 38, 405-406 (in Russian). Measure, in BSE 38, 831-832 (in Russian). Multidimensional space, in BSE 39, 577-578 (in Russian). Probability theory and its applications, in Mathematics and Natural Sciences in the USSR (GONTI, Moscow), pp. 51-61 (in Russian). On the section on information in the first issue of Uspekhi Mat. Nauk, Usp. Mat. Nauk No. 4, 326-327 (in Russian). A remark on the foundations of geometry, Usp. Mat. Nauk No. 4, 347-348 (in Russian). From the board of editors (Series of articles on the theory of random processes), Usp. Mat. Nauk No. 5, 3-4 (in Russian). Uber die analytischen Methoden in der Wahrscheinlichkeitsrechnung, Usp. Mat. Nauk No. 5, 5-41. Translated from German, Math. Ann. 104, 415-458 (1931). Ein vereinfachter Beweis der Birkhoff-Khintchineschen Ergodensatzes, Usp. Mat. Nauk No. 5, 52-56. Translated from German, Mat. Sb. 2, 367-368 (1937). (with G. M. Fichtengol'ts and I. M. Gel'fand) Some problems in the theory of functions of a real variable, Usp. Mat, Nauk No. 5, 232-234 (in Russian). On the solution of a problem in biology, Izv. Nil Mat. Mekh. Tomsk, Univ. 2:1, 7-12 (in Russian). Une generalisation de Finegalite de M. J. Hadamard entre les bornes superieures des derivees successives d'une fonction, C.R. Acad. Sci. Paris 207, 764-765. Editing the Russian translation of the book by H. Lebesgue: On Measurement of Quantities (Uchpedgiz, Moscow) (in Russian).
1939 103. (with P. S. Aleksandrov) Algebra, Chapter 1 (Uchpedgiz, Moscow) (in Russian). 104. Orientation, in BSE 43, 342-344 (in Russian). 105. On inequalities between least upper bounds of sequences of derivatives of an arbitrary function on an infinite interval, Uch. Zap. MGU Mat. 3:30, 3-16 (in Russian). 106. (with I. M. Gel'fand) On rings of continuous functions on topological spaces, Dokl. Akad. Nauk SSSR 22:1, 11-15 (in Russian). 107. Sur l'interpolation et extrapolation des suites stationnaires, C.R. Acad. Sci. Paris 208, 2043-2045. 1940 108. Surfaces, in BSE 45, 746-748 (in Russian). 109. Curves in a Hilbert space, invariant under a one-parameter group of motions, Dokl. Akad. Nauk SSSR 26, 6-9 (in Russian). 110. Wiener spirals and some other interesting curves in Hilbert space, Dokl. Akad. Nauk SSSR 26, 115-118 (in Russian). 111. (with V. L. Goncharov) On the 60th birthday of S. N. Bernshtein, Izv. Akad. Nauk SSSR Ser. Mat. 4, 249-260 (in Russian).
149 112. V. I. Glivenko, Usp. Mat. Nauk No. 8, 379-383 (in Russian). 113. Review of V. I. Romanovsky, Mathematical statistics, Usp. Mat. Nauk No. 7, 327-329 (in Russian). 114. On a new argument for Mendel's laws, Dokl. Akad. Nauk SSSR 27, 38-42 (in Russian). 1941 115. Stationary sequences in Hilbert space, Bull. MGU Mat. 2:6,1-10 (in Russian). 116. Interpolation and extrapolation of stationary random sequences, Izv. Akad. Nauk SSSR Ser. Mat. 5, 3-14 (in Russian). 117. Points of local topological character of countably multiple open mappings of compacta, Dokl. Akad. Nauk SSSR 30, 477-479 (in Russian). 118. Local structure of turbulence in an incompressible viscous fluid for very large Reynolds numbers, Dokl. Akad. Nauk SSSR 30, 299-303 (in Russian). 119. On the lognormal distribution for particle sizes under grinding, Dokl. Akad. Nauk SSSR 31, 99-101 (in Russian). 120. On degeneration of isotropic turbulence in an incompressible viscous fluid, Dokl. Akad. Nauk SSSR 31, 538-541 (in Russian). 121. Energy scattering for local isotropic turbulence, Dokl. Akad. Nauk SSSR 32:1, 19-21 (in Russian). 122. (with P. S. Aleksandrov) Properties of inequalities and the notion of approximate computations, Mat. Shk. No. 2, 1-12 (in Russian). 123. (with P. S. Aleksandrov) Irrational numbers, Mat. Shk. No. 3, 1-15 (in Russian). 124. Confidence limits for an unknown distribution function, Ann. Math. Stat. 12:4, 461-463. 1942 125. Definition of the scatter centre and measures of exactness using a finite number of observations, Izv. Akad. Nauk SSSR Ser. Mat. 6, 3-32 (in Russian). 126. The equation of turbulent motion of an incompressible fluid, Izv. Akad. Nauk SSSR Ser. Fiz. 6:1-2, 56-58 (in Russian). 1943 127. (with P. S. Aleksandrov) Nikolai Ivanovich Lobachevskii, 1793-1843 (Gostekhizdat, Moscow) (in Russian). 128. A great Russian scientific innovator: on the 150th birthday of N. I. Lobachevskii, Izvestiya (2 September 1943) (in Russian). 1945 129. The number of hits for several shots, and general principles for estimating the efficiency of shooting systems, Trudy Mat. Inst. Akad. Nauk SSSR 12, 7-25 (in Russian). 130. Artificial scatters in the case of hitting in one shot and scatter in one measurement, Trudy Mat. Inst. Akad. Nauk SSSR 12, 26-45 (in Russian).
150 1946 131. On the problem of the law of resistance in a turbulent flow in smooth tubes, Doki Akad. Nauk SSSR 52, 669-671 (in Russian). 132. On the substantiation of the method of least squares, Usp. Mat. Nauk 1:1, 57-70 (in Russian). 133. On the foundations of the theory of real numbers, Usp. Mat. Nauk 1:1, 217-219 (in Russian). 134. Newton and modern mathematical thought, in Moscow University - In Commemoration of Isaac Newton, 1643-1943 (Moscow Univ. Press), pp. 27-43 (in Russian). 135. Review of M. A. Velikanov's article "Transfer of suspended pumps by a turbulent flow", Izv. Akad. Nauk SSSR OTN 5, 781-784 (in Russian). 1947 136. Development of mathematics in the USSR, in BSE, Vol. "SSSR", 1318-1323 (in Russian). 137. Average quantities, in BSE 52, 508-509 (in Russian). 138. (with A. A. Petrov and Yu. M. Smirnov) A formula of Gauss from the theory of the least squares method, Izv. Akad. Nauk SSSR Ser. Mat. 11, 561-566 (in Russian). 139. (with N. A. Dmitriev) Branching random processes, Dokl. Akad. Nauk SSSR 56, 7-10 (in Russian). 140. (with B. A. Sevast'yanov) Computation of final probabilities for branching random processes, Dokl. Akad. Nauk SSSR 56, 783-786 (in Russian). 141. Statistical theory of oscillation with continuous spectrum, in Collection of Articles Dedicated to the 30th Anniversary of the Great October Socialist Revolution, Vol. 1 (USSR Acad. Sci. Press), pp. 242-252 (in Russian). 142. The role of Russian science in the development of probability theory, in The Role of Russian Science in the Development of World Science and Culture, Vol. 1 (Moscow Univ. Press), pp. 53-64 (Uchen. Zap. MGU, No. 91). 1948 143. Statistical theory of oscillation with continuous spectrum, in General Meeting of the USSR Acad. Sci. Dedicated to the 30th Anniversary of the Great October Socialist Revolution (USSR Acad. Sci. Press), pp. 465-472 (in Russian). 144. (with B. V. Gnedenko) Probability theory, in Mathematics in the USSR During Thirty Years, 1917-1947 (Gostekhizdat), pp. 701-727 (in Russian). 145. On two theorems regarding probabilities: comments, in P. L. Chebyshev, Complete Collection of Works, Vol. 3, Mathematical Analysis (Gostekhizdat), pp. 404-409 (in Russian). 146. Construction of complete metric Boolean algebras, Usp. Mat. Nauk 3:1, 212 (in Russian). 147. Notes regarding the Chebyshev polynomials of least deviation from a given function, Usp. Mat.Nauk 3:1, 216-221 (in Russian). 148. E. E. Slutskii, Usp. Mat Nauk 3:4, 143-151 (in Russian). 149. Editing and writing the foreword to the Russian translation of the book by H. Cramer: Mathematical Methods of Statistics (Inost. Lit., Moscow) (in Russian).
151 150. Algebres de Boole metriques completes, in VI Zjazd Mat. Polskich, pp. 22-30. 1949 151. (with B. V. Gnedenko) Limit Distributions for Sums of Independent Random Variables (Gostekhizdat) (in Russian). 152. On the problem of 'geometrical selection' of crystals, Dokl. Akad. Nauk SSSR 65, 681-684 (in Russian). 153. Solution of a problem in mathematical statistics, related to the problem of the mechanism of fibre-formation, Dokl. Akad. Nauk SSSR 65, 793-796 (in Russian). 154. (with Yu. V. Prokhorov) On sums of a random number of random terms, Usp. Mat. Nauk 4:4, 168-172 (in Russian). 155. A local limit theorem for classical Markov chains, Izv. Akad. Nauk SSSR Ser. Mat. 13, 281-300 (in Russian). 156. Basic problems of theoretical statistics, in Proc. 2nd USSR Conf. Math. Statistics, Tashkent, 27 Sept.-2 Oct., 1948, Uzbekgosizdat, Tashkent, pp. 216-220 (in Russian). 157. The true meaning of results by dispersion analysis, in Proc. 2nd USSR Conf. Math. Statistics, Tashkent, 27 Sept.-2 Oct., 1948, Uzbekgosizdat, Tashkent, pp. 240-268 (in Russian). 158. On the breakage of drops in a turbulent flow, Dokl. Akad. Nauk SSSR 66:5, 825-828 (in Russian). 159. Absolute quantities, in BSE-2 1, 32 (in Russian). 160. J. Hadamard, in BSE-2 1, 388 (in Russian). 161. Additive quantities, in BSE-2 1, 394 (in Russian). 162. Axioms, in BSE-2 1, 613-616 (in Russian). 163. Axonometry, in BSE-2 1, 617 (in Russian). 1950 164. Unbiased estimators, Izv. Akad. Nauk SSSR Ser. Mat. 14, 303-326 (in Russian). 165. On the problem of determining the coefficient of temperature-conductivity of soil, Izv. Akad. Nauk SSSR Ser. Geogr. Geofiz 2, 97-98 (in Russian). 166. Algebra in secondary school, in BSE-2 2, 61-62 (in Russian). 167. Algebraic expressions, in BSE-2 2, 64 (in Russian). 168. Algorithms, in BSE-2 2, 65 (in Russian). 169. Euclid's algorithm, in BSE-2 2, 65-67 (in Russian). 170. A. D. Aleksandrov, in BSE-2 2, 83 (in Russian). 171. R S. Aleksandrov, in BSE-2 2, 84 (in Russian). 172. Asymptote, in BSE-2 3, 238-239 (in Russian). 173. Asymptotic expressions, in BSE-2 3, 239 (in Russian). 174. N. I. Akhiezer, in BSE-2 3, 565 (in Russian). 175. S. Banach, in BSE-2 4, 183 (in Russian). N. K. Bari, in BSE-2 4, 245 (in Russian). 176. S. N. Bernshtein, in BSE-2 5, 52 (in Russian). 177. Infinitely large, in BSE-2 5, 66-67 (in Russian). 178. (with V. F. Kagan) Infinitesimal, in BSE-2 5, 67-71 (in Russian).
152 179. (with B. V. Delone) Infinitely distant elements, in BSE-2 Russian). 180. Infinity (in mathematics), in BSE-2 5, 73-74 (in Russian). 181. Biharmonic function, in BSE-2 5, 159 (in Russian). 182. Bilinear form, in BSE-2 5, 167 (in Russian). 183. Law of large numbers, in BSE-2 5, 538-540 (in Russian).
5, 71-72 (in
1951 184. On the problem of differentiability of transition probabilities in timehomogeneous Markov processes with countable number of states, Uch. Zap. MGU 148, Mat. 4, 53-59 (in Russian). 185. Generalizations of Poisson's formula to the case of selection from a finite sample, Usp. Mat. Nauk 6:3, 133-134 (in Russian). 186. I. G. Petrovskii, Usp. Mat. Nauk 6:3, 161-164 (in Russian). 187. Statistical quality control for admissible number of defective articles equal to zero, Leningrad, pp. 1-24 (in Russian). 188. (with S. A. Yanovskaya) L. E. J. Brouwer, in BSE-2 6, 62 (in Russian). 189. Variational series, in BSE-2 6, 641 (in Russian). 190. (with S. A. Yanovskaya) H. Weyl, in BSE-2 7, 106 (in Russian). 191. Quantity, in BSE-2 7, 340-341 (in Russian). 192. Probable deviation, in BSE-2 7, 507 (in Russian). 193. Probability, in BSE-2 7, 508-510 (in Russian). 194. (with T. I. Kozlov) Sampling method, in BSE-2 9, 417-418 (in Russian). 195. (with A. N. Khovanskii) Introduction and commentary to the book: N. I. Lobachevskii, Complete Collected Works, Vol. 5 (Gostekhizdat), pp. 329-332; 342-348 (in Russian). 1952 196. On the problem of pressure in the velocity profile under turbulent flow in tubes, Dokl. Akad. Nauk SSSR 84, 20-30 (in Russian). 197. Gaussian distribution, in BSE-2 10, 275 (in Russian). 198. Geodesic curvature, in BSE-2 10, 481 (in Russian). 199. D. Hilbert, in BSE-2 11, 370-371 (in Russian). 200. Histogram, in BSE-2 11, 447 (in Russian). 201. B. V. Gnedenko, in BSE-2 11, 545 (in Russian). 202. Homeomorphism, in BSE-2 12, 21 (in Russian). 203. Homotopy, in BSE-2 12, 35 (in Russian). 204. Motion (in geometry), in BSE-2 13, 447 (in Russian). 205. Two-term, in BSE-2 13, 518 (in Russian). 206. Real numbers, in BSE-2 13, 570 (in Russian). 207. Division, in BSE-2 13, 628 (in Russian). 208. Discreteness, in BSE-2 14, 425 (in Russian). 209. Variance, in BSE-2 14, 438 (in Russian). 210. Distributivity, in BSE-2 14, 479 (in Russian). 211. Distributive operator, in BSE-2 14, 479 (in Russian). 212. Differential, in BSE-2 14, 498-499 (in Russian). 213. (with B. P. Demidovich and V. V. Nemytskii) Differential equation, in BSE-2 14, 520-526 (in Russian).
153 214. Confidence probability, in BSE-2 14, 616 (in Russian). 215. Confidence bound, in BSE-2 14, 617 (in Russian). 216. (with I. G. Bashmakova and A. P. Yushkevich) Mathematical symbols, in BSE-2 17, 115-119 (in Russian). 217. Significant digits, in BSE-2 17, 135 (in Russian). 218. (with V. I. Bityutskov) Isomorphism, in BSE-2 17, 478-479 (in Russian). 219. Isotropic line, in BSE-2 17, 509 (in Russian). 220. Concrete number, in BSE-2 17, 557 (in Russian). 221. V. G. Imshenetskii, in BSE-2 17, 607 (in Russian). 222. On the Profession of a Mathematician: A Guide for Those Entering Colleges, (Sovetskaya Nauka) (in Russian). 1953 223. On the notion of algorithm, Usp. Mat. Nauk 8:4, 175-176 (in Russian). 224. Some papers of the last decade on the limit theorems of probability theory, Vestn. MGU 10, 29-38 (in Russian). 225. On dynamical systems with an integral invariant on a torus, Dokl. Akad. Nauk SSSR 93, 736-766 (in Russian). 226. Mathematical induction, in BSE-2 18, 146 (in Russian). 227. (with V. I. Glivenko) Integral, in BSE-2 18, 250-253 (in Russian). 228. Probability integral, in BSE-2 18, 253 (in Russian). 229. Interpolation, in BSE-2 18, 304-305 (in Russian). 230. Intuitionism, in BSE-2 18, 319 (in Russian). 231. Elimination of variables, in BSE-2 18, 483 (in Russian). 232. Testing, in BSE-2 18, 604 (in Russian). 233. Method of exhaustion, in BSE-2 19, 50-51 (in Russian). 234. Quadrant, in BSE-2 20, 434 (in Russian). 235. Compactum, in BSE-2 22, 282 (in Russian). 236. Constant, in BSE-2 22, 416 (in Russian). 237. Continuum, in BSE-2 22, 454-455 (in Russian). 238. Coordinates, in BSE-2 22, 524-525 (in Russian). 239. Correlation, in BSE-2 23, 55-58 (in Russian). 1954 240. On the preservation of conditionally periodic motions under small variations of the Hamilton function, Dokl. Akad. Nauk SSSR 98:4, 527-530 (in Russian). 241. The general theory of dynamical systems and classical mechanics, in Proc. Int. Math. Congress, Amsterdam, 1954, Review lectures (USSR Acad. Sci. Press, 1961), pp. 187-208 (in Russian). 242. Line (curve), in BSE-2 25, 167-170 (in Russian). 243. Law of small numbers, in BSE-2 26, 168 (in Russian). 244. A. A. Markov, in BSE-2 26, 294 (in Russian). 245. Mathematics, in BSE-2 26, 464-483 (in Russian). 246. Mathematical statistics, in BSE-2 28, 485-490 (in Russian). 247. Mathematical physics, in BSE-2 26, 490 (in Russian). 248. R. von Mises, in BSE-2 27, 414 (in Russian). 249. Multidimensional space, in BSE-2 27, 660 (in Russian). 250. (with P. S. Aleksandrov) Set theory, in BSE-2 28, 14-17 (in Russian).
154 251. (with S. V. Fomin) Elements of Theory of Functions and Functional Analysis: A Course of Lectures. No. 1: Metric and Normed Spaces (Moscow Univ. Press) (in Russian). 1955 252. Estimates of the minimal number of elements of £-nets in various function classes and their application to the problem of representing functions of several variables by superpositions of functions of a smaller number of variables, Usp. Mat. Nauk 10:1, 192-194 (in Russian). 253. Orientation, in BSE-2 3 1 , 188-189 (in Russian). 254. Foundations of geometry, in BSE-2 3 1 , 296 (in Russian). 255. (with L. A. Skornyakov) Surface, in BSE-2 33, 346-347 (in Russian). 256. Ordinal number, in BSE-2 34, 238 (in Russian). 257. Statistical quality control, in BSE-2 34, 498-499 (in Russian). 258. Estimates of the minimal number of elements of e-nets in various function classes and their application to the problem of representing functions of several variables by superpositions of functions of a smaller number of variables, Dokl. Akad. Nauk SSSR 101, 192-194 (in Russian). 1956 259. On Skorokhod convergence, Teor. Veroyainosi. Primenen. 1, 239-247 (in Russian). 260. Two uniform limit theorems for sums of independent terms, Teor. Veroyainosi. Primenen. 1, 426-436 (in Russian). 261. (with Yu. V. Prokhorov) Zufallige Funktionen und Grenzverteiilugssatze, in Bericht iiber die Tagung Wahrscheinlichkeitsrechnung und Mathematische Statistik, Berlin, pp. 113-126. 262. Some principal problems in the approximate or exact representation of functions of one or several variables, in Proc. 3rd USSR Math. Congress, Vol. 2 (Moscow Univ. Press), pp. 28-29 (in Russian). 263. On the Shannon theory of information transmission in the case of continuous signals, IEEE Trans. Inform. Theory, IT-2, 102-108. 264. On the representation of continuous functions of several variables as superpositions of continuous functions of a smaller number of variables, Dokl. Akad. Nauk SSSR 108:2, 179-182 (in Russian). 265. On some asymptotic characteristics of totally bounded metric spaces, Dokl. Akad. Nauk SSSR 108, 385-388 (in Russian). 266. (with I. M. Gel'fand and A. M. Yaglom) On a general definition of amount of information, Dokl. Akad. Nauk SSSR 111, 745-748 (in Russian). 267. Probability theory, in Mathematics, Its Content, Methods, and Significance, Vol. 2 (USSR Acad. Sci. Press), pp. 252-284 (in Russian). 268. (with S. B. Stechkin) S. M. Nikol'skii: on his 50th birthday, Usp. Mat. Nauk 11:2, 239-244 (in Russian). 269. E. E. Slutskii, in BSE-2 39, 378 (in Russian). 270. N. V. Smirnov, in BSE-2 39, 406 (in Russian).
155 1957 271. Theory of information transmission, in USSR Acad. Sci. Session on Scientific Problems in Automatization of Industry, 15-20 October 1956: Plenary meetings (USSR Acad. Sci. Press), pp. 66-69 (in Russian). 272. On the representation of continuous functions of several variables as superpositions of continuous functions of one variable and addition, Dokl. Akad. Nauk SSSR 114:5, 953-956 (in Russian). 273. On the foundation of the theory of real numbers, Mat. Prosveshchenie 2, 169-173 (in Russian).
274. 275.
276. 277. 278. 279. 280. 281. 282.
283. 284. 285. 286. 287. 288.
1958 On the Profession of a Mathematician (2nd ed.) (Moscow Univ. Press) (in Russian). (with I. M. Gel'fand and A. M. Yaglom) The amount of information and entropy for continuous distributions, in Proc. 3rd USSR Math. Congress, Vol. 3 (USSR Acad. Sci. Press), pp. 300-320 (in Russian). Sufficient statistic, in BSE-2 51, 106 (in Russian). Information, in BSE-2 51, 129-130 (in Russian). Cybernetics, in BSE-2 51, 149-151 (in Russian). A new metric invariant of transitive dynamical systems and automorphisms of Lebesgue spaces, Dokl. Akad. Nauk SSSR 119, 861-864 (in Russian). Sur les proprietes des fonctions de concentrations de M. P. Levy, Ann. Inst. Henri Poincare 16:1, 27-34. On the linear dimension of topological vector spaces, Dokl. Akad. Nauk SSSR 120, 239-241 (in Russian). (with V. A. Uspenskii) On the definition of an algorithm, Usp. Mat. Nauk 13:4, 3-28 (in Russian). 1959 On entropy in unit time as a metric invariant of automorphisms, Dokl. Akad. Nauk SSSR 124, 754-755 (in Russian). (with V. M. Tikhomirov) e-entropy and e-capacity of sets in function spaces, Usp. Mat. Nauk 14:2, 8-86 (in Russian). Transition of branching processes into diffusion processes and related problems of genetics, Teor. Veroyatnost. Primenen. 4, 233-236 (in Russian). Notes on the works of R. A. Minlos and V. V. Sazonov, Teor. Veroyatnost. Primenen. 4, 237-239 (in Russian). Probability theory, in Mathematics in the USSR During 40 Years, Vol. 1, Fiz-matgiz, Moscow, pp. 781-795 (in Russian). Foreword to the Russian translation of the book by W. R. Ashby: Introduction to Cybernetics (Inost. Lit., Moscow) (in Russian).
1960 289. (with O. V. Sarmanov) On Bernshtein's work in probability theory, Teor. Veroyatnost. Primenen. 5, 215-221 (in Russian). 290. On the classes $( n ) of Ford and Blanc-Lapierre, Teor. Veroyatnost. Primenen 5, 373 (in Russian).
156 291. Random functions of several variables, almost all realizations of which are periodic, Teor. Veroyatnost. Primenen. 5, 374 (in Russian). 292. (with B. V. Gnedenko, Yu. V. Prokhorov and 0 . V. Sarmanov) On N. V. Smirnov's work in probability theory: on his 60th birthday, Teor. Veroyatnost. Primenen. 5, 436-440 (in Russian). 293. (with Yu. A. Rozanov) On strong mixing conditions for Gaussian stationary processes, Teor. Veroyatnost. Primenen. 5, 222-227 (in Russian). 294. (with B. V. Gnedenko) A. Ya. Khinchin, Usp. Mat. Nauk 15:4, 97-110 (in Russian). 295. On the Profession of a Mathematician, 3rd ed. (Moscow Univ. Press) (in Russian). 296. (with S. V. Fomin) Elements of the Theory of Functions and Functional Analysis, No. 2: Measure, Lebesgue Integral and Hilhert Space (Moscow Univ. Press) (in Russian). 1961 297. Automata and life, Mash. Perevod Prikl. Lingvistika 9, 3-8 (in Russian). 298. Automata and life, Tekh. Molodezhi, No. 10, 16-19; No. 11, 30-33 (in Russian). 299. Note on the paper by V. K. Lezerson, Teor. Veroyatnost. Primenen. 6, 367 (in Russian). 300. (with P. S. Aleksandrov) Properties of inequalities and the notions of approximate computations, in Questions of Mathematics Teaching in School (Uchpedgiz, Moscow) (in Russian). 301.
302. 303. 304. 305.
306.
1962 (with M. Arato and Ya. G. Sinai) On an estimate of parameters of a complex stationary Gaussian Markov process, Dokl. Akad. Nauk SSSR 146, 747-750 (in Russian). (with A. M. Kondratov) The rhythm of Mayakovskii's poem, Vopros. Yazykoznaniya, No. 3, 62-74 (in Russian). (with L. S. Pontryagin and E. F. Mishchenko) On a probabilistic problem in optimal control, Dokl. Akad. Nauk SSSR 145, 993-995 (in Russian). On the work of B. V. Gnedenko in probability theory, Teor. Veroyainost. Primenen. 7, 323-329 (in Russian). A refinement of the conception of the local structure of turbulence in an incompressible viscous fluid at large Reynolds numbers, in Mecanique de la Turbulence: Colloq. Int. CNRS, Marseille, Aout-Sept 1961, Paris, pp. 447-458. A refinement of the conception of the local structure of turbulence in an incompressible viscous fluid at large Reynolds numbers, J. Fluid Mech. 13:1, 82-85.
1963 307. On approximating the distribution of a sum of independent terms of infinitely divisible distributions, Trudy Moskov. Mat. Obshch. 12, 437-451 (in Russian). 308. Discrete automata and finite algorithms, in Proc. 4th USSR Math. Congress, Vol. 1 (Leningrad Univ. Press), p. 120 (in Russian).
157 309. Various approaches to estimating the difficulty of approximately specifying and computing' functions, in Proc. Int. Congress Math. Stockholm, pp. 369-376 (in Russian). 310. On tables of random numbers, Ind. J. Stat. Ser. A 25:4, 369-376. 311. (with A. V. Prokhorov) On the down-to-earthness of modern Russian poetry: a general characteristic, Vopros. Yazykoznaniya No. 6, 84-95 (in Russian). 312. On the study of Mayakovskii's rhythm, Vopros. Yazykoznaniya No. 4, 64-71 (in Russian). 313. (with A. V. Prokhorov) Statistics and probability theory in the study of Russian poetry compositions, in Abstracts and Annotations Symp. on Complex Investigation of Artistic Creative Work (Nauka), p. 23 (in Russian). 314. How I became a mathematician, Ogonek No. 48, 12-13 (in Russian). 315. Foreword to the Russian translation of the book by C. A. Shannon: Works on Information Theory and Cybernetics (Inost. Lit., Moscow) (in Russian). 1964 316. (with A. V. Prokhorov) On the down-to-earthness of modern poetry: a statistical characteristic of this with Mayakovskii, Bagritskii, Akhmatova, Vopros. Yazykoznaniya No. 1, 75-94 (in Russian). 317. On the metre in Pushkin's "Songs for Eastern Slaves", Rus. Lit. No. 1, 98-111 (in Russian). 318. Foreword to the Russian translation of the book by W. Feller: Introduction to Probability Theory and Its Applications (2nd ed.) (Mir, Moscow) (in Russian). 1965 319. Three approaches to the definition of "amount of information", Prohlemy Peredachi Inform. 1:1, 3-11 (in Russian). 320. Geometric education in geometry courses at school, Mat. Shk. No. 2, 24-29 (in Russian). 321. (with I. M. Yaglom) On the contents of the school course of mathematics, Mat. Shk. No. 4, 53-62 (in Russian). 322. Functions, graphs, continuous functions, Mat. Shk. No. 6, 12-21 (in Russian). 323. Notes on the analysis of the rhythm of Mayakovskii's "Verses on a Soviet passport", Vopros. Yazykoznaniya No. 3, 70-75 (in Russian). 1966 324. Introduction to Analysis (Moscow Univ. Press) (in Russian). 325. P. S. Aleksandrov and the theory of 6s-operations, Usp. Mat. Nauk 21:4, 275-278 (in Russian). 326. On the school definition of identity, Mat. Shk. No. 2, 33-35 (in Russian). 327. Geometry on the sphere and geology, Nauka Zhizii, No. 2, 32 (in Russian). 1967 328. (with Ya. M. Bardzin) On the realization of sets in three-dimensional space, Probt. Kibnernetiki 19, 261-268 (in Russian). 329. New programmes and some basic questions on the advanced course in mathematics at secondary school, Mat. Shk. No. 2, 4-13 (in Russian).
158 1968 330. Some theorems on algorithmic entropy and algorithmic amount of information, Usp. Mat. Nauk 21:2, 201 (in Russian). 331. On the study of the exponential function and logarithms in the 8-year school, Mat. Shk. No. 2, 23-25 (in Russian). 332. Generalizations of the notion of power and exponential functions, Mat. Shk. No. 1, 24-32 (in Russian). 333. Introduction to probability theory and combinatorics, Mat. Shk. No. 2, 63-72 (in Russian). 334. (with S. V. Fomin) Elements of the Theory of Functions and Functional Analysis (2nd ed. revised and extended) (Fizmatgiz, Moscow) (in Russian). 335. (with A. V. Prokhorov) On the foundations of classical Russian metre, in Cooperation of Sciences and Mystery of Creative Work (Iskusstvo, Moscow), pp. 397-432 (in Russian). 336. Example of the study of metre and its rhythmic variants, in Prosody (Nauka), pp. 145-167 (in Russian). 1969 337. On the logical foundations of information theory and probability theory, Problemy Peredachi Inform. 5:3, 3-7 (in Russian). 338. S. N. Bernshtein, Usp. Mat. Nauk 24:3, 211-218 (in Russian). 339. Scientific foundation for a school course in mathematics, Mat. Shk. No. 3, 12-17; No. 5, 8-17 (in Russian). 340. New material in school mathematics, Nauka Zhizri No. 3, 62-66 (in Russian). 1970 341. V. N. Zasuhin: to the memory of a mathematician who died in World War II, Usp. Mat. Nauk 25:3, 243 (in Russian). 342. G. A. Seliverstov, Usp. Mat. Nauk 25:3, 244-245 (in Russian). 343. Generalizations of the notion of number, a non-negative rational number, Mat. Shk. No. 4, 27-32 (in Russian). 344. What is a function?, Kvant No. 1, 27-36 (in Russian). 345. What is the graph of a function?, Kvant No. 2, 3-13 (in Russian). 346. Tilings from regular polyhedra, Kvant No. 3, 24 (in Russian).
347. 348. 349. 350. 351. 352. 353.
1971 Quantity, in BSE-3 4, 456-457 (in Russian). N. Wiener, in BSE-3 5, 72 (in Russian). D. Hilbert, in BSE-3 6, 519 (in Russian). On a system of basic notions and notation for a school course of mathematics, Mat. Shk. No. 2, 17-22 (in Russian). Modern mathematics and mathematics at modern school, Mat. Shk. No. 6, 2-3 (in Russian). (with V. A. Gusev, A. B. Sosinskii and A. A. Shershevskii) A Course of Mathematics for Physico-Mathematical Schools (Moscow Univ. Press) (in Russian). Summer School at Rubskoe Lake, Prosveshchenie, Moscow (in Russian).
159 1972 354. Complexity of specification and complexity of construction of mathematical objects, Usp. Mat. Nauk 27:2, 159 (in Russian). 355. Integral, in BSE-3 10, 300-302 (in Russian). 356. Method of exhaustion, in BSE-3 10, 586 (in Russian). 357. (with S. V. Fomin) Elements of the Theory of Functions and Functional Analysis (3rd revised ed.) (Fizmatgiz, Moscow) (in Russian). 358. Scientific handbook, in L. Neiman, Delight of Discovery (Detskaya Literatura, Moscow), pp. 160-164 (in Russian). 359. There is no substitute for teachers, Koms. Pravda (19 January 1972) (in Russian). 360. A qualitative study of mathematical models of population dynamics, Probi. Kibernetiki 25:2, 101-106 (in Russian). 1973 361. Continuum, in BSE-3 13, 64 (in Russian). 362. Semilogarithmic and logarithmic sets, Kvant No. 4, 2 (in Russian). 363. On the profession of the mathematician, Kvant No. 4, 12 (in Russian). 1974 364. Basic Notions of Probability Theory (2nd ed.) (Nauka) (in Russian). 365. (with P. S. Aleksandrov and O. A. Oleinik) To the memory of I. G. Petrovskii (18 January 1901-15 January 1974), Trudy Moskov. Mat. Obshch. 31, 5-16 (in Russian). 366. I. G. Petrovskii, Usp. Mat. Nauk 29:2, 3-5 (in Russian). 367. A. A. Markov, in BSE-3 15, 379 (in Russian). 368. Mathematics, in BSE-3 15, 467-478 (in Russian). 369. (with Yu. V. Prokhorov) Mathematical statistics, in BSE-3 15, 480-484 (in Russian). 370. Multidimensional space, in BSE-3 16, 372 (in Russian). 371. Orientation, in BSE-3 18, 508-510 (in Russian). 372. (with I. T. Tropin and K. V. Chernyshev) Concern about adequate stock replacement, Vestn. Vyssh. Shk. No. 6, 26-33 (in Russian). 373. New programmes: specialized schools, in Mathematics Education Today (Prosveshchenie, Moscow), pp. 5-12 (in Russian). 374. The sieve of Eratosthenes, Kvant No. 10, 2 (in Russian). 1975 375. (with Yu. K. Belyaev) Statistical quality control, in BSE-3 20, 572-573 (in Russian). 376. Elements of combinatorics, Mat. Shk. No. 2, 16-25 (in Russian). 377. (with O. S. Ivashev-Musatov) Real numbers, infinite sequences and their limits, Mat. Shk. No. 2, 25-35 (in Russian). 1976 378. (with S. V. Fomin) Elements of the Theory of Functions and Functional Analysis (4th revised ed.) (Nauka) (in Russian). 379. (with S. I. Shvartsburd) Trigonometric functions, their graphs and derivatives, tenth grade school fashion, Mat. Shk. No. 1, 10-25 (in Russian).
160 380. (with G. A. Galperin) Moscow 38th Math. Olympiad, Mat. Shk. No. 4, 68-72 (in Russian). 381. Integral, tenth grade school fashion, Mat. Shk. No. 6, 15-17 (in Russian). 382. Transformation groups, Kvant No. 10, 2-5 (in Russian).
383. 384. 385. 386.
1977 Infinity, in Math. Encyclopedia, Vol. 1, 455-458 (in Russian). Quantity, in Math. Encyclopedia, Vol. 1, 651-653 (in Russian). Probability, in Math. Encyclopedia, Vol. 1, 667-669 (in Russian). (with V. V. Vavilov) The physics-mathematics school at MGU, Kvant No. 1, 56-57 (in Russian).
1978 387. On education on the levels of mathematics and physics of a dialecticalmaterialistic world view, Mat. Shk. No. 3, 6-9 (in Russian). 388. (with I. G. Zhurbenko) Estimates of the spectral functions of random processes, in 11th European Conf. Statistics, Abstract of Papers. 14-18 August 1978, Oslo, p. 36 (in Russian). 1979 389. (with A. V. Bulinskii) Linear sample estimates of sums, Teor. Veroyatnost. Primenen 24, 241-251 (in Russian). 390. (with V. V. Vavilov and I. T. Tropin) Fiz.-Mat. Shkola at MGU: 15 years, Kvant No. 1, 55-57 (in Russian). 1980 391. The dialectical-materialistic world view in school courses of mathematics and physics, Kvant No. 4, 15-18 (in Russian). 1981 392. (with S. V. Fomin) Elements of the Theory of Functions and Functional Analysis (5th ed.) (Nauka) (in Russian). 393. (with V. V. Vavilov and I. T. Tropin) Physico-Mathematical School at Moscow University (Moscow Univ. Press) (in Russian); (Mat. Kibemetika, No. 5). 394. (with A. F. Semenovich and P. S. Cherkasov) Geometry for 6-Sth Forms: Textbook (3rd ed.), Prosveshchenie, Moscow (in Russian) (1st ed., 1979). 395. On the notion of vector in courses of secondary school, Mat. Shk. No. 3, 7-8 (in Russian).
396. 397. 398. 399. 400.
1982 (with A. G. Dragalin) Introduction to Mathematical Logic (Moscow Univ. Press) (in Russian). (with I. G. Zhurbenko and A. V. Prokhorov) Introduction to Probability Theory (Nauka) (in Russian); (Bibliotechka "Kvani", No. 23). Mathematics, in Math. Encyclopedia Vol. 3, pp. 560-564 (in Russian). (with Yu. V. Prokhorov) Mathematical statistics, in Math. Encyclopedia Vol. 3, pp. 576-581 (in Russian). Newton and modern mathematical thought, Mat. Shk. No. 6, 58 (in Russian).
161 1983 401. (with A. M. Abramov, B. E. Veits, O. S. Ivashev-Musatov and S. I. Shvartsburd) Algebra and Elements of Calculus: Textbook for the 9th and 10th Forms (4th ed.) (Prosveshchenie, Moscow) (in Russian) (1st ed., 1980). 402. On the science textbook "Geometry 6-10", Mat. Shk. No. 2, 45 (in Russian). 403. Combinatorial foundations of information theory and computation of probabilities, Usp. Mat. Nauk 38:4, 27-36 (in Russian).
162 OBITUARY M R A N D R E I KOLMOGOROV Giant of Mathematics Mr Andrei Kolmogorov, Russian mathematician, died in Moscow on October 20. He was 84. His unique distinction was to have been the living embodiment of the theory of mathematical probability, and to have had countless pupils and disciples the world over others. Kolmogorov was born at Tambov on April 25, 1903. He was educated at Moscow University, where he subsequently taught as an instructor from 1925 until 1931, in which year he was made professor. He had held the chair of theory of probability since 1937. Much important work in probability had already been done without benefit of any foundations, but his little book, published in German, "Foundations of the Calculus of Probabilities" (1933), immediately became the definitive formulation of the subject. This was no mere codification of the findings of others. It is studded with beautiful theorems now bearing his name: the famous Strong Law of Large Numbers is just one of these. It further provided the essential basis for the study of stochastic ("random") systems dispersed in space and/or developing in time, and set the stage for the fruitful concept of a filtration of sigma-algebras representing the progressive swelling of the corpus of available information. This lies at the heart of the modern theory of random processes; it is an essential concept in the very idea of control theory, and it plays a vital role in Kolmogorov's later synthesis of information theory and ergodic theory. Another later theme was a penetrating analysis of randomness itself from the standpoint of the theory of the complexity of algorithms. This has become a key idea hinting at a link between classical and statistical mechanics. The theory of the smoothing and prediction of stationary time-theories is usually associated with the name of Norbert Wiener, another scholar of universal scope, but in fact it was developed simultaneously and independently by Wiener and by Kolmogorov during the Second World War. Those random processes called "Markovian" have a specially simple predictive structure. Kolmogorov created the basic theory for these and also constructed pathological examples which are still receiving study. He was interested in every branch of science; he and his pupils wrote about crystal growth, about the geometry of the interactions of plants, and also made significant contributions to "birth and death" processes and to genetics. One of these led to a head-on confrontation with Lysenko. Another joint work (with Piscounov and Petrovsky) treated the rate of advance of an advantageous gene in a linear environment (a topic studied independently by R. A. Fisher, for whom Kolmogorov had a high regard). This was later adapted to describe spreading of epidemics of innovations, and of rumours.
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In 1941, Kolmogorov made a contribution of the greatest importance to the theory of turbulent fluid-flow; in two short papers he proposed hypotheses putting into concise mathematical form ideas about the structure of the small-scale components of turbulent motion latent in earlier experimental work, particularly by G. I. Taylor. According to these, the small-scale structure has a universal form determined by two parameters: one was the rate of dissipation of energy in the fluid and the other (relevant only to the smallest scales in which dissipation occurs) was the fluid viscosity. These hypotheses imply many qualitative results (following from dimensional arguments) that are widely applicable — what goes on, for example, within the turbulence that occurs in the wake of a jet aircraft. Subsequent research has shown the hypotheses to be valid to a good approximation, though subtle refinement is needed because of the highly non-uniform or "spotty" nature of the energy dissipation in the fluid, as he himself showed two decades later. In the long (and unfinished) history of research on turbulence there has perhaps been no contribution more important than these two papers published in English in 1941 in the comptes rendus (record of proceedings) of the Soviet Academy and (remarkably) delivered to libraries in Britain during the war. In 1954, Kolmogorov made a seminal contribution to the fundamental problem of classical dynamics identified some fifty years earlier by the French mathematician Henri Poincare in his study of the motion of the planets round the sun. The motion of a single planet (neglecting the others) provides an example of an "integrable" problem and is well understood. The small effects associated with gravitational interaction between the planets introduces a profound qualitative change related to the fact that the governing equations are now "non-integrable". The phase trajectories of the unperturbed integrable system lie on a family of nested toroidal (ring-shaped) surfaces (or "tori") in the phase space. Kolmogorov's great achievement in 1954 was to demonstrate that a large set of these tori (the so-called non-resonant tori) survive the perturbation. This was a breakthrough in the study of general Hamiltonian systems in which the "K. A. M. tori" (named after Kolmogorov, his brilliant student V. I. Arnold, and J. Moser) play a pivotal role. Subsequent computational studies aptly confirm Kolmogorov's insights and have opened up the enormously fruitful field of "chaos in dynamical systems" which is currently attracting much attention. These studies could lead, for example, to more accurate weather forecasting. Kolmogorov enjoyed the company of young people, and encouraged them by pointing out possibilities latent in their scientific ideas. He devoted much of his time to improving the teaching of mathematics in secondary schools in the Soviet Union, and in providing special schools for the mathematically gifted. His dacha near Moscow became a meeting place for mathematicians. One of them said: "It is just like Oberwolfach (a mathematical institute in the Black Forest), except that here Kolmogorov buys all the drinks."
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For many years he attempted to capture in quantitative form some aspects of Russian poetry, especially that of Pushkin. To hear him lecture about this, with examples, was fascinating whether one understood Russian or not. It is, perhaps, a matter of regret that none of his work in this area has been translated into English. Kolmogorov was loved and admired by colleagues in all parts of the world. For two generations, each probabilistic compass needle has pointed to Moscow. The shock of losing him will be tempered only by gratitude for having known him and by the pleasure of recalling his memory with others.
Wolf Prize in Mathematics, Vol. 2 (pp. 165-217) eds. S. S. Chern and F. Hirzebruch © 2001 World Scientific Publishing Co. Integr. equ. oper. theory 30 (1998) 123 - 134 © Birkhauser Verlag, Basel, 1998
Integral Equations and Operator Theory
MARK GRIGORIEVICH KREIN: RECOLLECTIONS
ISRAEL GOHBERG
1. C U R R I C U L U M VITAE Mark G. Krein was born into a Jewish family of modest means in Kiev,on April 3, 1907. His father was a lumber merchant. As a youngster he exhibited a talent for mathematics. At the age of 14 he was already attending research seminars. He never obtained an undergraduate degree. In 1924 he ran away from home to Odessa and in 1926 he was accepted for doctoral studies by N.G. Chebotarev at Odessa University. He completed the degree requirements in 1929. M. G. Krein was an excellent and enthusiastic teacher. He attracted many students. In the thirties he created at Odessa University one of the strongest centers of functional analysis throughout the world. His interests included matrices and integral equations, geometry of Banach spaces, moment problems, spectral theory of linear operators, extension problems and applications. Many of the results of his outstanding students of this period (A.B. Artemenko, M.S. Livsic, D.P. Milman, M.A. Naimark, V.P. Potapov, M.A. Rutman, V.L. Shmuljan), as well as joint results together with his friends and colleagues (N.I. Achiezer, F.R. Gantmacher), are now characterized as classical and appear in textbooks on functional analysis. During World War II, from 1941 to 1944, M. G. Krein held the Chair of Theoretical Mechanics at the Kuibyshev (now Samara) Industrial Institute. M. G. preferred this position over the Chair in Mathematics because he thought that in technical institute of higher education it offered more interesting work and a wider set of possibilities and responsibilities. In 1944 he returned to Odessa. Soon he and his closest friend, B.Ja. Levin, were dismissed from Odessa University. This step was a direct result of the antisemitic Communist Party policy and corruption of the administration of the university (by the
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way t h e letter of dismissal was h a n d e d to M . G . on his b i r t h d a y ) . Also their s t u d e n t s h a d to leave. This was t h e end of t h e famous center of Functional Analysis at Odessa University. T h e a d m i n i s t r a t i o n of Odessa University replaced M. G. Krein by more "reliable" m a t h e m a t i c i a n s , such as N. A. Lednev, a n d N. I. Gavrilov.
T h e former became
famous for his Marxist critique of Einstein's T h e o r y of Relativity, while t h e l a t t e r claimed publicly t h a t he h a d resolved a n u m b e r of t h e o u t s t a n d i n g open problems in M a t h e m a t i c s including t h e R i e m a n n hypothesis. E a c h time elementary mistakes were found, a n d he tried t o force t h e acceptance of his a r g u m e n t s u n d e r t h e pressure of t h e C o m m u n i s t P a r t y a d m i n i s t r a t i o n . F u r t h e r c o m m e n t s , I t h i n k are superfluous. ^From 1944 to 1952 M . G . K r e i n held a p a r t - t i m e position as head of t h e Departm e n t of Functional Analysis a n d Algebra at t h e M a t h e m a t i c a l I n s t i t u t e of t h e Ukrainian A c a d e m y of Science in Kiev. He was dismissed from this post in 1952. T h e official reason given was t h a t he was not a p e r m a n e n t resident of Kiev. T h e real reason is easy to guess, it h a p p e n e d soon after the tragedy of t h e Jewish physicians. In t h e period 1944-1954 M . G . was a professor a t t h e Chair of Theoretical Mechanics a t t h e Odessa M a r i n e Engineering I n s t i t u t e . For reasons t h a t a r e stiE unclear he did not t r y to extend his contract with this institution, instead he moved to a less prestigious i n s t i t u t e - Odessa Civil Engineering I n s t i t u t e . Here he held the Chair of Theoretical Mechanics till his retirement. During t h e last period of his life he was a consultant to the I n s t i t u t e of Physical Chemistry of t h e Ukrainian A c a d e m y of Sciences. 2.
ACHIEVEMENTS
One of t h e most eminent m a t h e m a t i c i a n s of our t i m e , M a r k Grigorievich Krein is t h e a u t h o r of m o r e t h a n 270 p a p e r s a n d m o n o g r a p h s of u n s u r p a s s e d b r e a d t h and quality. His work opened up new areas of m a t h e m a t i c s a n d greatly enriched t h e more traditional on.es. He e d u c a t e d dozens of brilliant s t u d e n t s in t h e U S S R a n d inspired t h e work of m a n y m a t h e m a t i c i a n s , engineers a n d physicists all over t h e world. A list of themes where M . G . Krein's research was f u n d a m e n t a l , a n d in m a n y cases even d e t e r m i n e d t h e future of t h e field, includes: oscillating (totally positive) kernel functions =and matrices; p r o b l e m of m o m e n t s , orthogonal polynomials, a n d approximation theory; cones a n d regular convex sets in B a n a c h spaces; t h e theory of gaps between subspaces a n d operators in spaces with two n o r m s ; the extension theory of Hermitian o p e r a t o r s , continuation of H e r m i t i a n positive definite functions a n d helical arcs, entire o p e r a t o r s ; integral o p e r a t o r s , direct a n d inverse spectral problems for nonhomogeneous strings a n d Sturm-Liouville equations; t r a c e formula a n d scattering theory; m e t h o d of directing functionals; stability theories for differential equations; Wiener-Hopf,
Toeplitz
a n d singular integral operators; o p e r a t o r theory in spaces with indefinite metric, indefinite
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extension problems; nonselfadjoint o p e r a t o r s , characteristic operator-functions a n d triangular models; p e r t u r b a t i o n a n d F r e d h o l m theories; interpolation a n d factorization theories; prediction theory for s t a t i o n a r y stochastic processes; problems in elasticity theory, and ship waves a n d water resistance. A profound intrinsic unity is characteristic of Krein's work. Interlacing of general a b s t r a c t a n d geometric ideas with concrete a n d analytical results a n d applications are also characteristic of his work. Krein was a very fine p e d a g o g u e a n d lecturer. He would always share his new ideas a n d plans w i t h his s t u d e n t s a n d colleagues. He was known for his scientific generosity a n d e n t h u s i a s m , as well as his kindness a n d a t t e n t i o n to young m a t h e m a t i c i a n s . In 1982 M . G . Krein was a w a r d e d the prestigious i n t e r n a t i o n a l Wolf Prize in M a t h e m a t i c s in J e r u s a l e m . T h e citation to this prize reads in p a r t as follows: "His work is t h e culmination of t h e noble line of research begun by Chebyshev, Stieltjes, S. Bernstein a n d Markov a n d continued by F . Riesz, B a n a c h a n d Szego. Krein b r o u g h t t h e full force of m a t h e m a t i c a l analysis to bear on p r o b l e m s of function theory, o p e r a t o r theory, probability a n d m a t h e m a t i c a l physics. His contributions led to i m p o r t a n t developments in the applications of m a t h e m a t i c s to different fields ranging from theoretical mechanics to electrical engineering. His style in m a t h e m a t i c s a n d his personal leadership a n d integrity have set s t a n d a r d s of excellence." A m o n g his h o n o r a r y a w a r d s , he was elected corresponding m e m b e r of the U k r a i n i a n Academy, 1939; h o n o r a r y m e m b e r of the American A c a d e m y of Arts and Sciences, 1968; Foreign M e m b e r of t h e National Academy of Sciences of t h e United States of America,1979. He was also awarded t h e S t a t e prize of Ukraine, 1988. 3.
OUR FIRST
MEETING
I m e t M a r k Grigorievich Krein in the a u t u m n of 1950. M . G . Krein was already t h e n a world renowned m a t h e m a t i c i a n with a list of publications containing m o r e t h a n 75 very i m p o r t a n t p a p e r s a n d books. He was considere'd to be one of t h e strongest experts in functional analysis a n d its applications. His m a i n position at t h a t time was in t h e Marine Engineering I n s t i t u t e in Odessa at t h e Chair of Theoretical Mechanics. I was a t t h a t t i m e a s t u d e n t a t t h e Kishinev University, in m y fifth a n d final year.
O u r meeting was at m y initiative, a n d my a i m was to discuss with him m y first
results in o p e r a t o r theory. T h e y were a b o u t Fredholm theory a n d index. After I met him he invited m e to listen to his seminar talk a b o u t e x t r e m a l distributions of t h e m a s s of a string a n d inverse problems. T h e talk was a masterpiece a n d m a d e a great impression on m e .
He invited m e to his h o m e in t h e evening a n d we h a d detailed discussions on
m a t h e m a t i c s , m y future education a n d other subjects.
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He drew my attention to two topics, Toeplitz operators and normed commutative rings. He presented the topics in such an interesting and attractive way that I already fell in love with them, and planned to spend a lot of time studying them. During that evening M.G. told me that his first impression was that I am not Jewish, and he had planned to take me on as a graduate student. He explained to me that antisemitism is very strong and that the administration is not allowing acceptance of Jewish graduate students. He also told me that he had recently been dismissed from Odessa University, which had become very antisemitic, and relations towards him were personally hostile and extended also to his family and his friends. M.G. invited me to write to him and to visit him. After this conversation it became clear to me that my dream of becoming his graduate student would never be realized. It was difficult to imagine then that I would one day be privileged to have M.G. Krein as a teacher, coauthor and friend. Joint work with M.G. Krein was an outstanding university in the best and most friendly atmosphere. I will always remember M.G. Krein with gratitude, affection and admiration. 4. J O I N T W O R K For more than twelve years I worked intensively with M.G.. During those years we wrote twenty papers and two books. A third book was in our plans, but we never finished it. For the remaining twelve years M.G. was mostly busy with other projects and our joint work was less intensive. At other times I remember him to be busy with large joint projects with V.M. Adamjan and D.Z. Arov; and with each of the following colleagues: M.Sh. Birman, Yu.L. Daletsky, I.S. Iokhvidov, V.A. Jakubovich, I.S. Kac, H. Langer, G.Ya. Liubarsky, A.A. Nudelman, Yu.L. Shmuljan, and I.Spitkovsky. Over a long period of time I spent two or three months a year in Odessa for joint work. Most of the time we worked in Krein's apartment; a small room there got the official title of Izea's office. At the beginning of our joint work we worked late into the night. In fact we often stopped working in the early hours of the morning, and that would cost me a ruble to get the porter of the house to open the door, which was already locked for the night. Later we changed our hours of work for more convenient ones. Some of the time was during my official vacation when we would work on M.G.'s dacha on the shores of the Black Sea. On occasions we would even allow ourselves an hour off for a swim. Mark Grigorievich was extremely demanding of himself and of everyone working with him. If, by a total change in the writing of a paper or a chapter he could improve it by even a small fraction, he would not think twice. It was never a question of whether it was worthwhile or not. Papers which were almost ready would lie for years until they attained perfection. He maintained that the writing of a paper is no less scientifically important than
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proving t h e result. As a rule t h e polishing process leads to new proofs, new connections a n d new m a t h e m a t i c s . At t h e beginning of our collaboration M . G . himself wrote t h e formulas in by h a n d in t h e first copy of t h e p a p e r , he did not t r u s t m e enough t o let m e d o this. W h e n I asked if this was so, he answered t h a t he likes to do it himself because this gives h i m a n o p p o r t u n i t y t o think again over t h e proofs a n d t h e whole m a t e r i a l . T h e writing of t h e joint books took u p m o r e a n d m o r e t i m e . E a c h t i m e m y wife would ask, "Is it finished?" a n d each t i m e I would reply t h a t a little m o r e needs to be done. It is still told in Odessa, now as a joke, t h o u g h at t h e time it was t r u e , how L.A. Sakhnovich m e t m e a n d asked, "How is t h e book going?" "Well, it is 85 percent ready," I replied. " T h e n why do you look so sad? T h a t is wonderful." "Yes," I answered, " b u t if you h a d asked m e yesterday I would have said it was 95 percent ready." O u r s t u d e n t s a n d friends m a d e fun of us; I.S. Iokhvidov even wrote a little p o e m on t h e eve of t h e new year in 1963 ( t h a n k s to C. Davis for t r a n s l a t i n g it into English). A r o u n d t h e festive table, all our friends Have come to m a r k our new book's publication. T h e fresh a n d shiny volume in their h a n d s , T h e y offer Izia a n d m e congratulations. T h e long awaited h o u r is here at last. T h e sourest sceptic sees he was mistaken, A n d , smiling, comes to cheer us like t h e rest, A n d I ' m so delighted,....I awaken.
W h e n m a k i n g selections or discussing expositions or proofs we sometimes h a d differences of opinion, a n d this often led t o an a r g u m e n t . In general M . G . was a fighter a n d could m a k e himself very convincing. B u t he would also listen attentively to m y a r g u m e n t s . D u r i n g such discussions we sometimes exchanged positions; I would a d o p t his position a n d he mine. B u t we always ended up on a cheerful n o t e with a c o m m o n point, jokes often being used to illustrate a r g u m e n t s . After we wrote the first joint p a p e r I suggested t h a t his n a m e a p p e a r before m i n e on the m a n u s c r i p t . My a r g u m e n t was based on t h e fact t h a t this would be to the benefit of t h e p a p e r , which would be m o r e appreciated a n d a t t r a c t m o r e a t t e n t i o n with his n a m e first. M . G . refused, maintaining t h a t he would not write joint work with a person w h o could not be an equal p a r t n e r , a n d h e always used t h e alphabetical order of n a m e s in joint p a p e r s . Any change m a y cause speculations in the m i n d s of t h e reader. After a while I drew his a t t e n t i o n to a p a p e r by " M . G . Krein a n d M.A. Krasnoselsky". He laughed a n d r e m a r k e d t h a t t h e difference is in t h e t h i r d letter. M . G . considered joint work as t h e work of a t e a m a n d after deciding t h a t we would work together on certain topics thereafter we
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would not distinguish who did what in these topics. M . G . would c o m p a r e joint work with a t e a m of s p o r t s m e n , especially in a long t e r m collaboration. E a c h player passes t h e ball on to a n o t h e r player t o t h e best of his ability, a n d it is impossible to delineate which pass was t h e most i m p o r t a n t . T h e s e principles worked very well a n d during twenty four years of our collaboration we never h a d any m i s u n d e r s t a n d i n g . Our joint work usually s t a r t e d with plans, discussions, p r e p a r a t i o n s of ingredients for proofs a n d changes, changes, changes. In such discussions a p p e a r e d also t h e basis a n d topic for a new p a p e r . T h e n t h e joint work was realized in different ways. Sometimes one of us would write t h e first draft of t h e m a n u s c r i p t a n d t h e second would t h e n polish the m a n u s c r i p t a n d m a k e additions. This process would continue a n d t h e m a n u s c r i p t would change h a n d s m a n y times until it was considered ready for publication. Sometimes each of us would p r e p a r e a p a r t of t h e m a n u s c r i p t a n d we would p u t t h e p a r t s together at a very late stage. Sometimes we would write t h e m a n u s c r i p t together, discussing t h e presentation on t h e spot. Usually t h e final step in t h e process of p r e p a r a t i o n of t h e m a n u s c r i p t was its joint reading. We continued thinking a b o u t a p a p e r even after it was considered ready for publication. M a n y times this led to surprising results a n d to m a n y last m i n u t e changes. Often we continued our m a t h e m a t i c a l discussions d u r i n g walks in town. P a r t of t h e walk would take us along t h e m a i n street - Deribasovskaya . It was especially pleasant on a s u n n y day.
T h e street was very crowded a n d usually M . G . m e t m a n y people he
knew. Sometimes he would stop for a short conversation a b o u t t h e latest Odessa news. Next we would visit t h e m a i n bookstore. M.G. would check t h e newly arrived books in m a t h e m a t i c s , mechanics, physics a n d astro-physics. He usually m a d e an order for a book in advance a n d his copy would be waiting for him. O u r next stop would be the office supplies store. Here he would b u y p a p e r , ink, p e n s , pencils, glue a n d o t h e r accessories which we used in our work. L a t e r I found out t h a t M.G. liked to meet a young saleswoman in this store. She was really beautiful.
She h a d a biblical look, with d a r k hair a n d black eyes.
She always welcomed M . G . with a m o d e s t smile. We would also visit a second-hand store in t h e neighborhood. M . G . was looking for a typewriter with L a t i n T e t t e r s . He planned to replace his old one (in t h e usual stores one could buy only typewriters with Russian lettering). He never found one. Sometimes we would also visit a h a r d w a r e store. M.G. liked to b u y new original tools for t h e household. He greatly a d m i r e d t h e ingenuity of such tools. M . G . a n d his family were very kind to m e . This wonderful family always insisted t h a t I eat with t h e m . Once M . G . i n t r o d u c e d m e to a nice looking girl who was a friend of his d a u g h t e r . M u c h later I learned t h a t M.G. t h o u g h t we would suit each other a n d t h a t we would m a k e t h e perfect couple. At t h e time I did n o t u n d e r s t a n d his intentions a n d I p r o b a b l y disappointed him. Working with M . G . was a joy. I was never a formal s t u d e n t of his, but for m e
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he was m u c h m o r e t h a n an o r d i n a r y instructor. W h e n I s t a r t e d to work with M.G. I was warned by a friend t h a t I wo'uld get little credit for this joint work. I h a d also h e a r d m a n y tales a b o u t w h a t a difficult person M.G. Krein was, b u t during t h e twenty four years in which we worked together I found Krein to be otherwise. He set very high s t a n d a r d s for himself a n d I found his d e m a n d s on others to be fair a n d reasonable. 5.
SEMINARS AND
LECTURES
For m a n y years M.G. Krein r a n a m a t h e m a t i c a l seminar at t h e Scientists House in Odessa. This house used to be a n old palace, b u t since t h e revolution it served as a kind of club for scientists. T h e r o o m s with high ceilings were very luxurious, covered with expensive m a r b l e a n d wood paneling. Missing was only a good blackboard. T h e meetings were always in a very pleasant a t m o s p h e r e . M.G. would usually open with an interesting i n t r o d u c t i o n to t h e subject u n d e r discussion, a n d from this i n t r o d u c t i o n one could feel how rich a n d wide was his experience. During t h e talk he would ask i m p o r t a n t questions a n d s o m e t i m e s a d d r e m a r k s with a n historical perspective, or other approaches a n d proofs. As a rule he would present details, closing t h e talk with illuminating c o m m e n t s . All was done in a very friendly m a n n e r a n d with a great respect for t h e lecturer a n d audience. At t h e e n d of such seminars I felt myself enriched. M . G . would leave t h e Scientists House for his h o m e a c c o m p a n i e d by a large group of his a d m i r e r s . O u t s t a n d i n g m a t h e m a t i c i a n s p a r t i c i p a t e d in these seminars, some of t h e m m e m bers of t h e famous older generation of M . G . ' s s t u d e n t s a n d colleagues, a n d also t h e younger generation. A m o n g t h e m was V.M. A d a m j a n , D.Z. Arov, M.L. Brodsky, J u . P. Ginsburg, I.S. Iokhvidov, V . A . J a v r j a n , I.S. K a c , K . R . Kovalenko, H. Langer, F.E.Melik-Adamjan, S.M. M k h i t a r j a n , B.R. M u k m i n o v , A.A. N u d e l m a n , I. Ovcharenko, G . J a . P o p o v , Sh.N. Saakjan, L.A. Sakhnovich, P.A. S h w a r t s m a n , J u .
L. Schmulyan, V.G. Sizov, a n d I.
Spitkovsky. M . G . Krein also regularly held other interesting seminars on a smaller scale in t h e institutions where he was working. For instance he r a n a seminar in H y d r o d y n a m ics in t h e Marine Engineering I n s t i t u t e . A.A. Kostiukov, V . G . Sizov a n d Yu.L. Vorob'ev were p a r t i c i p a n t s in this seminar. In Kiev were Yu.M. Berezansky, B.I. K o r e n b l u m , M.A. Krasnoselsky, a n d S.G. Krein (M.G.'s younger b r o t h e r ) , in Kuibyshev, G.Ya. Liubarsky, A.V. S t r a u s s a n d O.V. Svirsky. I recollect M.G.'s seminars a n d lectures as one of t h e best I ever h e a r d .
His
lectures a n d talks were always well p r e p a r e d with historical r e m a r k s a n d examples. Almost every year M . G . gave courses of lectures for experts a n d P h . D . s t u d e n t s . Usually these lectures were based on his recent results.
Many of t h e m were even unpublished.
His
lectures on t h e theory of entire operators (prepared for publication by the Gorbachuks) will a p p e a r only now in print, in Birkhauser Verlag.
Lectures on t h e theory of cones
were circulating a m o n g experts for m a n y years a n d were never published. Lectures on t h e
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theory of the string exist only in rough handwritten notes. A manuscript of a book on group representations (written together with M.S. Brodsky) is not finished. As a matter of fact, almost no one from Odessa University attended.M.G. Krein's seminars and lectures. They probably did not even realize how much they were missing. 6. TRAVEL A B R O A D M.G. Krein never traveled abroad. The reason for this was that he would not have been granted an exit visa had he apphed. An exit visa for a trip abroad was considered by the authorities as a special privilege given only to the very "reliable". He received many invitations, then he would apply for an exit visa, entailing the filling in of thick official forms. He would then get the inevitable negative reply, or no reply at all. He even developed a sort of allergy to the filling in of forms, and during his last years it was very difficult to get him to undertake such a task, even on his wife's insistence. He already knew it would be of no use, so why bother. In one case was he given permission to travel abroad. That was in 1970 to attend a conference in Tihany on the Balaton Lake in Hungary. It probably worked out this time because he used a private invitation which did not have to go through the high official channels. But this time he could not use the visa because precisely at that time there was an epidemic of cholera in Odessa and no one was allowed to leave Odessa. I took part in this conference and at the request of M.G. I gave Professor B. Sz.-Nagy (the organizer of the conference) regards from M.G. and told him the reason why M.G. could not come. Professor Sz.-Nagy smiled and answered, "So it's now called cholera, is it?" In the West people were already used to the various reasons that were invented to justify the absence of M.G. Krein. But this was the only time that the reason given was the true reason. M.G. liked very much to meet and to receive foreign colleagues. However those meetings always brought repercussions. He would have to answer unpleasant questions put to him by KGB agents (in the best case). The small technical institutions that he worked for did not have the official facilities to receive foreigners and to organize such visits (usually they existed only in large universities or research institutes). For instance, in the case of the one year visit of H. Langer an unusual solution was found. Langer was well established at Odessa University. There was his residence and there he received his salary, but scientifically he was connected only with M.G., with his seminars and his School. Very often Mark Grigorievich would avoid a meeting with a colleague in order to prevent the inevitable repercussions and difficulties. In order to avoid similar difficulties and problems with the administration he did not send preprints and reprints abroad.
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Unfortunately foreign colleagues did not always understand these sad facts of real life in the USSR. 7. T H E R E S I D E N C E M.G. Krein loved Odessa. He was well acquainted with the history of the town and even the history of the main streets. He liked especially the satirical books written about Odessa. Often he read out the funny bits for his guests. All his mature life (with a relatively short exception during World War Two) the Kreins lived in Odessa in an old apartment house on 14 Artema Str., App. 6. This was the apartment in which his wife and her parents lived previously. The apartment building, which had been built at the beginning of this century, was luxurious. It had an elevator, heating, bathrooms, and every apartment had many large high ceilinged rooms, and separate quarters for servants. The owner of the building used to send a cleaner every week to polish the parquet flooring and to take care of other small things. In general he wanted to make sure that the house would be suitably maintained as such a building deserves. After the revolution normal life in such apartments became extremely difficult and in many cases impossible. First of all the houses were nationalized and the newly appointed caretakers were more involved in watching the occupants and reporting on their doings than in maintaining the building. In each apartment new families were installed. Very often each family had only one room, sharing the use of the kitchen and other facilities. Very soon the elevator and heating broke down and the water failed to reach the upper floors. Everyone knew what was going on in each of the neighboring rooms and what was being served for dinner in each room. Situations arose which were described with great humor in Russian literature of the post revolution period. All of this was the lot also of the Krein's apartment. There were times when this a-partment accommodated six families (according to the number of rooms). Sometimes running water did not reach the apartment (certainly not in the summer). Hot water they never had. Each morning there was a queue for the toilet. For many years heating was a serious problem. I remember one occasion during lunch or dinner while a hvely discussion was taking place one of the members of the family made a sign, and the topic under discussion was automatically changed. The sign meant that someone from another room w,as passing by their room on the way to the bathroom or to the telephone and their discussion could be overheard. People from other rooms would often call in on the Kreins during a visit by colleagues. Television was introduced to the Krein family relatively late. The family would watch together different programmes. They liked theater, opera, classical music, as well as other programmes. The Kreins liked the summer when they moved to the dacha on the shores of the Black Sea (in Arcadia) without the television.
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Mark Grigorievich was happily married to Raisa L'vovna Romen. She was. an expert in naval architecture and worked in the Marine Engineering Institute. They had only one child. Their daughter, Irma Markovna Krein (Kozdoba), has a Ph.D. in Philology and is an expert in cybernetics. They also had a grandson Aleosha and and a great grandson, also called Mark. 8. B A T T L I N G HOSTILITIES In general, M.G. Krein was a fair, very amiable and kind person. However, all of his life he battled against mediocrity. After the Second World War he had to contend with hostile elements which fought fiercely against him using the officially supported antisemitism which was rife in the Ukraine, and especially so in Odessa. He was accused of Jewish nationalism, presumably for having had too many Jewish students before the War. This accusation was certainly entered into his classified file and was held against him all of his fife. Presumably, it played a significant role in his two dismissals which were mentioned earlier. He was not allowed to have Jewish students and was deprived of a university base. All attempts on the part of various societies, academies and individuals in the Soviet Union to gain for him some measure of the official recognition which he so richly deserved, were unsuccessful. Remarkably enough, he was never elected as a full member of the Ukranian Academy of Sciences. Their standards must have been very high. Worse than that, there were times when his friends feared that he was in serious danger of arrest. In 1948, in an article in a local Odessa newspaper, M.G. Krein was called a "rootless cosmopolitan." He was accused of too often quoting foreign mathematicians and following too much their ideas and ignoring the achievements of Russian and Soviet mathematicians. The official establishment considered this to be a crime. This accusation was supported publicly by a professor of the University in Astronomy. The latter based his conclusions on the famous paper of M.G. Krein and M.A. Rutman (Linear operators leaving invariant a cone in Banach space, Uspekhi Mat.Nauk 3,No 1 (1948), 3-95; English translation: Amer. Math.Soc. Transl. (1), 10 (1962), 199-325). I guess he could understand only the general introduction to this paper and the fist of references; remarkable that he did not need to know any more in order to make the accusation. This was the usual way of starting many of the campaigns against Jewish scientists, writers and cultural activists which often led to arrests. M.G.was lucky, he escaped without an arrest. M.G. Krein gave a plenary talk at the Mathematical Congress held in Moscow in 1966. It was a very deep talk with a detailed analysis of the state of affairs in operator theory and its applications. Special attention was paid to the theory of nonselfadjoint operators. In this talk M.G. introduced, for convenient references, a mathematician Gokr (as a shortened form of Gohberg-Krein). My friend V.G. Boltjansky was working in the Press Committee of the Congress and mentioned this talk and Gokr in the public press.
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After the Congress he had serious problems with the mathematical authorities. He was accused of passing to the press information which had a "wrong political orientation." In the second half of the sixties the situation worsened. In particular two very important and influential committees became almost openly antisemitic. I have in mind the Committee of Experts (Chairmen V.A. Il'in, V.S. Vladimirov) which dealt with approval of degrees in mathematics in USSR, and the Publication Committee (chairman L.S. Pontrjagin) which over-saw all publications in mathematics in USSR. Approval of the degree of "Doctor of Sciences" was absolutely refused for Jews, and projects for books by Jewish authors were rejected. Laughable reasons were given to justify all of this. M.G. Krein responded to these hostile surroundings in the only way open to him, by deep research and hard work. He and many of his students were protected by virtue of his outstanding achievements. In retrospect, it seems clear that he won this very difficult struggle. Firstly, he was able to devote all his life to mathematics (teaching and research), the work he loved so much. Secondly, he was able to spend most of his life in Odessa, a town which he had always regarded with love and affection (some of his friends thought that his life would have been much easier in Moscow or Leningrad). Thirdly, he was always the leader of a strong and dedicated group of colleagues and followers who loved and respected him. (This group existed almost on a private basis, holding many of its meetings in his house, or at the Scientists Club.) Fourthly, he had a great impact on the development of mathematics and its applications throughout the world. Even though he was never allowed to travel abroad, his brilliant work knew no borders. In spite of all the difficulties that surrounded him, M.G. was cheerful and optimistic. He liked to tell stories and jokes, some of which he invented himself. At one time he went to the rector of his institute and asked if there was any danger that he could be accused of Armenian nationalism since he had four graduate students who were Armenians. The rector did not understand the joke and tried, in all seriousness, to explain that in this case there was no danger. 9.
EPILOGUE
This fight took a heavy toll on his health, and towards the end of his life he suffered from depression. This condition worsened after the tragic loss within one year of his wife, and his only grandson, Aleosha. On October 17, 1989, M.G. Krein died in Odessa (USSR). There he is buried. The passage of time brought about considerable changes. In the Institute of Mathematics of the Ukrainian Academy and the Institute of Mathematics of Odessa University some of M.G. Krein's followers and outstanding former students are now working. Preparation of an international conference on the occasion of the ninetieth anniversary of
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the birth of M.G. Krein is in progress. This conference is being organized by the two above mentioned institutes, which just half a century earlier had dismissed M.G. The conference is even partially supported by the city of Odessa. The conference will take place in Arcadia (in the suburbs of Odessa) on the shores of the Black Sea, not far from the Krein family dacha, where M.G. liked so much to work. There he spent many hours working with students and friends. And there he also spent the last days of his life. Unfortunately the dacha no longer belongs to the Krein family. It was taken away from them recently. We hope that this conference will help everyone to understand in full measure who was Mark Grigorievich Krein, and also to understand the bitter and tragic mistakes and injustices of the past and the present. 10.
REFERENCES
The following sources were used in these recollections: 1. M.G. Krein. Autobiography, 1966. Private communication (Russian). 2. I. Gohberg and M. Kac. Biography of M.G. Krein. Topics in functional analysis. Essays dedicated to M.G. Krein on the occasion of his seventieth birthday, I.Gohberg and M.Kac editors, Academic Press, 1979. 3. I. Gohberg, Mathematical Tales. Gohberg Anniversary Collection, vol. 1, edited by H. Dym, S. Goldberg, M.A. Kaashoek and P. Lancaster. Birkhauser Verlag, 0 T 40, 1989, pp. 17-56. 4. I. Gohberg. Mark Grigorievich Krein (1907-1989), Notices of the American Mathematical Society, vol. 37, no. 3, 1990, pp. 284-285.
Raymond and Beverly Sackler Faculty of Exact Sciences School of Mathematical Sciences Tel-Aviv University Ramat-Aviv, 69978 Tel-Aviv, Israel
MSC 1991: primary 01A70, secondary 01A99
Submitted: June 15, 1997
177 Ukrainian Mathematical Journal, Vol. 46, No. 3, 1994
MARK GRIGOR'EVICH KREIN (RECOLLECTIONS) B. JA. LEVIN From 1924 and until his death in October 1989, Mark Grigor'evich Krein—one of the most brilliant and famous mathematicians of our century—lived and worked in Odessa. I had a great opportunity to work with him for many years. He greatly influenced my scientific interests, directions, and the character of my investigations. For the first time, I met M. Krein in June 1934 at the Second Mathematical Congress in Leningrad. I attended his talks at this Congress devoted to loaded integral equations and the L-moment problem. In the second talk, he presented his work with N. I. Akhiezer. His talks always made a strong impression on the audience not only by their important and profound contents but also by their brilliant form. M. Krein.often said: "I want to be understandable." His concern for those who listened to him was always evident: He tried to make his presentations as clear and well thought-out as possible, to create a festive atmosphere on his talks and lectures. The same was my impression about his talks at this Congress. There or, more precisely, as we used to say, "in the couloirs of the Congress" I found the opportunity to read a manuscript of an unpublished article by M. G. Krein and M. A.Naimark, "A Method of Symmetric and Hermitian Forms in the Theory of Separation of Roots of Algebraic Equations," which was quite close to my scientific interests. This article was written for the encyclopedia. In 1936, it was published as a separate paperback edition and very soon became a bibliographic rarity. I decided to meet M. Krein and discuss with him the problems we were both interested in. We got acquainted but found no possibility for discussion. There were too many people who wanted to speak with Krein but absolutely no time for this 1 . After I had told him that I often came to Odessa, M. Krein invited me to visit him during my next visit. This happened a year later, when I came to Odessa for summer vacations from Rostov-on-Don, where I was completing my post-graduate studies. M. Krein asked me about my scientific activities. At that time, I was carried away by the construction of the theory of entire functions of completely regular growth and was very happy to have the opportunity to speak about my ideas and results with a great scientist. M. Krein listened to me very carefully, rarely interrupting with questions. Some of these questions were so important that I undertook a serious additional investigation and when they were finally solved, I suddenly understood that this investigation was indeed necessary to make the entire theory more precise and complete. That summer we met quite often, spoke about mathematics, went for walks near the sea. At that time, M. Krein studied, together with F. R. Gantmakher, oscillatory matrices and kernels and his talks about these problems were very exciting. I was deeply surprised by his observation that a simple mechanical fact made it possible to extract an important class of matrices about which it was clear that it would play an important role in algebra, analysis, and other fields of mathematics. This was the following: Consider an elastic continuum, e.g., a string in tension, and apply to it n transverse forces lying in the same plane; then the deflection curve of this continuum has at most n - 1 knots. (A knot is defined as a point where the ordinate changes its sign.) Later, I often noticed in Krein's works how the ideas of the theory of oscillatory matrices penetrate in profound problems in the theory differential and integral operators, and then understood that this method of investigation represents the general style of Krein's scientific work. For example, his well-known theory of extensions of semibounded operators appeared in connection with certain problems in the theory of bending of plates, and then found numerous applications both in Krein's works and in the works of other mathematicians. Large series of his investi1
"These congresses and conferences are so efficient in separating mathematicians" — Alexander Kurosh once joked on a conference.
Translated from Ukrainskii Matematicheskii Zhumal, Vol. 46, No. 3, pp. 305-309, March, 1994. 318
© 1995
Plenum Publishing Corporation
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gations were often stimulated by the problems in mechanics, electromechanics, the theory of communications, and the theory of shipbuilding. Mathematics is a science dealing with mathematical structures. Formally speaking, one absolutely arbitrarily chooses an independent system of axioms that describes an object and then, according to the rules of logic, the properties of the indicated object are derived from these axioms. This is just a description of a mathematical structure. In principle, one can construct a great number of these structures but their scientific importance will be. as a rule, quite low. It is probable that many structures of this sort give about the same amount of knowledge about the world and are of the same level of importance in cognition of the truth as chess-playing. It seems to be quite a difficult problem to select in this collection of structures those that are indeed useful and important. To solve this problem, the scientist must have high scientific culture, profound knowledge, and great intuition. (I do not even mention such necessary requirements as desire and ability to be objective.) M. Krein possessed in a remarkable degree the ability to understand and evaluate the importance of various structures. His strong mathematical intellect allowed him to reveal the finest and deepest relations between different objects and this often determined his interests and directed his investigations. Let us present several examples. At the beginning of the 1930s. M. G. Krein and N. I. Akhiezer performed an extensive and very fine investigation of the trigonometric L-moment problem. More than 30 years later, this great abstract investigation found, in the works of N. N. Krasovskii, important applications to the theory of optimal control. At first sight, the abstract of the Krein- Mil'man theorem on extreme points published in 1940, became one of the most important in many branches of contemporary mathematical analysis. This list of examples can easily be continued but I present only one example. In 1943, M. G. Krein posed and studied the problem of extension of helix lines in Hilbert spaces 2 and soon applied the results obtained in the study of important problems of the theory of predictions. The high practical value of this work was mentioned by N. Wiener in his book "I Am a Mathematician." This ability to discover general fundamental ideas in particular problems and construct, on this basis, theories that are very abstract on the one hand and, on the other hand, deeply penetrating in various fields of mathematical science was a distinctive feature of Krein's creative work. "Certainly",— M. Krein said one day — "science preserves only simple rough results and all fine points are forgotten little by little but how can one determine what is rough and what is fine? For example, could you say when it became clear that L2-metric is simpler than uniform?" In his works, M. Krein originated many new trends in science which were than developed and applied in the works of other mathematicians. At the same time, he paid great attention to the development of old classical fields. Being a qualified expert in the works of the Petersburg school of mathematicians and related researches, he devoted many of his works to the development of this important and extensive field of science. Let us now recall the characteristics of this part of his creative activities, given in 1983 when he was awarded the International Wolff Prize in mathematics: "His work represents a culmination of the noble series of investigations initiated by P. L. Chebyshev, T. Stieltjes, and S. N. Bernstein and then continued by F. Riesz, S. Banach, and H. Szego...". A major part of these results can be found in his books and surveys. His last work published after his death also belongs to this series. He established the priority of A. M. Lyapunov in the theory of stability regions for the Hill equation. Later, he generalized the entire Lyapunov theory for a large class of systems, namely, for the so-called canonical systems of differential equations of an arbitrary order with periodic coefficients. Note that in this case, a decisive role in the selection and study of these systems was also played by the mechanical considerations. M.Krein always gave extremely high praise to the results of A.M.Lyapunov; once, he even said that Lyapunov's works in the theory of stability of the shape of rotating liquid are "of genius." He did not make this remark, 2
By the way, it is worth noting that in the 1920s Krein was a post-graduate student of N. G. Chebotarev and often said that he was indebted to Nikolai Grihor'evich for the idea of studying general problems of extension with preservation of a certain property. Thus, as an example of a problem of this sort, one can mention the problem (posed by Chebotarev) of extension (i.e., of adding higher powers) of a polynomial whose roots are all real with preservation of this property. M. Krein also regarded as problems of this type the following problems posed and studied by him: On the extension of a Hermitian positive function defined on an inten-al to a Hermitian positive function on the enure real axis and on the extension of a helix line in a Hilbert space or in a Lobachevskii space to a complete helix line in the corresponding space.
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made during one of our long walks around the town, more precise. All these walks were, as a rule, finished in a bookstore on the famous Deribasovskaya street. M. Krein loved these walks around Odessa very much, he loved his sunny town with fresh air and a wonderful mixture of sea and steppe smells. (As it was in the 1930s.) He liked the irregular but very expressive and vivid "Odessa" language and rendered with great humor the picturesque Odessa phrases he had heard in the streets. He had a great skill of Ukrainian, enjoyed this language and often gave lectures in it. In the 1930s, M. Krein could never imagine himself without Odessa and the Odessa University. The first edition of his book with F. R. Gantmakher "Oscillatory Matrices and Kernels" published in 1940 was dedicated to the 75 th anniversary of the Odessa University. For M. Krein, it was always interesting to share his ideas and results with his mathematician friends, colleagues, post-graduate students, and students. He was an excellent narrator able to create an impression that new ideas and thoughts came to his mind just in a precise finished form. This feature distinguished him from the majority of people whom I know. Usually, a new thought first is not absolutely clearly connected with complicated processes in the subconscious and observed as an obscure "distant prospect of a free novel," only far later the picture acquires clear contours, and it becomes possible to distinguish principal facts and exact arguments. For Krein, this period was always a "thing in itself," when listening to him, you could never get rid of the impression that these new results appeared in his mind just as they were delivered to us—in a precise form and separated into lemmas and theorems. This was always interesting. We felt his enthusiasm, a specific atmosphere of creation, of finding the truth. This feature of Krein was very attractive. He was always surrounded by young mathematicians. A picture of slowly moving groups of young people speaking with enthusiasm on a mathematical language hardly understandable for normal people was typical of the Paster street in the 1930s. These were Krein's students and post-graduate students accompanying their professor from the university after a lecture or a seminar. Walking near the sea in the summer of 1935, we spoke not only about mathematics. We were of the same age, less than 30, and both were going in for sport. Once, Krein told me that, at the age of 17-18, he tried to get a job in the circus as an acrobat but failed for some reasons. This was a great event for mathematics, since he was born as a mathematician. In 1935,1 finished my post-graduate studies in the Rostov University and moved to Odessa and, in September, began my career as a lecturer at the Physico-Mathematical Department of the Odessa University. By that time, I had already got certain experience as a lecturer in high schools. To understand the situation in the Odessa University at that time, one should make a small excursion into the near past. At the end of the 19 th and the beginning of the 20th century, Odessa or. more precisely, Novorossiiskii University played an important role in the scientific life of the country. The Department of Natural Sciences of the Physico-Mathematical Faculty was famous for the prominent scientists working there. It suffices to mention M. I.Sechenov, I.I. Mechnikov, a famous physicist N. A. Uraov, and the lectures of a prominent zoologist A. O. Kovalevskii. All these scientists caused a great influence on the spiritual life of the university and created good scientific traditions. Of great importance for the development and direction of the mathematical investigations at the faculty, were the activities of Prof. I. V. Sleszczynski who was a highly educated mathematician and one of the founders of the "Mathesis" Publishing House, which played an important role in the dissemination of mathematical knowledge in Russia. This Publishing House published the translations of the famous works of Bolzano, G. Cantor, Dedekind, and other classics of contemporary mathematics. P. S. Aleksandrov was sure that the ideas of the theory of sets came to Russia through Odessa. S. O. Shatunovskii and V. F. Kagan who worked at the faculty took part in the activities of the "Mathesis" Publishing House. Mathematics was taught at a very good level. Samuil Osipovich Shatunovskii was a brilliant lecturer and a talented mathematician who often left time behind in his lectures and works. As early as at the end of the 19th century, i.e., before the appearance of Gilbert's book "Foundations of Geometry," Shatunovskii gave, in his course of lectures, "Introduction to Analysis," an axiomatic definition of the notion "magnitude" and verified the independence of these axioms by using models. Later, this course was published. He also gave a definition of the limit of a function given on a semiordered set. (The Moore-Shatunovskii limit.) Shatunovskii often criticized in his lectures the unlimited use of the law of excluded middle and not only criticized but also gave in his master's thesis (in 1917) a constructive, i.e., without the law of excluded middle, presenta-
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tion of the Galois theory. N. G. Chebotarev specially emphasized in his book "Galois Theory" that the presentation is based on Shamnovskii's ideas. The lectures of a prominent geometrician Veniamin Fedorovich Kagan were also of great scientific importance. These lecturers, their colleagues and disciples guaranteed a permanently high level of the faculty. But 1920 was marked with great trouble. As a result of the so-called "administrative rapture" of bureaucratic 'reformers' all the universities in the Ukraine were closed by a special order of the Ukrainian People's Commissariat of Education. Hard times lasted until 1933, when the "disfavor" of the universities ceased and they were restored. (In Russia, the same "reform" was undertaken later, was milder, and did not touch the Moscow and Leningrad Universities.) For these 13 years, many professors moved to other places. University or, as they are sometimes called now, "fundamental" sciences were kept down. Traditions were lost. It was necessary to begin from the very beginning: to invite and teach new experts, form new educational plans and programs of lecture courses, take care of the quality of lectures, and so on. This work aimed at the restoration and modernization of the faculty was headed (not formally but in fact) by M. Krein, who worked much charging his colleagues, disciples, and students with his enthusiasm. Parallel with compulsory courses, many special courses were delivered to the students including Functional Analysis, Hilbert Spaces and the Theory of Operators, the Method of Fixed Points and Its Applications in Analysis, and Harmonic Analysis on Groups. There were also classical courses: Equations of Mathematical Physics, Abstract Theory of Groups, Galois Theory, Theory of Algebraic Functions, Almost Periodic Functions, Theory of Entire Functions, etc. The students also had the opportunity to attend special courses in mechanics, also patronized personally by M. Krein. An incredible role in raising the scientific level of the faculty was played by Krein's scientific seminar. It deserves to be described in more detail. The subjects of the seminar were not restricted at all. Sometimes, the talks were suggested by the members of the seminar, sometimes they were offered by Krein. Many talks were delivered by him personally. M. Krein listened to all the talks very carefully and understood the authors even in the situations when everybody caught nothing; as a rule, his brief summaries after the talk made the entire picture much clearer. New problems were also posed in these brief summaries and it often happened that their solutions were also later presented at the seminar. Krein created there an atmosphere of free scientific discussion. It looked like a group of young people gathered to discuss something interesting for everybody. In 1936, the first issue of the journal "Uspekhi Matematicheskikh Nauk" appeared; it was mainly devoted to the problems of functional analysis and played a significant role in the popularization of this science. The survey article of L. A. Lyustemik published in it was, in fact, the first manual in functional analysis. The attention of mathematicians was also attracted by the work of V. V. Nemyckii dealing with the method of fixed points in analysis. Krein's scientific interests, and then the interests of the other members of the seminar, moved toward the theory of Banach spaces. This very soon resulted in the appearance of a series of works devoted to cones in Banach spaces, positive functionals, operators preserving an invariant cone, to the theorems on regularly closed sets, and many other functional-analytic papers. Among the members of the seminar one should mention M. A. Naimark, V. L. Shmul'yan, D. P. Mil'man, M.A. Rutman, V.P. Potapov, S.A.Orlov, A. P. Artemenko, M. S. Livshits, I.M. Glazman, and others. The works and names of these scientists then became widely known. It was, in fact, a new mathematical center in the south of the USSR. This fact was recognized by the mathematical community and mentioned in many surveys. In 1939, Krein was elected a corresponding member of the Ukrainian Academy of Sciences and soon, after the annexation of the Western Ukraine, he was sent, together with M. A. Lavrentyev, to the Lvov University to establish scientific relations with mathematicians living in Lvov. There, at the seminar of S. Banach, he presented his results and results of his colleagues in the theory of Banach spaces. Stefan Banach highly appreciated his talk and said: "Your talk is a progress." We were witnesses of the rapid progress of the Physico-Mathematical Faculty of the Odessa University both in the scientific and pedagogic senses, proud of its successes and considered ourselves as cocreators of this process. Beginning with 1936, M. Krein was a collaborator of the Institute of Mathematics at the Kharkov State University; in 1940, he occupied a position (not leaving his work in Odessa) of the Head of the Department of Functional Analysis at the Institute of Mathematics of the Ukrainian Academy of Sciences in Kiev. His regular visits to Khar-
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kov and then to Kiev, talks delivered there, discussions with mathematicians and common works greatly contributed to the development of functional analysis in these scientific centers. During the Second World War, many employees of the university including colleagues and followers of M. G. Kxein went to the regular army, the others were evacuated to various places in the East of the country. Krein was evacuated to Kuibyshev (now Samara). There, he worked as a professor in the Kuibyshev Pedagogical Institute, taught post-graduate students, and was one of the organizers of the Aviation Institute. In these three years, M. Krein published his first works devoted to the theory of entire operators, to representations of functions by the FourierStieltjes integrals, and to the solution of Kolmogorov's problem of prediction. In 1944, M. Krein returned to Odessa after it was liberated from Nazi occupation. One after one, his disciples and colleagues also returned to Odessa. But some of them did not return. V. L. Shmul'yan, always merry and tireless, perished near Warsaw; even in the regular army, he found possibilities to continue extensive studies in mathematics and published several articles during the war. A. G. Okun' was lost near Korsun-Shevchenkovsky, V. N. Shmushkovich also died, extraordinary gifted A. P. Artemenko was missing. But the others returned. The university also returned to Odessa. But they got into unexpected trouble. Neither M. Krein, nor his intimate colleagues were allowed to return to the university. An attempt undertaken by Mark Grigor'evich to convince the Rector that this was not reasonable failed. M. Krein occupied a position at the Chair of Mechanics of the Odessa Institute of Navy Engineers (OENE); his colleagues had to work in different high schools of Odessa. But what had happened? What was the reason to keep a recognized mathematical school separate from the mathematical youth? Why it was necessary to weaken the Physico-Mathematical Faculty of the Odessa University once again? No reasonable answer was received. The situation was so unnatural and weird that the university authorities were forced to give explanations and, in particular, they alluded to the orders "from above." Certainly, such orders indeed existed at that time, but I think, that the university authorities were too active in realizing these general rules. Scientific community of Odessa and many Soviet mathematicians tried to protest against this unnatural situation by the sole method possible at that time, namely, by writing letters of protest to higher authorities, which, of course, sent inquiries and the university authorities, to defend themselves, replied with various accusations against M. G. Krein. This caused the appearance and propagation (among the Communist Party activists and philistines) of a fairy tale that Krein, in their frame of references, is a "Zionist." Of course, this version had nothing common with reality. As the flow of accusations increased, Krein's secret "dossier" lying somewhere in the KGB increased as well. Since that time, M. Krein lived under the permanent pressure of the authorities combined with an unfriendly attitude to him by some mathematicians. (The accusation against Krein became evident every time when he tried to go abroad, or when the problem of awarding him a prize was discussed, etc.). In 1952. he was dismissed from the Kiev Institute of Mathematics with a simple motivation that he did not live in Kiev permanently. This weight pressed M. Krein uninterruptedly for the second half of his life, beginning in 1948. Only once he complained to me: "I read sometimes in books that somebody was slandered but never could imagine what it means in reality." What a powerful spirit he possessed and how strong was his love of mathematics to continue in that situation, as he did, extensive studies, undertook all new researches, making one discovery after another. This permanent and successful work supported his high life tone for many years. In this period of his life, Krein carried out thorough investigations of the regions of stability of canonical systems of differential equations with periodic coefficients and the Wiener-Hopf equation, classical investigations in the theory of operators, wrote a large series of works in the spectral theory of a string, the works in the theory of operators in spaces with indefinite metric, works devoted to Hankel operators, etc. Even a superficial analysis of Krein's scientific heritage would cover many pages. In the last period of his life, he wrote six books and more than 150 articles. He delivered large lecture courses in the mathematical schools. His investigations caused more and more profound influence on various branches of mathematics and, correspondingly, his results were more and more highly appreciated throughout the world. In 1968, he was elected an Honored Member of the American Academy of Science and Arts; in 1987, he became a foreign member of the National Academy of the USA, and, in 1987, an Honored Member of the Moscow Mathematical Society. In a well-known book by P. D. Lax and R. S. Phillips "Scattering Theory for Automorphic Functions," the introduction is completed by the words: "This book is dedicated to Mark Krein, one of the mathematical giants of our
182 MARK GRIGOR'EVICH KREIN (RECOLLECTIONS)
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century, as a recognition of his extremely versatile and deep contribution to mathematics. As any analyst, we are indebted to him very much." This wide appreciation of his great role in the world of science was his spiritual victory over his opponents. M. Kxein was always friendly and enjoyed personal contacts with different people being interested in their problems. At the same time, he was an interesting narrator himself and often told about his meetings with interesting people and about interesting events. Scientific contacts were necessary for him. Probably this was one of the main reasons for which he always organized seminars that attracted many participants. In addition to the seminar in the institute where he worked, Krein organized a city seminar in the House of Scientists, his seminar in hydromechanics organized at the OINE was of great importance for shipbuilding engineers. Among the participants of this seminar, one should mention A. A. Kostyukov, V. G. Sizov, Yu. L. Vorob'ev. and others. As before, gifted young people were attracted by Mark Grigor'evich, many students and post-graduate students of the university and pedagogical institute attended his seminars. Young scientists came to him for consultations from other cities and republics. Some universities sent their graduates to Krein for post-graduate studies. He always worked much. Krein's life was especially intense in the summer on his dacha in Arcadia (near Odessa). A major part of the day he worked at the table on a large veranda. This was also a place for receiving visitors. Once Boris Nikolaevich Delone delivered a lecture entitled "What is necessary for anybody who wants to become a good scientist ?" Prof. Delone told about many important things in his traditional witty manner but finished his lecture quite unexpectedly. He said: "However, everything that I have already told you is not so important, the most important problem is to marry successfully." Krein's wife, Raisa Lvovna Romen was an attractive educated lady and good witty company. All her life she was dedicated to her husband, created good conditions for his scientific work; in difficult situations, she tried to protect him against all troubles; in hard times, she never allowed him to give up. She always deserves to be mentioned when anybody speaks about Mark Krein. Raisa Lvovna died in May, 1989. Mark Krein died five months later.
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ABSTRACT to papers by M.G.Krein submitted by Moscow Mathematical Society and Department of Mathematics and Mechanics of Moscow State University for the Lenin Prize on 1963.
In accordance to the presentation of the mentioned departments for the Lenin Prize on 1963 my works in 10 fields of mathematical analysis are submitted. A short description of these works in each field separately is given below.
1. The theory of oscillatory matrices, oscillatory integral and differential operators On this subject the following papers were published: 1.1 Sur quelques applications des noyaux de Kellog aux problemes d'oscillation, Soobshch. Nauchn.-Issled. Inst. Mat. Mekh. Khar'kov. Gos. Univ. i Khar'kov. Mat. o-va (4) 11 (1935), 3-19. 1.2 Sur les matrices oscillatoires, C.R. Acad. Sci. Paris, 201 (1935), 577-579 (with F.R. Gantmakher). 1.3 On oscillation differential operators, Dokl. Akad. Nauk SSSR 4 (1936), 379-382. 1.4 Sur les vibrations propres des tiges dont l'une des extremites est encastree et I'autre libre, Zap. Khar'kov. Mat. O.-va (4) 12 (1936), 3-11. 1.5 On a special class of determinants in connection with Kellog's integral kernels, Mat. Sb., 42 (1935), 501-508 (with F.R. Gantmakher). 1.6 Sur quelques proprietes de noyaux de Kellog, Zap. Khar'kov. Mat. O.-va (4) 13 (1936), 15-28. 1.7 On some properties of the Kellog's resolvent kernel, Zap. Khar'kov. Mat. O.-va (4) 14 (1937), 61-73. 1.8 Sur les matrices complement non-negatives et oscillatoires, Comp. Math., 4 (1937), 445-476 (with F.R. Gantmakher). 1.9 On totally nonnegative Green functions of ordinary differential operators, Dokl. Akad. Nauk SSSR 24 (1939), 220-223 (with G.M. Finkel'shtein). 1.10 On nonsymmetric oscillating Green functions of ordinary differential operators, Dokl. Akad. Nauk SSSR 25 (1939), 643-646. 1.11 Oscillation theorems for ordinary linear differential operators of arbitrary order,
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Dokl. Akad. Nauk SSSR 25 (1939), 717-720. 1.12 Oscillation Matrices, Kernels a n d Small Oscillations of Mechanical Systems, 2nd edition, Moscow, Leningrad (1950) (with F.R. G a n t m a k h e r ) . 1.13 Oszillationsmatrizen, Oszillationskerne u n d kleine Schwingungen Mechanischer Systeme, Akademie Verlag, Berlin, (1960) (with F . R . G a n t m a k h e r ) . T h e m a t h e m a t i c a l theory of oscillatory properties of small vibrations of mechanical systems is presented in t h e books 1.12 and 1.13. In t h e papers 1.9, 1.10, 1.11 the a u t h o r showed t h a t in this way one can obtain oscillatory theorems for b o u n d a r y value problems, selfadjoint and non-selfadjoint, moreover of any order (even or o d d ) .
2. G e o m e t r y o f B a n a c h s p a c e s a n d its a p p l i c a t i o n s in a n a l y s i s On this subject the following papers were published: 2.1 On several questions concerning the geometry of convex sets belonging to a linear n o r m e d a n d complete space, Dokl. Akad. Nauk SSSR 14 (1937), 5-7. 2.2 On positive additive functionals in linear n o r m e d spaces, Zap. Khar'kov. M a t . O.-va (4) 14 (1937), 227-237. 2.3 O n positive functionals in linear n o r m e d spaces, In the book "Some questions in t h e Theory of M o m e n t s " , Gos. Nauchn. Tekhn. Izd-vo Ukr., Kharkov (1938), (with F . R . G a n t m a k h e r ) ; English Transl.: Transl. M a t h . M o n o g r a p h s , Vol. 2, A m e r . M a t h . S o c , Providence, R.I. (1962). 2.4 On the linear operators t h a t leave invariant a conic set, Dokl. Akad. Nauk SSSR 23 (1939), 749-752. 2.5 Basic properties of n o r m a l conic sets in B a n a c h space, Dokl. Akad. Nauk SSSR 28 (1940), 13-17. 2.6 On the minimal decomposition of a linear functional into positive c o m p o n e n t s , Dokl. Akad. Nauk SSSR 28 (1940), 18-22. 2.7 O n an intrinsic characteristic of t h e space of all continuous functions denned on a bicompact Hausdorff space, Dokl. Akad. Nauk SSSR 27 (1940), 427-430 (with S.G. Krein). 2.8 On regularly convex sets in t h e space conjugate to a B a n a c h space, Ann. M a t h . 41 (1940), 556-583 (with V.L. S h m u l ' a n ) .
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2.9 On extreme points of regularly convex sets, Studia Math. 9 (1940), 133-138 (with D.P. Mil'man). 2.10 Linear operators leaving invariant a cone in Banach space, Usp. Mat. Nauk 3, no. 1 (1948), 3-95 (with M.A. Rutman); English transl.: Amer. Math. Soc. Transl. (1), 10 (1962), 199-235. Offprints of the papers 2.3, 2.9, 2.9 (4 copies) and 2.4, 2.5, 2.6 (4 copies) were submitted to the Lenin committee. The notion of the "cone" in Banach space (as the term itself) were introduced in the papers 2.2, 2.3. In the papers 2.2 and 2.3 and 2.5-2.7 the geometry of these cones, and in the paper 2.4 and in the joint paper 2.10 the theory of operators leaving a cone invariant with its applications were developed. These papers were continued and used by soviet and foreign mathematicians. At the last time the theory of cones got a series of essential applications in nonlinear functional analysis in papers by the group of Voronezh mathematicians headed by M.A.Krasnosel'skii. In the papers 2.1, 2.8, 2.9 convex sets in Banach spaces are studied. In 2.8 using using results of the paper 2.9 the theorem on extreme points of a regularly convex set which is known in the mathematical literature as the "Krein-Mil'man" theorem was established. Thanks to the well known paper by I.M.Gel'fand and D.A.Raikov this theory got important applications in the theory of representations of unitary groups. Thanks to papers by N.N.Bogol'ubov and N.N.Krylov it is used in the theory of dynamical systems. This theorem is presented in various papers and monographs.
3. Theory of Hermitian operators with finite deficiency numbers and method of directing functionals On this subject the following papers were published: 3.1 On the continuation problem for Hermitian-positive continuous functions, Dokl. Akad. Nauk SSSR 26 (1940), 17-22. 3.2 On Hermitian operators with deficiency indices equal to one, I, Dokl. Akad. Nauk SSSR 43 (1944), 323-326. 3.3 On Hermitian operators with deficiency indices equal to one, II, Dokl. Akad. Nauk SSSR 44 (1944), 131-134. 3.4 On a remarkable class of Hermitian operators, Dokl. Akad. Nauk SSSR 44 (1944), 191-195. 3.5 On a generalized moment problem, Dokl. Akad. Nauk SSSR 44 (1944), 239-243.
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3.6 On the logarithm of an infinitely decomposable Hermitian-positive function, Dokl. Akad. Nauk SSSR 45 (1944), 99-102. 3.7 On t h e continuation problem for helical arcs in Hilbert space, Dokl. Akad. N a u k SSSR 45 (1944), 139-142. 3.8 On a generalization of investigations of G.Szego, V.I.Smirnov and A.N. Kolmogorov, Dokl. Akad. Nauk SSSR 46 (1945), 91-94. 3.9 On an extrapolation problem of A.N.Kolmogorov, Dokl. Akad. N a u k SSSR 46 (1945), 306-309. 3.10 On t h e resolvents of a Hermitian operator with deficiency index ( m , r a ) , Dokl. Akad. Nauk SSSR 52 (1946), 657-660. 3.11 On a general m e t h o d of decomposing positive-definite kernels into elementary p r o d u c t s , Dokl. Akad. Nauk SSSR 53 (1946), 3-6. 3.12 On Hermitian operators with directing functionals, Sb. Trudov Inst. M a t . Akad. Nauk Ukr. SSR 10 (1948), 83-106. 3.13 F u n d a m e n t a l aspects of t h e representation theory of Hermitian matrices with deficiency index
(m,m),
Ukr. M a t . Zh., 1, no.2 (1949), 3-66; English t r a n s l : A m e r . M a t h . Soc. Tranl. (2) 97 (1970), 75-143. 3.14 Infinite J - m a t r i c e s a n d t h e m a t r i x m o m e n t problem, Dokl. Akad. Nauk SSSR 69 (1949), 125-128. 3.15 On a one-dimensional singular b o u n d a r y value problem of even order on t h e interval (0,oo), Dokl. Akad. N a u k SSSR 74 (1950), 9-12. 3.16 Analogue of t h e Chebyshev-Markov inequalities in a one-dimensional b o u n d a r y value problem, Dokl. Akad. Nauk SSSR 89 (1953), 5-8. 3.17 On the theory of entire functions of exponential t y p e , Izv. Akad. Nauk SSSR Ser. M a t . , 11 (1947), 309-326. Offprints of the p a p e r s 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 2.10, 2.11, 2.12, 2.13, 2.14, 2.15, 2.17 (4 copies) were s u b m i t t e d to t h e Lenin committee. In t h e papers of this set the a u t h o r developed t h e theory of representations of of Hermitian operators (an a b s t r a c t analogue of Fourier transforms) which allowed to consider a new class of "entire" Hermitian operators. In t h e theory of entire operators it was a success to unite a n d to elaborate t h e solution of such problems as t h e problem of continuation of helical lines in Hilbert spaces, t h e problem of filtration a n d extrapolation of stationary casual processes, t h e problem of description of all spectral functions of t h e
187 non-determined Sturm-Liouville problem, - it was a success also to carry over the famous Chebyshev-Markov inequalities on boundary value problems. These researches are supplemented by elaborating of the method of directing functionals thanks to which the gap which was existed between abstract spectral theory and spectral theory of concrete differential operators was removed. The method of directing functionals allowed to obtain by the unite method for spectral partitions assertions of the Plancherel theorem type and of the Bochner-Khintchine type. Strictly analytic papers of the set 5 on inverse problems are in close connection with these papers. All this papers were a channel for penetration of different methods of the analytic functions theory to the operator theory and new facts from the functions theory were required; one of them was obtained in the paper 3.17. This work found applications in the theory of non-selfadjoint operators (see 10.4). It was a starting point in new researches of young soviet mathematicians I.V.Ostrovsky, V.I.Matsaev and A.A.Gol'dberg.
4. Theory of semibounded Hermitian operators On this subject the following papers were published: 4.1 On self-adjoint extensions of bounded and semibounded Hermitian operators, Dokl. Akad. Nauk SSSR 48 (1945), 323-326. 4.2 The theory of self-adjoint extensions of semibounded Hermitian operators and its applications. I, Mat. Sb., 20 (62) (1947), 431-495. 4.3 The theory of self-adjoint extensions of semibounded Hermitian operators and its applications. II, Mat. Sb., 21 (63) (1947), 365-404. 4.4 The basic theorems concerning the extension of Hermitian operators and some of their applications to the theory of orthogonal polynomials and the moment problem, Usp. Mat. Nauk 2, no.3 (1947), 60-106 (with M.A.Krasnosel'skii). Offprints all of these papers were presented to the Lenin committee, 4 copies of 4.2. In the papers 4.1, 4.2 the complete description of all positive selfadjoint extensions of a positive selfadjoint operators is given, the complete solution of the Newman-Friedrichs on existence criterion for positive selfadjoint extension is given, it was proved that the "Friedrichs extension" is the "most hard", the "most soft" extension is found. In the papers 4.3 and 4.4 an illustration of applications of these results to ordinary boundary value problems and to the moment problem is given.
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5. I n v e r s e p r o b l e m s for t h e s p e c t r a l t h e o r y o f differential o p e r a t o r s On this subject the following papers were published: 5-1 Solution on t h e inverse Sturm-Liouville problem, Dokl. Akad. Nauk SSSR 76 (1951), 21-24. 5.2 Determination of t h e density of a nonhomogeneous symmetric string from its frequency spectrum, Dokl. Akad. Nauk SSSR 76 (1951), 345-348. 5.3 On inverse problem for a nonhomogeneous string, Dokl. Akad. N a u k SSSR 82 (1952), 669-672. 5.4 On a generalization of investigations of Stieltjes, Dokl. Akad. Nauk SSSR 87 (1952), 881-884. 5.5 On some new problems of t h e oscillation theory for S t u r m i a n systems, Prikl. M a t . Mekh. 14 (1952), 555-568. 5.6 On t h e transfer function of a one-dimensional second-order b o u n d a r y value problem, Dokl. Akad. Nauk SSSR 88 (1953), 405-408. 5.7 On some cases of effective determination of t h e density of a nonhomogeneous string from its spectral function, Dokl. Akad. N a u k SSSR 93 (1953), 617-620. 5.8 On inverse problems of t h e theory of filters a n d A-zones, Dokl. Akad. N a u k SSSR 93 (1953), 767-770. 5.9 A fundamental approximation problem in t h e theory of extrapolation a n d filtration stochastic processes, Dokl. Akad. N a u k SSSR 94 (1954), 13-16; English transl.: Selected Transl. in M a t h . Statistics and Probability, Inst, of M a t h . Statistics and Amer. M a t h . Soc. 4 (163), 127-131. 5.10 On a m e t h o d of effective resolution of an inverse b o u n d a r y value problem, Dokl. Akad. Nauk SSSR 94 (1954), 987-990. 5.11 On integral equations t h a t generate second-order differential equations, Dokl. Akad. Nauk SSSR 97 (1954), 21-24. 5.12 On the determination of the potential of a particle from its ,9-function, Dokl. Akad. N a u k SSSR 105 (1955), 433-436. 5.13 Continual analogues of propositions on polynomials orthogonal on t h e unit circle, Dokl. Akad. N a u k SSSR 105 (1955), 637-640. 5.14 On t h e theory of accelerants a n d 5-matrices of canonical differential systems, Dokl. Akad. Nauk SSSR 111 (1956), 1167-1170.
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5.15 On a continual analogue of a formula of Christoffel from t h e theory of orthogonal polynomials, Dokl. Akad. N a u k SSSR 113 (1957), 970-973. Offprints of these papers are presented to t h e Lenin committee in 4 copies. In the p a p e r 5.1 of t h e enclosed list for t h e first time was shown a chain of analytic operations which allows to solve inverse problem for t h e S t u r m - Liouville equation in t h e Borg formulation defined more precisely by V.A. Marchenko. In t h e same p a p e r for t h e first time a transfer function for solution of inverse problem was introduced. Mechanical sense of this function a n d simultaneously a series of its basic properties was t h e n clarified in 5.3 and 5.6. In 5.2, 5.3 conditions of uniqueness a n d solvability of inverse problems for nonhomogeneous string are established. In 5.5 results of t h e p a p e r s 5.2, 5.3 are illustrated by example of mechanical systems with a finite n u m b e r degrees of freedom. In 5.8 an inverse problem for t h e Hill equations is solved, i.e. t h e problem of reconstruction of a periodic coefficient in equation by its A-zones of stability (on t h e physical language by zones of letting pass of corresponding
filter).
After papers 5.1, 5.2 a p p e a r e d p a p e r s by I.M.Gel'fand a n d B.M.Levitan in which they starting from t h e same transfer function obtained simple integral equations which allow t o recover one-dimensional radial Schroedinger equation from its spectral function. Uniqueness of solution for this problem was established earlier by G.Borg a n d V.A.Marchenko. In 5.6, 5.11 a gap in necessary a n d sufficient conditions of solvability of t h e problem for recovering of a potential by spectral function which took place in p a p e r s by I.M.Gel'fand a n d B.M.Levitan is removed. Simultaneously was obtained a simplification of their integral equation which allowed to see a series of new cases when the inverse problem is solved effectively. Different cases of effective solving of inverse problems of classical a n d q u a n t u m mechanics are shown in 5.7, 5.10, 5.15. In 5.15 it is shown how to reduce t h e solving of inverse problem for radial s t a t e equation of a particle with an a r b i t r a r y value of the q u a n t u m n u m b e r to the corresponding problem for its s t a t e . In 5.12 t h e p r o b l e m of d e t e r m i n a t i o n of t h e potential of a particle by its limit p h a s e is solved. In 5.15 this result is generated on t h e case of a system of differential equations (the case of tensor forces in inverse problems of q u a n t u m mechanics). In 5.13 it is found out t h a t a n a t u r a l generalization a n d in certain cases also a simplifica-
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tion of previous researches is obtained by replacing of equation of the radial Schroedinger equation t y p e by a certain system of equations of t h e first order (of radial Dirac equations type). In 5.4 it is shown t h a t t h e successive generalization of well-known researches by Stieltjes in chain fractions leads to complete theory of spectral functions of regular and singular structure. In 5.9 it is established t h a t the problem of extrapolation a n d filtration for real casual process by its d a t a for finite time interval is reduced to solving of a problem of the foundation of the density of nonhomogeneous string by its spectral function.
6. T h e o r y o f linear H a m i l t o n i a n s y s t e m s o f differential e q u a t i o n s w i t h p e r i o d i c coefficients O n this subject t h e following papers were published: 6.1 Generalization of a certain investigations of A.M.Lyapunov concerning differential equations with periodic coefficients, Dokl. Akad. Nauk SSSR 73 (1950), 445-448. 6.2 On t h e application of an algebraic proposition in t h e theory of m o n o d r o m y matrices, Usp. M a t . N a u k 6, n o . l (1951), 171-177. 6.3 T h e basic propositions of the theory of A-zones of stability of a canonical system of linear differential equations with periodic coefficients, Memorial volume dedicated to A.A.Andronov, Moscow (1955), 413-498; English transl.: Topics in Differential a n d Integral E q u a t i o n s and O p e r a t o r Theory, O p e r a t o r Theory: Advances a n d Applicatins, Vol. 7, Birkhauser, Basel (1983), 1-70. 6.4 On tests for stable boundedness of solutions of periodic canonical systems, Prikl. M a t . Mekh., 19 (1955), 641-680; English transl.: A m e r . M a t h . Soc. Transl. (2) 120 (1983), 71-110. 6.5 Hamiltonian systems of linear differential equations with periodic coefficients, P r o c . I n t e r n a t . S y m p . Linear Vibrations, Kiev, (1961) (with V.A.Yakubovich). 6.6 Analytic properties of multipliers of periodic canonical differential systems of positive type, Izv. Akad. Nauk SSSR Ser. Mat., 26 (1962), 549-572 (with G.Ya.Lyubarskii). English transl.: Amer. M a t h . Soc. Transl. (2) 89 (1970), 1-20. T h e papers 6.3, 6.4, 6.5 (4 copies) are s u b m i t t e d to t h e Lenin committee. T h e m a i n p a p e r is 6.3 in which there are established t h e classification of multiplicators, their rules of motion on t h e unit circle, existence of central zone of stability for systems
191 of positive type and a basic rule for determination of critical frequencies in phenomenon of parametric resonance. After the papers 6.1 and 6.2 for the first time became possible the extension of classical results by A.M.Lyapunov obtained at 90-th of the past century for a single equation on systems. Some of results of the author were rediscovered later by foreign physicists in connection with theory of sinchrotrons with strong focusing (Courant and Snider, Seiden, Luders and others).
7. Generalization of L.S.Pontrjagin's duality principle On this subject the following papers were published: 7.1 On positive functionals on almost-periodic functions, Dokl. Akad. Nauk SSSR 30 (1941), 9-12. 7.2 The duality principle for a bicompact group and square block-algebra, Dokl. Akad. Nauk SSSR 69 (1949), 726-728. 7.3 Hermitian-positive kernels on homogeneous spaces, II, Ukr. Mat. Zh., 2, no.l (1950), 10-59; English transl.: Amer. Math. Soc. Transl. (2) 34 (1963), 109-164. Offprints of the papers 7.2, 7.3 are submitted to the Lenin committee (4 copies). In the paper 7.1 was obtained a partial generalization of the classical L.S.Pontrjagin's duality principle, obtained by him for locally-compacted commutative groups on, the case of compact noncommutative groups. This result as it turned later was obtained somewhat earlier by Japanese mathematician Tanaka. In 7 years in 7.2 the author obtained complete generalization of the principle containing the precise description of duality object which formed the most difficult part in the establishing of the new principle. The detailed exposition with certain further development is given in 7.3.
8. Spectral theory of operators in spaces with indefinite metric On this subject the following papers were published: 8.1 Lorentz transformation in a real Hilbert space, Chapter 5 of the paper 2.10. 8.2 Helical lines in an infinite-dimensional Lobachevsky space and Lorentz transformations, Usp. Mat. Nauk 3, no.3 (1948), 158-160. 8.3 On an application of the fixed point principle in the theory of Hnear transformations of spaces with an indefinite metric,
192
Usp. Mat. Nauk 5, no.2 (1950), 180-190; English transl.: Amer. M a t h . Soc. Transl. (2) 1 (1955), 27-35. 8.4 Spectral theory of operators in spaces with an indefinite metric. 1,11, Tr. Mosk. Mat. O.-va 5 (1956), 367-432; 8 (1959), 413-496 (with I.S.Iokhvidov); English transl.: Amer. M a t h . Soc. Transl. (2) 13 (1960), 105-175; (2) 34 (1963), 283-373. 8.5 On an integral representation of a continuous Hermitian-indefinite function with a finite n u m b e r of negative squares, Dokl. Akad. Nauk SSSR 125 (1959), 31-34. Offprints of t h e papers 8.4, 8.5 are s u b m i t t e d to t h e Lenin committee (4 copies). T h e first fundamental result on the theory of self-adjoint operators in spaces with an indefinite metric was obtained by L.S.Pontrjagin. To the a u t h o r belongs t h e elaboration of a new m e t h o d (8.1, 8.3) which allowed to obtain easier a n d at t h e same time to generalize t h e result of L.S.Pontrjagin. T h e essentially new application of t h e theory of spaces with an indefinite metric was the creation by t h e a u t h o r of foundations of the theory of helical lines in Lobachevsky spaces of infinite n u m b e r of dimensions and establishing of t h e t h e o r e m on integral representations of a Hermitian-indefinite function (a generalization of the Bochner-Khintchine t h e o r e m ) . Except of the last result of 8.5, all the others are presented in t h e joint s u m m a r y paper 8.4.
9. T h e o r y of singular integral e q u a t i o n s and s y s t e m s of such equations On this subject the following papers were published: 9.1 On a new m e t h o d of solving linear integral equations of first and second kind, Dokl. Akad. Nauk SSSR 100 (1955), 413-416. 9.2 Integral equations on t h e half-line with kernel depending on the difference of the arguments, Usp. M a t . N a u k 13, no.5 (1958), 3-120; English transl.: Amer. M a t h . Soc. Transl. (2) 22 (1962), 163-288. 9.3 On some basic propositions of the theory of systems of integral equations on the half-line with kernels depending on t h e difference of t h e a r g u m e n t s , P r o c . 3rd All-Union M a t h . Conf., Vol. 2 (1956), 37-38 (with I.C.Gohberg). 9.4 Systems of integral equations on t h e half-line with kernel depending on t h e difference of a r g u m e n t s , Usp. Mat. Nauk 13, no.2 (1958), 3-72 (with I.C.Gohberg); English transl.: Amer.
193
Math. Soc. Transl. (2) 14 (1960), 217-287. 9.5 On the stability of the systems of partial indices of the Hilbert problem with several unknown functions, Dokl. Akad. Nauk SSSR 119 (1958), 854-857 (with I.C.Gohberg). Offprints of the papers 9.1, 9.2, 9.4, 9.5 are submitted to the Lenin committee (4 copies). In connection with inverse problems the author elaborated (the paper 9.1) a new method for solution of Hnear integral equations of first and second kind in a finite interval. This method became to be used in contact problems of the elasticity theory (the papers by N.H.Arutjunjan, P.E.Prokopovich, G.Ya.Popov). In the paper 9.2 for the first time is established the structure of simple or generalized resolvent of Wiener-Hopf integral equation moreover in the unique assumption of belonging of the kernel function to the class Li[ — oo,oo). It allowed to give to all the theory the harmony and to formulate a series of final theorems. In joint papers 9.3, 9.4 based on the paper 9.2 for the first time the theory of systems of Wiener-Hopf integral equations was elaborated. Let note separately criterion of stability of the systems of partial indices of integral system which gave a reason for establishing of the corresponding result in the RiemannHilbert problem for several unknown functions.
10. Method of infinite determinants of a perturbation On this subject the following papers were published: 10.1 On a trace formula in perturbation theory, Mat. Sb., 33 (75) (1953), 597-626. 10.2 On perturbation determinants and a trace formula for unitary and self-adjoint operators, Dokl. Akad. Nauk SSSR 144 (1962), 268-271; English transl.: Soviet Math.. Dokl. 3 (1962), 707-710. 10.3 On the theory of wave operators and scattering operators, Dokl. Akad. Nauk SSSR 119 (1962), 475-478 (with M.Sh.Birman); English transl.: Soviet Math. Dokl. 3 (1962), 740-744. 10.4 On the theory of linear non-selfadjoint operators, Dokl. Akad. Nauk SSSR 130 (1960), 253-256; English transl.: Soviet Math.. Dokl. 1 (1960), 38-40. Offprints of all these papers are submitted to the Lenin committee (4 copies).
194 In 10.1 by this method is given a strict basis for a trace formula in perturbation theory derived for the first time (not perfectly strictly) for finite-dimensional perturbations by I.M.Livshits (in connection with problems of quantum statistics of solid solutions). In 10.2 further generalizations of this formula are established. Using of papers 10.1, 10.2 allowed to establish in the joint paper 10.3 a first theorem on structure of scattering matrix giving the Heisenberg scattering operator in elementary spaces. In the paper 10.4 the method of perturbation determinants is used to establish the completeness of system of the root subspaces for non-selfadjoint operator with an imaginary kernel component.
/M.G.Krein/ 10/XI.1962
195
SOLUTION OF T H E I N V E R S E STURM-LIOUVILLE P R O B L E M
M. G. KREIN
1. Let q(x) (0 < x < I, I < oo) be a real-valued function that is integrable in its interval of definition (0,1), and let Lq = —d2/dx2 + q(x) be the differential operator corresponding to q. We denote by Sq(a, (3) (—oo < a, (3 < oo) the spectrum, arranged as an increasing sequence of numbers, of the boundary value problem (1)
Lqy = Ay;
cos a y(0) + sin a y'(0) = 0, cos/3 y(l) + sin (3 y'(l) = 0.
As is known, the number I is completely determined by a given spectrum Sq(a, /3) = {-\j}i°i s m c e (2)
I2 = 7T2 lim ( n 2 / A „ ) . n—Voo
Borg [1] showed that if for two functions q\(x) and q[2(x) and some numbers a\, a.2, and (3 ( t a n a i ^ tan«2) we have Sqi(ai,f3) = Sq2(ai,(3) (i = 1,2), then almost everywhere qi(x) = q2(x) (0 < x < I).1 Recently, Marchenko [3], using other techniques, obtained a more complete result; namely, he showed that the function q(x), as well as the numbers t a n a i , tana2, and tan/3 ( t a n a i 7^ tan02) are uniquely determined by the corresponding spectra Sq(ai, (3) (i = 1,2). Moreover, changing slightly the setting of the question, Marchenko generalized this uniqueness theorem to a singular case where I is not necessarily finite and the function q(x) (0 < x < I) is absolutely integrable, but possibly in subintervals (0, a) with a < I only, and not in the entire interval (0,1). We call l-sequence any sequence {An} that satisfies condition (2) with a given l>0. As is known, the numbers of two distinct spectra Sq(oti,(3) (i = 1,2) strictly alternate. The above-mentioned important studies by Borg and Marchenko left the following basic problem unsolved: P r o b l e m A. Given two interlacing l-sequences of numbers, S = {A„} and S' =
(3)
Ai < A'i < A2 < A'2 < • • • ,
Doklady Akademii Nauk SSSR, vol. 76 (1951), pp. 21-24. ^ o r g assumed that one of the numbers a or a' equals zero, but it is easy to get rid of this restriction [2].
196
it is required to determine whether there exist a function q (0 < x < I) and numbers a, a', and (3 such that Sq(a,(3) = S and Sq(a',(3) = S', and if they do, to find them. In the sequel, we present a chain of analytic procedures that give a solution to this problem. The solution is based on the following idea: I. Just as to each Jacobian matrix J corresponds a certain power moment problem for every solution of which (mass distribution function) the matrix J itself is completely determined, so to a second order differential operator L with boundary condition on one end corresponds a "generalized" moment problem such that for every distribution function of it, is completely determined, together with the boundary condition, the operator itself (if the latter is given in a "canonical" form of some kind). For operators of "sufficiently regular type" this generalized moment problem is the problem of continuation of Hermitian-positive functions developed by the author [4-7]. This idea is applied here to the solution of Problem A. The Borg-Marchenko uniqueness theorem will be obtained incidentally. 2. We denote by Pa (0 < a < oo) the set of real-valued continuous even functions F(t) (\t\ < a) to which corresponds a positive definite kernel F(s — t) (0 < s,t < a). The central mass M of a function F(t) € Pa is the greatest number p for which F(t) — p G Pa. There are many ways to find M. In the most difficult case where a < oo, we can obtain M, for example, using the formula M
~
1 =
X>;(/
Xj(.s)ds)
,
where Xj(s) (i = 1,2,...) is a complete orthonormal system of fundamental functions of the integral equation X(s)
= p f F(sJo
t)X(t)dt,
and {MJ}I° is the corresponding sequence of characteristic numbers [4]. If F(t) £ Pa, then, considering F in a shorter interval (—x, x), we obviously have F(t) G Px (0 < x < a). To the function F as an element of Px (0 < x < a) will correspond a central mass M(x) (0 < x < a). The function M(x) (0 < x < a) so defined will be called the central function for F(t) € Pa- It turns out that this function is always lower semicontinuous and nonincreasing as x grows; moreover, M(x) < F(0) (0<x +0. 3. When solving Problem A, it can obviously be assumed without loss of generality that all the numbers in (3) are positive. To every two infinitely increasing interlacing sequences 5 = {Aj}J° and {\'j}f (0 < Ai < \[) corresponds an absolutely convergent expansion OO
3= 1
-i
\
OO
l\l
'
J
where 7 > 0, and rrij > 0 (j = 1,2,...).
3= 1
J
197 The existence of the expansion (4) for the product on the left-hand side can be proved by elementary means, but it also results directly from the following general proposition: In order that a function $(z) defined for all z £ (0, oo) be a regular analytic function that assumes only nonnegative values for negative z and maps the upper half-plane into its part, it is necessary and sufficient that $(z) have, for each z £ (0, oo), the representation [°° do-(X)
da(X)
where 7 > 0 and cr(\) (0 < A < 00) is a nondecreasing function. We now use (4) to construct the function 00
F(t) = S2 -T1 COS y/Xjt 3= 1
(-00 < t < OO)
J
that obviously belongs to the class Poo. Its central function will be denoted by M{x;S,S') (0
M (2a:) - M(2l) = const (^
- ^^]
forO<x
where
4—^)
(0<x
we arrive at the following rule of recovering the operator Lq from its two spectra S and S'. We construct the function M(x) = M(x;S,S') (0 < x < I), find the function (p(x) from formula (6), and using the
198 Now, constructing the solution x(x> -M that satisfies the conditions x(^>^) = sin/3, X'(l; A) = — cos/3, we obtain
<*>
'*•""'—m u=^->-
Thus, for the solvability of Problem A, the following conditions are necessary: 1°. The function M(x) = M(x; S, S') (0 < x < 21) is absolutely continuous, together with its first two derivatives. 2°. The derivative dM/dx (0 < x < 21) is different from zero. If these conditions hold, we can use formula (6) to construct the function
G. Borg, Acta Math. 78 (1946), 1-2. L. A. Chudov, Mat. Sbornik 25 (67) (1949), no. 3. V. A. Marchenko, Dokl. Akad. Nauk SSSR 72 (1950), no. 3. M. G. Krein, Dokl. Akad. Nauk SSSR 26 (1940), no. 1. , Dokl. Akad. Nauk SSSR 4 4 (1944), no. 5. , Dokl. Akad. Nauk SSSR 4 5 (1944), no. 3. , Ukrain. Mat. Zh. 1949, no. 2.
I N S T I T U T E O F M A T H E M A T I C S , A C A D E M Y OF SCIENCES O F T H E U K R A I N .
SSR
199
ON A FUNDAMENTAL APPROXIMATION PROBLEM IN THE THEORY OF EXTRAPOLATION AND FILTRATION OF STATIONARY RANDOM PROCESSES M. G. KREIN Let o{\) (- M < A < m) be a nondecreasing function which satisfies the condition —oo
Denote by Ac the Hilbert space of functions F{X) (- oo < A < oo) a-measurable and a-integrable, together with their squares, with the usual definition of the scalar product (F, G) {F, G G Aoc) as the integral over the whole axis of F(A)GOOrfoU). Let / = (a, b) (- °o < a < b < °°) be some interval of the real axis - o° < t < «>. Denote by A/ the closed linear manifold in A^ of the family of functions of argument A: exP (it2A) - exp (itlA)
=
,2
e x p (-A5)
^
(
Note that if a (A) is a function of bounded variation,
j
b J2
g ^_
then A/ can be defined
more simply, viz. as the closed linear manifold in A,*, of the family of functions exp (itX) {t G / ) . Two questions can be posed: I.
Under what criteria A/ = Aoo?
II.
/ / A / ^ A^, then how to express analytically
the orthogonal projection
P[F of any element F G A,*, on A/? For the case of / = (- oo, a) the first question was already answered by die author in 1944 [1] and (when a (A) is of bounded variation) he also obtained PjF for F(A) = exp(itA) {t > a) *.
For special cases of functions a and F the question
of the effective determination of P[F for / = (- oo, o) was studied by N. Wiener [3] who explained its applications in technology (see also [4]). The analytically formulated problems I and II can be reformulated as problems in the theory of prediction (extrapolation) and filtration of stationary random processes from their observation in the time interval /. Naturally problems I and II are of greatest interest when / is a finite interval. However their investigation is particularly difficult in this case.
Unaware of the applications which problems I and II have, the
author had already obtained in 1940 [5] and then in 1943—1944 [6] a series of essential results directly connected with them. Certain results have been obtained 'These results were obtained by the author in answer to the question raised by A. N. Kolmogorov [1]; K. Karhunen L^J repeated them in 1950. Dokl. Akad. Nauk SSSR 94 (1954), 13-16, Selected translation in Math., Statistics and Probab., Inst, of Math. Statistics and AMS 4 (1963), 127-131. 127
200 128
M. G. KREIN
in recent times which give the complete solution of t h e s e problems. It i s derived below for an odd function a{\) b e c a u s e the investigation of the solution in the most general c a s e would require u s to undertake much preliminary work.
Besides
in problems of the theory of real stationary random p r o c e s s e s only this c a s e occurs. In considering a finite interval /, it is possible without loss of generality to assume that / = (— a, a) and then it will be convenient to write A a and Pa respectively for A/ and Pi. 1. Solution
of problem I. Thus l e t cr{— A) = - a(\)
Assume r(A) = 2a(yJ\-
(0 < X < C«D; a ( 0 ) = 0).
0) (0 < A < <*,; r(0) = 0). By virtue of (1) DO
f dr{\)/{l + .X)
and sufficient
that
r — —
t(x) = / y/M'{s)ds
the
function M (x) will have a derivative almost everywhere equal to zero but this property no longer holds in any larger interval containing the interval {xa_ Q, xa ( Q)Let us find die solution of the integral equation
- s) 6 is; X)dM{s) (0 < x < I).
Denote by L(2) (0, f ) , where 0 < £ < I, the set of all functions f{x) (0 < x <
which are M-measurable and which satisfy the condition
/l/(x)|2^f(.x) <„,. 0 Theorem 2. Entire functions F{X) of form *a-0 F(X) = / f(x)
all even functions
Entire functions
of
Aa.
G(A) of form
iJV
(o, *„_<,))
201 APPROXIMATION PROBLEM IN RANDOM PROCESSES G(A) = T
J
£*)
A „
exhaust
*a+0 / f{x) 0
F{\) of form.
all even entire functions
FiX) satisfying
/|F(A)|2,fo(A)
lim
'
( / G ' L ( ^ ( 0 , *a+0)) the '
conditions (£=A+ip).
(2)
G(A) of form i *a+0
G(A) = i exhaust
d*
of Ka.
Theorem 3. Entire functions
exhaust
|g(»)|2rfx
0
all odd functions
F(A) =
i 7
129
7
*a+0
g(x) <£' U ; X 2 ) ^ ( J" | g U)|2rf a ; <
0 all odd entire functions
GiX) satisfying
0 condition
(2).
F o r c/a = dX/rr we w i l l h a v e : Z < °o,
cos
X*> <£'(*; A 2 )/A =
- sin Xx, and Theorems 2, 3 transform into the well known Paley— Wiener [9] theorem. 3.
Solution of problem II. Let F G Aoo. Form the functions
fix) = l.i.m. / F{X) $ ix; X2)doiX),
gix) = l.i.m. / FiX) 0 ' ix;
/v->oo -yv
X2)-j^;
/v-*oo -yv
the limit in the first e x i s t s in view of convergence in L ( 2 ) ( o , I) and in the second in view of convergence in L
(0, I), i.e., convergence in mean square in the in-
terval (0, I). The solution of problem II for I = i— a, a) is given by the formula X
PaFW
=
a-Q / fix) 0
. *a-0 $ ix; X2)dMix) + Y f gix)
Let us also derive the formula for the square of the distance of F from A a :
f \FiX) - PaFiX)\2do{X) -°°
= ;
\fix)\2dMix)
r
+
*a-0
\g{x)\2dx.
*a_0
The family of projective operators Pa (a > 0, Prj = /) form some spectral decomposition of identity; here the operator Pa is a weakly continuous function, from the left, of the argument o: P a _ o = Pa {a < 0). The point a > 0 is a point of weak continuity on the right:
Pa+o = Pa if and only if Mixa
+
Q + 0) = M r a _ 0 - 0), i-e.,
when the points x a _o and xa+Q are the points of continuity of the function
Mix)
and between them the function Mix) is of constant value. 4.
Let I* (< Z) be the greatest of the points of growth of the function Mix).
the function Mix) lor x > + 0 and < /* has two absolutely continuous derivatives
If
202 130
M G. KREIN
where M {x) > 0 for x > + 0 then taking ; T = fjM'(x)dx,
4 $ U ; A) = V « ' U ) <M*;A)
x
t = fy/M'{s)ds, (0
<x
we get $ U ; A) as the normalized solution (
q{t)u + Au = 0,
u'(0) + U A - A) u (0) = 0 (0 < t < T),
where q(t) =
(3)
and
/ <MA) (> 0). —oo
In this c a s e for any F G Aoo we will have* i a
a
PaF(X)
= ffU)${t;k2)dt 0
°°
+ ±fg(t)yHt;>a)dt+m A 0
f F(\)do{\), -~>
(4)
where V{t, X) = *Ct; 0)-£-[<&(«; A)/<3(«; 0)] / W = l.i.m. fK\)$U;\*)M\),
g(0=l.i.m.
/
(0 < t < I ) ,
(5)
F(A) * (t; A2) - ^ l i ,
(6)
where the limits are understood in the s e n s e of convergence in mean square in the interval (0, 7). B e s i d e s this we have f | F ( A ) - PaF{\)\2do{\) —°°
T = f\f{t)\2dt
T + f \ g{t)\*dt
a
(0
T).
a
Conversely if for some system (3) (with m > 0, — °° < A < °o) the spectrum is positive and $ ( j ; A) is the solution of the system (3), normalized, for example, by the condition
II is solved by the formulas (4),
(5) and (6). Thus the effective solution of problems I and II depends on the effective construction of the functions M(x), and <£>{x; A) (and in some c a s e s only on one function $ U ; A)) corresponding to the s e n s e of the given function CT(A) indicated above. The several c a s e s where it is possible to make such effective construction have been mentioned in the preceding note [8]. In a following note we will show that this is p o s s i b l e in all c a s e s where the function a is analytic and da/dX is a *For m = 0 the last term on the right side of (4) is assumed to be zero quite tndependendy of whether the integral multiplying m has sense or not.
203 APPROXIMATION PROBLEM IN RANDOM PROCESSES
131
rational function and a l s o in many other c a s e s (for example when da/d\
can be ob-
tained by dividing unity by a trigonometric polynomial).
BIBLIOGRAPHY [l] M. G. Krein, On a problem of extrapolation Nauk SSSR 46 (1945), 306-309.
(Russian)
[2] K. Karhunen, Uber die Struktur station'drer (1950), 141-160. [3J N. Wiener, Extrapolation,
of A. /Y. Kolmogoroff, Dokl. Akad.
interpolation
zufdlliger
Funktionen,
and smoothing
Ark. Mat. 1
of stationary
time
series,
Wiley, New York, 1949. [4] A. Jaglom, Introduction
to the theory of stationary
Mat. Nauk 7 (1952), no. 5 (51), 3-168.
random functions,
Uspehi
(Russian)
[5] M. G. Krein, Dokl. Akad. Nauk SSSR 24 (1940), no. 1. [6]
, ibid. 44 (1944), no. 5; 45 (1944), no. 4; no. 5.
[7J
, On a generalization 884.
[8]
of investigations
of Stieltjes,
ibid. 87 (1952), 881-
(Russian) , ibid. 93 (1953), no. 4.
[9] N. I. Ahiezer, Lectures (Russian)
on the theory of approximation,
Kharkov, 1940.
Translated by: D. V. Thampuran
204 O N T H E D E T E R M I N A T I O N OF T H E P O T E N T I A L OF A PARTICLE F R O M ITS
S-FUNCTION
M. G. K R E I N
Results of the paper [1] allow one to indicate a simple criterion that a given function S(k) (—00 < k < 00) be the S'-function (S'-matrix) of s-state of a particle in a centrally symmetric field with a certain potential V(r), and also to obtain a simple rule of recovering the potential from the function S(k). 1. Let H(t) (0 < t < 2T) be a real-valued measurable locally integrable function that satisfies the condition (Y) For each r (0 < r < T) the equation (1)
q{t;r)+
f
H(\t-s\)q(s;r)ds
= l
{~r
has a unique bounded (and therefore continuous) solution
q(t;r).
For any complex k we set (2)
x(r;k)
=k
q(s; r) cos ks ds Jo Directly from the results of [1] follows
(0 < r < T).
Theorem 1. The function x(r', k) (0 < r < T) is the solution of the differential system (3)
Tr(9{rr2%)+k29{rr2x
= Q]
x(0;*) = o, x'(Q;k) = k,
where g(r) = q(r; r) (0 < r < T). We set ip(r; k) = x{r\ k)/g{r) (0 < r < T); then ip is the solution of the system (4)
{al~ig~){di
+ i
g-)i,
+ k2 P
'
= 0''
^(0; A;) = 0,
^'(0;k)
= k.
The function g(r) (0 < r < T) is always absolutely continuous and positive. If H(t) (0 < t < 2T) is absolutely continuous, then the first derivative g'(r) is also absolutely continuous, in which case system (4) can be rewritten in the form (5) (6)
V" - V(r)V> + fc2V = °; V
V'(0;A;) = 0,
( r ) = s ( r ) ^ ^
ip'(0;k)=k,
(0
We will associate system (5) to a function H(t) (0 < t < 2T) in all cases, thus allowing V(r) (0 < r < T) to be a generalized function of the type D\ (derivative of locally integrable function). Doklady Akademii Nauk SSSR, vol. 105 (1955), no. 3, pp. 433-436.
205
2. Later on, we will be interested in the case where T = oo. In this case, system (5) has a completely determined spectrum, which is nonnegative. Integrating (3) by parts we obtain for tp the following expression: V(r; k) = Im leikr
(7)
(l - J * V2r(s)e-iks
ds) j ,
where T2 r (s) denotes a solution of the integral equation /•2r
T2r(t)+
(8)
Jo
H(\t-s\)T2r(s)ds
= H{t)
(0<*<2r).
It is natural to assume that under certain conditions the asymptotic formula for ip(r; k) as r —• oo will include a function T(t) (0 < t < oo) that is a solution of the integral equation /•OO
V(t) +
(9)
H(\t-s\)T{s)ds
= H{t)
(0
Jo
Consider the case where H(t) € £i(0, oo). As is easily seen, in this case the condition (Y) is equivalent to the condition of nonnegativity of the function /•oo
(10)
p(k) = 1 + 2
H(t) cosktdt (-co < k < oo). Jo It turns out that if p{k) > 0 (0 < k < oo), then equation (9) has one and only one (real-valued) solution Y £ £i(0, oo), which can be obtained as follows. By virtue of the general Wiener-Levy theorem (see [2], Chapter VI, subsection 101), there is a real-valued function 7 G la(0, 00) such that (11)
\ogp(k) = 2
/*oo
~/(t)cosktdt
(-00 < k < 00).
Jo Then T(t) can be found from the relation (12)
1-
T(t)e-ikt
[
dt = exp (-
f
It follows from (11) and (12) that p(k) =
ikt ikt )e-— dt r(t)<
/
(Imfc < 0).
2
/f•O OO O
(13)
e-iktj(t)dt\
( — 00 < k < OO),
Jo
so that we can set /•OO
(14)
1- /
T(t)e-ikt
dt = p'll2{k)eiS^
(-00 < k < 00).
If we normalize the continuous function d(k) by the condition 6(0) = 0, then (15)
S(~k) = -6{k),
6(±oo) =
lim 5(k) = 0. k—>±oo
The second equality follows from the fact that £(—00) — J(+00) = 2S(—00) is the number of zeros of the function (14) inside the lower half-plane Im/e < 0, which is obviously equal to 0.
206 Theorem 2. If the conditions (16)
H{t) £ Za(0,oo),
p(k) > 0
(0 < fc < oo)
are satisfied, then the following asymptotic formula holds: (17)
\im{i(;(r;k)-p(k)-1/2sm(kr
+ 5(k))} = 0
(-co < k < oo).
r—^oo
3. Consider now the 5-function S(fc) = exp(2i<S(fc)). Obviously, 1) |5(fc)| = 5(0) = 1; 2) S(-jfe) = 5(fc) ( - o o < fc < oo);
3) argS(k) —>• arg 5(0) as A; —» oo. Since, according to (12) and (14), (18)
S(k) = exp ( 2i /
sinfct7(£)d£ 1
(—oo < k < oo),
we have for S(k) also the representation 4) S(k) = 1 + / ^ e ifct s(i) di, where s(t) € Li(0, oo). Theorem 3. / / 5(&) (—oo < k < oo) is a function with properties l)-4), then there is always an ordinary or generalized function V(r) (of the type D\) such that the spectrum of system (5) will be nonnegative and, for S(k) = arg S(k) and the corresponding definition of p{k) — p(—k) (> 0), the solution ip of system (5) will satisfy the asymptotic relation (17). Indeed, as is easily seen, the properties l)-4) imply the existence of a real-valued function 7 G £i(0,00) for which (18) holds, and then (10) and (11) yield a function H(t) G Li(0, 00) from which the desired potential V(r) will be found using formula (6). As is easy to see, a rational function S(k) (5(0) = 1) satisfies conditions l)-4) if and only if a) all zeros and poles of S(k) are nonreal, b) for every zero of S(k) of certain multiplicity there is a pole of the same multiplicity, symmetric to the former with respect to the real axis, c) for a zero (a pole) of S(k) that does not lie on the imaginary axis, there is a zero (a pole) of the same multiplicity, symmetric to the former with respect to the imaginary axis, and d) the function S(k) has equal numbers of zeros and poles (counted with multiplicities) inside the upper (and hence the lower) half-plane. We note that, for a rational function S(k) with given zeros and poles, the function p(k) is also rational and can be found directly. After that, H(i) (0 < t < 00) is obtained as a sum of products of exponential functions by polynomials, and the solution of equation (1), and thereby the function V(r) will be expressed in explicit form [3, 4]. 4. Now let Si(k) (—00 < k < 00) be a function that satisfies conditions 1), 2), and 4), but not 3). Then there is an integer m (> 0 or < 0) such that 3') arg S(k) -> arg 5(0) - 2mn as k -» 00. Suppose m > 0. We choose arbitrary positive numbers K\ < K2 < • • • < Km and make up a function m
/h —'
\
2
207 It is easily seen that the function S(k) satisfies all conditions l)-4); therefore, a system (5) corresponds to it. Using a formula by Jost and Kohn [5], V(r) can be transformed into a new potential Vi(r) for which the system (5) has exactly m negative eigenvalues Ej = —K2 (j = 1,2,..., m) with their respective eigenfunctions i>i{r; IKJ) {j = 1,2,..., m) satisfying ipi(r; iKjf dr = p- l
(j = 1, 2 , . . . , m),
Jo
where pj (j = 1,2,..., m) are given positive numbers. It turns out that if V(r) is replaced with Vi(r), the function S\(k) given before corresponds to system (5). For instance, if m = 1 («i = K, p\ = p), then the formula of transition from V(r) to V\{r) has the form Vl{r) =
dP V{r)-2—,
where P(r) = pip(r;iK,)2/(l+p
ip(s;iK)2 ds)
We note also that in this transformation of the potential V(r), the function xjj changes according to the formula •>Pi(r;k) = ip(r;k) + (k2 + / t 2 ) - 1 P ( r ) ^ ( r ; ZK) — Mr; k)/tp(r;iK)]. dr It seems that this result has not been noticed by the authors. By the way, it remains true also for complex «, as well as after the special initial conditions in (5) changed to the general initial conditions ^(0; k) = CQ, ij)'(Q; k) = c\. 5. It remains to observe how general is the class of potentials V(r) that are obtained by the above-described procedures. One of the results in the paper [6] allows us to assert that this class contains every function that satisfies the following two conditions: I. rV{r) e Xi(0,oo). II. The solution (p of the system
M. G. Krein, Dokl. Akad. Nauk SSSR 9 7 (1954), no. 1. N. I. Akhieser, Lectures on approximation theory, 1947. L. A. Zadeh and J. R. Ragazzini, J. Appl. Phys. 2 1 (1950), no. 1. M. G. Krein, Dokl. Akad. Nauk SSSR 9 4 (1954), no. 1. R. Jost and W. Kohn, Kg. Danske Videnskab Selsab, Mat.-fys. Medd. 2 7 (1953), no. 9. , Phys. Rev. 8 7 (1952), no. 6. ODESSA HYDROTECHNICAL INSTITUTE
208 CONTINUAL ANALOGUES
OF
P R O P O S I T I O N S O N POLYNOMIALS ORTHOGONAL O N T H E U N I T CIRCLE
M. G.
1. Let (i > 0, and let H(t) = H(-t) in every interval (—r,r) (r
KREIN
( - T < t < T; T < oo) be a function integrable
If for each continuous function
da{\) o c l + A2
^
i»-"<" 4 =/,'^-^ T + ( " - H ' +
(2)
1 + A2
, do-(X) ;' A2
( - T < t < T), where 7 is a reo^ constant. This proposition, in a slightly different formulation, was established by the author back in 1944 [1, Theorem 6]. For the sake of simplicity, in what follows, only the case where \x = 1, T = 00 will be considered. In this case the spectral function a in (2) is determined uniquely. If, in addition, it is assumed that the equality in (1) is possible only for ip = 0, then condition (1) (for fi = 1, T = 00) means that, for any positive r, the Hermitian kernel H(t — s) (0
(0 < s, t < r).
According to the general formula (4) in [2], we have (3)
dTr(t, s)/dr = -V(r, s)Tr(t,r)
(0 < r < 00; 0 < s, t < r).
It is evident also that (4)
rr(t,s)=rr(r-s,r-t). Doklady Akademii Nauk SSSR, vol. 105 (1955), no. 4, pp. 637-640.
209 2. We set (0 < r < oo) (5) (6)
P(r; A) = eiXr (1 - / Tr(s, 0)e~iXs ds P*(r;A) = l -
rr(0,s)eiAsds.
/ Jo
With the help of (3) and (4) we easily obtain {A(r) = Tr(0,r)) dP(r; X)/dr = iXP(r; A) - A(r)P.(r; A),
(7)
dP*(r;X)/dr =
-A(r)P(r;X).
From this, without any difficulty, we get the relation | P * ( r ; A ) | 2 - | P ( r ; A ) | 2 = 2ImA f
(8)
\P(r;X)\2 ds
(0 < r < oo),
Jo
which obviously implies that for any r > 0 all the zeros of the entire function P(r; A) lie inside the upper half-plane Im A > 0. It turns out that for any finitely supported (terminating) function f(r) G £2(0, oo) we have /•OO
/
/•OO
\f(r)\2dr=
\F(X)\2da(X),
JO
J -00
where
/•oo
F(X)=
f(r)P(r;X)dr. Jo Thus, the correspondence / ( r ) —> F(X) generates a unitary mapping Up of the entire space £2(0) 00) onto a part of L£ • From an old result of the author [3] follows Theorem 1. The mapping Up is a unitary mapping of the entire 1/2(0, 00) onto the entire L^ if and orih) tf the integral
/ J — 00
loga'(X)dx + A2
T x
diverges (equals —00). By means of other techniques, the following theorem can be established. Theorem 2. The following assertions are equivalent: I. Integral (9) has a finite value. II. The integral JQ \P(r; A)|2 dr has a finite value for at least one A (Im A > 0). III. The function P*(r; A) (0 < r < 00) is bounded for at least one X (Im A > 0). IV (V). On every bounded closed set of points A of the open half-plane Im A > 0 integral (10) is uniformly convergent (there exists a uniformly convergent limit) (10)
n ( A ) = lim P*(r;A)). r—>oo
210 We note that the following relations also hold under conditions I-V: POO
n(A)n(/x) = (A - p) /
(11>
P(r; X)P(r;(i)dr
(Im A > 0, I m p > 0);
^HMk:^w-+<^4
where a € (—oo, oo) and (3 > 0. Formula (11) is interesting in that it shows that the function 11(A) does not depend on the functions CT,J(A) and as(X) in the decomposiotion a = aa + ad + <JS of a into its absolutely continuous part aa, jump function <Jd, and singular part as. According to (11), at every point A at which the derivative cr'a(X) exists, we have 27rcra(A) =lim|II(A + i e ) r 2
as e | 0.
We note also that conditions I-V are satisfied if A{r) £ 1/2(0, oo) or A(r) € Li(0, oo), and in this case we also have />oo
n(A) = l - / Jo
(i2)
A(s)P(s; A) ds
(ImA>0),
nW-«,(-jf 4 . ) ^ * = P,(l;r)+o(
f
\A(r)\dr\
(ImA > 0, I -» oo).
Therefore, for A{r) € Li(0,oo) the function 11(A) (ImA > 0) is continuous and upper and lower bounded: (13)
^(A)^1 <expf /
\A(r)\dr)
(ImA>0).
Moreover, in this case the function cr(X) is continuously differentiable and 2na'(X) = |II(A)|- 2 (-oo < A < o o ) . The propositions and relations given in subsections 2 and 3 are continual analogues of the results obtained for orthogonal polynomials by Szego [4], Kolmogorov, Geronimus [5, 6], and others. In particular, compare Theorem 2 with Theorem 21.1 in [5], and formulas (12) and (13) with Geronimus' formulas (26.7) and (26.6) in [5]. Also the original work by Kalafati [7] has led us to studying the functions P(r; A). 3. We set S(r; A) = e-iXrP{2r;
A) = $(r; A) + M(r; A),
£(-r; X) = 5(r; A) = $(r; A) - M(r\ A). Using the first equation (7) we easily get d$/dr = - A * - a(r)§ + b(r)V, ^
d$/dr=
A$ + 6(r)$ + a(r)tf,
$(0;A) = 1, tf(0;A)=0,
where a(r) — 2ReA(2r) and b(r) = 21mA(2r) are locally integrable functions.
211 Investigation of differential system (14) allows us to assert that the correspondence oo
/
f(r)£(r;X)dr, -oo
defined on compactly supported functions f(r) £ Li{—00,00), generates a unitary operator Us that always maps the entire L,2(—00,00) onto the entire L2 . The functions £(r; A), $(r; A), and $(r; A) are similar to the functions exp(iAr), cosAr, and sinAr, respectively. Naturally, this property of the generalized Fourier transform has a consequence that if !F(X) = Usf and .T^A) (—00 < A < 00) is a compactly supported function in 4 , then f(r) is the integral of Jr(X)£(x; A) da(X) from —00 to 00. Here we have started with the kernel H(t — s) (or the spectral function u(A) generating it) and arrived at system (14). We could proceed vice versa: take arbitrary locally integrable measurable functions a(r) and b(r) and prove directly the existence of a spectral function cr(A) for which the closure of transformation (15) (as an operator) will give a unitary operator that maps the entire Z/2(—00,00) onto the entire L 2 . Formula (2) and the relaton a(r) + ib(r) = 2r2 r (0, 2r) solve the problem of recovering system (14) from its spectral function a(X). We note that if the function 27rcr(A) is an integral of a rational function -R(A) = 1 + 0(1/A) as |A| —> 00, then system (14) (that is, functions a(r) and b{r)) is obtained by rational operations, provided the poles and zeros of the function R(X) are determined. Here we have obtained a generalization and at the same time, a simplification of the conclusions of the paper [2]. To understand this better, consider also the case where the function H(t) is real-valued, that is, <J(A) is an odd function. In this case b(r) = 0. Then any of the functions $,\& can be easily eliminated from equations (14). Assume for simplicity that H(t), and therefore the function a{r) = 2r2 r (0, 2r) are absolutely continuous; then it is not difficult to find that ^ ( r ; A) and $(r; A) are definite solutions of second order differential systems $ " - V ( r ) * + XH = 0, *(0;A) = 0, ^'(0;A) = A; $ " - V i ( r ) $ + A 2 $ = 0,
$(0;A) = 1,
2
$'(0; A) + a(0)$(0; A) = 0,
2
where V(r) = a (r) + a'(r), V^r) = a (r) - a'(r) (0 < r < 00). Thus, the function £(r; A) is constructed from the solutions $, ty of different equations. As is easily seen, \I>(r; A) = - A _ 1 $ ( r ; 0 ) [ $ ( r ; A)/$(r; 0)]'. These facts were already established by the author before, [8]. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
M. G. Krein, Dokl. Akad. Nauk SSSR 4 5 (1944), no. 3. , Dokl. Akad. Nauk SSSR 9 7 (1954), no. 1. , Dokl. Akad. Nauk SSSR 46 (1948), no. 8. G. Szego, Orthogonal Polynomials, 1939. Ya. L. Geronimus, Zapiski Kharkov. Mat. Obshch. Ser. 4, 1948, 19. , The theory of orthogonal polynomials, 1950. P. D. Kalafati, Dokl. Akad. Nauk SSSR 105 (1955), no. 4. M. G. Krein, Dokl. Akad. Nauk SSSR 9 4 (1954), no. 1. ODESSA HYDROTECHNICAL INSTITUTE
ANALYTIC PROBLEMS AND RESULTS IN THE THEORY OF LINEAR OPERATORS IN HILBERT SPACE (Abstract)
M. G. KREIN In the last 25 years the influence of the theory of linear operators in Hilbert space has been greatly extended, partly because of its various applications to many wide-ranging branches of mathematics (the theory of differential equations, group representation, probability, harmonic analysis and so forth), and partly because of the exceptional role that this theory has played in contempotary quantum p h y s i c s . In the same period the theory has been enriched by the creation of basic new s u b d i v i s i o n s . Among them let us mention the following. I. Theory of canonical r e p r e s e n t a t i o n s of Hermitian operators Here we particularly have in mind the new results and problems that have arisen in dealing with a question which had seemed to be very simple and to have received a definitive answer, namely the extension of Hermitian operators. The theory of generalized resolvents and of resolvent matrices has been worked out. A truly remarkable c l a s s of operators, namely the entire Hermitian operatots, has been discovered and studied. Representations of Hermitian operators by means of canonical differential operators have been obtained. Connected with these results are the extensive studies on the so-called inverse problems of the spectral theory of differential operators. In many of these studies the final goal was reached by the methods and ideas of the abstract theory of operators. The theory of entire Hermitian operators and their canonical representations has provided an answer to several difficult questions in the theoty of interpolation and extrapolation of stationary stochastic p r o c e s s e s . The theory of canonical representation of Hermitian operators throws new light on the theoty of generalized eigenelements of selfadjoint operators with its numerous results in a n a l y s i s . An important role in the c l a s s i f i c a t i o n of entire Hermitian operators has been played by the entire operators corresponding to the indefinite case in the problem of moments and in the problem of the extension of positive Hermitian functions. Amer. Math. Soc. Transl. (2) 70 (1968), pp. 68-72. 68
LINEAR OPERATORS IN HILBERT SPACE
69
In the theory of entire H e r m i t i a n o p e r a t o r s , and e s p e c i a l l y in the g e n e r a l theory of canonical representations of Hermitian and selfadjoint operators, certain fundamental questions have remained unanswered up to the present time. It turned out very unexpectedly that these questions can be reformulated as problems in the theory of nonselfadjoint operators. II. Theory of perturbations of operators Even in the theory of perturbations of operators of the simplest c l a s s , namely the selfadjoint operators, mathematicians and physicists have discovered much that is new and sometimes paradoxical from the point of view of linear algebra (usually these paradoxes arise from the possibility of the existence of an absolutely continuous spectrum for a selfadjoint operator, a phenomenon that has no analog in linear algebra). For scattering problems in quantum m e c h a n i c s , the theory of quantized fields, and quantum s t a t i s t i c s , p h y s i c i s t s have developed a very extensive formalism in perturbation theory. By now this formalism, at least in many c a s e s , has been put on a rigorous b a s i s by mathematicians and has been greatly extended. A general and rigorous theory has been constructed for wave operators, and for the operators and suboperators of scattering in the quantum problem of one-channel scattering; and a number of exact results have been obtained in the many-channel scattering problem, in various questions of second quantization, etc.
New mathematical interrelations have been discovered
between scattering theory and the theory of stationary stochastic p r o c e s s e s , and between the theory of dilatations and the characteristic functions of contraction operators. Studies have been made to generalize these results to the case of nonselfadjoint perturbations of selfadjoint operators. In the present lecture no d i s c u s s i o n will be given of perturbations that can be treated in the broader framework of the general theory of operators in Banach s p a c e s
( a l t h o u g h here also many significant advances have been
made ). III. Theory of nonselfadjoint operators. For completely continuous operators and for unbounded operators with discrete spectrum, methods have been worked out to establish theorems on the completeness of the system of eigenvectors and a s s o c i a t e d vectors, to obtain generalized s p e c t r a l e x p a n s i o n s , and to sum these expansions. The analogous questions have been worked out for s h e a v e s of polynomial operators in certain c l a s s e s . The theory of section III here has been developed in close contact with the theory of section II. Satisfactorily complete results are
70
M. G. KREIN
obtainable only for nonselfadjoint operators that are 'close in some sense to selfadjoint, unitary or more generally normal operators. Powerful criteria have been established for the e x i s t e n c e , for a given linear operator, of maximal invariant s u b s p a c c s corresponding to various parts of its spectrum. I'rom amongst these investigations it will be desirable to single out the following theory. IV. The theory of operators in s p a c e s with an indefinite metric The geometry of t h e s e spaces-, in particular of the so-called / - s p a c e s , has been highly developed. Various theorems have been established on the existence of invariant maximal definite subspaccs for operators of various c l a s s e s . It is customary to emphasize the role of set-theoretic and algebraic ideas in the creation of functional a n a l y s i s . But in the theory of operators at the present time increasing use is made of ideas, methods and apparatus from the theory of analytic functions of one or several complex variables. Many important objects in the theory of operators have by definition a " h y b r i d " character; the analytic
functions with operator value's (generalized resolvent,
resolvent matrix, characteristic operator-function, scattering suboperator and s o forth). A discussion of the role of t h e s e objects and of the interrelations among them will be an important chapter in the present lecture. Spectral expansions for /-selfadjoint and /-unitary operators have been obtained under certain r e s t r i c t i o n s . Many results have also been obtained in the theory of representations of groups and algebras in s p a c e s with an indefinite metric. The role of this theory in various questions of mathematical physics has also been clarified (the theory of stability, of damped oscillations, of canonical transformations, of the Hamiltonians of quantized fields and so forth). Since we will confine our attention here to the above sections I—IV, we must emphasize that one of the most important achievements in the theory of operators in Hilbert space has been the creation in the last few years of the theory of Banach algebras of operators (acting in a Banach s p a c e ) together with its numerous applications in the theory of representations (of groups and algebras) and in quantum p h y s i c s . This subject will be dealt with in two s p e c i a l lectures and therefore will not be discussed here. As mentioned above, in the theory of perturbations of selfadjoint operators we encounter facts that have no analog in linear algebra. On the other hand, it was noticed only recently that for certain well-known constructions in linear algebra no continual analogs have been found. Nevertheless
71
LINEAR OPERATORS IN HILBERT SPACE
development of the theory of such analogs is altogether necessary and opens up many new points of view. W,e have in mind the following constructions. A. Reduction to Jacobian form of a real symmetric matrix with simple spectrum by means of orthogonal transformations. D. Reduction to triangular form of an arbitrary matrix by a unitary transformation. C. Reduction of a Hermitian form to a sum of squares by the method of Lagrange (or the more general problem of representing a matrix as the product of triangular matrices). The development of continual analogs for the construction in A leads to the theory of canonical representations of Hermitian operators and to the theory of entire Hermitian operators. The development of continual analogs for the construction in B leads to the theory of an abstract
triangular
integral.
Closely connected with this
integral is an entire calculus that plays an important role in the theory both of nonselfadjoint and of selfadjoint operators (the latter fact was particularly surprising). The investigations of an abstract triangular integral allow us in their turn to work out the continual analog of construction C, namely the theory of factorization of an operator with r e s p e c t to a given chain of projectors. The connecring link in the development of continual analogs for the - constructions A, B, C is the basic concept of the characteristic
operator-function
of a non-
selfadjoint operator. This function c h a r a c t e r i z e s the difference between an operator and its adjoint operator. Its v a l u e s are contraction operators, generally with respect to an indefinite metric. This concept is still being developed. An important role in the theory of operators is played by the question of all the possible multiplicative properties of the characteristic operaror-function. A short time ago a continual multiplicative representation of the characteristic operator-function was obtainable only on the basis of certain preliminary investigations that were difficult and of a purely analytic
nature,
from the theory of analytic matrix-functions. At the present time, on the other hand, many of the results of the theory of matrix-functions are obtained as corollaries of various general s t a t e m e n t s in the theory of operators. Tliis competition between the two fields is still continuing. An important role in this direction is played by the construction of a theory of the abstract triangular integral and of the factorization of an operator with respect to a given chain of projectors.
72
M. G. KREIN The characteristic operator-function plays an exceptional role in the
construction of continual analogs to the theory of elementary d i v i s o r s . The concept of a Jordan cell in linear algebra corresponds to the concept of a unicellular
operator. For operators belonging to certain important c l a s s e s
criteria for their unicellularity have been set up. For operators of certain c l a s s e s we have also been able to prove the possibility of their expansion into a "skew s u m " of unicellular operators. Several criteria have been set up for similarity of operators. Various criteria have also been found for the similarity of a contraction (dissipative operators) to a unitary (selfadjoint) operator. Some of t h e s e criteria are quite remarkable. Translated by: S. H. Gould
217
N O T E S B Y I. G O H B E R G I. Gohberg, who compiled this section of the book, makes the following comments. 1. The first paper is taken from Integral Equations and Operator Theory, Vol. 30, 1998. Four consecutive issues of IEOT, Vol. 30, No. 4, volume 31 issues 1 to 3, are dedicated to the memory of M.G. Krein on the occasion of the 90th anniversary of his birth. 2. The paper by B. Ya. Levin is a translation into English of the paper published in the Ukrainian Mathematical Journal, Vol. 46, 1994, No. 3. The first three issues of this volume are dedicated to the memory of M.G. Krein. They contain different papers about M.G. Krein and about his mathematical work. 3. The paper "M.G. Krein, Abstract" is published for the first time. It was written originally by M.G. Krein in Russian and it contained a description of a number of papers of M.G. Krein proposed by Moscow University and Moscow Mathematical Society for the Lenin Prize in 1963. The prize was not awarded to M.G. Krein. In his later years M.G. became allergic to writing this type of descriptions. He knew that it was futile and no one could convince him to do so. This paper which is published here in translation, is from the private archive of I. Gohberg. It is translated into English by Yu. Eidelman. 4. Paper number 8 in the list is an abstract of the plenary talk presented by M.G. Krein in 1966 to the International Congress of Mathematicians in Moscow. A more complete version of this talk appears later in the American Mathematical Society Translations, (2) 90, 1970, pp. 181-209. 5. Papers 4 to 7 are translations of original papers of M.G. Krein. 6. A complete list of publications of M.G. Krein can be found in Operator Theory: Advances and Applications, Vol. 118, Operator Theory and Related Topics, Birkhauser Verlag, 2000. 7. The books Operator Theory: Advances and Applications, Vols. 117 and 118, Birkhauser Verlag, 2000, are Proceedings of the Mark Krein International Conference on Operator Theory and Applications, Odessa, Ukraine, August 18-22, 1997. These books contain a wealth of material on the life and work of M.G. Krein.
Wolf Prize in Mathematics, Vol. 2 (pp. 219-262) eds. S. S. Chern and F. Hirzebruch © 2001 World Scientific Publishing Co.
Curriculum Vitae Born Budapest, (Hungary), 1 May 1926 Education 1947 New York University, AB 1949 New York University, Ph.D.
Positions Held 1945-46 Los Alamos Scientific Laboratory, Manhattan Project 1950 Staff Member, Los Alamos Scientific Laboratory 1951 Assistant Professor, New York University 1958 Fulbright Lecturer in Germany 1958- Professor, New York University 1972-80 Director, Courant Institute of Mathematical Sciences, New York University
Honors and Awards 1966, 1973 Lester R. Ford 1969 von Neumann Lecturer, S.I.A.M. 1972 Hermann Weyl Lecturer 1973 Hedrick Lecturer 1974 Chauvenet Prize, Mathematical Association of America 1975 Norbert Wiener Prize, American Mathematical Society and Society of Industrial and Applied Mathematics 1982 Member, National Academy of Sciences of the U.S.A. Member, American Academy of Arts and Sciences Honorary Life Member, New York Academy of Sciences 1983 Foreign Associate, French Academy of Sciences National Academy of Sciences, Award in Applied Mathematics and Numerical Sciences 1986 National Medal of Science 1987 Wolf Prize 1989 Member, Soviet Academy of Sciences 1992 Steele Prize 1993 Member, Hungarian Academy of Sciences Member, Academia Sinica, Beijing 1995 Distinguished Teaching Award, New York University Member, Moscow Mathematical Society
220 Honorary Doctoral Degrees 1975 Kent State University 1979 University of Paris 1988 Technical University of Aachen 1990 Heriot-Watt University 1992 Tel Aviv University 1993 University of Maryland, Baltimore Brown University Beijing University 2000 Texas A&M University
Professional Societies 1966-67 Board of Governors, Mathematical Association of America 1986-87 New York Academy of Sciences 1969-71 Member, Society of Industrial and Applied Mathematics Vice President, American Mathematical Society 1977-80 President, American Mathematical Society
Government Service 1977 President's Committee on the National Medal of Science 1980-86 National Science Board DOE Related: Theory Division, Advisory Committee, LANL Senior Fellow, Los Alamos Scientific Laboratory Review Committee, Oak Ridge National Laboratory
221
Short Mathematical Essays
In these pages I describe some of my mathematical investigations that did not fit into Peter Sarnak's (overly generous) article. 1. Numerical Solution of Linear Partial Differential Equations I became interested in these problems during my visits to the Los Alamos Laboratory. Under the leadership of von Neumann, the numerical solution of the equations of continuum mechanics became a cornerstone for weapons design, and for much more. So did the planning, construction and acquisition of electronic computers of ever increasing speed and memory, as well as programming aids, such as flow charts, subroutines, etc. Truly, Los Alamos was the birthplace of computational fluid dynamics. Here is an abstract formulation of a typical numerical problem: Let ut
Gu,
u{o) = u{o)
be a hyperbolic system of linear equations, G a first order operator. The initial value problem is properly posed for initial values uW in L 2 . Let «("+!) = C f c u < n \
u (o) given,
be a difference approximation, where u^ is an approximation to the G solution of the differential equation at time tn and h = tn+\ — tn. The difference equation has to be consistent with the differential equation, which means that, in the sense of the energy norm, 'Cfc-I - G lim = 0 /i-s-0
for all u in a dense subset of the domain of G. Solving the difference equations recursively gives
uW = cjy°). Equivalence theorem. Suppose the difference equation is consistent with the differential equation. Then tends to the solution u(t) of the differential equation as nh tends to t, for all L2 initial data, iff the scheme is stable in the sense that ||C£|| < const, for nh < 1. The proof, see [15],* is simple; the result is very general and has been a useful guide for those who work with difference equations. The necessity of stability is merely an instance of the principle of uniform boundedness. I included it in [15] because I heard someone claim in a lecture that the trouble with unstable schemes is that they magnify roundoff errors beyond all bounds; 'Number within brackets refers to that in the List of Publications (pp. 240-251).
222 if the arithmetic operations could be carried out exactly, unstable schemes would converge! The verification of stability is not an easy task. Friedrichs has made the important observation that if Ch is a difference operator with positive coefficients: (Chu)k — 22 BjUk-j , j
where uk denotes the value of the approximate solution at the fcth lattice point, k an integer vector, and the coefficients Bj are non-negative symmetric matrices, depending Lipschitz continuously on x, and EBj = I, then ||C fc || < l + 0 ( / i ) . This implies stability. The stability of schemes that are not positive is more difficult; when the coefficients Bj of Ch are constant, then expansion into Fourier series yields
v(0 = B(Qu(Q, where « ( 0 = S«*e*fc€,
« ( 0 = Sufceifc« ,
vk = (Chu)k,
and
£(£) = E B , e ^ .
Here stability means that the powers of B(£), called the amplification matrix, are bounded. This is the case when the eigenvalues of £>(£) do not exceed 1 (apart from some interesting complications when there are multiple eigenvalues). For schemes with variable coefficients, von Neumann suggested the power boundedness of the amplification matrix B(x, £) at each point a; as a criterion of stability. I have shown in [26] that von Neumann's criterion (with some provisos) is indeed sufficient for stability. Burt Wendroff and I devised and analyzed a difference scheme that is of second order accuracy see [23]. This scheme has been used successfully for a wide variety of initial value problems. Perhaps of greater interest than the linear case is the numerical solution of hyperbolic systems of conservation laws, discussed in the next section. 2. Hyperbolic Systems of Conservation Laws A hyperbolic system of conservation laws is of the form ut + fx + gv + hz = 0, where u is a vector valued function of x, y, z and t, and / , g, and h are vector valued functions of u. This system is hyperbolic if Uu + V 9u + C.K have real eigenvalues and a full set of eigenvectors for all u and all real choices of £, r], £; here the subscript u denotes the gradient with respect to u. The equations of compressible flow can be written as a hyperbolic system of conservation laws, the conservation of mass, moment and energy; so are the equations
223 of magneto-hydrodynamics. It is an ineluctable fact that solutions of such equations develop discontinuities sooner or later, no matter how smooth the prescribed initial values are. These discontinuities are called shocks, and they caused great discomfort to such distinguished 19th century mathematical physicists as Stokes and Airy. The matter was clarified by Riemann, who pointed out that discontinuous solutions make physical sense if we interpret them by integrating the conservation laws over arbitrary smooth domains G and demand that [ udV + [ F-nds = 0, (1) dl Ja JdG where the flux vector F is (/, g, h), and n the outward normal. Today we say that the differential equations should be satisfied in the weak sense, or in the sense of distributions. For the sake of simplicity we shall restrict the discussion to one space variable,
The integral form of the conservation law requires that the rate at which each component of u flows across a discontinuity be the same on both sides. This leads to the Rankine-Hugoniot jump conditions s
=
ti si IL-JL
(2)
for all components, i = 1 , . . . ,n, where s is the speed with which the discontinuity propagates, and ur, ui the states on the right and left side of the discontinuity. The differential equation ut + uux = 0 (3) can be written in many ways as a conservation law, for instance as
Uf+
G" 2 ) = 0 ,
(4)
and also as
These two conservation laws have the same smooth solutions, but a discontinuous solution of one is not a solution of the other, for the jump condition for the two are different. The class of all discontinuous solutions is too large, as may be seen from the following example. The function ' 0 u(x, t) = < x/t 1
for x < 0 for 0 < x < t for t < x
is a continuous solution of Eq. (3) with initial values „ , ( 0 for x < 0 , u(x,0)=:4 K ' 1 1 for 0 < x .
224 On the other hand, the function 0 1
v(x,t)
for x < t/2 . for t/2 < x ,
satisfies the conservation law (4), vt + ( \v2 I = 0 , because its discontinuity propagates with speed s = 1/2, satisfying the RankineHugoniot jump condition (2), with ue = 0, ur = 1, fa = 0, fr = 1/2. The functions u and i> have the same initial values, and they satisfy the same conservation law. We reject v in favor of the continuous solution u. The general principle, a kind of an entropy condition, goes as follows: Denote by \j(u) the eigenvalues of the matrix / „ , arranged as Ai < A2 < • • • < A„ a discontinuity traveling with speed s is admissible of type k, 1 < k < n, if \k-i{ui)
< s < Xk(ue)
and
Afc(ur) < s < Xk+i(ur).
(5)
Such a discontinuity is called a k-shock. These fc-shocks exist unless the fcth characteristic speed A*; is linearly degenerate in the following sense: grad Afc7-fc = 0 . Discontinuities of this type are called contact discontinuities, and they satisfy Afc(w^) =s = A fc (u r ). Yet another important type of waves are centered rarefaction waves, of the form u = u{x/t), corresponding to each k for which the fcth characteristic speed is nondegenerate. Given U£, there is a one-parameter family of states ur(s), s < \k(ue), to which it can be connected by a fc-shock. There is another one-parameter family of states ur(s) to which it can be connected by a centered fc-rarefaction wave, s > Afc(u^). The two families fit together smoothly, twice differentiably. Using these n oneparameter families we can solve the so-called Riemann initial value problem: f uQ { uu
for x < 0 , for x > 0 .
We can connect UQ to n one-parameter families of waves; the resulting n-parameter family includes u°, if u° is close enough to UQ. The resulting solution consists of the two states uo and u°, separated by n waves, shocks, rarefactions and contacts, and n — 1 intermediate states. The Riemann problem has been used by Godunov for numerical approximation, and by Glimm as a building block for proving the existence of solutions to the initial value problem, combined with a deep estimate and an ingenious random choice of parameters. The initial value problem for ut + (^w2) = 0 has been solved by Hopf and Cole as the limit of solutions of the viscous Burger's equation
i4 e) +u< e M, e) =«4x ) .
u(e)(*,0) =«„(*)
(6)
225
as e tends to zero. It is given by the explicit formula u(x,t) = —— ,
(7)
where y = y(x, t) is that value of y which minimizes Uo(y) + ^
^
;
(8)
here Uo(y) is the integral of the initial function uo{y): dllo/dy = uo(y). I have extended this formula to all single conservation laws "t + f(u)x = 0 , where / is a convex function of u, as follows:
u(x,y)=b(^y where b is the inverse function of a(u) = g£, and y = y(x,t) minimizes
U0(y)+tg(^-y
(7)' the value y that
(8)'
where g(u) is the integral of the function b(u): dg/du = b(u). The Hopf-Cole formula has been extended by Joseph in an interesting way to solutions of mixed initial-boundary value problems. When the flux function / is not convex, interesting new considerations enter. The entropy condition A(u+) < s < A(w_) no longer picks out only admissible solutions. Oleinik has derived the additional conditions necessary. Much of the theory outlined above is contained in the two papers [7] and [18]. In another paper [58], I have denned a concept of entropy as follows: Let ut + fx =ut + Aux = 0,
A = fu(u),
(9)
be a hyperbolic system of conservation laws. A convex function U(u) is an entropy for this system if all smooth solutions of (9) satisfy Ut + Fx = 0 ,
(9)'
where F(u) is some function of u. Clearly this is the case iff U and F satisfy the linear first order equation Uufu = Fu . (9)' For n = 1 this is an underdetermined system, for n = 2 a determined system, and for n > 2 overdetermined. Yet for the equations of compressible flow it has a solution. The significance of this concept can be seen when discontinuous solutions are regarded as limits of solutions of the viscous equation
226 as e > 0 tends to zero. Multiply this equation by Uu; we get Ut + Fx=
eUuuxx.
The right-hand side can be written as eUx x
£UXUuuV'X
•
Since U is assumed convex, the second term is negative, so we conclude that Ut + Fx< sUxx , where U and F are functions of u^E\ As e —> 0, u^ tends strongly to u, so U(u^) tends strongly to U(u). Since the limit in the distribution sense of a non-negative distribution is non-negative, we conclude that Ut+Fx<0. (10) For physical systems this expresses the principle of increase of entropy. For 2 x 2 systems one can construct a one-parameter family of entropies in the form of an asymptotic series. The numerical calculation of solutions of systems of conservation laws is a formidable task, calling for pinpointing shocks as they arise, and tracking them thereafter, through collisions with other shocks. Von Neumann had the brilliant idea, as early as 1944, to ignore discontinuities during the calculations, and capture them as very rapid transitions in the numerical approximation. The results that von Neumann obtained in 1944 showed some curious oscillations that von Neumann interpreted as the conversion of kinetic energy into internal energy. I have a different explanation, which will be presented in Sec. 3. In 1949 von Neumann and Richtmyer showed how to eliminate oscillations by introducing an artificial viscosity. My idea, a few years later, was to approximate differential equations in conservation form by discrete equations in conservation form, as follows in one space dimension: At
„,"+!_„»
ff(n)
Uk
/^x^k+l/2
,(n)
C11)
4-1/2)-
Here u™, /J 1 denote approximations to u and / at x = jAx, t = nAt. The numerical flux /fc+i/2 is a function of Uk and Uk+i, or more generally of Uk-i, Wfc, Ufe+i, Uk+2, etc. The numerical flux is required to be consistent with the physical flux, in the sense that when all variables in fk+1/2 a r e equal, the numerical flux equals the physical flux: fk+\/2{u,u,u,u) = f(u). The numerical flux may very well include a viscous term, also in conservation form. In [9] I proposed as numerical flux f
_ f(uk)
/fc+i/2 -
+ /(ufc+l)
g
Ufc+1 ~ "fc A t
2
A^'
called the Lax-Friedrichs flux. The resulting difference scheme is stable if the Courant-Friedrichs-Levy condition Ax/At > |A max | is satisfied, where A max denotes the largest eigenvalue of the matrix A = fu. Even more, it is a positive scheme in the sense explained in Sec. 1.
227 The Lax-Wendroff scheme mimics the Taylor polynomial u ("+!)
=
u
+ At ut + \{At)2uu
+ 0(Ai) 3 .
Using the formulas ut = -fx
= -Aux,
utt = -fxt
= -ftx
= {-Aut)x
= {A2ux)x
= (Afx)x
,
the Lax-Wendroff flux is . /(«*) + / ( « * + i ) Jfc+i/2 = 2
At J{uk+i) Y ( k+1'2' Ax
f(uk) '
It achieves the same order of approximation of u( n+1 > as the second order Taylor polynomial. It is stable if the CFL condition is satisfied. The two-space dimensional analogue of the LW scheme is stable under a condition somewhat more stringent than CFL. Recently, [137] and [141], Xu-Dong Liu and the author have constructed a new type of difference scheme for hyperbolic conservation laws, with numerical flux of the form /fc+1/2 = /fc+Cl/2 + sk+i/2
/fc+1/2 •
(12)
Here / j S ; 2 is some numerical flux of an accurate scheme, such as LW, and f^+i/^ is a numerical flux with a lot of viscosity, such as LF or upwind. The function Sfc+1/2 is a switch, that is nearly zero in a region of smooth flow, and nearly the identity in a region of rapid variation. The idea, going back to Harten and Zwas, is that in a region of rapid variation, the use of a highly accurate scheme is not called for, since the accuracy of such schemes is based on the accuracy of Taylor polynomials. The second idea is that with the right form of a switch the resulting difference scheme, expressing u£ + as a nonlinear function of Uk-j, j ranging over a finite set of indices, can be written as a linear combination of the u^-j with positive coefficients. These schemes work astonishingly well, giving accurate and detailed resolution of complicated two-dimensional flows such as the diffraction of a shock by a wedge, resulting in double Mach reflection, or the obstruction of a shock propagating in a channel that narrows suddenly, see Figs. 1 and 2. Numerical experimentation is the life-blood of the investigation of numerical methods. My first numerical studies of the LF scheme were done in 1952, before the advent of compilers, using von Neumann's MANIAC at the Los Alamos Laboratory, built under the tutelage of Nick Metropolis. It had a huge memory of 1,024 words and used punched cards, like the voters in Palm Beach County. The initial value problem for hyperbolic systems of conservation laws differs from its linear analogue in this important respect: whereas time is reversible for linear equations, it is not reversible for systems of nonlinear conservation laws; the entropy condition (10) points in one time direction. Even more is true: the mapping of initial data u(x, 0) onto data at u(x, t) at time t > 0 is a compact mapping. This means that a lot of information is lost with the passage of time. I derived this result for single conservation laws in one space variable, with convex flux, Luc Tartar in full generality. DiPerna proved compactness for systems of two conservation laws. It would be desirable to quantify this compactness, in terms of entropy or capacity functions, see [84].
228 Density. T = 0.2
0.5
/)))
J 0.5
PWuM
^ 1
1.5 2 2.5 Stepsize dx=dy=l/l20. dt/dx=0.02
3
3.5
Pressure. T = 0.1
0.5
1
1.5 2 2.5 Stepsize d.x=dy= 1/120, dt/dx=0.02
3.5
Fig. 1.
Density. T = 4 1
^
0.8'
"tl>
0.6 • 0.4-
•
0.2 • 0
M
BpsS?\^^^
^^\M>- r~~- ^ •"*J<
I / f r f 'W^^^^^^Sm^^^^^-^i^.^^'-d^
0
0.5
1
/
k (\ $rj***r?&=
%^
'.<
1.5
I
Stepsize dx=dy=1/80, dt/dx=0.15 Pressure, T = 4
1 1.5 2 Stepsize dx=dy=1/80, dt/dx=0.15
Fig. 2.
C- C \\ 1
fcs^ltll 4<&$fe&$h*i&!^hi'-:
'&i
229 The compactness of the set of solutions for positive times shows that it is easier to compute the solutions of systems of nonlinear conservation laws than of linear ones, for there is less to compute. 3. Dispersive Oscillations Take the simplest nonlinear hyperbolic equation ut + uux = 0 u(x, 0) = uo{x) prescribed.
(1)
As stated in the last section this initial value problem develops singularities in finite time, no matter how smooth the initial function, unless UQ{X) is monotonically increasing. Hopf and Cole have shown that the solution u^ of the modified equation u[e) + ii( e ) u W = eu<>>,
«<E>(x, 0) = u0(x),
(2)
tends uniformly as e —>• 0 to the solution of (1) as long as the solution is smooth. For t beyond the smooth region, u^ tends to a discontinuous solution of the conservation law
Ut+ u2
(l )
=
(3)
°'
and the limit satisfies the entropy condition described in the previous section. In [89] and [103] I have studied, with Leverrnore, the behavior of solutions u^ of u<e)We>i4e)+«41=0, u^(x,0)=u0(x). (4) Whereas the added term in Eq. (2) represents viscosity and dissipates energy, the added term in (4) is merely dispersive. What happens when e tends to 0? As long as the corresponding solution of (1) is smooth, v,(£' tends uniformly to that solution. For t beyond the smooth region, u^ becomes oscillatory in certain regions of space-time, on a wavelength proportional to e1'2. These oscillatory solutions converge weakly but not strongly to some limit u; so do their squares: w — lim u^2 = u2 . Taking the limit as e —» 0 in the sense of distribution of the dispersive equation (4) we obtain
=0.
(5)
Since u(E> tends weakly, not strongly, to u, it follows that u2 > u2 in those regions. In other words, u is not a solution in the distribution sense of the conservation law (3). One could express the relation between the weak limit of u and u2 as i 2 = u2 + w2 and rewrite Eq. (5) as '1
ut+[^u2\
+(^
2
)
=0-
(5)'
230 The extra term (^w2)x in Eq. (5)' is reminiscent of the appearance of so-called Reynolds stress in turbulence theory. Whereas Reynolds stress is derived from a stochastic model, as a difference of ensemble averages, the additional term in (5)' arises deterministically, as a difference of weak limits. It turns out that there is a 3 x 3 system of hyperbolic equations in terms of whose solutions one can express u and u 2 , at least for a limited time. Afterwards one has to go to a 5 x 5 system, and so on. The derivation of these remarkable properties of solutions of Eq. (4) is based on the complete integrability of the dispersive equation (4), which yields semi-explicit formulas for its solution. More about this in the next section. Jonathan Goodman and I have in [120] investigated another dispersive approximation of Eq. (1). Instead of adding a dispersive term as in (4), we discretized (1), replacing ux by a centered difference quotient: J2uk + Uk
2A
u
'
fc(°) = u o ( ^ A ) .
(6)
We denote the solution of this system of equations as « A , where UA is defined as constant between lattice point: uA(x,t)
= uk(t)
for ( f c - l / 2 ) A < x < (fc + l / 2 ) A .
Equation (6) is well-posed for positive initial values. Numerical results suggest strongly that the behavior of WA as A —> 0 is entirely analogous to the behavior of u(£) as e -> 0. In the smooth regime UA tends uniformly to the solution of (1). For t beyond the smooth regime, MA becomes oscillatory in space-time on a wavelength proportional to A. These oscillatory solutions converge weakly but not strongly to some limit u. For the model (6) one can give an intuitive argument why oscillations develop. Equation (6) can be written in conservation form: jtuk
+ (/fc+i/2 - A - i / 2 ) / A = 0,
(7)
where /fc+l/2 = -UfcWfc+1 •
(8)
Dividing (6) by uk leads to yet another conservation form: d_ logwfc + (sfc+i/2 - 5 / t - i / 2 ) / A = 0 , ~dt
(7)'
where uk + uk+i
fffc+i/2 =
^
,
'
v
^'
We claim that as A tends to zero, beyond the smooth regime the solutions UA of (7) cannot remain in a compact set of function space. For then there would be a strongly convergent subsequence it^. tending to some limit u, and / A 3 would tend to f(u) as Aj tends to 0. Equation (7) would, tend in the sense of distributions to
ut + fx =
ut+(\u2)x=Q.
231 The same argument applied to Eq. (7)' shows that the strong limit u of the
subsequence is a solution of the conservation law (log u)t+ux
=0.
But these conservation laws are incompatible; their discontinuous solutions satisfy different jump conditions, as explained in Sec. 2. So we conclude that the family UA contains no strongly convergent subsequence. This lack of compactness is the source of the oscillatory behavior of the u&; the approximate solutions are desperately trying to satisfy two or more incompatible conservation laws! It is a curious fact that the discrete system (6) can be written in infinitely many ways as a conservation law. In fact the transformation uk = vk turns (6) into the Kac-Van Moerbeke system, known to be integrable. This creates the possibility of a rigorous proof of the behavior of UA as A —> 0. A similar analysis can be made of dispersive approximations of some systems of conservation laws, such as the equations of one-dimensional gas dynamic: U
t+Px
= 0,
conservation of momentum,
(9)
and Vt — ux = 0,
conservation of mass,
(9)'
where u is velocity, p pressure and V specific volume. The equation of state is simplified to specifying p as a smooth, decreasing function of V. In his 1944 report von Neumann proposed and tried out the following numerical method for solving the initial value problem for the system (9), (9)'. Discretize the differential equations by setting uk(t) as an approximation of u(kA,t), and Pk+i/2(t) a nd V fc+1 / 2 (i) as approximations of p((k + 1/2)A, t) and V((k + 1/2)A, t). Replacing the space derivatives with centered differences turns (9) into |„fc +
ft^-ft-i/*
= 0
(10)
and (9)' into d Jtvk+1/2
uk+1 - uk _
n
£—-o.
(io)
As initial data von Neumann took gas flowing with negative velocity in a tube 0 < x < 1, closed at x = 0, open at x = 1. The subsequent flow contains a shock generated instantaneously at the closed end x = 0, and a rarefaction wave centered at the open end x = 1; eventually they interact. The numerical solution based on the approximations (10) and (10)' agrees, roughly, with the flow described above, with this difference: The velocity field oscillates in a neighborhood of the shock. Von Neumann's explanation of these oscillations was based on the irreversible nature of shock formation: it converts mechanical energy into heat energy, represented statistically by an oscillation in the velocity distribution around its mean. Furthermore, von Neumann believed that as A —> 0 the solutions of (10), (10)' tend weakly, but not strongly, to a solution of the continuum equations (9), (9)'. That, of course, cannot be, because if VA tends weakly but not strongly to V, a nonlinear function of VA like P(VA) may converge weakly, but not to p(V).
232
My explanation of the oscillation is as before: too many conservation laws: For if you multiply Eq. (10) by uk, and subtract from it Eq. (10)' multiplied by Pk+i/2: you get after a cancellation, d
uk-uk
d
-
Pk+l/2Uk+l
Pk+1/2-Vk+1/2
-Pk-l/2,Uk
+
= 0
which is a conservation law of the form j t (K\
+ Pfc+i/a) + (hfc+i/2 - hfc-1/2) / A = 0 ,
(10)"
where P(V) is defined by dP/dV = p(V), and hk+i/2 = Pk+i/2uk+i- If U&,VA were a strongly convergent subsequence, their limit u, V would satisfy the two conservation laws (9), and the additional conservation law \u2 + p)
+(up)x
= 0.
(11)
This third conservation law is incompatible with the conservation laws (9) and (9)', for a discontinuity that satisfies the jump conditions derived from (9) and (9)' will in general fail to satisfy the jump condition for (11). In [129], Tom Hou and I have explored these issues. In particular, as von Neumann has already observed, one can introduce an Eulerian coordinate X so that Xt = u, Xx = V , Eq. (9)' shows that these are compatible. Settings X in (9) gives Xu +p(Xx)x
= 0,
a quasilinear wave equation. One can proceed similarly for the difference equations; here the Euler coordinate Xk satisfies d v -Xk
= uk ,
Xk+\ — Xk T/ ^ = T4+1/2 •
Equation (10)' shows that these are compatible. Setting Xk in (10) gives
£MK^)->(*¥^))A=°-
(12)
Equations (12) can be interpreted as governing the motion of a chain of unit masses located at positions Xk, connected to their nearest neighbors by identical nonlinear springs. There is one spring law, the celebrated Toda chain, p(V) — e~v, which is completely integrable. For this case the limit A -> 0 has been studied by Percy Deift and Ken McLaughlin. It is natural to ponder if too many conservation laws, incompatible for nonsmooth solutions of the incompressible Euler equations, may be responsible for initiating fluid turbulence. When I mention this to turbulence experts, the reaction ranges all the way from polite scepticism to impolite scepticism.
233
4. Completely Integrable System When mathematicians declare that they have solved a problem, very often they mean that they have proved that a solution exists. It is otherwise for completely integrable ordinary and partial differential equation; for these one can describe all solutions in a (quasi) explicit fashion, and extract information about properties of solutions - though not without much hard work. The concept of completely integrable Hamiltonian systems in 2N variables is due to Liouville; the modern examples are all infinite dimensional. Completely integrable systems have played a significant role in the development of science. It was the complete integrability of the equations governing the twobody problem of gravitation that enabled Newton to derive Kepler's three laws of planetary motion. In the early days of quantum theory, integrable systems served as quantisable models. And in 1944, Onsager was able to exploit the complete integrability of the Ising model to demonstrate the existence of a critical temperature where phase change occurs. Other classical completely integrable system are geodesic flow on ellipsoids discovered by Jacobi, the motion under gravity of a particle confined to a sphere, investigated by Carl Neumann, and Sonya Kowalewski's celebrated top. The modern era began with Kruskal and Zabusky's astonishing discovery of the role of solitons for solutions of the Korteweg-deVries equation ut + uux + uxxx = 0 ,
(1)
and the even more astonishing explanation, given by Gardner, Greene, Kruskal and Miura, that the KdV equation is completely integrable, an idea developed further by Faddeev, Zakharov and Takhtajan. My first contribution to the subject was a scheme for the contraction of nonlinear evolution equations with infinitely many conserved quantities. These are of the form L t = [P,L],
(2)
where P and L are operators, [P,L] their commutator. When P is antisymmetric, U t = P U generates a one-parameter family of unitary operators U(£). It follows that the solution L(t) of (2) has the property that U*LU is independent of t. It follows that the eigenvalues of L are conserved quantities. Setting 6
ax 3
u some function of x and t, and P = 4d + ud + \ux, the right side of Eq. (2) becomes multiplication by \\uux + uxxx\; so Eq. (2) is the KdV equation, except for a rescaled time. In case the operator L acts on functions on the whole line as a self-adjoint operator on L 2 (R), and u tends to zero as |a:| —>• oo, L has a finite number of eigenvalues; in the periodic case L has infinitely many eigenvalues. All are invariant under the KdV flow (1). There is a sequence of differential operators P „ , of order 2n + 1, each antisymmetric, and such that its commutator with L is of order zero. For each, (2) yields an equation of the form ut+Kn(u) = 0, (3) K„ a quasilinear operator of order 2n + 1. They constitute the KdV hierarchy.
234 The operators P and L in Eq. (2) are called a Lax pair; they occur in surprisingly many contexts. In [72] and [73] I constructed finite dimensional invariant manifolds for the space periodic KdV equations, characterized as the functions minimizing the functional Fn(u) among all functions for which Fn-i(u),... ,F0(u) have prescribed values. Here Fn is the sequence of functionals, constructed by Gardner, Kruskal and Miura, which are invariant under the KdV flow, each of the form dnu)dx ,
Fn{u) = / pn(u, du,...,
p a polynomial, quadratic in its last argument. The solutions u of this variational problem satisfy an ordinary differential equation of order 2n. As already mentioned in Sec. 3, [103] is a joint study with Levermore of the family of equations ut + uux + euxxx = 0 ,
u(x,0) = u0(x).
(4)
Each is integrable on the whole real axis, and their solution can be expressed in terms of the eigenvalues of the operator L = ed2 + f, as well as the so-called norming constants of the eigenfunctions. The passage to the limit e —»• 0 can be evaluated, thanks to a sequence of miracles. 5. Mathematics Light I am delighted by and look for nontechnical mathematics that yields a surprising result, or opens a new window on a classical topic. Here are some examples: 5.1. The existence of Green's function for the Laplace equation Let G be a domain in the plane with a smooth boundary B. Denote by C the space of continuous functions on B, and by H the boundary values of harmonic functions in G. In [5] I have shown that H is all of C by arguing as follows: Fix some point q in G, and for any f in H define the linear functional iq as the value at q of the harmonic function h in G whose boundary value is / : £q(f) = h(q) . By the maximum principle, |^ q | ma x = 1- So by the Hahn-Banach theorem lq can be extended to all of C as a bounded linear functional. Let p be any point not on the boundary B. Define kp in C as kp(z) = log \z — p\,
z on B .
kp is a harmonic function of p in each component of the complement of B. Therefore 9(p,q) = tq(kp) is a harmonic function of p. For p not in G, kp belongs to H; therefore g(p, q) = %() = log \q — p\. Let p be a point in G near the boundary B; denote by p the reflection of p across B. It is easy to show that \kp - %| m ax tends to zero as p approaches the boundary
235 B; therefore so does g(p, q) - g(p,q). Since p lies outside G, g(p, q) = log \q - p\; therefore as p approaches the boundary point z, g(p,q) tends to \og\q — p\. Thus log \q — p\— g(q,p) is Green's function. 5.2. The factorization
of matrix
valued
functions
The solution of Wiener-Hopf integral equations and the inversion of Toeplitz matrices rely on factoring continuous complex valued functions on, say, the unit circle, whose change of argument is zero, as a product of the boundary values of an analytic and an antianalytic function that are invertible in the unit disk. For scalar functions this is easily accomplished by taking the logarithm of the function and decomposing it additively. This does not work in the matrix valued case. In [76] I described the following procedure. Let F be the matrix valued function on the boundary of the unit disk, such that the change of argument of det F is zero. Let F = AB be the desired factorization, A analytic, B antianalytic. Differentiate with respect to z and ~z: FZ = AZB,
FT = ABZ,
(1)
FZZ=AZBZ.
(1)'
Using (1) to express Az and Bj in (1)' gives &F = FZF~1FZ.
(2)
Here we have used the identity that d2/dzdz- is the Laplace operator. Since det F = det A det B can be obtained by a scalar factorization, we can express F _ 1 in (2) as a polynomial in F divided by d e t F . Equation (2) is a system of n 2 equations whose main term is the Laplace operator, coupled by lower order terms; the value of F is prescribed on the boundary of the unit disk. This Dirichlet problem may or may not have a solution, just as the factoring problem may or may not have a solution, see Gohlerg and Krein on partial indices. If it does have a solution, A and B can be determined by solving the ordinary differential equations AZ = FZF'XA, 5.3. The approximation
B? = BF~lF-z.
of measure preserving
transformations
Numerical experimentation with the iterates of a measure preserving transformation often reveals interesting properties. Because computers use finite digit arithmetic, say k decimals, one could think of the numerical experiment as iterating an approximation of the given transformation that maps squares (cubes, etc.) of edge length 10 -A: into squares etc. of the same size. Such an approximation is itself measurepreserving iff it is a permutation of the squares. Impose on the approximation the requirement that the approximate image of every square have a nonempty intersection with its exact image. In [59] I deduced from the combinatorial Marriage Theorem that a permutation consistent with this requirement exists. Steve Alpern has deduced the same conclusion from the Birkhoff-Konig theorem about doubly stochastic matrices, which says that the extreme points of this set of matrices are the permutation matrices.
236
5.4. Polya's
area
filling
curve
In 1914 Polya constructed a continuous curve that goes through every point of a given right triangle T, required to be not isosceles. In this case the altitude from the right angle divides the triangle into two unequal triangles, each similar to the original one. The construction of the function P(t), 0 < t < 1 goes as follows: Expand t in binary fraction, t =
hb2....
Each binary digit bi is a zero or a one. If &i = 1, choose the larger of the two triangles into which the altitude divides T; if b\ = 0, choose the smaller one. Then repeat the process with whichever triangle was chosen, using the second digit of t to make the decision. Continue this process add infinitum; the result is a nested sequence of triangles T = T\, T%, • • •, Tn,... all similar and whose size tends to zero. Their intersection consists of a single point, defined to be P(i). Some numbers t have two distinct binary expansions, such as .1000... = .011111. It is an amusing fact that using either expansion of such t yields the same point
Pit). The continuity of the function P is now easy to see. For if t\ and t 2 are near each other, then either their first n binary digits are equal, or there is a tm between them, ti < tm < ti, such that t\ and tm have n digits in common in one representation of tm and tm and t2 have n digits in common in the other binary representation of tm. In either case, P(t\) and P(*2) are near each other. That the image of P(t) fills the triangle T is clear; for every point Q of T is the intersection of a nested sequence of triangles of the kind constructed by repeatedly drawing altitudes. Polya's example is by far the simplest of the many Peano type curves; when I taught real variables, I always presented it to my class. One time I assigned as homework: prove that Polya's function is not differentiable. Since nobody handed in the homework, I started doing it myself, and was surprised to find the following: (a) When the smallest angle of T is greater than 30°, P(t) is nowhere differentiable. (b) When the smallest angle of T lies between 30° and 15°, P(t) is differentiable on a set of measure zero. (c) When the smallest angle of T is less than 15°, P(t) is differentiable almost everywhere. The proof is given in [67]. 5.5. Linear
spaces of nonsingular
matrices
Frank Adams had proved that the largest number of real n x n matrices all whose linear combinations are nonsingular equals the Radon-Hurwitz function R(n) defined as follows: let b be the largest power of 2 that divides n; write b as 4d + c, 0 < c < 4. Then R(n) = 2c + 8d.
237 Ralph Phillips and I had been investigating scattering theory for symmetric hyperbolic systems of the form ut
£ ^ , = 0 ,
(3)
where the Aj are real; symmetric n x n matrices. In order to have a scattering theory, all signals have to propagate to infinity. The signal speeds in the direction (£ii • • • i£fc) a r e the eigenvalues of J2i €jAj\ so we required all these to be nonzero. This led naturally to the question, what is the largest number k of real, symmetric n x n matrices, all of whose linear combinations are nonsingular. We submitted a paper on this question to the Proceedings of the AMS; we received a referee's report with so many excellent suggestions for improving our arguments, that we wrote to the editor offering to withdraw our paper in favor of one by the referee. The end result was a joint paper with the referee; thus did we become coauthors with the topologist Frank Adams. The paper is [36]; the maximum value of k is 1 for n odd, and R(n/2) + 1 for n even. 5.6. Multiple eigenvalues Another incursion into topology also came from the study of first order hyperbolic systems of form (3), in three variables: ut + Aux + Buy + Cuz = 0, A, B, C real symmetric n x n matrices. The signal speeds in the direction £, 77, £ are the eigenvalues of £,A + rjB + C,C. The presence of multiple signal speeds has a significant effect on the propagation of singularities, such as conical refraction. Therefore, it is important to know in any given case if there are multiple signal speeds present or not. I proved in [99] that for n = 2(mod 4) there always exist values of £, r], £, £2 + T]2 + £2 = 1, such that £A + r]B + C,C has a multiple eigenvalue. The proof is based on the fundamental group of the space TV of all real, symmetric n x n matrices with distinct eigenvalues Ai < A2 < • • • < A„. Let N(6) be a continuous, closed curve in N, N(2TT) = N(0). The eigenvalues Xj(9) are continuous functions of 6, and the normalized eigenvectors rj{6), \\rj\\ = 1, can be chosen to be continuous. We have then rj(27r) = TJTJ(O) ,
Tj = 1
or
-1.
Clearly the Tj are homotopy invariants of the closed curve. If a closed curve is contractible to a point, all Tj must be 1. Let TV be an odd closed curve, satisfying N(9 + TT) =
-N(0).
Here \j{6 + it) = -A n _ J + i(fl), and rj(6 + n) — (ijrn-j+i(6), rj{2ir) = fijrn-j+i(n) = /j,jfj,n_j+1rj(0), it follows that Tj = The ordered basis ri(0),...,rn(Q)
/J,J = 1 or —1. Since fj,j/j.n_j+1.
238 has the same orientation as ri(7r),...,r n (7r), which is the same as Atir n (0),...,/i n ri(0). So we conclude that when n = 2(mod 4), J}™ fij = —1. Since HjUn-j+i = TJ, it follows that J]™' Tj = — 1 - I n particular, some n = — 1; so it follows that an odd curve is not contractible to a point. N(0) = A cos 6 + B sin 6 is an odd curve. It follows that not all £A + rjB + (C, £ 2 + V2 + C2 = 1) belong to; Af, for then the curve N(0) above could be contracted to a point on the sphere £2 + rj2 + £2 = 1. Friedland, Robbin and Sylvester have shown that a multiple eigenvalue must occur when n ^ 0, 1, — l(mod 8). 5.7. Huygens'
principle
for the spherical
wave
equation
In [82], Phillips and I investigated the spherical wave equation governing the propagation of waves on the unit sphere. This has the form utt - Lw = 0,
(5)
where L = Ag — 1, As the spherical Laplacian on the 3-dimensional unit sphere. As is related to the Euclidean Laplacian A4 in R 4 by
A4 = dl + -dr + ~ .
(6)
r r2 Denote by hn(u>) a spherical harmonic of order n. Then rnhn(u>) is harmonic in R 4 , so it follows from (6) that (n 2 + 2n)hn + Ashn
= 0,
and therefore Lhn — — ( n + l)2hn . Every solution with finite energy of the spherical wave equation (5) can be expanded in terms of the eigenfunctions of L: u{w,t) = J2(*nei(n+1)t
+
bne-^n+1^)hn{uj).
Since hn(w) is even or odd, depending on the parity of n, we deduce that u(—w,t + ir) = —u(Lj,t).
(7)
Now take a solution u of (5) whose initial data U(LJ, 0), ut(u, 0) are supported in a ball of radius e around some point U>Q of S3. It follows from (7) that the initial data of u at time -K are supported in a ball of radius e around the antipodal point — WQ. Take any t between 0 and 7r. Since signals for the wave equation (5) propagate with speed < 1, it follows that the support of u(u>, t) is confined to the ball of radius t + £ around UJO, and therefore, u(u>,t) = 0 at all points LJ whose distance from UJQ exceeds t + e.
239 Applying the same argument to the wave equation (5), starting at time n and going backward by IT — t, we conclude that u(w,t) is zero at all points w whose distance from — u>o exceeds IT — t + e. It follows that the support of u(u>,t) is confined to the annulus consisting of points UJ whose distance from U>Q lies between t — e and t + e. Letting e —>• 0 we conclude that a signal starting at CJQ spreads in time t to the sphere of radius t centered at, U)Q and nowhere else. Replacing the unit sphere with a sphere of radius R and letting R —>• oo yields Huygens' principle in Euclidean space. Curiously, as shown in [82], the Euclidean wave equation can be obtained from the spherical wave equation by a change of variables!
240 List of Publications
1944 [1] Proof of a conjecture of P. Erdos on the derivative of a polynomial, Bull. Amer. Math. Soc. 50, 509-513. 1948 [2] The quotient of exponential polynomials, Duke Math. J. 15, 967-970. 1950 [3] Partial Differential Equations, Lecture Notes, NYU, IMS (1950-51). 1951 [4] A remark on the method of orthogonal projections, Coram. Pure Appl. Math. 4, 457-464. 1952 [5] On the existence of Green's function, Proc. Amer. Math. Soc. 3, 526-531. 1953 [6] Nonlinear hyperbolic equations, Comm. Pure Appl. Math. 6, 231-258. 1954 [7] Weak solutions of nonlinear hyperbolic equations and their numerical computation, Comm. Pure Appl. Math. 7, 159-194. [8] Symmetrizable linear transformations, Comm. Pure Appl. Math. 7, 633-648. [9] (with A. Milgram) Parabolic equations, Ann. Math. Studies 33 (Princeton) 167-190. [10] The initial value problem for nonlinear hyperbolic equations, Ann. Math. Studies 33 (Princeton) 211-229.
241 1955 [11] Reciprocal extremal problems in function theory, Comm. Pure Appl. Math. 8, 437-454. [12] On Cauchy's problem for hyperbolic equations and the differentiability of solutions of elliptic equations, Comm. Pure Appl. Math. 8, 615-633. [13] (with R. Courant) Cauchy's problem for hyperbolic differential equations, Ann. Mat. Pura Appl. 40, 161-166. 1956 [14] A stability theorem for solutions of abstract differential equations, and its application to the study of local behavior of solutions of elliptic equations, Comm. Pure Appl. Math. 9, 747-766. [15] (with R. D. Richtmyer) Survey of the stability of linear finite difference equations, Comm. Pure Appl. Math. 9, 267-293. [16] (with R. Courant) The propagation of discontinuities in wave motion, Proc. Nat. Acad. Sci. 42, 872-876. 1957 [17] A Phragmen-Lindelof theorem in harmonic analysis and its application to some questions in the theory of elliptic equations, Comm. Pure Appl. Math. 10, 361-389. [18] Hyperbolic systems of conservation laws, II, Comm. Pure Appl. Math. 10, 537-566. [19] Remarks on the preceding paper, Comm. Pure Appl. Math. 10, 617-622. [20] Asymptotic solutions of oscillatory initial value problems, Duke Math. J. 24, 627-646. 1958 [21] Differential equations, difference equations and matrix theory, Comm. Pure Appl. Math. 11, 175-194. 1959 [22] Translation invariant spaces, Acta Math. 101, 163-178. 1960 [23] (with B. Wendroff) Systems of conservation laws, Comm. Pure Appl. Math. 13, 217-237.
242 [24] (with R. S. Phillips) Local boundary conditions for dissipative symmetric linear differential operators, Coram. Pure Appl. Math. 13, 427-455. 1961 [25] Translation invariant spaces, Proc. Int. Symp. on Linear Spaces, Israeli Acad. of Sciences and Humanities, Jerusalem (1960) (Pergamon), 299-307. [26] On the stability of difference approximations to solutions of hyperbolic equations with variable coefficients, Comm. Pure Appl. Math. 14, 497-520. 1962 [27] (with R. S. Phillips) The wave equation in exterior domains, Bull. Math. Soc. 68, 47-79.
Amer.
[28] A procedure for obtaining upper bounds for the eigenvalues of a Hermitian symmetric operator, Studies in Mathematical Analysis and Related Topics (Stanford Univ. Press), 199-201. 1963 [29] On the regularity of spectral densities, Teoriia Veroiatnosteii i ee Prim. 8, 337-340. [30] An inequality for functions of exponential type, Comm. Pure Appl. Math. 16, 241-246. [31] (with C. Morawetz and R. Phillips) Exponential decay of solutions of the wave equation in the exterior of a star-shaped obstacle, Comm. Pure Appl. Math. 16, 477-486. [32] Survey of stability of difference schemes for solving initial value problems for hyperbolic equations, Symp. Appl. Math. 15, 251-258. 1964 [33] Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys. 5, 611-613. [34] (with B. Wendroff) Difference schemes for hyperbolic equations with high order of accuracy, Comm. Pure Appl. Math. 17, 381-398. [35] (with R. S. Phillips) Scattering theory, Amer. Math. Soc. Bull. 70, 130-142. 1965 [36] (with J. F. Adams and R. S. Phillips) On matrices whose real linear combinations are nonsingular, Proc. AMS 16, 318-322; correction, ibid. 17 (1966) 945-947.
243 [37] Numerical solution of partial differential equations, Amer. Math. Monthly 72 II, 74-84. [38] (with K. O. Priedrichs) Boundary value problems for first order operators, Comm. Pure Appl. Math. 18, 355-388. 1966 [39] (with R. S. Phillips) Analytic properties of the Schrodinger scattering matrix, in Perturbation Theory and its Application in Quantum Mechanics, P r o c , Madison, 1965 (John Wiley & Sons), 243-253. [40] Scattering theory; remarks on the energy theory, and on scattering theory for a geometrical optics model, in Proc. Conf. on Dispersion Theory, Cambridge, MA, 1966, M 11, pp. 36-39, 40-42. [41] (with L. Nirenberg) On stability for difference schemes; a sharp form of Garding's inequality, Comm. Pure Appl. Math. 19, 473-492. [42] (with J. P. Auffray) Aspects mathematiques de la mecanique de phase, Acad. Sci. Paris, Compt. Rend. (B) 263, 1355-1357. 1967 [43] (with J. Glimm) Decay of solutions of systems of hyperbolic conservation laws, Bull. AMS 73, 105. [44] Hyperbolic difference equations: a review of the Courant-Friedrichs-Lewy paper in the light of recent developments, IBM J. Res. Develop. 11(2), 235-238. [45] (with R. S. Phillips) Scattering Theory (Academic Press). [46] (with R. S. Phillips) The acoustic equation with an indefinite energy form and the Schrodinger equation, J. Func. Anal. 1, 37-83. [47] (with R. S. Phillips) Scattering theory for transport phenomena, in Functional Analysis, Proc. of Conf. at Univ. of Calif., Irvine, 1966, ed. Gelbaum (Thompson Book Co.), 119-130. [48] (with K. O. Friedrichs) On symmetrizable differential operators, Symp. on Pure Math., Singular Integrals, AMS 10, 128-137. 1968 [49] Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21, 467-490. 1969 [50] Nonlinear partial differential equations and computing, SIAM Rev. 11, 7-19.
244 (with R. S. Phillips) Decaying modes for the wave equation in the exterior of an obstacle, Comm. Pure Appl. Math. 22, 737-787. Toeplitz operators, in Lecture Series in Differential Equations 2, ed. A. K. Aziz (Van Nostrand), 257-282. 1970 (with J. Glimm) Decay of solutions of systems of nonlinear hyperbolic conservation laws, Memoirs of the AMS 101. (with R. S. Phillips) The Paley-Wiener theorem for the Radon transform, Comm. Pure Appl. Math. 23, 409-424; Errata ibid 24 (1971) 279. 1971 Nonlinear partial differential equations of evolution, Proc. Int. Conf. of Mathematicians, Nice, September 1970 (Gauthier-Villars), 831-840. (with R. S. Phillips) Scattering theory, Rocky Mountain J. Math. 1, 173-223. (with H. Brezis, W. Rosenkrantz and B. Singer) On a degenerate ellipticparabolic equation occurring in the theory of probability, Comm. Pure Appl. Math. 24, 410-415; appendix by P. D. Lax. Shock waves and entropy, Contributions to Functional Analysis, ed. E. H. Zarantonello (Academic Press), 603-634. Approximation of measure preserving transformations, Comm. Pure Appl. Math. 24, 133-135. (with K. O. Friedrichs) Systems of conservation equations with a convex extension, Proc. Natl. Acad. Sci. 68, 1686-1688. (with R. S. Phillips) A logarithmic bound on the location of the poles of the scattering matrix, Arch. Rational Mech. Anal. 40, 268-280. 1972 (with R. S. Phillips) On the scattering frequencies of the Laplace operator for exterior domains, Comm. Pure Appl. Math. 25, 85-101. The formation and decay of shock waves, Amer. Math. Monthly 79, 227-241. (with R. S. Phillips) Scattering theory for the acoustic equation in an even number of space dimensions, Ind. U. Math. J. 22, 101-134. [65] Exponential modes of the linear Boltzmann equation, in The Boltzmann Equation, ed. F. A. Grunbaum, NYU, CIMS, 111-123.
245 1973 [66] Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Conf. Board of the Mathematical Sciences, Regional Conf. Series in Appl. Math. {SIAM), 11. The differentiability of Polya's function, Adv. Math. 10, 456-465. (with R. S. Phillips) Scattering theory for dissipative hyperbolic systems, J. Func. Anal. 14, 172-235. 1974 Invariant functionals of nonlinear equations of evolution, in Nonlinear Waves, eds. S. Leibovich and R. Seebass (Cornell Univ. Press), 291-310. Applied mathematics and computing, Symp. on Appl. Math. (AMS) 20, 57-66. Periodic solutions of the KdV equations, in Nonlinear Wave Motion, ed. A. C. Newell, Lectures in Appl. Math. 15, 85-96. 1975 Periodic solutions of the KdV equation, Comm. Pure Appl. Math. 28, 141-188. Almost periodic behavior of nonlinear waves, Adv. Math. 16, 368-379. 1976 Almost periodic solutions of the KdV equation, SIAM Rev. 18, 351-375. (with A. Harten, J. M. Hyman and B. Keyfitz) On finite-difference approximations and entropy conditions for shocks, Comm. Pure Appl. Math. 29, 297-322. On the factorization of matrix-valued functions, Comm. Pure Appl. Math. 29, 683-688. (with R. S. Phillips) Scattering Theory for Automorphic Functions, Ann. Math. Studies 87 (Princeton Univ. Press and Univ. of Tokyo Press). (with S. Burstein and A. Lax) Calculus with Applications and Computing, Undergrad Texts in Math. 1 (Springer). 1977 [79] The bomb, Sputnik, computers and European mathematicians, the Bicentennial Tribute to American Mathematics, San Antonio, 1976 (Math. Assoc, of America), 129-135.
246 [80] (with R. S. Phillips) The scattering of sound waves by an obstacle, Comm. Pure Appl. Math. 30, 195-233. 1978 [81] (with M. S. Mock) The computation of discontinuous solutions of linear hyperbolic equations, Comm. Pure Appl. Math. 31, 423-430. [82] (with R. S. Phillips) An example of Huygens' principle, Comm. Pure Appl. Math. 3 1 , 415-421. [83] (with R. S. Phillips) The time delay operator and a related trace formula, Topics in Functional Analysis, Adv. Math. Suppl. Studies, 3, eds. I. T. Gohberg and M. Kac (Academic Press), 197-215. [84] Accuracy and resolution in the computation of solutions of linear and nonlinear equations, Recent Adv. in Numer. Anal., Proc. of Symp., Madison (Academic Press), 107-117. [85] (with A. Lax) On sums of squares, Linear Algebra Its Appl. 20, 71-75. [86] (with R. S. Phillips) Scattering theory for domains with nonsmooth boundaries, Arch. Rational Mech. Anal. 68, 93-98. [87] Chemical kinetics, in Lectures on Combustion Theory, eds. S. Burstein, P. D. Lax and G. A. Sod, NYU, COO-3077-153, 122-136. 1979 [88] (with R. S. Phillips) Translation representations for the solution of the nonEuclidean wave equation, Comm. Pure Appl. Math. 32, 617-667. [89] (with C D . Levermore) The zero dispersion limit for the Korteweg-de Vries equation, Proc. Natl. Acad. Sci. 76, 3602-3606. [90] Recent methods for computing discontinuous solutions — a review, in Computing Methods in Applied Sciences and Engineering, 1977, II; 3rd Int. Symp. IRIA, eds. R. Glowinski and D. L. Lions, Springer Lecture Notes in Physics, 91, 3-12. 1980 [91] (with R. S. Phillips) Scattering theory for automorphic functions (AMS), Bull. of C.N.S. 2, 161-195. 1981 [92] On the notion of hyperbolicity, Comm. Pure Appl. Math. 33, 395-397. [93] (with A. Harten) A random choice finite difference scheme for hyperbolic conservation laws, SIAM J. Numer. Anal. 18, 289-315.
247 [94] (with R. S. Phillips) The translation representation theorem, Integral Equations and Operator Theory 4, 416-421. [95] (with R. S. Phillips) Translation representations for the solution of the nonEuclidean wave equation, II, Comm. Pure Appl. Math. 34, 347-358. [96] Applied mathematics 1945 to 1975, Amer. Math. Heritage, Algebra & Appl. Math., 95-100. [97] Mathematical analysis and applications, Part B, essays dedicated to Laurent Schwartz, ed. L. Nachbin (Academic Press) Advances in Math. Suppl. Studies, 7B, 483-487.
1982 [98] The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces, J. Func. Anal. 46, 280-350. [99] The multiplicity of eigenvalues, AMS Bull, 213-214. [100] (with R. S. Phillips) A local Paley-Wiener theorem or the Radon transform of L2 functions in a non-Euclidean setting, Comm. Pure Appl. Math. 35, 531-554.
1983 [101] Problems solved and unsolved concerning linear and nonlinear partial differential equations, Proc. Int. Cong. Mathematicians, 1 (North-Holland), 119-137. [102] (with A. Harten and B. van Leer) On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SI AM Rev. 25, 35-61. [103] (with C. D. Levermore) The small dispersion limit of the Korteweg-de Vries equation, Comm. Pure Appl. Math. 36, I 253-290, II 571-593, III 809-930.
1984 [104] On a class of high resolution total-variation-stable finite difference schemes, Ami Harten with Appendix by Peter D. Lax, SIAM 2 1 , 1-23. [105] (with R. S. Phillips) Translation representations for the solution of the non-Euclidean wave equation, Comm. Pure Appl. Math. 37, I 303-328, II 779-813, III 38 (1985), 179-813. [106] Shock Waves, Increase of Entropy and Loss of Information, Seminar on Nonlinear Partial Differential Equations, ed. S. S. Chern (Math. Sci. Res. Inst. Publ.), 129-171.
248 1985 Large Scale Computing in Science, Engineering and Mathematics, Rome. (with R. J. Leveque and C. S. Peskin) Solution of a two-dimensional cochlea model using transform techniques, SIAM J. Appl. Math. 45, 450-464. (with R. Phillips) Translation representations for automorphic solutions of the wave equation in non-Euclidean spaces; the case of finite volume, Trans. AMS 289,715-735. (with P. Constantin and A. Majda) A simple one-dimensional model for the three-dimensional vorticity equation, Comm. Pure Appl. Math. 38, 715-724. 1986 On Dispersive Difference Schemes, 1985, Kruskal Symposium, Physica 18D, 250-255. Mathematics and computing, J. Stat. Phys. 43, 749-756. (with A. Jameson) Conditions for the construction of multipoint total variation diminishing difference schemes, App. Numer. Math. 2, 335-345. Mathematics and its applications, Math. Intelligencer 8, 14-17. Hyperbolic systems of conservation laws in several space variables, Current Topics in Partial Differential Equations, papers dedicated to Segeru Mizohata (Tokyo Press), 327-341. The Soul of Mathematics, Studies in Mathematics and its Applications 16, Patterns and Waves (North-Holland). 1988 Oscillatory solutions of partial differential and difference equations, Mathematics Applied to Science (Academic Press), 155-170. (with R. J. Leveque and C. Peskin) Solution of a two-dimensional cochlea model with fluid viscosity, SIAM J. Appl. Math. 48, 191-213. The flowering of applied mathematics in America, AMS Centennial Celebration P r o c , 455-466; SIAM Rev. 3 1 , 65-75. (with J. Goodman) On dispersive difference schemes I, Comm. Pure Appl. Math. 4 1 , 591-613. 1989 Science and computing, Proc. IEEE 77. Writing mathematics well, Leonard Gillman review, Amer. Math. Monthly 96, 380-381.
249 [123] From cardinals to chaos: Reflections on the life and legacy of Stanislaw Ulam, reviewed in Phys. Today 42, 69-72; Bull. AMS 22 (1990) 304-310, St. Petersburg Math. J. 4 (1993) 629-632. [124] Deterministic turbulence, Symmetry in Nature, Volume in Honor of Luigi A. Radicati di Brozolo II (Scuola Normale Superiore), 485-490. 1990 [125] (with R. Phillips) Translation representations for automorphic solutions of the wave equation in non-Euclidean spaces, IV, Comm. Pure Appl. Math. 43. [126] Remembering John von Neumann, Proc. Symp. Pure Math. 50. [127] The ergodic character of sequences of pedal triangles, Amer. Math. Monthly 97, 377-381. 1991 [128] Deterministic analogues of turbulence, Comm. 1047-1055.
Pure Appl.
Math.
44,
[129] (with T. Hou) Dispersive approximations in fluid dynamics, Comm. Pure Appl. Math. 44. 1993 [130] (with C. D. Levermore and S. Venekides) The generation and propagation of oscillations in dispersive IVP's and their limiting behavior, in Important Developments in Soliton Theory 1980-1990, eds. T. Fokas and V. E. Zakharov (Springer- Verlag). [131] The existence of eigenvalues of integral operators, in Honor of C. Foias, ed. R. Teman, Ind. Univ. Math. J. 42, 889-991. 1994 [132] Trace formulas for Schrodinger operator, Comm. Pure Appl. Math. 47, 503-512. [133] Cornelius Lanczos and the Hungarian phenomenon in Science and Mathematics, Proc. Lanczos Centennary Conf. (N.C. State University Press). 1995 [134] Computational fluid dynamics at the Courant Institute 1-5, Computational Fluid Dynamics Review, eds. M. Hafez and K. Oshima (John Wiley & Sons). [135] A short path to the shortest path, Amer. Math. Monthly 102, 158-159.
250
1996 [136] Outline of a theory of the KdV equation, Lecture Notes in Mathematics, Recent Mathematical Methods in Nonlinear Wave Propagation (Springer) 1640, 70-102. [137] (with Xu-Dong Liu) Positive schemes for solving multidimensional hyperbolic conservation laws, Comp. Fluid Dynamics J. 5, 133-156. [138] The Old Days, A Century of Mathematical Meetings (AMS), 281-283. [139] (with A. Harten, C. D. Levermore and W. J. Morokoff) Convex entropies and hyperbolicity for general Euler equations, SIAM J. Numer. Anal. 1997 [140] Linear Algebra, Pure and Applied Math. Series (Wiley-Interscience). 1998 [141] (with Xu-Dong Liu) Solution of two-dimensional Riemann problem of gas dynamics by positive schemes, SIAM J. Sci. Comput. 19, 319-340. [142] Jean Leray and Partial Differential Equations, Introduction to Volume II, Selected Papers of Jean Leray (Springer-Verlag). [143] On the discriminant of real symmetric matrices, Comm. Pure Appl. Math. 51, 1387-1396. [144] The beginning of applied mathematics after the Second World War, Quart. Appl. Math. 56, 607-615. 1999 [145] A mathematician who lived for mathematics, Book Review, Phys. Today, 69-70. [146] The mathematical heritage of Otto Toeplitz, in Otto Toeplitz, Bonner Mathematische Schriften, 319, 85-100. [147] Mathematics and computing, in Useful Knowledge, ed. A. G. Beam (Amer. Philos. Soc), 23-44. [148] Change of variables in multiple integrals, Amer. Math. Monthly 105, 497-501. 2000
[149] Mathematics and computing, in IMU, Mathematics: Frontiers and Perspectives (AMS), 417-432.
251 2001 [150] Change of variables in multiple integrals II, Amer. Math. Monthly 108, 115-119. [151] On the accuracy of Glimm's scheme, Math. Appl. Anal, to appear. [152] Functional Analysis (John Wiley and Sons), to appear. [153] The Radon transform and translation representation.
252 Proceedings of Symposia in Pure Mathematics Volume 65, 1999
A Sample of Lax's Contributions to Classical Analysis, Linear Partial Differential Equations and Scattering Theory Peter Sarnak
1. Classical Analysis and P D E In this lecture I will describe a small subset of Peter Lax's profound contribution to analysis and partial differential equations. His works, especially on hyperbolic equations, constitute some of the most influential works in the subject in the last 50 years. My own interests have been so to speak at the boundary of his work (though Peter has certainly influenced my work both directly and indirectly) so my angle on some of the topics below may be somewhat different from the mainstream views. My aim is to give you a glimpse of Peter's brilliance and originality. Let me begin naturally enough with Peter's first paper [LI] written in his teens, in which he resolved a conjecture of Erdos. If P(z) is a polynomial of degree n, then a well-known inequality of Bernstein asserts that (1)
max|P'(z)|
|z|
equality holding if and only if P{z) = az . Erdos conjectured that if P(z) has no zeros in \z\ < 1, then (1) should hold with n replaced by n/2 (and this is sharp, as (zn + l)/2 shows). In [LI] Peter gives an elegant proof of this conjecture. An impressive start for a kid, and I have no doubt that at that time the older Hungarian did his best to lure the younger one into a field like combinatorics. Luckily for us, Peter was drawn into the fields of analysis and partial differential equations. Indeed, some years later (after the war) we find that Peter is fully involved in PDE. In [L2] he establishes a very interesting extension of the Phragmen-Lindelof principle. Recall that a form of this principle asserts that if (2)
S={(x,t)
:0<x<
1,2/>0}
and if u{x, y) is harmonic in S, vanishes on x = 0 and x = 1, and is bounded, then u decays exponentially as y —> oo. It takes quite some insight to see that the root of this phenomenon is a spectral one which allows for the following generalization due to Peter. Let V0 be a compact subset of Rn and let V = V0 x [0, oo) C K n + 1 . Let L be an elliptic differential operator of order 2m with coefficients independent © 1999 American Mathematical Society
169
253 170
PETER SARNAK
of y. Suppose that u satisfies Lu ~ 0 ,
u = 0
on dV0 x [0, o o ) .
Then if the Dirichlet integral of u over V is finite (that is m derivatives are in Z,2 (£>)), then u decays exponentially as y —> oo (the decay rate being independent of u). His proof of this is ingenious and elegant (typical of much of Peter's work) and it introduces some themes which persist in his later works. These being; compactness and translation invariance with exponential decay as a consequence. These arise in his analysis of the last problem as follows: Let H be the set of functions uy{x), where y > 0 and uy(x) — u(x,y), u(x,y) satisfying the equation and the boundary conditions. For w > 0 the operator Tw : H —> H, with appropriate norms of H, defined by Twvy = vy+w satisfies: Tw is compact for w > 0, ||7i|| < 1, ||T„|) 1 / / n = ||rpn||l/n _^ a < i ; as n —• oo. The theorem follows easily from these. The above led Peter to think about translation invariant spaces in general. Much had been written about these for scalar valued functions. In [L3] he found the following striking and simple description of such spaces for vector valued functions. Let S be a finite dimensional inner product space. Let H$ be the space L 2 ( R > 0 , S) of square integrable functions from R>o to S. If T is a closed translation invariant subspace of L2(M.>0, S) (i.e., g & T ==> gt G r for t > 0 where gt(x) = g(x +1)) and R is the space of Fourier Transforms of members of r , then R = FHT, where T is an inner product space of dimension at most dim S, F(z) : T —* S is defined in 0>(z) > 0, ||F(z)|| < 1, F(z) is an isometry onto its image if z £ M. and F(z) is holomorphic for $s(z) > 0. Moreover this representation is unique except for multiplication of F on the right by a constant unitary matrix. In 1956 Peter wrote a paper [C-L] with Courant on propagation of discontinuities in wave motion. This was followed by one of Peter's seminal papers "Asymptotic solutions of oscillatory initial value problems" [L4]. This paper is the genesis of the theory of Fourier Integral Operators (see [HI] and [H2]). Even with all the excellent modern treatments of this subject, I still recommend to t h e new-comer t o this theory to first read Peter's paper. Moreover, as I indicate in more detail below, for certain purposes, the techniques introduced in this paper are still at the forefront of what we know today. The paper is concerned with the strictly hyperbolic system (3)
Mu = 0 .
M being a first-order differential operator acting on vectors u(t,x)
= I • I
(t,x),
x € R m , by the rule m
(4)
Mu = ut + Y^ Aj uXj + Bu
Aj and B being n x n matrix functions. Assume that in some neighborhood of t = 0 (which is assumed to be a space-like hypersurface) (5)
Pi M + VI M H
+PmA,
A SAMPLE OF LAX'S CONTRIBUTIONS TO CLASSICAL ANALYSIS
171
has real and distinct eigenvalues for all p e R m , this being the condition of strict hyperbolicity. One seeks oscillatory solutions
4>fr) = elie{x
+ v, /£ + •••}
for a suitable phase function £(x,t). Substituting (6) in (3) and solving recursively for the coefficients of £ ~ m one is led to the well-known eikonal equation from geometric optics. If X(t, x,pi,... ,pm) is one of the eigenvalue branches of (5) then we get the nonlinear first-order (scalar) equation d£ { ')
rv,
A\t,X\,
. . . , Xm,
t X j , . . . , tXrn ) -
One solves this equation for small t, \t\ < to (in particular before any caustics can develop) by the method of characteristics — these being the bi-characteristics of the original equation (3). The Vj's are determined by solving the corresponding 'transport equations.' In this way one gets a formal asymptotic series in f for such a 4>{x) with highly oscillatory initial data. Most importantly, Peter goes on to make this analysis precise and rigorous. In particular he develops a Fourier synthesis in £ which allows him to develop an explicit approximate geometric construction of the fundamental solution to (3), for \t\ < to- His 'parametrix' differs from the fundamental solution to any order of smoothness that one desires. Before this paper there were the constructions of parametrices by Hadamard [HA] and Riesz [RI] for second-order equations. As mentioned above, Peter's general method above led to the (local) theory of Fourier Integral Operators. To illustrate the power of the above method, consider the wave equation on a domain or a Riemannian manifold. The construction yields, for \t\ < to and any integer JV, an explicitly described distributional kernel KN(t,x,y) which differs from the fundamental solution K(t,x,y) by a CN+1 function, in all the variables. It can be used among other things to study the spectrum of the Laplacian. For example, if X is a smooth compact Riemannian manifold of dimension v and A is the Laplace-Beltrami operator acting on functions on X, then there is an orthonormal basis of eigenfunctions <j>j and eigenvalues kj such that (8)
- A ^ = ^ . .
For the (hyperbolic) wave equation (9)
utt = Aw
on 1 x I , with initial conditions w(0, x) = 6(x,y) fundamental solution
and ut{0,x)
= 0, we have the
oc
(10)
K(t,x,y)
= ^coB(fcjt)0j(i)0j(j/). 3=0
The Lax construction gives the singular part of K(t,x,y) one to investigate asymptotically the sums (11)
^
4>j(x)4>j(y)
as
for small t. This allows
A -* oo.
A
This range, X < kj < A + 1, might be called the "Lax range" since it comes from his parametrix. I mention a couple of my favorite applications:
172
PETER SARNAK
(A) Weyl's Law with Remainder (Hormander [HI])
N(X) := J2 1 = (27r)"" c" Vo1 (X) X" + ° (A"_1) fcj
where c„ is the volume of the unit ball in W'. (B) Bounding the eigenfunctions in Lp (Seeger and Sogge [S-S]), If 6(p) = max
{"
1 1 p~2
UA^
\, o | for
then ?i^±il
These results are sharp for the i/-sphere with its round metric. However if X has negative curvature for example (in which case the bicharacteristic Hamilton flow which is the geodesic flow cmT*X, is "chaotic") then (A) and (B) are no doubt very far from being sharp. Recently there has been a lot of interest in such fine spectral questions for such X's (the subject going by the name Q u a n t u m Chaos, see [G-V-Z], [SA]). One problem being to understand the sums in (11) but in shorter ranges X < Kj < \ + 77(A), with 77(A) of size A _ Q for some a > 0. Unfortunately, in general no progress has been made on such short sums. The global theory of Fourier Integral Operators allows one to deal with 77(A) is small as 1/ log A and this already is quite deep, see Duistermaat and Guillemin [D-G] and Berard [BE]. The exponential proliferation of periodic orbits of the bicharacteristic flow for such an X creates a fundamental barrier to analyzing sums substantially shorter than the Lax range. It seems fair to say that this is where we are stuck on such problems today. The fine structure of the spectrum of the Laplacian has been attacked by many people and with varying approaches such as the heat equation, the Schrodinger equation, and the wave equation as above. In this connection Peter once remarked that it is surely better to throw light rather than heat or uncertainty on the matter. Indeed, today the wave equation offers the best approach to these problems. To end Section 1, let me mention what appears to be the unique joint paper by Lax and Nirenberg [L-N]. It concerns the sharp Garding inequality. Recall that in the theory of pseudo differential operators, if a(x,£) is a zeroth order nonnegative symbol, then Garding showed that for e > 0 there is K{e) such t h a t (12)
Re(Au, u) > -e\\u\\20 - K{e) \\u\\\/2
.
For some applications of this inequality one would like to take e = 0 and still have K finite. Such an inequality is known as a sharp Garding inequality and was established by Hormander [H3], Fefferman and Phong [F-P], and others. It is natural to ask whether such sharp inequalities hold for matrix symbols a(x,^). Indeed, such a result has applications to some numerical difference schemes, see [L-N]. Lax and Nirenberg show that there is a matrix version of the sharp inequality. Precisely if a(x,£) is nonnegative and Hermitian, then (12^ holds with e = 0 and a suitable K. It appears that in this matrix case their result is still the best known. 2. S c a t t e r i n g T h e o r y I turn now to Peter's joint papers with Ralph Phillips (who unfortunately could not be present on this special occasion). This collaboration is the only one that I can think of that rivals that of Hardy and Littlewood. They, Peter and Ralph,
A SAMPLE OF LAX'S CONTRIBUTIONS TO CLASSICAL ANALYSIS
173
that is, have developed a general time dependent approach to scattering theory and applied it in a number of settings. I discuss two cases, one of the exterior of an obstacle in R n and the other a hyperbolic manifold with ends. Let CI C K" be a compact smooth subdomain. Consider the wave equation (13)
u „ - Au = 0 n
on R \ $1 with u satisfying either Dirichlet boundary conditions, U\QQ = 0, or Neumann conditions dnu\an = 0. Scattering theory is concerned with the relation between the behavior of the solutions u(x, t) to (13) as t —* ±oo and in particular the effect of the obstacle on waves' which hit the obstacle. Now as Rellich showed that the spectrum of A on L2 (R n \Q) is continuous. So an analysis of the scattering via A requires analyzing the resolvent (A — A ) - 1 and its meromorphic continuation. This is one approach to scattering theory. The Lax-Phillips theory is time dependent and proceeds much more geometrically. For simplicity, assume that n is odd so that there is a Huygens principle (i.e., sharp propagation of signals). There are solutions u(x,t) which vanish for all x, for which d(x,Q.) < t, (t > 0). For positive times these do not interact with fi and are called outgoing solutions. Similarly, if u(x, t) vanishes for all x with d(x, t) < —t for t < 0, we call u incoming. Let U(t) be the unitary group which gives the solution operator to the wave equation (13) for all t. T h a t is for initial conditions, / = ( y 1.
where u(x,0) (15)
= f\ and u t ( x , 0) = fa- The energy E(f)
:= [
( ( / 0 £ + (h)2x)
dx
is preserved by the evolution operator U(t). Abstracting this set up (it is clear that Peter likes to abstract to a point that is elegant and insightful but without straying from the concrete — he seems to be able to find the perfect balance) we get the Lax-Phillips axioms [ L - P l ] : U(t) acts unitarily on a Hilbert space H and assume t h a t there are orthogonal subspaces £>_ and U+ of H such that !/(<)£>_ C D _ U(t) V+ c V+
n W
for for
(ii)
()U(t)V-
(iii)
\JU(t)V-=\JU(t)T>+ = H. t
= {0} =
t<0 t > 0. {)U(t)V+
t
n
For K \ f2 it is (iii) that is the most difficult to establish (with D_ and V+ the incoming and outgoing subspaces respectively). Indeed, establishing it is what put the theory in motion. Given U(t), X>_, X>+ as above, they prove a representation theorem (much like the usual spectral theorem) which asserts that there is a Hilbert space ./V and isometries / and O from H to L 2 (R, N) such t h a t (I)
7(D_) = L 2 ( R _ , A 0 ,
R- =
{x\x<0}
174
PETER SARNAK
(II)
0(V+) =
L2(R+,N)
(III)
U(t) is conjugated to translation by t.
In particular the diagram defines S : L2(R,N)
L2{R,N)
* S
-> L2(R,N)
and is called the
L2(R,N)
scattering operator. Clearly S commutes with translation and hence acts by convolution by a function denoted by S(t). Furthermore S is unitary and S(t) = 0 for t > 0 (causality). Its Fourier transform S(z) is a multiplication operator (giving the spectral representation of 5) and is holomorphic in $s(z) < 0. S(z) is called the scattering matrix. The abstract theory revolves around a semigroup and its infinitesimal generator. Let P+ and P_ be the projections which annihilate 23+ and 23_ respectively. For t > 0 set Z(t) := P+ U(t) P_ So Z{t) : K —• K where K = H 0 (23- © 23+). It satisfies ||Z(t)|| < 1 for t > 0 and forms a semigroup. Denote its infinitesimal generator by B. They show that B has discrete spectrum in C (in fact they prove that Z(\)(\Q — B ) - 1 is compact) and that its eigenvalues A* satisfy (16)
Re(Xk)<0.
The importance of B lies in the following fundamental result [L-Pl]: S(z) has a menomorphic continuation to C with poles at i\k, where {A*} is the spectrum of B. With this data (i.e., the poles) coming from M.n \ H they ask the fundamental question about the location of the poles of S(z) (or the eigenvalue of B) and its relation to the classical mechanics on R n \ U (i.e., rays ... ).. It is remarkable that Lax-Phillips are asking these questions about the influence of periodic or trapped rays on the poles, as early as 1962. This is long before such questions were addressed in connection with the eigenvalues of Laplacians by physicists such as Gutzwiller [GU] or mathematicians Chazarain [CH] and Colin de Verdiere [CVl]. The first result relating the geometry of K" \ fi and the poles is due to Lax, Morawetz, and Phillips [L-M-P]. It asserts that if SI is star shaped then there is an £o > 0 such that Re(Xk) < —£o for all k. A striking corollary to this result is that when SI is star shaped then locally solutions to (13) decay exponentially with a fixed rate as |£| —• oo. Such a result is ample proof of the power of the Lax-Phillips method. After this they were led to a fundamental conjecture. They say f2 is nontrapping, if there is a ball B D ft and an £ < oo such that any unit speed ray starting in
A SAMPLE OF LAX'S CONTRIBUTIONS TO CLASSICAL ANALYSIS
175
B \ Q and obeying the law of reflection on dCl, will remain in B \ CI for a time of at most L If 0, does not satisfy the above then it is called trapping. The conjecture mentioned above (see [ L - P l ] , 1967) asserts that a) If O is nontrapping then Z(t) is eventually compact (which then has many implications on the poles of B). b) If Q. is trapping then \\Z{t)\\ = 1 for all t > 0. The above far-reaching insight led to lovely results by a number of different authors. Today, the conjecture is essentially settled. In 1979, Ralston [RA] showed that if fl has a trapped ray (and he makes a certain technical assumption about the order of tangency not being infinite) then (b) holds. The proof of (a) had to await developments in microlocal analysis. Its proof by Melrose [Ml] is one of the major results in scattering theory. The study of the distribution of the poles was of course also initiated by Lax and Phillips. For example, they show that if Z(to) is compact for some to then for some b > 0 Re(Xk) < a b\og\Q{\k)\. In connection with the behavior of the continuous spectrum, Majda and Ralston [M-R] developed an analogue of the Weyl law for the winding number of the scattering matrix determinant, which gives the asymptotics of J_^ log det S(a) da, as A —» oo. Other results on the distribution of poles have been established especially by Melrose [M2] and Zworski [Z]. This is still a very active area of research. The second setting in which I describe the Lax-Phillips scattering theory, is for hyperbolic manifolds. T h a t their approach is well suited to this setting was first noted by Faddev and Pavlov [FA-P]. Let H n + 1 be hyperbolic n + 1 dimensional space, that is {{y,x)\y > 0, x G R™} equipped with the line element (17)
d
ds> =
-^±^ . y2 Let A be the corresponding Laplacian. Also let T be a discrete subgroup of the isometry group of H n + 1 ( = 0^(n + 1,1)) and let Xr = r \ H n + 1 be the corresponding quotient. Xr is a hyperbolic manifold and we let A r denote the corresponding Laplacian on L2(Xr)Lax and Phillip's work is concerned with the case that Xp is geometrically finite. T h a t is to say that there is a fundamental domain for Xr in H n + 1 bounded by a finite number of geodesic hyperplanes. For scattering to take place we should have either: (A) Xp has finite volume (but not compact). (B) Xr has infinite volume. The study of the spectrum of A r in the case (A) was pioneered by Selberg in the 50's [SE]. His two major achievements were the analytic continuation of the Eisenstein series (which furnish the continuous spectrum for X r ) and the celebrated trace formula. In this finite volume case, Xr looks like a compact piece K together with a finite (disjoint) union of cusps C i , . . . ,CT. The Lax-Phillips axioms may be modified to apply to the wave equation on Xr'u\
=
Kut)t
(18)
where
Lu
=
[0 \L (
A
I\
(u 0)\ut/
+ (?)
)
259 176
P E T E R SARNAK
FIGURE 1.
The shift by (^) ensures a Huygens principle in odd dimensions but it also renders the corresponding energy form to be indefinite. However, if one projects out the finite number of positive eigenvalues, then the energy form is nonnegative. There may also be an infinite dimensional space of bound states (i.e., square integrable eigenfunctions) within the continuum. The scattering theory takes place on the orthogonal compliment of all bound states. The spaces X>_ and T>+ correspond to special functions supported in the cusps. Examples of members of V+ are solutions to the wave equation initially supported in C\ which are functions of y alone and which move out into the cusp as t increases (t > 0). The scattering process corresponds to what comes back into the cusps when sending a wave into the compact part. The theory is modified to take these new features into account. One defines a corresponding semigroup with infinitesimal generator B. Again the eigenvalues Afc of the operator B (whose resolvent is compact) for which Re(Xk) < 0, correspond to poles of the scattering operator S by the relation Afc —»iX).. These poles can also be identified with poles of the Eisenstein series. In fact, their set up yields a natural and elegant new proof of the analytic continuation of the Eisenstein series [L-P2]. A variation of their proof using their "cut-off Laplacian" was given by Colin de Verdiere [CV2]. His proof was exploited very profitably in Mueller's proof of the trace class conjecture for higher rank symmetric spaces [MU]. The Lax-Phillips set up also leads very naturally to a derivation of the trace formula [L-P2]. One might ask if their approach yields further insights or techniques not offered by Selberg's method. There is at least one fundamental problem for which (at the present time) their approach is indispensable. This concerns deformations of Xr or of T. The spectrum of B in C identifies both the poles of the Eisenstein series and the discrete spectrum of L. It allows one to study the behavior of these quantities as we deform Y along a curve Tt in the Teichmuller space of such T's. This was used crucially by Phillips and myself [P-S] in our work which showed that the bound states are very fragile and are probably present in abundance only for arithmetical T's. In this connection I would like to acknowledge Peter, who pointed us to the advantage of using real analytic curves in the parameter t. We turn to the case (B) above when Xp has infinite volume. In this case, before their work, much less was known even about the nature of the spectrum of
A SAMPLE OF LAX'S CONTRIBUTIONS TO CLASSICAL ANALYSIS
177
Xr- Patterson [PA] had some definitive results but only for n = 1 (i.e., Xr of dimension 2). The two basic results which Lax and Phillips [L-P3] established are that the spectrum of A r in [ ( f ) 2 , oc) is continuous (i.e., there are no bound states with eigenvalues in this range) and that in [0, ( f ) 2 ) the spectrum of Ap consists of a finite number of points. The proof of there being no bound states in [ ( f ) 2 , oo) is based on a clever use of Volterra integral equations to show that there are no such square-integrable eigenfunctions in any sector of H n + 1 which contains a piece of dW1+x (note that infinite volume, geometrically finite Xp's contain such a piece). Using the non-Euclidean Radon transform Lax and Phillips go on to develop a translation representation for the wave operator on an Xr of infinite volume. This in turn leads to a Fourier decomposition of the general function of Xr- Their analysis includes such Xp's which have cusps of intermediate ranks. A development of a complete trace formula for such Xp's remains an open problem, though Patterson and Perry [P-P] develop properties of the Selberg Zeta Function for such X r , which are closely related. Lax and Phillips have applied their analysis of the wave equation on Xr to the problem of counting lattice points in H n + 1 [L-P3]. Their method is simple and effective and it yields the sharpest results known. Precisely let T be as above. For z,w e Mn+1 and R>0 let N(z, w, R) = # { 7 £ T| din, z, w) < R} where d(z, w) is the non-Euclidean distance from z to w. The problem is to determine the asymptotics of N as R —> oo. Let bo = ^ / ( f ) 2 — Ao where Ao is the smallest eigenvalue of Ap. For simplicity, we assume t h a t 0 < Ao < ( f ) 2 with corresponding eigenfunction >o(z) and that there are no other eigenvalues of Ap in [0, ( f ) 2 ) (otherwise these enter in a simple way). Their result asserts that Nr(z,w,R)
andc(A0)
=
=
c(\0)
^ ( ^ , ( ^ 0 - 1 ) !
This result is the best known even when Xr is compact (in which case Ao = 0), when the main term in the asymptotics goes back to Delsarte [DE] and the above sharp remainder term is due to Selberg. To end let me mention a couple of papers which do not strictly fall under the title of this lecture. These are papers by Peter — on sums of squares. In [L-L] he and Anneli Lax show that F{x\,... ,15) = J2i=\ YljM(xi ~~ xj)' which while 5 being nonnegative for x £ R is nevertheless not a sum of squares of polynomials in x. Examples of such polynomials were already found in Hilbert [HI] disproving a conjecture of Minkowski. However the above is a very pleasing example which, by the way, arose in a mathematical olympiad where it was asked to show that F(x\,X2,- • • ,£5) > 0. Recently in [L5] (between the time of the Florence conference and the writing of this lecture) Peter gave a beautiful proof of a result of Ilyushechkin [IL], that the discriminant D(a,ij) = rL<j(^> ~ ^ J ) 2 °f a r e a ' symmetric matrix A = ( a ^ ) (where A ; . . . An are its eigenvalues) which is plainly nonnegative, is a sum of squares of polynomials in a , j . He further discusses the
178
P E T E R SARNAK
interesting question as to the minimal such representation. Clearly the master has not lost his touch! References [BE]
P. H. B e r a r d , On the wave equation on a compact Riemannian manifold without conjugate points, M a t h Z. 1 5 5 (1977), p p . 2 4 9 - 2 7 6 . [CH] J. C h a z a r a i n , Formules de Poisson pour les varietes riemannienes, Inent. M a t h . 2 4 (1974), p p . 6 5 - 8 2 . [C-L] R. C o u r a n t a n d P. Lax, The propagation of discontinuities in wave motion, P r o c . N a t . Acad. Sci. U.S.A. 4 2 (1956), p p . 8 7 2 - 8 7 6 . [CV1] Y. Colin-de-Verdiere, Sur le spectre des operateurs elliptiques a bicharacteristique toutes periodiques, C o m m e n t . M a t h . Helv. 5 4 (1979), p p . 5 0 8 - 5 2 2 . [CV2] Y. Colin-de-Verdiere, Une novelle demonstration du prolongement meromorphe des series de Eisenstein, C. R. A c a d . Sci. P a r i s Serie I, 2 9 3 (1981) p p . 3 6 1 - 3 6 3 . [DE] J. Delsarte, Sur le gitter fuchsien, C. R. A c a d . Sci. P a r i s 2 1 4 (1942), p p . 147-149. [D-G] H. D u i s t e r m a a t a n d V. Guillemin, The spectrum of positive elliptic operators and periodic ^characteristics, Invent. M a t h . 2 9 (1975), p p . 3 9 - 7 9 . [FA-P] L. Faddeev a n d B . Pavlov, Scattering theory and automorphic functions, P r o c . Steklov Inst. M a t h . 2 7 (1972), p p . 1 6 1 - 1 9 3 . [F-P] C. Fefferman a n d D. P h o n g , The uncertainty principle and sharp Garding inequalities, C o m m . P u r e A p p l . M a t h . 3 4 (1981), p p . 2 8 5 - 3 3 1 . [GU] M. Gutzwiller, Phase integral approximation in momentum space and the bound states of an atom, J. M a t h e m a t i c a l P h y s . 8 (1967), p p . 1 9 7 9 - 2 0 0 0 . [G-V-Z] M. G i a n n o n i , A. Voroz, a n d J. Z i n n - J u s t i n , eds., Chaos and Quantum Physics, Les Houches, 1991. [HA] J. H a d a m a r d , Le Probleme de Cauchy, H e r m a n n , P a r i s , 1932. [HI] D. Hilbert, Uber die Darstellung definiter Formen als Summen von Quadraten, Math. A n n . 3 2 (1888), p p . 3 4 2 - 3 5 0 . [HI] L. H o r m a n d e r , Fourier intergral operators I, A c t a M a t h . 1 2 7 (1971), p p . 7 9 - 1 8 3 . [H2] L. H o r m a n d e r , The specturm of a positive elliptic operator, A c t a M a t h . 1 2 1 (1968), p p . 193-218. [H3] L. H o r m a n d e r , Pseudo-differential operators and non-elliptic boundary problems, Ann. of M a t h . 8 3 (1966), 129-209. [IL] N. V. Ilyushechkin, The discriminant of the characteristic polynomial of a normal matrix, M a t . Z a m e t k i 5 1 (1992), p p . 1 6 - 2 3 . [LI] P. Lax, Proof of a conjecture of P. Erdbs on the derivative of a polynomial, Bull. Amer. M a t h . Soc. 5 0 (1944), p p . 5 0 9 - 5 1 3 . [L2] P. Lax, A Phragmen-Lindelof theorem in harmonic analysis and its application to some questions in the theory of elliptic equations, C o m m . P u r e Appl. M a t h . 1 0 (1957), p p . 361-389. [L3] P. Lax, Translation invariant spaces, A c t a M a t h . 1 0 1 (1959), p p . 163-178. [L4] P. Lax, Asymptotic solutions of oscillatory initial value problems, Duke M a t h . J. 2 4 (1957), p p . 6 2 7 - 6 4 6 . [L5] P. Lax, On the discriminant of real symmetric matrices, p r e p r i n t , 1997. [L-M-P] P. Lax, C. M o r a w e t z , a n d R. Phillips, Exponential decay of solutions of the wave equation in the exterior of a star-shaped obstacle, C o m m . P u r e A p p l . M a t h . 1 6 (1963), p p . 4 7 7 486. [L-Pl] P. Lax a n d R. Phillips, Scattering Theory, 2nd ed., A c a d e m i c P r e s s , San Diego, 1989. [L-P2] P. L a x a n d R. Phillips, Scattering Theory for Automorphic Functions, A n n a l s of M a t h e m a t i c a l Studies, No. 8 7 , P r i n c e t o n , 1976. [L-P3] P. L a x a n d R. Phillips, The asymptotic distribtuion of lattice points in Euclidian and [L-L] [L-N] [Ml]
non-Euclidian spaces, J. F u n c t . A n a l . 4 6 (1982) p p . 2 8 0 - 3 5 0 . A. Lax a n d P. Lax, On sums of squares, Linear A l g e b r a a n d A p p l . 2 0 (1978), p p . 7 1 - 7 5 . P. Lax a n d L. N i r e n b e r g , On stability for difference schemes; A sharp form of Garding's inequality, C o m m . P u r e Appl. Math- 1 9 (1966), p p . 4 7 3 - 4 9 2 . R. Melrose, Singularities and energy decay in acousticle scattering, Duke M a t h . J. 4 6 (1979), p p . 4 3 - 5 9 .
A SAMPLE OF LAX'S CONTRIBUTIONS TO CLASSICAL ANALYSIS [M2] [MU] [M-R] [PA] [P-P] [P-S] [RA] [RI] [SA] [SE] [S-S] [Z]
179
R. Melrose, Polynomial bound on the number of scattering poles, J. Funt. Anal. 53 (1983), pp. 287-303. W. Muller, The trace class conjecture in the theory of automorphic forms, Ann. of Math. 130 (1989), pp. 473-529. A. Majda and J. Ralston, An analogue of Weyl's formula for unbounded domains, Duke Math. J. 45 (1978), pp. 183-196. S. Patterson, The Laplace operator on a Riemann surface I, Compositio Math. 31 (1975), pp. 83-107. S. Patterson and P. Perry, Divisor of the Selberg zeta function for Kleinian groups, I, preprint, 1997. R. Phillips and P. Sarnak, Perturbation theory for the Laplacian on automorphic functions, J. Amer. Math. Soc. 5, (1992), pp. 1-32. J. Ralston, Solution of the luoue equation with localized energy, Comm. Pure Appl. Math. 22 (1969). pp. 807-823. M. Riesz, L'integrale de Riemann-Liouville et le probleme de Cauchy, Acta Math. 81 (1949), 1-223. P. Sarnak, Arithmetic quantum chaos, The Schur Lectures, Israel Math. Conf. Proc. Vol. 8, Ramat Gan, 1995, pp. 183-236. A. Selberg, Collected Works, Vol. 1, Springer-Verlag, 1988. A. Seeger and C. D. Sogge, Bounds for eigenfunctions of differential operators, Indiana Univ. Math. J. 38 (1989), pp. 669-682. M. Zworski, Sharp polynomial bounds on the number of scattering poles, Duke Math. J. 59 (1989), pp. 311-323.
PRINCETON UNIVERSITY, PRINCETON, NJ 08544
E-mail address: sarnakOmath.princeton.edu
Wolf Prize in Mathematics, Vol. 2 (pp. 263-310) eds. S. S. Cfaem and F. Hirzebruch © 2001 World Scientific Publishing Co.
I»
Courtesy of Mrs Helen Lewy
264 HANS LEWY
1. Curriculum Vitae Mathematicae by David Kinderlehrer
Hans Lewy was born on October 20, 1904 in Breslau, Germany (now Wroclaw, Poland), the first son and second of three children of Max and Greta Lewy. As a child he was drawn to science and to music. Gottingen, in 1922 when Hans Lewy matriculated, was among the premier mathematical establishments in the world. With Klein and Hilbert still in residence, as well as Runge, Prandtl, Landau, Herglotz and Emmy Noether, Courant, of course, and many others it radiated irresistable scientific excitement. We are told that Lewy's gymnasium professor suggested to his father that the young man should travel to Gottingen. In this the professor was not only prescient but, perhaps, well informed. Not only Courant, but also Born, Hellinger and Toeplitz had come to Gottingen via Breslau. Friedrichs also arrived in 1922 and their life-long friendship took hold immediately. Unfortunately, 1922 marked the beginning of the Weimar hyper-inflation period, which ended at the end of 1923. The situation led to considerable conflict between Lewy and his family, who understandably wanted the eighteen-year-old to return to the security of his home, and eventually led to a temporary rupture of relations. In this period he took a railroad job, doing maintenance on the tracks and ties; neither his personality nor his physique made this easy for him. Courant wrote several times to Lewy's parents expressing great satisfaction and extolling the achievements of his student, perhaps in part to relieve this family pressure. He completed his thesis in 1926 with Courant and became, together with Friedrichs, Courant's Assistant and a Privatdozent. In the next few years he, and Friedrichs, pursued new methods for understanding elliptic and hyperbolic problems. He solved the initial value problem for general nonlinear hyperbolic equations in two independent variables [5]. He used this to give a new proof of the analyticity of solutions of elliptic equations in two variables which far exceeded the known proof in its elegance and simplicity, [9]. He and Friedrichs developed energy methods for wave equations. Most well known, in 1928, the joint work of Courant, Friedrichs, and Lewy [7] was published. About this, Lax has written, the onset of the Second World War brought an unprecedented (and unabated) pressure on the technological centers to provide numerical solutions of partial differential equations; here the ideas introduced by C-F-L; the famous stability condition, energy inequalities, and the leap frog difference method, turned out to be basic. This is an outstanding example of research undertaken for purely theoretical purposes turning out to be of immense practical importance. Lewy's work attracted the attention of Hadamard who included a long special appendix about it in the second edition, which was the later To be published in Collected Works of Hans Lewy (Birkhauser).
265 French edition, of his book. 1 To paraphrase Leray, he opened a world to us whose richness continues and will continue to amaze us. These years in Gottingen also marked the period of the "boy physics". In 1929-1930, under the sponsorship of the Rockefellar Foundation, he studied in Rome where he was especially influenced by Tullio Levi-Civita. He also met many younger Italian mathematicians, and in particular Beniamino Segre who would later have an influence on his work. The fellowship then took him to Paris, 1930-31, where he was active in Hadamard's seminar. Lewy left Germany quite soon after Hitler assumed power in 1933. He went first to Italy and then to Paris where Hadamard had managed to obtain for him a year's support. At this time, Lewy introduced Jean Leray and Juliusz Schauder. Later, during the war when Leray was taken prisoner, Lewy communicated with him via the Red Cross, and after the war the families remained quite close. In Paris, on the recommendation of Hadamard, he was offered a one to two year position at Brown, funded by the Duggan Foundation. In the fall of 1933, Lewy was in Providence, where he completed his first work on the Monge-Ampere Equation [14, 16, 17]. On the invitation of G.C. Evans, Lewy went to Berkeley in 1935. Courant, visiting Berkeley in 1932, had spoken enthusiastically of Lewy's work on that occasion. Courant himself was, in fact, offered a position at Berkeley. Evans arrived in Berkeley in 1934, chosen to build up the department. C.B. Morrey arrived in 1933 and Lewy in 1935. A.P. Morse, Jerzy Neyman and Raphael Robinson came at the end of the 1930s. Lewy's work on Minkowski's Problem was completed here [18, 19]. Erhard Heinz remarked that, "His estimates are of local type. It is remarkable that he did not make use of the Weyl estimates. Thus he went far beyond the original goal." Although the events in Europe suggest that it was natural for America to welcome scientists from the fascist countries, it is well to keep in mind that the country remained in the Depression and that there was considerable concern, by all parties, that emigres might be assuming positions that could be held by natives and that this could lead to a negative environment. With the outbreak of the war, Lewy took flying lessons and obtained a solo license in the hope of offering his services, but was soon called to Aberdeen Proving Grounds as part of the University of California contingent. He also worked halftime with the Office of Naval Research in New York. Here he became interested in water waves and the Dock Problem, resuming his collaboration with Friedrichs, [24]. The explicit solution in [23] also gave rise to a proof of the Law of Quadratic Reciprocity. At the war's conclusion, Lewy, then a Full Professor, was invited to return to Gottingen as Privatdozent! In 1947 he and Helen Crosby, an artist, writer and translator, were married. Their honeymoon included a trip around the world, beginning with a return to Europe. Also included was a two-month stay in Chengtu, Szechuan (Chengdu, Sichuan) where Lewy gave a course on water waves and a month visiting other 1
Le probleme de Cauchy et les equations aux de.rive.es partielles lineaires hyperboliques. & Cie, Paris 1932.
Hermann
266
institutions in China. Still under Nationalist rule, China was rarely visited by westerners in these years. The analytic continuation of minimal surfaces bounded by analytic arcs or satisfying free boundary conditions was given in the early 1950s, [28, 29]. These and related results had a formative impact on the study of free boundary and regularity problems in variational principles. Also in this decade he devoted considerable attention to the issue of the smoothness of solutions of equations and systems at the boundary and their asymptotic development near change of regime boundary points [26, 27, 32]. During the Loyalty Oath controversy in the state of California, Lewy was part of the group dismissed for refusal to sign the oath. Having seen Fascism at first hand in Italy, and watched its rise in Germany, he was wary of cooperating with any totalitarian tendencies in his new country. The details of those events, insofar as he is involved, are chronicled in Constance Reid's article. 2 He was on the faculty at Harvard in the fall of 1952 and then at Stanford in 1952 and 1953. This was the genesis of the work on Riabouchinsky flow with Garabedian and Shiffer, [33]. In the settlement of the Loyalty Oath dispute by the courts, the professors were reinstated and Lewy returned to Berkeley. In 1955, the Lewys' son Michael was born. Lewy's completely unexpected example of a linear equation with no solution, and with it a homogeneous equation with only the trivial zero solution, appeared in 1957, [38], one of the major events in the modern development of PDE theory. This example arose in the context of Lewy's study of Cauchy-Riemann structures in several complex variables and pseudoconvex domains, [36, 37, 43]. Both simple and general, it illustrates many different aspects of differential equations. We teach, for example, that real first order linear equations admit solutions by the elementary method of characteristics. But complex ones may not admit any solutions. It put an end to speculation about finding a version of the fundamental solution for a general linear equation, even with analytic coefficients, although this had recently been established for equations with constant coefficients. He received the Steele Prize of the American Mathematical Society for this work in 1979. In 1959-1960 Lewy accepted an invitation from the Scuola Normale Superiore and the University of Pisa. In these years, Pisa was achieving stature as a vigorous world center with Andreotti, De Giorgi, and Stampacchia among its leaders. Here, Lewy helped creating the emerging area of variational inequalities with Stampacchia. Their work was fundamental to the growth of the subject, introducing new types of free boundary problems. Lewy published in this area through the 1980s. In 1969-1970, he returned to Rome on the invitation of the Accademia dei Lincei. He was elected a Foreign Member in 1972. He also retired in 1972, continuing his research with undiminished vitality. On retirement he insisted that available funding should be dedicated to the support of younger scientists and declined to apply for any more grants. In 1973 he joined Ahlfors as one of the first two Ordway Professors at the University of 2
Hans Lewy, 1904-1988, Miscellanea mathematica (P. Hilton, F. Hirzebruch, R. Remmert, editors) Heidelberg, Springer, 1991.
267 Minnesota. Always attentive to the encouragement of young people, he returned to Minneapolis to participate in the 1984/1985 Institute for Mathematics and its Applications Program on Continuum Physics and Partial Differential Equations. In 1985 in Trento there was a meeting in his honor. He shared the 1984/1985 Wolf Prize with K. Kodaira for initiating many, now classic and essential, developments in partial differential equations. Subsequently he wrote frequently to the Wolf Foundation in support of other scientists, especially De Giorgi, who received the 1990 prize. He was awarded an honorary doctorate from Bonn University in 1986. In his acceptance speech Krisen in der Mathematik he spoke, characteristically, about the future. Hans Lewy died on August 23, 1988 in Berkeley.- He and Helen had recently returned from a trip to Europe where he delivered his last paper, [74], in honor of Ennio De Giorgi. There are many scientists whose objective is to bring order to chaos, to organize a disparate subject and bring it unity of theme and method. In an arcane way, Lewy was their antithesis: with a profoundly innovative point of view, he illustrated and revealed to us fundamental principles and ideas with the simplest of tools. Hans Lewy was a pillar of science in the twentieth century. His work touched nearly every significant area of analysis and endures in the canon that we pass on to our children.
268
HANS LEWY (gezeichnet von Helen Lewy)
269 2. The Music in Hans Lewy's Life by Helen Lewy
Hans Lewy always shunned the intrusion of the personal into his professional life. His private life was his own domain. Yet he was an eminently social being with so many friends and colleagues across Europe and Asia, and in the U.S., that a great deal about his personal life, - his work habits, his personality, his roots, his friends and "hang-outs" - is already well known. During his extended stays in Europe, especially in Italy, it was his custom to spend a great deal of his time socially, as well as mathematically (if the two can be separated) with the students. Like all scholars, he was always energized by the marvelous curiosity of the young. I have been asked to write a few words about Hans Lewy for this volume. His family and I are, for the most part, content to let his legacy rest on his mathematical works, for that's where it should be. I will not try to evoke the man behind the math. But, I do not feel he would object, if, to honor the students he loved to be with, I pass along some facts about the music which was always the companion of the math in his life. The combination of mathematics and music, while not uncommon, especially in the Europe of Hans Lewy's time, was, in his case, serious enough to force him to make, in his teens, a difficult career choice. He played the violin, the viola and the piano with mastery, and, at times in his life also the clarinet and the cello. He composed many string quartets, which unfortunately were lost, and at least one string trio (this was found by Lori Courant after the death of her mother; it was composed when Hans Lewy was 20 years old, and all three parts had been laboriously hand-copied by Nina Courant (this was long before photocopies - or even mimeographs). Over a period of years during the 1990s, Lori Courant, with fellow musicians, in a labor of love, edited, corrected, copied and brought to life the faded sheets of the score). According to his mother, Hans Lewy's love for music began with the piano lessons of his older sister when she was about 10 years old. At that particular time, she had little interest in learning to play the piano, but her little brother, five years her junior, did; he managed to be present during the lessons and soon he was outperforming his sister. This situation led to so much "mischief in the family", according to his mother, that a music teacher for little Hans was promptly acquired, for lessons not on the piano, but on the violin. Hans took to his lessons with passion, ("ferocity", was the word his mother used): he soon had outgrown his first two teachers, and, at length, as a pre-teen he was studying under a well-known concert violinist. By then the violin had become the mode through which Hans' native ebullience and independence found expression. He told me, sixty years later, of his joy in playing even as a child (but added that he had not always enjoyed "being produced" for family occasions). By the time he reached his teens, he was already playing regularly with chamber-music groups. To be published in Collected Works of Hans Lewy (Birkhauser).
270 His parents would not permit the unusual skills of their son to be exploited, until, under pressure from his teacher, they agreed when he was 16 to allow him to perform in public, for one time only: the concert took place in Bautzen in July 1920. One can sense the excitement of the young man in this partial translation of the letter he wrote to his parents two days later: My dear ones: Saturday was THE day: an orchestra, a male chorus, a woman soloist and me. I played the Chaconne by Bach and Devil's Trills by Tartini, accompanied by a very nice elderly gentleman, Prof. Crusl. During the first piece I still felt a little tight, but the second I could hardly wait to start playing. The reviews are here today; I quote them verbatim: Bautzner Tageblatt "The talented young Hans Lewy did the ... "Devils Trills" sonata by Tartini. The youthful musician amazed the audience with his splendid technique. Doublestopping, octaves and staccato don't seem to give him any trouble. That he also has a mature understanding of the character of the music was apparent when he played Bach's "Chaconne" for violin only. The audience thanked him with thundering applause." Bautzner Nachrichten "The youthful violinist, Hans Lewy, from Breslau, a very promising high school student there, played the "Chaconne" by Bach and the "Devils Trills" sonata by Tartini with admirable precision and technical expertise." Actually the teenager was allowed another performance before the public; it took place also in Bautzen, in October of the same year. This time he was soloist for Mozart's D-sharp violin concerto. The reviews were, to say the least, enthusiastic. Here are excerpts, in translation, from two of the reviews: Bautzner Tageblatt "The spirit and solid technique so important in Mozart's works were present. The bowing is light and elegant, and, on the other hand, does not lack energy. The left hand is excellently schooled and establishes full success in passage-work in all ranges. Nevertheless, Mozart's splendid work was interpreted with classic poise and superior precision, and the youthful violinist received rich applause that came from the heart." The Bautzner Nachrichten review hailed him as a "Virtuoso " and "Wunderkind with true Mozartian charm" and predicted for him a great future as a musician. After these successes, the teenaged Lewy, of course, had to consider a career as a professional musician. The choice would have been easy if it had not been for that other passion, mathematics. His father favored mathematics; his mother was not sure; and he, of course, wanted both. In the end, mathematics won, and he soon left home for his studies in Gottingen. There he played with the city orchestra and, during all his years as student and Privatdozent, participated in weekly musical evenings at the home of Helene
271 Wintgen, the wife of a professor of Chemistry, and a friend and contemporary of Nina Courant. Frau Wintgen, as an octogenarian, described for me her first meeting with my husband. The quartet needed "a violin" and a newly arrived student had been recommended. It was raining heavily that evening when the new student knocked on her door, and "there he was", she said, smiling, "the 17-year-old Hans, standing there forlorn and drenched: he wore no hat, no raincoat, and carried no umbrella but the violin was carefully wrapped in a towel!" After his move to the U.S., Lewy's instrument of preference gradually became the piano. (It is my impression that the making of "salon music" in the Gottingen of Lewy's time, was very informal-improvisational, rollicking, if you will. In those days, the teaching of music in Germany placed more emphasis on sight-reading than was customary in the U.S., and perhaps this had something to do with the "freewheeling" character of the musical evenings in Gottingen described by my husband . . . . At any rate, from the stories that reached my ears, it was evident that the players in his circle enjoyed lustily, not only the challenges and the riches of the music - but also each others' company and wit - and, at times, even some nonsense; I remember my husband telling of one "session" that included a very accomplished Russian violist, and a good friend. They were about to begin playing, when his Russian friend put down his bow and said solemnly: "Let's try it first without sharps and flats!") But had mathematics won? That is, had music lost? My husband said later in his life that he had never regretted his decision. He made music almost every day of his life, (exceptions were extended foreign stays when no piano was at hand). But his seminal decision had made it possible for him to make music, and live, as a free spirit, - free of the constraints of a concert performer's life, especially, he told me more than once, the constant travelling, and the uprooted life style . . . Mathematics and music remained the twin passions of his life, and, in the end, I believe he felt he had been lucky enough to have had the best of both worlds.
272
3. Crises in Mathematics by Hans Lewy
Your Magnificence, Colleagues of the Science Faculty, Ladies and Gentlemen, I am very grateful to you for the honor you have conferred to me. The title of my lecture is "Crises in Mathematics". I will not define the word "crises"; I hope to make clear what is the point in question. Our century has seen scientific discoveries in abundance, and in many cases mathematics provided essential assistance. Often these discoveries have overthrown older notions and conceptions and were replaced by new notions, expressing them in a mathematical way in order to formulate them as incontestable as possible. For, one may think, mathematics expresses eternal truths since it only states the logical consequences of a few principles, and these principles or axioms are of such simplicity and immediacy that to deny them would only lead to fruitless sophistry. But is mathematics truly always the "resting pole admidst the whirl of phenomena", or have there been bitter differences of opinion also in mathematics? Do they exist even today? Are there, possibly unspoken, opinions underlying our mathematical thinking, views which cannot be defended by logic alone? The first attempt towards an axiomatization of mathematics goes back to the ancient Greeks and is presented in Euclid's books, which were thought to be basic for almost two thousand years. Why? It is known that many mathematicians of the previous generations had studied in Babylon where they had learned their algebra. In Babylon one considered as numbers the so-called natural numbers; 1, 2, 3, etc., and the fractions of these as quantities which we nowadays denote as positive rational numbers. But then came the discovery that there are geometric quantities which are not fractions of integers. Namely, in a rectangular triangle with two sides of length one, the third side has a length which is not a fraction of integers. For this fact we say today that the square-root of two is irrational. How exciting this discovery must have been for the natural philosophers of that time who had learned to consider the natural numbers as a very successful foundation for all contemporary descriptions of nature! The echo of this crisis is reflected in Plato's utterance that a human being who is not shaken in his heart by learning that the square-root of two is not a rational number resembles a stupid beast. And undoubtedly the geometric expression of all axioms in Euclid's books can be attributed to the desire to give these axioms a generality which could not be achieved by the algebra of the time. Since Descartes, the modern conception takes the opposite way and reduces all geometric notions to algebraic ones. But the endeavor to formulate a sufficiently general concept of quantities algebraically requires the notion of the so-called actual infinity, thereby generating a deeper rooted crisis the offshoots of which have had an influence up to the recent past. Today we learn in school that each quantity is an infinite decimal fraction, English translation of Hans Lewy's lecture Krisen in der Mathematik at Bonn University on 28 November 1986 when Hans Lewy was awarded an honorary doctorate by the Science Faculty.
273 say 1.2357 . . . ad infinitum. This concept of quantities requires the specification of infinitely many numbers from 0 to 9. Such a definition would have been unbearable for Euclid. If he speaks of the infinity of the set of all prime numbers, that is, of all those numbers having no divisors except themselves and the unit, he expresses and proves this by: There exist more prime numbers than in any given set. The need to include infinity into mathematics became of immediate interest only with the invention of the infinitesimal calculus. The success of the infinitesimal calculus and its overwhelming fertility in solving old problems and creating new questions led in the 18th century to the well-known slogan: "En avant, toujours en avant, et la foi vous viendra." (Literally translated: "Forwards, always forwards, and belief will follow.") But was it really easy to believe? For example, the infinite series 1 — 1/2 + 1/3 — 1/4 + . . . converges to natural logarithm of 2. However, if one orders the terms of this series in a different way, then one can obtain any positive or negative number as sum, or we can let the series diverge, just as we wish. This example must have made a devastating impression upon the contemporaries. In fact, the foundation of the infinitesimal calculus was, even in the 19th century, a topic of vivid discussion, and this not only in mathematical circles. So for instance Karl Marx wrote an essay about the foundation of the infinitesimal calculus, and his concept of this field has been taught in the schools of the Soviet Union and of China up to our time. I myself witnessed the lecture of a Russian mathematician entitled "Marx's foundation of the calculus" held in the course of the International Congress of Mathematicians in Zurich, 1932. The inclusion of this lecture of 45 minutes in the program of the meeting was a precondition for the participation of the Russian colleagues in the congress. The speaker characterized Weierstrass's e — (5-definition of convergence as a static and bourgeois concept, whereas Marx's definition was seen to be something fluent and vivid. Weierstrass's concept of convergence rested on preparatory work by Cauchy and Abel and allowed a clarification of all seeming contradictions of the infinitesimal calculus. But this long desired goal had scarcely been attained when in the secondhalf of the 19th century there arose a completely new conception of infinity in the set theory of Georg Cantor. Cantor introduced the notion of denumerability of a set and showed that there are infinite sets which are not denumberable, i.e. sets where it is impossible to furnish each of its elements with a natural number in such a way that two different elements obtain different numbers. The simplest example is the set of all decimals between zero and one. For us, having grown up with the notions of denumerability and nondenumerability of infinite sets, it is hard to understand any longer why so many outstanding mathematicians of those days fought with such hostility against these new concepts. Weierstrass's point of view is interesting: The famous Gottingen mathematician David Hilbert, at that time a young man, visited Weierstrass in Berlin and asked him for his opinion about the new set theory. Weierstrass told him that he would accept this theory as soon as it would be able to prove the existence of transcendental numbers, that is, the existence of numbers which do not satisfy any algebraic equation with integers as coefficients. However, this was proved by Cantor in the following year, actually by verifying a stronger result:
274 The set of transcendental numbers is not only infinite, but even nondenumerable. Further successful applications of the set theory to other branches of mathematics which previously seemed to have nothing to do with it, secured a firm place for set theory in mathematics. But with the new attitude against the actual infinity new clouds arose in the firmament of mathematics. Certain paradoxes such as the set of all sets containing themselves showed that even the axioms of logic have to be revised if infinite sets are taken into consideration. In particular the so-called axiom of choice by Zermelo in its unrestricted application to infinite sets made many mathematicians feel ill at ease. A radical cure for all doubts was suggested by so-called intuitionism. Essentially, intuitionism expelled all statements from mathematics containing notions which cannot be defined in a finite number of steps. Particularly, for intuitionists the principle of the tertium non datur is not applicable to infinite sets. For example, let us take the statement: In the decimal representation of it = 3.1415... there is an uninterrupted sequence of ten Nines. The intuitionist says: As long as there is no finite proof of this statement or of its reverse, one is not allowed to say that this statement be either correct or false. This was the example cited by the well-known topologist L.E.J. Brouwer in a great lecture at Gottingen, about 1930. According to Brouwer most of the mathematics taught at the time was false. In the discussion following his lecture he was asked as to whether this statement should also be extended to his own papers that had made him famous. Yes, he answered, it's all false. The majority of mathematicians rejected intuitionism, mainly for the reason that intuitionism was not only extensively restricting the area of allowed questions, but would also make all proofs uncommonly complicated, even in those cases where it led to the same results as ordinary mathematics. Still one has to admit that the problem of consistency within mathematics is, in the broadest sense, unsolved. Intuitionists would cripple mathematics in order to make sure that there is no contradiction. In any case, logicians had step by step erected a building of mathematics, beginning with the natural numbers, when in the thirties Goedel's discovery struck like a lightening bolt that, expressed briefly, for any given finite system of axioms there is always an assertion formulable by means of the notions of this system whose correctness cannot be decided by these axioms alone, and also that the consistency cannot be proved with this system of axioms alone. Now I turn to another crisis in which present-day mathematics finds itself. We older mathematicians and the majority of our younger colleagues believe it to be absolutely unavoidable to convince ourselves personally of the correctness of the results of others if we want to use them. But today there exist published papers based on results whose examination would require the study of more than 5,000 printed pages, a task which is admittedly beyond human capability. In all experimental sciences it is of course impossible to check personally the validity of all experiments preceding one's own research and forming its foundation. But should we mathematicians, so far having been proud to recognize no other authority than our own intellect, also resign ourselves to accept upon trust the printed word as
275 the truth? And here is another source of discomfort - the reliance placed on the computer. A celebrated, more than a hundred years old problem is the so-called four-color-problem. This is the question: whether it is always possible to paint a globe with many countries by only four colors in such a way that any two countries with a common border line have different colors. Two American mathematicians have reduced this problem to another one, approachable by the computer, and after the work of many hours the computer had decided: Yes, it is possible. But do I have to accept it thereby to be proved that four colors suffice? What is a mathematical proof? The answer to this question contains a personal element, even if one rejects placing trust in the computer and the acceptance of unchecked results. I know of a paper which was published prior to the invention of the computer, and whose aim was to prove a certain conjecture by Leonhard Euler. This paper makes a distinction of several hundred cases and investigates each case separately, thereby arriving at the desired result. Even if I had succeeded in examining all these cases, I would not take this as a proof of the correctness of this assertion, but only as a confirmation of my inability to discover a mistake. It seems to me that one has to ask more of a proof, namely, an understanding of the connections which underlie the truth of an assertion. The nonmathematician envies us for the incontestability of our results but the questions of conscience may well escape his notice that, as in the past, so also today have contributed to a crisis in mathematics.
276 4. Further Reading
1. C. Reid, Courant in Gottingen and New York (Springer, 1976). 2. M. Protter, J.L. Kelley, T. Kato and D.H. Lehmer, in Memoriam Hans Lewy (University of California Obituary 1988). 3. C. Reid, Hans Lewy, in More Mathematical People, eds. D.J. Albers, G.L. Alexanderson and C. Reid (Harcourt Brace Jovanovich Publ., 1990), pp. 180-194. 4. C. Reid, Hans Lewy. 1904-1988, in Miscellanea Mathematica, eds. P. Hilton, F. Hirzebruch and R. Remmert (Springer, 1991), pp. 259-267. 5. R.E. Rider, An opportune time: Griffith C. Evans and Mathematics at Berkeley, in A Century of Mathematics in America, Part II, ed. P. Duran (AMS, 1989). 6. S. Hildebrandt, F. Hirzebruch, F. John, R. Leis, H. Lewy and J. Tits, Mathematische Betrachtungen, Bonner Akademische Reden, Nr. 68 (Bouvier Verlag, 1988).
277 5. Bibliography by David Kinderlehrer
1925 [1] Uber einen Ansatz zur numerischen Losung von Nachrichten Ges. Wiss. Gottingen, 18-21.
Randwertproblemen,
1927 [2] Verallgemeinerung der Riemannschen Methode auf mehr Dimensionen, Nachrichten Ges. Wiss. Gottingen, 118-123. [3] Uber den analytischen Charakter der Losungen elliptischer Differentialgleichungen, Nachrichten Ges. Wiss. Gottingen, 178-186. [4] Uber die Methode der Differenzengleichungen zur Losung von Variations- und Randwertproblemen, Math. Ann. 98, 107-124. [5] Uber das Anfangswertproblem bei einer hyperbolischen nichtlinearen partiellen Differentialgleichung zweiter Ordnung mit zwei unabhangigen Veranderlichen, Math. Ann. 98, 179-191. [6] (with K. Friedrichs) Uber die Eindeutigkeit und das Abhangigkeitsgebiet der Losungen beim Anfangswertproblem linearer hyperbolischer Differentialgleichungen, Math. Ann. 98, 192-204. 1928 [7] (with R. Courant and K. Friedrichs) Uber die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann. 100, 32-74. [8] (with K. Friedrichs) Das Anfangswertproblem einer beliebigen nichtlinearen hyperbolischen Differentialgleichung beliebiger Ordnung in zwei Variablen. Existenz, Eindeutigkeit und Abhangigkeitsbereich der Losung, Math. Ann. 99, 200-221. 1929 [9] Neuer Beweis des analytischen Charakters der Losungen elliptischer Differentialgleichungen, Math. Ann. 101, 609-619. 1930 [10] Sulla unicita della soluzione del problema di Cauchy per un'equazione ellitica del secondo ordine in due variabili, Rend. Accad. Lincei, XI.6, 162-165.
To be published in Collected Works of Hans Lewy (Birkhauser).
278 1931 [11] Eindeutigkeit der Losung des Anfangswertproblems einer elliptischen Differentialgleichung zweiter Ordnung in zwei Veranderlichen, Math. Ann. 104, 325 339. 1932 [12] (with K. Friedrichs) Uber fortsetzbare Anfangsbedingungen hyperbolischer Differentialgleichungen in drei Verander lichen, Ges. Wiss. Gottingen, Math.-Phys. Klasse., 135-143. 1935 [13] (with C.R. Adams) On convergence in length, Duke Math. J. 1, 19-26. [14] A priori limitations for solutions of the Monge-Ampere equations, Trans. A.M.S. 37, 417-434. 1936 [15] Generalized integrals and differential equations, Proc. Nat. Acad. Sci. 22, 377-381. [16] On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. A.M.S. 42, 689-692. 1937 [17] A priori limitations for solutions of the Monge-Ampere equations II, Trans. A.M.S. 41, 365-374. 1938 [18] On the existence of a closed convex surface realizing a given Riemannian metric, Proc. Nat. Acad. Sci. 24, 104-106. [19] On differential geometry in the large I (Minkowski's problem), Trans. A.M.S. 43, 258-270. [20] Generalized integrals and differential equations, Trans. A.M.S. 43, 437-464. [21] A property of spherical harmonics, Amer. J. Math. 60, 555-560. 1939 [22] Aspects of the Calculus of Variations (University of California Press, 96 pp.), notes by J.W. Green.
279 1946 [23] Water waves on sloping beaches, Bull A.M.S. 52, 737-775. 1948 [24] (with K. Friedrichs) Solution of the dock problem, Comm. Pure Appl. Math. 1, 135-148. [25] On the convergence of solutions of difference equations, Studies and Essays. Courant Anniversary Volume, 211-214. 1950 [26] Developments at the confluence of analytic boundary conditions, in Int. Cong. Mathematicians, Vol. 1, 601-605. [27] Developments at the confluence of analytic boundary conditions, in Univ. Calif. Publications New Series Vol. 1, 247-280. 1951 [28] On the boundary behavior of minimal surfaces, Proc. Nat. Acad. Sci. 37, 103-110. [29] On minimal surfaces with partially free boundary, Comm. Pure Appl. Math. 4, 1-13. 1952 [30] A note on harmonic functions and a hydrodynamical application, Proc. A.M.S. 3, 111-113. [31] A theory of terminals and the reflection laws of partial differential equations, Technical Report. [32] Asymptotic developments at the confluence of boundary conditions, Nat. Bureau Standards, Ser. 18, 255-256. [33] (with P. Garabedian and M. Schiffer) Axially symmetric cavitational flow, Ann. Math. 56, 560-602. [34] On steady free surface flow in a gravity field, Comm. Pure Appl. Math. 5, 413-414. 1955 [35] Extension of Huyghens's principle to the ultrahyperbolic equation, Ann. Mat. Pura Appl. (4) 39, 63-64.
280 1956 [36] On the local character of the solutions of an atypical linear differential equation in three variables and a related theorem for regular functions of two complex variables, Ann. Math. 64, 514-522. [37] On the relations governing the boundary values of analytic functions of two complex variables, Coram. Pure Appl. Math. 9, 295-297. 1957 [38] An example of a smooth linear partial differential equation without solution, Ann. Math. 66, 155-158. [39] On linear difference-differential equations with constant coefficients, J. Math. Mech. 6, 91-108. 1958 [40] Introduction, in the Collected Works of Bernhard Riemann, 8 pp. (Dover). 1959 [41] Composition of solutions of linear partial differential equations in two independent variables, J. Math. Mech. 8, 185-192. [42] On the reflection laws of second order differential equations in two independent variables, Bull. A.M.S. 65, 37-58. 1960 [43] On hulls of holomorphy, Coram. Pure Appl. Math. 13, 587-591. 1961 [44] Atypical partial differential equations, in Partial Differential Equations and Continuum Mechanics, ed. Rudolph E. Langer (Univ. of Wisconsin Press), 171-175. [45] (with R.S. Lehman) Uniqueness of water waves on a sloping beach, Coram. Pure Appl. Math. 14, 521-546. 1964 [46] On the analytic continuation of minimal surfaces and similar problems, in Proceedings of the Fourth All Union Mathematics Congress, Vol. 2, 233-236 (in Russian). [47] Sul prolungamento delle funzione armoniche, Rend. Sem. Mat. Torino 24, 27-29.
281 1965 [48] On the definiteness of quadratic forms which obey conditions of symmetry, Ann. Scuola Normale Superiore Pisa 19, 577-591. [49] On the extension of harmonic functions in three variables, J. Math. Mech. 14, 925-928. [50] The wave equation as limit of hyperbolic equations of higher order, Comm. Pure Appl. Math. 17, 5-16. 1966 [51] Sulla riflessione delle funzione armoniche di tre variabili. Convegno sulle Equazioni alle Derivate Parziali, (Nervi). 1968 [52] On a variational problem with inequalities on the boundary, J. Math. Mech. 17, 861-864. [53] On the nonvanishing of the Jacobian of a homeomorphism by harmonic gradients, Ann. Math. 88, 518-529. 1969 [54] About the Hessian of a spherical harmonic, Amer. J. Math. 9 1 , 505-507. [55] (with Guido Stampacchia) On the regularity of the solution of a variational inequality, Comm. Pure Appl. Math. 22, 153-188. 1970 [56] On a minimum problem for superharmonic functions, in Proc. Int. Conf. on Functional Analysis and Related Topics (Univ. of Tokyo Press), 283-284. [57] On a refinement of Evans' law in potential theory, Rend Accad. Naz. Lincei 48, 1-9. [58] On the partial regularity of certain superharmonics, in Studies and Essays presented to Yu-Why Chen on His Sixtieth Birthday, 169-171. [59] (with Guido Stampacchia) On the smoothness of superharmonics which solve a minimum problem, J. d'Anal. Math. 23, 227-236. 1971 [60] Obobscenie teoremy o prostranstvennom ugle (Generalization of the spatial angle) (translated from the English by Ju. V. Egorov), Usp. Mat. Nauk 26, 199-204. [61] (with Guido Stampacchia) On existence and smoothness of solutions of some non-coercive variational inequalities, Arch. Rational Mech. Anal. 4 1 , 241-252.
282 1972 [62] On the coincidence set in variational inequalities, J. Diff. Geom. 6, 497-501. 1975 [63] On the nature of the boundary separating two domains with different regimes, Rend. Accad. Naz. Lincei 217, 181-188. 1976 [64] On analyticity in homogeneous first order partial differential equations, Ann. Scuola Norm. Sup. Pisa 4, 719-723. 1977 [65] On the boundary behavior of holomorphic mappings, Accad. Naz. Lincei. Contributi del Centro Linceo Interdisciplinare di Scienze Matematiche e Loro Applicazioni, No. 35, 1-5. [66] On the minimum number of domains in which the nodal lines of spherical harmonics divide the sphere, Comm. P.D.E. 2, 1233-1244. 1978 [67] On the relation between analyticity in one and in several complex variables, in Complex Analysis and its Applications (Russian). 1979 [68] An inversion of the obstacle problem and its explicit solution, Ann. Scuola Norm. Sup. 6, 561-571. [69] Expansion of solutions of 't Hooft's equation. A study in the confluence of analytic boundary conditions, Manuscripta Math. 26, 411-421. 1980 [70] On conjugate solutions of certain partial differential equations, Coram. Pure Appl. Math. 33, 441-445. 1981 [71] Uber die Darstellung ebener Kurven mit Doppelpunkten. Nachrichten der Akad. der Wiss. Gottingen, Mathematisch-Physikalische Klasse Nr. 4, 109-130.
283 1985 [72] (with Tang Zhiyuan) On free boundary problems in two dimensions, Rend. Circ. Mat. Palermo (Ser. II) 34, 325-336. 1987 [73] (with U. Dierkes and S. Hildebrandt) On the analyticity of minimal surfaces at movable boundaries of prescribed length, J. Reine Angew. Math. 379,100-114. 1989 [74] On atypical variational problems, in Partial Diff. Equ. and Calc. of Variations. Essays in Honor of Ennio De Giorgi, eds. F. Colombini, A. Mariano, L. Modica and S. Spagnolo (Birkhauser), 697-702.
6. Publications Dedicated t o Hans Lewy
1. In 1978 the Scuola Normale Superiore di Pisa published a collection of 27 papers dedicated to Hans Lewy which has appeared in the Annali della Scuola Norm. Sup. di Pisa in the same year. The title of the collection is: Raccolta degli scritti dedicati a Hans Lewy. 2. Manuscripta mathematica (Vol. 28, Fasc. 1-3, 1978) published 15 papers dedicated to Hans Lewy and C.B. Morrey. (The drawing of Hans Lewy by his wife Helen is taken from this issue of Manuscripta Math., frontispiece.) 3. In 1986 a conference on Calculus of Variations and Partial Differential Equations was held in Trento in Honor of Hans Lewy, and 23 papers covering the lectures of this conference are published in Vol. 1340 of Lecture Notes in Mathematics, eds. S. Hildebrandt, D. Kinderlehrer and M. Miranda (Springer, 1988).
Neuer Beweis des analytischen Charakters der Losungen elliptischer Differentialgleichungen*). Von
Hans Lewy in Gottingen.
In einer Note in den Gottinger Nachrichten 1927 Heft 2 habe ich einen neuen Beweis des analytischen Charakters der Losungen elliptischer analytischer Differentialgleichungen in einem Spezialfall ausgefiihrt. Hier folgt der Beweis des allgemeinen Theorems fiir Differentialgleichungen zweiter Ordnung in zwei unabhangigen Variablen, der wie jener darauf beruht, die vorgelegte Losung u einer elliptischen analytischen DifEerentialgleichung zunachst fiir komplexe Werte der unabhangigen Variablen x und y zu ermitteln. Diese Fortsetzung ins Komplexe wird hier aufgefaBt als Losung eines Anfangswertproblems eines „kanonisch-hyperbolischen Systems" von Differentialgleichungen, d. h. eines solchen, wie es z. B. auch zur Losung des Anfangswertproblems einer hyperbolischen Differentialgleichung verwendet wird. Es wird sodann nachgewiesen, dafi in einem vierdimensionalen Gebiet der Variablen x1,x^,yl,y.i (Real- und Imaginarteile von x und y), das das reelle Ebenenstiick x., = 0, y.2 = 0, in welchem die Losung u urspriinglich vorliegt, im Inneren enthalt, die CauchyRiemannschen Differentialgleichungen beziiglich x1, a;3 und bezuglich yx, y„ erfiillt sind. Das bedeutet aber, daB u analytisch von x und y in jenem Gebiet, also auch schon in seinem urspriinglichen Definitionsbereich abhangt. §1Charakteristisclie Differentialgleichungen und Fortsetzung ins Komplexe. Sei F(x, y,u,p,q,
r,s, t) = 0
eine analytisch von den Argumenten abhangige DifEerentialgleichung zweiter J
) Beziiglich der alteren Beweise sei auf den Enzyldopadiebericht von Lichtenstein 2 C 12, S. 1320 und auf die jiingst erschienenen Arbeiten von T. Rado, Math. Zeitschr. 25, S. 514 ff., und S. Bernstein, Math. Zeitschr. 'M, S. 330 ff., verwiesen. Mathematische Annaien. 101.
H. Lewy.
610 Ordnung, in der P = ux,
? = «„, r = uxx,
s = uxy,
t = uyy
wie iiblich zur Abkiirzung gesetzt sind. Fiir eine vorgelegte Losung u sei die DifEerentialgleichung in einem Gebiet der x, y-Ebene vom elliptischen Typus, d. h. es sei dort or
dt
\dsj
woraus insbesondere — 4= 0, ^— 4= 0 folgt. Man verstehe nun unter den dr ' ' at ' ° stetigen Funktionen Q1(X, y) und Qi{x, y) die beiden sicher verschiedenen komplexen Wurzeln der Gleichung dF 2 a dF . dF n
^Q -^Q
+ -gI = °->
unter den Differentiationsoperatoren ' bzw. ' die folgenden linearen Kombinationen der Symbole — und — : J
dx
i
dy
£._(_„ J L
»
? i
~dx~f~Qldy'
JL
~dx^~Q*cy'
Dann folgt y' —
Q±X'
= 0
und
i/ — £>„ x' = 0.
Aus dem Bestehen dieser Gleichungen folgen jetzt in bekannter Weise rein algebraisch noch weitere charakterische Differentialgleichungen fiir die GroBen x, y, u,p, q, r, s, t. Diese sowohl wie die beiden schon genannten y'—gix'=0 und y —Q„ #' = 0 lassen sich nach geeigneter Auswahl in die folgende Form setzen: U^kVl^O
(» = 1 , 2 , . . . , 6 ) ,
iait
(.- = 7,8),
:i) 1=1
in der die
Analytischer Charakter der Losungen elliptisoher Diflerentialgleichungen.
611
den nach der Voraussetzung des elliptischen Charakters erfiillten Ungleichungen * dr dt
\ds)
+
U )
dr
+ U
'
dt
+ U
-
Um die in der Einleitung angekiindigte Fortsetzung ins Komplexe auszufuhren, werden wir alle Groflen cp als Funktionen von vier unabhangigen Variablen |,,f 9 , Vi> Vi angeben, die wir so wahlen, daB fiir f3 = rjz = 0 die Ebene | x , rj1 mit der x, y-Ebene zusammenfallt, d. h. so, daB * ( f l , 0 , i 7 1 , 0 ) = 9P 1 (| 1 , 0 , ^ , 0 )
=^,
y(Slt 0, Vi> ° ) = V. (fi. °> fi» ° ) = >?i ist. Dadurch sind in der Ebene £2 = »?2 = 0 die samtlichen 99, die vorher Funktionen von x, y waren, nun als dieselben Funktionen von S± und rjl bekannt. Um nun diese
2aik
(» = 7 , 8 ) ,
t
geben aber den Operatoren nun die Bedeutung 3 )
Man hat hier ein Anfangswertproblem fiir Funktionen der beiden unabhangigen Variablen £lt ?;2 vor sich, dessen Losbarkeit von mir in der zitierten Arbeit iiber das hyperbolische Anfangswertproblem bewiesen wurde. Allerdings werden bei dieser Fortsetzung die Funktionen cp moglicherweise komplex werden, da schon fiir £, = >/3 = 0 nicht alle aik reell sind. Die Bedeutung der ailc fiir komplexe Werte der cp anzugeben, hat keine Schwierigkeit, da die aik analytisch von den cp abhangen; wir erklaren die aik fiir komplexe Werte der cp einfach als analytische Fortsetzungen der 3 ) Wesentlich an dieser Festsetzung der Bedeutung der Operatoren ist nur, daB sie Differentiationen in zwei versehiedenen Bichtungen der fl( rj2-Ebene bedeuten und daB keine dieser Richtungen parallel zur „Anfangsgeraden" »?2 = 0 ist.
H. Lewy.
612
aiu fur reelle Werte der
) H. Lewy, loc. cit. I m folgenden werden noch die folgenden Eigenschaften v o n Losungen eines solchen „kanoniseh hyperbolischen" Systems benutzt, die ich hier zusammenstellen m o c h t e : 1. Man kann eine etwa aus zwei kongruenten Trapezen (vgl. Figur) bestehende beiderseitige Umgebung des Geradenstiickes »;2 = konst. angeben, i n d e r die Funktionen cp existieren u n d d e n obigen Differential/ S. gleichungen gehorchen; die GroCe der Trapeze hangt KrVt^nst/
/
\t*r,fkonst.
°
6
6
'
/
6
\7 /^i-V!Hl"'st-
dabei, w a s die Anfangswerte betnfft, n u r a b dtp ^r"%"Jmi^\ 2 «) v o n oberen Schranken der - — ,t —— auf dem 0Vi herausgegriffenen Geradenstiick; ft) v o n oberen u n d unteren Schranken fiir die cp,, ?>, selbst; y) v o n einer unteren Schranke fiir den Abstand d e r durch die Anfangswerte
) Denn sonst wiirde ein m i t d e r neuen Matrix gebildetes linear homogenes Gleichungssystem eine nicht triviale Losung besitzen, also auch ein linear homogenes Gleichungssystem mit der alten Matrix, was unmoglich ist, weil dessen Determinante nicht verscliwindet.
Analytischer Charakter der Losungen elliptischer Differentialgleichungen.
613
Das Verfahren der Fortsetzung, das soeben fur ein Geradenstiick rj1 = konst. der Ebene fa = tj9 = 0 als Anfangskurve beschrieben wurde, wende man nun auf alle parallelen Geradenstiicke rjx = konst. dieser Ebene, soweit sie in den Definitionsbereich der Losung u fallen, an. Man bestimmt so die Funktionen
(* = 1 , . . . , 6),
k
Uaik
(» = 7,8)
k
benutzt, jetzt aber den Operatoren die folgende Bedeutung gibt: , _
l**\
_d
d_
' =+ — + — -
" '
^ dg, ^ dVl §2. Riickkehr zu den alten Yariablen. Ein Hauptpunkt der folgenden Untersuchung ist der Nachweis, daB man anstatt £x, £>, TJ1, J;9 auch xx, a;3, yx, y2 als unabhangige Variabe einfiihren kann, wenn man sich auf eine hinreichend kleine vierdimensionale Umgebung eines Punktes x, y des urspriinglichen Definitionsbereiches von « beschrankt, was wir natiirlich tun wollen. Es geniigt dazu wegen der StetigkeitT), das Nichtverschwinden der Determinante
in dem betrefEenden Punkte der „Anfangsebene" nachzuweisen. Da dort die £x, f2, rj1, tj„ bzw. mit den xx, or3, yx, y„ iibereinstimmen, hat man dort die Gleichungen fur acht von den in Frage kommenden Ableitungen:
(2)
af,
l
'
££i = n Bit ' •) Vgl. die FuBnote 4). ') Vgl. Anna. *}.
dSi~u'
dgl-v'
i^ —o sVl~v>
^i_i dVl~ '
d^-"' ?y* — (\ dVl~"-
614
H. Lewy.
Spaltet m a n weiterhin die Funktionen Q1 u n d e 2 , vgl. S. 610, in dem betreffenden P u n k t e nach Real- u n d Imaginarteil, also Qi=ai+
e» = ai — * ai
i °2>
o, + 0,
so liefern die Gleichungen y'— Q1X'=Q u n d «/' — Q^X' = 0 in den verschiedenen Bedeutungen (*) u n d (**) der Differentiationsoperatoren die folgenden vier Gleichungen (unter Benutzung von (2))
4* + t - | * _ ( o + 1 - 0 a ) ( (3) 3%
dyi;
i + | * + ,•*&) = 0 ,
h - M - ' + S + 'S)- 0 '
dj? a
+ ••#-1-(«. " ^ + l '••». "") U® f , '+' 9'ft) * . = «.
3fa^*3fa
(4)
8fi Die ersten beiden liefern durch Subtraktion
2
dXs
«i + 2 »«.(l5 + » 8*13
0*, , . aa;2
. a,,
3xt
"T" l ~~ > oa
"5 T * "5— = 3i?2 9i? 4
"5— a^
0;
:
o, dx.,
al
Weiter erhalt m a n d
V\ _J_ i 1^? _ ( g i-r*' g a)( g « + *'gi)
»(g? + g|). oa '
*?_! — $• 8T]2 '
8y
* = 3<7a
"{+"•' a2
Ahnlich entsteht aus den Gleichungen ( 4 ) durch Subtraktion as,
1
c| 8
woraus dann <>Vi _i_ v 9% di-i o$3
-, , (Qi + tq a )-t _ • < » ! . 3?! _ oa <j3 3fa
n
3ya _ o, of,
folgt. Die Matrix der Ableitungen der x1 x„ yx y^ nach den | t f 9 ^ J?2 sieht also in dem ins Auge gefafiten P u n k t e der Anfangsebene | 9 = »?9 = 0 so aus: *1
X.,
2*1
y-i
f,
1
0
0
0
f-2
0
Vi
0
Vt
0
1 g
0
2
0
1
o.3
0
0 °2
°3
291 Analytischer Charakter der Losungen elliptischer Difierentialgleichungen. 6 1 5
Wie man sieht, ist die Determinante ungleich null, namlich gleich 1; man kann also in einer vierdimensionalen Umgebung des betreffenden Punktes der x, y-Ebene die x1, x2, y1, t/2 als neue Koordinaten einfuhren. Allgemein werden die DifFerentiationssymbole nach £±, fa, r}x, rj2 lineare Kombinationen derer nach xx, x 2 , yx, y2. Schreibt man
m wird .va ( i
+ 5 L)(, 1 +
, - j , , ) _ 8 l ( ^ + jL)( I l + ,-«,)-0 wegea
der Tatsache, daB der Operator
\- i —— null ergibt, wenn er auf eine
analytische Funktion von x1-\-ix2
(hier x1-{-ix2
selbst) ausgeiibt wird,
und wegen der entsprechenden Tatsache fur den Operator B1-glA1
f- i —
= 0,
und Ax ist ungleich null, weil sonst der Operator (5) nicht reell sein kann, aufier wenn Cx und Dx auch verschwinden; das ist aber unmoglich, weil nicht -r— = — —- ist. Man findet demnach (6)
-^ + ~ = A1^ + QlA1^-
+ Cx(^- + i^-)+D1(^
+ i^-),
und ahnlich
(7) K
'
-Jz+&-±A+^k+
c?i
"3*i
9
a
^2/i
- \5x t '
3x s /
3
\a2/i
cz/.2
Alle diese GroBen Ax, Gx, Dx, ..., H„ sind nichts weiter als lineare Kombinationen der Differentialquotienten der xx, x.,, yx, t/a nach den fi; £2, rjx, J?2; z. B. ist offenbar
*C,-( ir + Ah-
iD
>~( £ + £)*•
616
H. Lewy.
und es ist auBerdem wichtig, nochmals hervorzuheben, daB A1} A2, Et, E.2 samtlich ungleich null sind. Fiir dasFolgende wesentlich sind dieWerte der C,, Dt, C 3 , Z>3, Gx, G„ Hx, H2 in der Anfangsebene x<1 = yi = 0. Man hat dort unter Benutzung unserer Differentiationstafel 1
°2
°2
°2
(10)
°2
•». = ^ '
°2
»'»= = t -
i0
* = T> °2
§3. In der Anfangsebene gelten nun die charakteristischen Differentialgleichungen (1) in dreierlei Form, entsprechend den drei Bedeutungen des Operators ' bzw. ' . Die erste auf S. 610 erwahnte Bedeutung von ' bzw.' lautete
' = J-_L
J_
, __ J_ ,
g
die zweite war die von (*), die dritte die von (**). Man subtrahiere nun von dem System (1) in der zweiten Bedeutung das mit Ax bzw. At multiplizierte der ersten Bedeutung und erhalt so unter Beriicksichtigung von (6) und (7) und von (10) 8
wobei der Index i von 1 bis 8 lauft. Aus dem Nichtverschwinden der Determinante \aik[ in der Anfangsebene folgt nunmehr
(11)
{«(J- + iJJ)+°J±?i(J-
v
i_ai\cxl
I
ox^j
a„
+
\dyx
iJl-Y\
(* = 1,...,8). v
Ahnlich folgert man, wenn man das charakteristische System in seiner ersten Bedeutung mit EL bzw. E.s multipliziert und von dem charakteristischen System in der dritten Bedeutung (**) abzieht und wiederum die Werte (10) der Koeffizienten Gx, G.2> Hx, Hx in der Anfangsebene heranzieht:
293 Analytischer Charakter der Losungen elliptischer Differentialgleichungen.
617
Fur jedes einzelne
a
? + g2
folgt. (Das darf nicht verwundern, denn die Bedeutung der Operatoren ' bzw. ' war natiirlich dementsprechend festzulegen.) Somit ist das Erfulltsein der Cauchy-Riemannschen Differentialgleichungen in der Anfangsebene x„ — y2 = 0 , oder um in der Sprache der £i>£i> Vi> Vi z u reden, in der Ebene f9 — r\„ = 0 gesichert. Wir weisen zunachst ihre Giiltigkeit in jenem dreidimensionalen Raumstiick f3 = 0 nach, in das wir zuerst vermoge der Bedeutung (*) der Operatoren ' bzw.' fortgesetzt haben. Wir setzen zur Abkiirzung
Wir dividieren die Gleichungen des charakteristischen Systems in der besagten Bedeutung bzw. durch At und A2, die ja beide nicht verschwinden (vgl. S. 615) und wenden auf das so umgeformte System die Operatoren Vx und Vy an. Da die aik analytisch von den q>1,...,q>s abhangen, ist
*,*» = 2j*r,
* \axl ^ ^1 dy^ i-
Ai
>x ~h Ai yy)
8
'
*Al
xP
«
'
x
fk
A l y ^cyL
ZJ
8 Ai
\yx
. . ^ vx
Vl.
dVlxV<-
H. Lewy.
618
Wir sehen also, daB die Anwendung des Operators Vx auf das durch At bzw. A^ dividierte Difierentialgleichungssystem zu dem folgenden Differentialgleichungssystem fiihrt: 2aik(Vx
(12)
k
+ ciliVy(pk) = 0
(»=1,...,6),
+ cikVy
(» = 7 , 8 ) .
k
Eine ahnliche Anwendung des Operators Vy bewirkt die Gleichungen 2««(*>*)'+ (13)
2{dikVx
Uaik(Vy
+ eikVy
(»•= 1, . . . . 6), (*=7,8).
k
Dieses System von 16 Differentialgleichungen fur die 16 Eunktionen Vx
in dem zu betrachtenden vierdimensionalen Gebiet, indem man sich erinnert, daB die Fortsetzung der
) Ein solcher Eindeutigkeitssatz steht in meiner zitierten Abhandlung. Es laCt sich die Eindeutigkeit schon unter schwacheren Voraussetzungen als den dort angegebenen beweisen, wenn man die in der Theorie der gewohnlichen Differentialgleichungen ublichen Methoden benutzt; vgl. hierzu auch O. Perron, „tJber Existenz und Nichtexistenz von Integralen partieller Differentialgleichungen im reellen Gebiet", Math. Zeitschr. 27, S. 549 ff., sowie Haar, C. R. 187 (2.6. 1928), und meine anfangs zitierte Abhandlung in den Gottinger Nachrichten 1927, S. 184 ff. Hier liegen die Dinge wegen der Linearitat der Differentialgleichungen ubrigens besonders einfach.
Analytischer Charakter der Losungen elliptischer Differentialgleichungen.
619
§4. Zum SchluB noch ein Wort iiber die Differenzierbarkeitsvoraussetzungen, denen die vorgelegte Losung u(x,y) gehorchen mufi. Zur Fortsetzung in den vierdimensionalen Raum ist es hinreichend, die Funktionen
104
MATHEMATICS:
H.
LEWY
PROC. N . A.
S.
ON THE EXISTENCE OF A CLOSED CONVEX SURFACE REALIZING A GIVEN RIEMANNIAN METRIC B Y HANS LEWY DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA AT BERKELEY
Communicated January 14, 1938
Let (To be the surface of the unit sphere and (dso)2 the first fundamental form of Differential Geometry on a0. Denote by (ds ) 2 another given positive definite form defined at each point of
G(dvy.
VOL.
24, 1938
MATHEMATICS:
H.LEWY
105
Suppose that the Gaussian curvature K\ of the metric (ds{)2 is positive. Weyl 1 has attacked the following problem: Does there exist a closed convex surface a\ which may be mapped in a one-to-one way on
+ 2Fxdudv + Gx(dv)2
in the parameters (u, v) on
106
MATHEMATICS:
H.LEWY
P R O C . N . A. S.
i) I Pxi |> I Pxi |> • • • a r e uniformly bounded. This is a consequence of Bonnet's theorem which states t h a t the diameter of a closed convex surface is bounded from above b y a n u m b e r depending only on a (positive) lower bound of t h e Gaussian c u r v a t u r e .
dpx,
dpx,
dpx,
ii) —— I, ——- ,i ——^ , ... are uniformly bounded. This follows from 5M ou bv bv 5M (rfpx)2 ^ (dsx)\ 52Px 5 2 p x /52px\2 iii) T h e coefficient of - — 2 -——2 — I —— J in Darboux's equation can be H ' 5M 5Z/ \budvJ assumed to b e = 1 while t h e other coefficients are bounded for complex d/0x 5px values of u, v, px, ——, —— in a bounded neighborhood of their values under ou ov consideration. T h e c o n t r a r y assumption is ruled o u t b y a simple geometrical consideration showing t h a t the radii of the m a x i m u m spheres inscribed in
5?
+
. 52M
5^
, [fbuV
= h
T\ba)
+
.
fdu\2~\
\W J
.
, (bu
+ K
\bab-a
bv
+
. bu
bv\
WW
4(0+(S)"]+*<
x
+
b(u, v) 5(a, ff)
S+g-4Cs),+Cs)"]+-+i*^-"' where /fix, thx, • • • , &ix, &2x> • • • depend on (u, v), b u t not explicitly on px bp\ bp\ nor — nor — . I n our case a + iff and a — i/3 are parameters on the bit bv asymptotic lines a n d our equations (1) are identical with equations a l r e a d y found b y Darboux. 5 1
Hermann Weyl, "Uber die Bestimmung einer geschlossen konvexen Fluche durcli ihr Linienelement." Vierteljahrsschrift der naturf. Ges. Zurich, 61, 40-72 (1915). 5 1 have been in possession of this proof for several years, cf. Bull. Am. Math. Soc, 42, 824 (1936). The preparatory investigations of Monge-Ampere equations were undertaken mainly for this purpose and published in Trans. Am. Math. Soc, 37, 417-434 (1935) and 41, 365-374 (1937). 3 G. Darboux, Theorie des surfaces, vol. 3, p. 260 (Paris, 1894). 1 Loc. eit., 41, 373 (1937); also Bull. Am. Math. Soc, 42, 689-692 (1936). 5 Loc. eit., p. 290.
VOL.
ON
37, 1951
THE
MATHEMATICS:
BOUNDARY
H.
BEHAVIOR
LEWY
OF MINIMAL
103
SURFACES
B Y HANS LEWY UNIVERSITY OF CALIFORNIA
Communicated by G. C. Evans, December 16, 1950
This paper deals with the analytic extension of a minimal surface across its boundary. LEMMA 1: Let U (x, y) be harmonic and absolutely bounded in the half plane y > 0. Assume U (x, y) to possess1 continuous boundary values U(x, 0) on y = 0 which admit limits Z7(°°, 0) and U{— °°, 0). Assume furthermore that the total variation from x = — <& to x = » o/ U(x, 0), denoted by T1* [U(x, 0)], is bounded. Denote similarly by Tb[U(x, y)]the total variation of U (x, y) as x ranges from a to b where a and b > a are any two numbers including ± °o, and y is kept constant. Then, for y > 0,
T1„[U{x,y)\
<
T!„[U(x,0)],
and lim Tba[U(x,y)] y — 0
= Tba[U(x, 0)].
104
MATHEMATICS:
Proof:
H. LEWY
PROC. N. A. S.
U(x, y) = U(P) is for y > 0 expressed by the formula U(P) = T-'SQ"-
-~f(Q)d6(PQ)
where 6{PQ) is the angle from the negative y-axis to the vector from P to Q, and f(Q) = U(Q) for Q on the x-axis. Integration by parts yields U{P) = - * - ! fe(PQ)df(Q)
+ (/(co) _ / ( - » ) ) / 2 .
(1)
Differentiating with respect to a; we find (b/dx)U(x,
y) = T - ^ - £
0(PQ)df{Q),
(2)
which we also write with 0r instead of {b/bx)0(PQ) and where we note that — 8X is positive. Thus Tlm[U(x,y)] Now S-"
= y _ - . | d £ / ( * , y ) | < f f'm
- 0,
(3)
— Oxdx = x/2 — (— x/2) = x. Hence the desired result Tlm[U{x,y)]
<
nm[U(x,0)].
Similarly 7*[U{x, y)] < T^S'-WPIQ) - 8(P£)]\df{Q) \, P , = P{a,y), P 2 = P(b, y). The bracketed expression is bounded by x and tends uniformly to x as y -*• 0, for a + t < Q < b — t, and to 0 for Q > b + t or for Q < a — «if e > 0 is assigned sufficiently small. For the two missing intervals from a — e to a + e and from b — e to b + t we gather from the continuity of f(Q) which entails (as is familiar) that of Txa[f{Q)), that their contribution to the last integral becomes for ally > 0 less than an arbitrary small t' > 0 provided t is sufficiently small. Thus lim J"-,, [6(PiQ) — e(P*Q)]\df(Q)\
< r fa"+:\df(Q)\ b
+ « ' < Trfab\df(Q)\
lhn T 0[U(x, y)] ^
+ « ' , whence
T*[U(x,0].
y—0
But since also lower semicontinuity holds, i.e., J i m Tb[U(x, y)] > Tba + \{U(x, 0)] y—0
no matter how small e, hence also for e = 0, we have the desired continuity of Tb[U(x,y)]asy-*0. A theorem for the circle analogous to Lemma 1 was proved by Evans. 2 LEMMA 1':
/ / Ui(x, y), Ut(x, y), U3(x, y) are harmonic and absolutely
bounded in y > 0 and assume on y = 0 continuous boundary values of bounded total variation, then the length L(y) of the curve of coordinates Ui(x, y), Ui(x, y), U3(x, y) for y fixed and — =° < x < °° tends to the length L0 of the curve Uy{x, 0), U2{x, 0), U,(x, 0) = /,((?), MQ),MQ), asy^O. Proof: rL(y) = x f°„dx[(dUl(x,y)/dxy + (bU2/bXy + (bU3/bx)*]Vt
VOL.
37. 1951
MATHEMATICS:
H. LEWY
105
= y_". dxKS-- - 0* dMQ))2 + ••• ]Vt < f~„ dx[{f~„ - ex\dMQ)|y + . . . ] ' ' • since — 0X > 0. Now [(/_". - ex\dfi(Q)\)2 + • • - ] ' A < •/*-"- - MI4A(C)I2 + .. . ] v * because this formula expresses the fact that a straight line joining the origin to the point with coordinates yi"„ — Sx\dfi(Q)\, ./*_"„ — 9x\df2(Q)\, f-„ — Ox\dfi(Q) | is the shortest line of this property. Hence *L(y) < y j " . dx f-m
-
9,(\dfi\*
+ .. .)*
and by Lemma 1, lira rL(y)
< lim f-m
dx f°„
- ff,(|i/i(0) |» + .. - ) ' A = T / - " . [ | 4 f i ( 0 1 ! + • • • l' A = T^O.
But the semicontinuity of the length requires lim L(y) > La. Thus L(y) is continuous as y —*• 0, q. e. d. LEMMA 2: Under the hypotheses of Lemma 1, the total variation of U on segments perpendicular to the x-axis is at most T1„ [/((?) ]• Proof: It suffices to prove the case when the segment ends on the x-axis, at the origin. Write U = U+ + U-, where U+(P) = v-*fo"
- S(PQ)df(Q) + const., U- = x-iy_°„ - 6(PQ)df(Q) + const.
We have, with dP = dy > 0, dP8(PQ) = 0udy < 0,U Q > 0; 0vdy>0, if Q < 0; |
S°. dP0(PQ)\df(Q)\ Thus, integrating over the segment in question,
f\dU+(P)\
< r"» So" 8(OQ)\df(Q)\ = (V.) fo" \df{Q)\
and similarly
f\dU-(P)\ Hence f\du{P)\
< S\du+(P)\
< 'A/- 0 . \df{Q)\. + f\dU-{P)\
< (V 2 )r-„[/(0].
LEMMA 3: Let S be a minimal surface of the three-dimensional £, 17, f space, and let S be bounded by a rectifiable Jordan curve C of length L. Take the conformal mapping of S on the upper half plane y > 0 of an x, y-plane which establishes a continuous one-to-one correspondence a between C and the
106
MATHEMATICS: H. LEWY
PRoe. N. A. S.
x-axis and for which £, t\, f become bounded harmonic functions of x, y in y > 0 with boundary values according to a, for which iny > 0 the two relations hold: £* + Vx + f* = $y + Vy + f y, && + VxVv + fxfv = 0 with £z = (d/dx)£, . . . . Denote by £*(x, y), T)*(X, y), f *(x, y) conjugate harmonics of £(x, y), 17, f a»
|!*(«. h) - ?(a, h')\ < fj %(a, y)\dy< fht {£ + v? + tf)l/'dy < SkH£ + fi + &l/'dy < SJtfy + vl + W'dy. Now by Lemma 2, Jhm\S,(a, y)\dy < nm[Z{x,
0)] < i ( 0 ) .
Hence J"° \£y(a, y) \dy converges and is <3Z,(0), thereby insuring that £* tends to a limit as y —*• 0. Similar facts hold for rj*, f*. T h a t these boundary values of £*, 17*, f * are continuous follows easily from i r ( o . 0) - {*(&, 0) I < T"a [{*(*, 0)] < Km inf T06 [{*(*, y)]
< length of C from x = a to x = 6 which also shows that the total variation of the boundary values of £* is continuous and remains bounded. Hence Lemma 1' applies to £*, 17*, f* as well as to £, 77, f; Lemma 3 follows. LEMMA 3 ' : Under the hypotheses of Lemma 3, we have more generally Lb(y) = L*b(y) -+ Z»(0) = L*o\0). Proof: Because of semicontinuity the only other possibility could be Z,£(0) < lim inf Lba(y). But then L(0) = L"_-(0) + Lj(0) + L6-(0) < Hm inf L l „ ( y ) , which contradicts Lemma 1'. Analogous considerations exclude L*ba(0) < lim inf L*ba(y), as y —* 0 through an arbitrary sequence.
303 VOL. 37, 1951
MATHEMATICS:
H. LEWY
107
LEMMA 4: Denote by L(0, y) the length of the boundary curve of the surface with coordinates £ cos y + | * sin y, ij cos y + v* sin y, f cos y + f * w i 7r ^ » L * ( 0 ( T) = £*(0, 0) = Lj(0). Proof: An elementary computation shows Z,(y, 7) = L(y, 0), and the passage y —* 0 gives, in view of Lemma 3', the desired relation. LEMMA 5: Denote by s the length on C. Excepting a set of measure zero with respect to s, we have on C 0 = (dS/ds)(dS*/ds) + (dv/ds)(dv*/ds)
+ (dt/ds)(dt*/ds).
(4)
Proof: Since ia(0, 7) = Lj(0, 0) = f,hm
a
[((£ COS 7 + f* Sin 7)/
we find by differentiation with respect to 7, setting 7 = 0, 0 = fxb- a m/dsKdp/ds)
+
...]ds
which is the integrated form of Lemma 5, from which it follows by differentiation with respect to 5 = s(b). THEOREM : Under the assumptions of Lemma 3 about S and its coordinates £» V, f(*> y)f suppose furthermore that the boundary curve contains an analytic arc'^4 which, without loss of generality, may be taken in the form V =
<M\S\.
108
MA THEM A TICS: H. LEWY
PROC. N. A. S.
Take e so small that M e < 1/8 and 4 e < p. We determine a function E(z) as solution of the differential equation dZ = 2[1 + *>'2(E) + *'ȣ)]-*{<&. +
*>'(E)<*M
+
with E(0) = 0.* Successive approximations give in the usual manner E(z) by integrating over straight segments from 0 to iy to x + iy for all x + iy = z of q. To show this, we observe, calling E* the &th approximation, that Eo = 0, and \dZi\ < 2d(r, |Ei| < 2
For a certain ilf*, to be de-
l^lHjt+2 - E*+i| | < |rf(Et+2 — E*+i)| < M*|E t+ i - E
G « = Gi(«V0O. G « = &(*)*'(*). Now set S =
lim E*. We have in q the integrated form of (5),
S = 2 / [ l + „"(E) + r*(3)]-i{d\
+
since the Stieltjes integral on the right hand is the limit as k —* °° of the analogous integral with E replaced by E*. That the construction yields an analytic function of z near all z = iy, y > 0, follows from the fact that (i) there is for arbitrary initial xo + iyo, yo > 0, a. solution of (5) which is analytic, according to the classical existence theorem and (ii) that there is no more than one solution of (5). For call 5S the difference of two solutions of (5). We have
VOL.
37, 1951
MATHEMATICS:
|5S| <
H. LEWY
109
Aff\5Z\d
with 5E vanishing initially. This inequality yields 5E = 0 in familiar manner. That the solution E is continuous in the whole of q follows from the same property of the S*. The latter fact follows from the convergence of the total variations of X, n, v over parallel segments which approach the x-axis normally. It is true that a priori (5) will hold on y = 0 only except for a set of measure zero relative to the variable s, but the integrated form holds everywhere. I say that E coincides with l-(x, 0) for y = 0. In fact, we have on y = 0 because ij = *>(£), f = ^(£), that almost everywhere in the sense mentioned, in view of (4),
2[i + w) + r\m-iUk/2 +
M(Z) = -J^S) + ^E(i)), *(z) = -Hi) + £(E(I)). First of all, these new definitions agree with the previous ones for n, v on y = 0, since there
- M * ) + tfS(*)) - -Of - «?*)/2 + *($) = (i, + *V)/2 = /•(«) and similarly
-Hx) + M£W) = »(*)• Secondly, the same argument as before yields the fact that for y < 0, fi and v are regular functions of z, thus becoming regular also on the x-axis.
110
MATHEMATICS:
PUTNAM
AND WINTNER
PROC N. A. S.
The Theorem is proved. The idea underlying the analytic extension, of forming a differential equation connecting the extension with the given function, is capable of application in many other situations. We have treated similarly the problem of the "free boundary surface" for minimal surfaces. Other problems concerning analytic boundary conditions come under the same heading. We intend to return to the subject elsewhere. 1
It suffices to demand assumption of boundary values upon normal approach to the boundary. * Evans, G. C , The Logarithmic Potential, Am. Math. Soc. Colloquium Publica-. tions VI, New York, 1927, pp. 35ff. 3 Courant, R., Dirichlet's Principle, Conformal Mapping and Minimal Surfaces, Interscience Publishers, 1950, esp. p. 118. 4 More precisely a solution of the integrated equation (6).
ANNALS OF MATHEMATICS
Vol. 66, No. 1, July, 1957 © 1957 The Johns Hopkins University Press. Reprinted with permission.
AN EXAMPLE OF A SMOOTH LINEAR PARTIAL DIFFERENTIAL EQUATION WITHOUT SOLUTION BY HANS
LEWY
(Received F e b r u a r y 10, 1957)
Introduction In dealing with the existence of solutions of partial differential equations it was customary during the nineteenth century and it still is today in many applications, to appeal to the theorem of Cauchy-Kowalewski which guarantees the existence of analytic solutions for analytic partial differential equations. On the other hand a deeper understanding of the nature of solutions requires the admission of non-analytic functions in equations and solutions. For large classes of equations this extension of the range of equation and solution has been carried out since the beginning of this century. In particular much attention has been given to linear partial differential equations and systems of such. Uniformly the experience of the investigated types has shown that—speaking of existence in the local senSe—there always were solutions, indeed, smooth solutions, provided the equations were smooth enough. It was therefore a matter of considerable surprise to this author, to discover that this inference is in general erroneous. More precisely, there exist linear partial differential equations with coefficients in C°° which possess not a single smooth solution in any neighborhood. The example to be presented in this paper is an equation of first order in three independent variables with complex-valued coefficients and unknown function, or, what amounts to the same, a system of two equations of first order for two functions of three variables, all real. A theorem We begin the discussion of this example by first deriving the following THEOREM. Let Xi, x>, f/i be independent real variables, u a dependent complexvalued variable and \p(yi) a real function of C . Consider the linear equation (1)
[-(d/dxi)
- iid/dxt) + 2i(xi + ixiXd/duMu
=
f'bii).
Assume that there exists a solution u(xi , x>, iji) of (1) in a neighborhood N = iV(0, 0, y\) of the point (0, 0, y\), with u in Cl. Then ^(//i) *s analytic at yx = y\. PROOF. Integrate (1) over a circle in N of equations xl + xi = const. = 1/2,
iji
= const.
On introducing the angle d by Xi -f- ix2 = yle'" we find (2)
d/dXi + id/dx2 = a(d/d log 7/2 + id/dd)
with a determined by applying (2) on log y\. Therefore a — (xt + ix2)y^1 and
308 156
HANS LEWY j.2r
/
Jo
.2ir
eieyV\{d/d log y\)u + i(d/d0)u] d6
(d/dxi + id/dx2)u de = /
Jo 2ir
/
e^ylHid/d log y\)u + u] d0
= (d/dyi)2 / Jo
e'ylu dB.
Setting (3)
U(yi,y,)
= i f
e"y\udB
Jo
we obtain from (1) (4)
dU/dyx + tdU/dy* = T^foi).
Note that (3) implies U(yi , 0 ) = 0. Furthermore, V(yi, 2/2) = V(y) = Ufa,
2/2) - WfoO
1
is in C and satisfies d P / t y i + idV/dyi
= 0
which states that V(y) is an analytic function of y whose domain of existence certainly includes all those points (2/1, 2/2) for which y2 > 0 is sufficiently small and 2/1 is such that {x\, x2,2/1) lies in N for xi + x2 ^ 2/2 • As V = — ir^ on 2/2 = 0 with ^ real by hypothesis, V can be continued across 2/2 = 0 as analytic function of y. Therefore V is analytic on 2/2 = 0. Hence ^(2/1) is analytic at and near 0
>A = 2/iWe apply the Theorem in a negative sense. Suppose we take an equation (1) in which ^(2/1) is real and in C°°, but not analytic at 2/1 = J/i, then the equation (1) can have no solution u which is in C in any .V = iV(0, 0, 2/1). It becomes desirable to remove the restriction to special neighborhoods which occurs in this example. Example of an equation without solution With the aid of a periodic real function ^(2/1.) of C°°, which is analytic in no 2/i-interval, we can construct a function F(x\, x 2 , iji) of C°° such that (5)
[-d/dxi
- id/dXi + 2i(xi + ix2)d/dyi}u = F(xi , x 2 , 2/1)
has no H -solution, no matter what open (x\, x 2 , >j\)-set is taken as domain of existence. A function is said to be in H* if its first partial derivatives satisfy a Holder condition with positive exponent, provided the distance of the points involved does not exceed 1. Choose a countable set of points Pi ,P*, • • • , which is dense in the (xi, x 2 , iji)space, and positive radii pi, p 2 , • • • with lim,,^ p„ = 0 and denote by N j the sphere of radius p, about P , . An arbitrary open set always contains some
309 A DIFFERENTIAL EQUATION WITHOUT SOLUTION
157
A^j . Designate by py and q, the x\ and .r2-coordinates of Pj and put Cj = m a x [j, | pj |, | gy |].
Consider the sequences e of real numbers £i, £2, • • • with (6)
l . U . b . j _ i . . . . . . I £j\
< CO
= £ r efi?W(yi
+ 2ffix, - 2p,x 2 ).
and set Ft(Xl, x2,
Vl)
F r is itself in C°°, for if we formally construct a j- t h derivative Z>" by termwise differentiation D'Fc(Xl,
xt, 2/0 = E " - i * * 7 V ' + u ( l / i + 2 ^ , -
2p J :r 2 )?; i (-p,r2' 1 + ' 2 > »"1 +
^2 = V
the series of the right member converges absolutely and uniformly as ip(y+>) is bounded on account of the periodicity of ^ and ^27=i CJC' < °° • Recall that the sequences f with norm (6) form a complete metric (Banach) space and consequently this space is not exhausted by a countable sum of nondense sets. This fact enables us to demonstrate the existence of a sequence £* in this space such that for F = Fc* there is no open set of points (x,, .T2 , 2/1) on which (5) has an H -solution. l PROOF. Denote by H nm the property of a function of having first partial derivatives satisfying a Holder condition of exponent \/n and constant m, where n, m = 1, 2,3, • • • . Evidently Hl = ^Z„,m F ^ a n d all functions which vanish at Pj and satisfy Hnm are compact so that the functions of H vanishing at Pj and existing in Nj are all contained in a countable sum of compact sets. Let Ejnm be the set of sequences E such that (5) with F = Fr has a solution existing in Nj and belonging there to Hlnm . Einn is closed for the following reason: we can always suppose that the solutions vanish at Pj and out of any sequence of solutions belonging to various e of Eilim tending to a limit sequence we can select a suitable subsequence of sequences e and corresponding solutions u of (5) with F = Fc such that the solutions converge in .V, together with their first partials. The limit function then satisfies the limit equation in A'; and lies h\H„,„ , proving that Ej„m is closed. But Ejnm is nondense. In view of the closure property of Ejlim it suffices to ascertain that every sequence of Ejlim is limit of sequences of the Banach space none of which is in Ej„m . Observe that if a and /3 arc elements of EjHm so is (a — /3)/2. Consider the particular sequence 5'' = (8\ , 52 , • • • ) where 5^ is the Kronecker symbol. It will be seen in the last § that (5) has no Cl-solution in Nj for F = Fit . Hence since (5) is linear, none of the sequences c + \8', X ^ 0, e in Ej,im , can be Ejnm . Taking j X j arbitrarily small, we obtain the sequence E of Ej„m as limit of sequences not in Ejnm . It follows that• ^ j , „ . m Ej„m does not exhaust the space of all sequences E
310 158
HANS LEWY
with (6). Let £* be a sequence not in X ^j»« • Then (5) with F = Fc. has no H -solution in any Nj, j = 1, 2, • • • or any other open set. The only thing remaining is the proof that Lu = [ — d/dxi — id/dXi + 2i(xi + ix^)d/dy^\u (7) = f'iy!
+ 2qjXi -
2pjX2)
has no C -solution in Nj. Consider the transformation Xi = xi - pj,
X2 = x2 — qj,
Fi = i/i -f- 2g,-a:1 — 2pyx2
whose inverse is xi = Xi + PJ ,
x2 = X 2 + qj,
?/i = Yx - 2qjXi + 2p,X 2
and for which d/dxi = (d/dXt) + 2qj(d/dY1),
d/dxi = (d/dX2) -
d/dyi =
2pj(d/dY1),
d/dYi.
We verify that (7) is transcribed into (8) Lu = [-d/dXi - id/dX, + 2i(X1 + iX,)d/dYAu = *'(Ki). If (7) had a C1 -solution in Nj, (8) would have a C l -solution in the neighborhood of the transform of the center of Nj, whose new coordinates are X, = X, = 0,
Fi = Y\ = 2 / ^ ) .
But here our Theorem applies that for the solution to exist i£( Fi) would have to be analytic at this F?, contrary to the hypothesis about \f/. U N I V E R S I T Y O F C A L I F O R N I A AT B E R K E L E Y
Wolf Prize in Mathematics, Vol. 2 (pp. 311-358) eds. S. S. Chern and F. Hirzebruch © 2001 World Scientific Publishing Co.
Curriculum Vitae
Born: March 9, 1948 in Budapest, Hungary Family. Married, 4 children Citizenship: Hungarian. Permanent resident in the US. Degrees: Dr.Rher.Nat., Eotvos Lorand University, Budapest, Hungary, 1971 Candidate of Math. Sci., Hungarian Academy of Sciences, Budapest, Hungary, 1970 Dr.Math.Sci., Hungarian Academy of Sciences, Budapest, Hungary, 1977 Elected memberships and positions in scientific societies: Hungarian Academy of Sciences, corresp. member, 1979; reg.member, 1985 European Academy of Sciences, Arts and Humanities, 1981 Academia Europaea, 1991 Nordrhein-Westfalische Akademie der Wissenschaften, corresponding member, 1993 Presiduum of the Hungarian Academy of Sciences, 1990-1993 Executive Committee of the International Mathematical Union, 1986-1994 Positions held: Research Associate, Eotvos Lorand University, Budapest, 1971-75 Docent, Jozsef Attila University, Szeged, 1975-78 Professor, Chair of Geometry, Jozsef Attila University, Szeged, 1978-82 Professor, Chair of Computer Science, Eotvos Lorand University, Budapest, 1983-93. Professor, Dept. of Computer Science, Yale University, 1993-99 Senior Researcher, Microsoft Research, 1999Visiting positions: Vanderbilt University, 1972/73 University of Waterloo, 1978/79 Universitat Bonn, 1984/85 University of Chicago, Spring 1985 Cornell University, Fall 1985 Mathematical Sciences Research Institute, Berkeley, Spring 1986 Princeton University, Fall 1987, Spring 1989, 1990/91, 1992/93
312 Honorary Degrees and Positions: Adjunct Professor, University of Waterloo, Waterloo, Ontario, Canada, 1980A.D.White Professor-at-Large, Cornell University, Ithaca, NY, 1982-1987 Honorary Professor, Universitat Bonn, 1984 John von Neumann Professor, Universitat Bonn, 1985 Honorary Professor, Academia Sinica, 1988 Doctor Honoris Causa, University of Waterloo, Ontario, Canada, 1992 Doctor Honoris Causa, University of Szeged, Hungary, 1999 Awards: Griinwald Geza Prize, Bolyai Society, 1970 George Polya Prize, Soc. Ind. Appl. Math., 1979 Best Information Theory Paper Award, IEEE, 1981 Ray D.Fulkerson Prize, Amer. Math. Soc.-Math. Prog. Soc, 1982 State Prize, Hungary, 1985 Tibor Szele Medal, Bolyai Society, 1992 Brouwer Medal, Dutch Matematical Society Royal Netherlands Academy of Sciences, 1993 National Order of Merit of Hungary, 1998 Bolzano Medal, Czech Mathematical Society, 1998 Wolf Prize, 1999 Knuth Prize, ACM SIGACT 1999 Editorial Boards: Combinatorica (Editor-in-Chief), Advances in Mathematics, J. Combinatorial Theory (B), Discrete Math., Discrete Applied Math., J. Graph Theory, Europ. J. Combinatorics, Discrete and Computational Geometry, Random Structures and Algorithms, Acta Mathematica Hungarica, Acta Cybernetica, Electronic Journal of Combinatorics Field of research: Combinatorial optimization, graph theory, theoretical computer science
313 List of Publications Books Kombinatorika (Pelikan J. es Vesztergombi K.) Tankonyvkiado, Budapest, 1977 (German translation: Teubner, 1977; Japanese translation: 1985). Algoritmusok (Gacs P.), Miiszaki Konyvkiado, Budapest, 1978; Tankonyvkiado, Budapest, 1987. Combinatorial Problems and Exercises, Akademiai Kiado - North-Holland, Budapest, 1979 (Japanese translation: Tokai Univ. Press, 1988; Hungarian translation: Typotech, 1999). Matching Theory (M. D. Plummer), Akademiai Kiado - North-Holland, Budapest, 1986 (Russian translation: Mir, 1998). An Algorithmic Theory of Numbers, Graphs, and Convexity, CBMS-NSF Regional Conference Series in Applied Mathematics 50, SIAM, 1986. Geometric Algorithms and Combinatorial Optimization (with M. Grotschel and A. Schrijver), Springer, 1988; Chinese edition: World Publishing Corp., Beijing, 1990. Greedoids (B. Korte and R. Schrader), Springer, 1991. Research papers 1965 Fiiggetlen koroket nem tartalmazo grafokrol (On graphs containing no independent circuits), Mat. Lapok 16, 289-299. 1966 On decomposition of graphs, Studia Math. Hung. 1, 237-238. 1967 On connected sets of points, Annates Univ. R. Eotvos 10, 203-204. Uber die starke Multiplication von geordneten Graphen, Acta Math. Hung. 18, 235-241. Operations with structures, Acta Math. Hung. 18, 321-328. 1968 Graphs and set-systems, in Beitrage zur Graphentheorie, Teubner, Leipzig, 99-106. On chromatic number of graphs and set-systems, Acta Math. Hung. 19, 59-67.
314 On covering of graphs, in Theory of Graphs (eds. P. Erdos and G. Katona), Akad. Kiado, Budapest, 231-236. 1969 Kapcsolatok polinomoknak es helyettesitesi ertekeiknek szamelmeleti tulajdonsagai kozott, Mat. Lapok 20, 129-132. 1970 Generalized factors of graphs, in Combinatorial Theory and its Applications, Coll. Math. Soc. J. Bolyai 4, 773-781. Subgraphs with prescribed valencies, J. Comb. Theory 8, 391-416. A generalization of Konig's theorem, Acta Math. Hung. 21, 443-446. A remark on Menger's theorem, Acta Math. Hung. 21, 365-368. The factorization of graphs, in Combinatorial Struc. AppL, Gordon and Breach, 243-246. Representation of integers by norm-forms II (with K. Gyory), Publ. Math. Debrecen 17, 173-181. 1971 On the cancellation law among finite relational structures, Periodica Math. Hung. 1, 145-156. On finite Dirichlet series, Acta Math. Hung. 22, 227-231. On the number of halving lines, Ann. Univ. Eotvos 14, 107-108. 1972 Normal hypergraphs and the perfect graph conjecture, Discrete Math. 2, 253-267; reprinted Ann. Discrete Math. 21 (1984) 29-42. On the structure of factorizable graphs, Acta Math. Hung. 23, 179-195. The factorization of graphs II, Acta Math. Hung. 23, 223-246. On the structure of factorizable graphs II, Acta Math. Hung. 23, 465-478. Direct product in locally finite categories, Acta Sci. Math. Szeged 23, 319-322. A characterization of perfect graphs, J. Comb. Theory 13, 95-98; reprinted in Classic Papers in Combinatorics (eds. I. Gessel and G. C. Rota), Birkhauser, 1987, 447-450. A note on the line reconstruction problem, J. Comb. Theory 13, 309-310; reprinted in Classic Papers in Combinatorics (ed. I. Gessel and G. C. Rota), Birkhauser, 1987, 451-452. A note on factor-critical graphs, Studia Sci. Math. 7, 279-280.
315 1973 On the eigenvalues of trees (with J. Pelikan), Periodica Math. Hung. 3, 175-182. Antifactors of graphs, Periodica Math. Hung. 4, 121-123. Dissection graphs of planar point sets (with P. Erdos, G. J. Simmons and E. G. Strauss), in A Survey of Comb. Theory (ed. S. Srivastava), Springer, 139-149. A note to a paper of Dudley (with P. Major), Studia Sci. Math. 8, 151-152. Permutation groups and almost regular graphs (with L. Babai), Studia Sci. Math. 8, 141-150. Finite homeomorphism groups of the 2-sphere (with L. Babai and W. Imrich), in Topics in Topology, Coll. Math. Soc. J. Bolyai 9, 61-75. Connectivity in digraphs, J. Comb. Theory 15, 174-177. Independent sets in critical chromatic graphs, Studia Sci. Math. 8, 165-168. On the sum of matroids (with A. Recski), Acta Math. Hung. 24, 329-333. Coverings and colorings of hypergraphs, in Proc. Ath Southeastern Conf. on Comb., Utilitas Math., 3-12. Factors of graphs, in Proc. Ath Southeastern Conf. on Comb., Utilitas Math., 13-22. 1974 Valencies of graphs with 1-factors, Periodica Math. Hung. 5, 149-151. Minimax theorems for hypergraphs, in Hypergraph Seminar (eds. C. Berge and D. K. Ray-Chaudhuri), Lecture Notes in Math. 411, Springer, 111-126. Every directed graph has a semi-kernel (with V. Chvatal), in Hypergraph Seminar (eds. C. Berge and D. K. Ray-Chaudhuri), Lecture Notes in Math. 411, Springer, 175. Applications of product coloring (with D. Greenwell), Acta Math. Hung. 25, 335-340. A family of planar bicritical graphs (with M. D. Plummer), in Combinatorics, London Math. Soc. Lecture Notes 13, 103-108. 1975 Problems and results on 3-chromatic hypergraphs and some related questions (with P. Erdos), in Infinite and Finite Sets, Coll. Math. Soc. J. Bolyai 11, 609-627. On bicritical graphs (with M. D. Plummer), in Infinite and Finite Sets, Coll. Math. Soc. J. Bolyai 11, 1051-1079. A characterization of cancellable fc-ary structures (with R. Appleson), Periodica Math. Hung. 6, 17-19. A family of planar bicritical graphs (with M. D. Plummer), Proc. London Math. Soc. 30, 160-176.
316
Three short proofs in graph theory, J. Comb. Theory 19, 269-271. Spectra of graphs with transitive groups, Periodica Math. Hung. 6, 191-195. 2-matchings and 2-covers of hypergraphs, Acta Math. Hung. 26, 433-444. On the ratio of optimal fractional and integral covers, Discrete Math. 13, 383-390. A kombinatorika minimax teteleirol (On the minimax theorems of combinatorics), Mat. Lapok 26, 209-264. 1976 The number of values of Boolean functions (with D. E. Daykin), J. London Math. Soc. 30, 160-176. On two minimax theorems in graph theory, J. Comb. Theory B21, 93-103. On graphs of Ramsey type (with S. A. Burr and P. Erdos), Ars Combinatoria 1, 167-190. On some connectivity properties of Eulerian graphs, Acta Math. Hung. 28, 129-138. Covers, packings and some heuristic algorithms, in Combinatorics, Proc. 5th British Comb. Conf. (eds. C. St. J. A. Nash-Williams and J. Sheehan), Utilitas Math., 417-429. On the number of complete subgraphs of a graph (with M. Simonovits), in Combinatorics, Proc. 5th British Comb. Conf. (eds. C. St. J. A. Nash-Williams and J. Sheehan), Utilitas Math., 439-441. A forbidden substructure characterization of Gauss codes (with M. Marx), Bull. Amer. Math. Soc. 82, 121-122. A forbidden substructure characterization of Gauss codes (with M. Marx), Acta. Sci. Math. Szeged 38, 115-119. Chromatic number of hypergraphs and linear algebra, Studia Sci. Math. 11, 113-114. 1977 Some remarks on generalized spectra (with P. Gacs), Zeitschr. Math. Logik Grundlagen Math. 23, 547-554. Certain duality principles in integer programming, Ann. Discrete Math. 1, 363-374. A homology theory for spanning trees of a graph, Acta Math. Hung. 30, 241-251. On minimal elementary bipartite graphs (with M. D. Plummer), J. Comb. Theory B23, 127-138. Flats in matroids and geometric graphs, in Combinatorial Surveys, Proc. 6th British Comb. Conf., Academic Press, 45-86. Polynomes de la matrice des distences d'un arbre (with R. L. Graham), in Problemes Combinatoires et Theorie de Graphes, CNRS, 189-190.
317 1978 Distance matrices of trees (with R. L. Graham), in Theory and Appl. of Graphs, Lecture Notes in Math. 642, Springer, 186-190. Distance matrix polynomials of trees (with R. L. Graham), Adv. Math. 29, 60-88. Mengerian theorems for paths with bounded length (with M. D. Plummer and V. Neumann-Lara), Periodica Math. Hung. 9, 269-276. Some finite basis theorems in graph theory, in Combinatorics, Coll. Math. Soc. J. Bolyai 18, 717-729. Restricted permutations and the distribution of Stirling numbers (with K. Vesztergombi), in Combinatorics, Coll. Math. Soc. J. Bolyai 18, 731-738. Kneser's conjecture, chromatic number, and homotopy, J. Comb. Theory A25, 319-324. 1979 Topological and algebraic methods in graph theory, in Graph Theory and Related Topics, Academic Press, 1-14. Strong independence of graphcopy functions (with P. Erdos and J. Spencer), in Graph Theory and Related Topics, Academic Press, 165-172. Graph theory and integer programming, Ann. Discrete Math. 4, 141-158. On the Shannon capacity of a graph, IEEE Trans. Inform. Theory 25, 1-7. An algorithm to prevent the propagation of certain diseases at minimum cost (with A. Hajnal), in Interfaces between Computer Science and Operations Research, Amsterdam Math. Centr. Tract 99, 105-108. Grafelmelet es diszkret programozas (Graph theory and discrete programming), Mat. Lapok 27, 69-86. Determinants, matchings, and random algorithms, in Fundamentals of Computation Theory, FCT'79 (ed. L. Budach), Akademie-Verlag Berlin, 565-574. Random walks, universal travelling sequences, and the complexity of maze problems (with R. Aleliunas, R. M. Karp, R. J. Lipton and C. W. Rackoff), Proc. 20th IEEE Ann. Symp. on Found, of Comp. Scl, 218-223. 1980 On a product dimension of graphs (with A. Pultr and J. Nesetril), J. Comb. Theory B29, 47-67. Selecting independent lines from a family of lines in a space, Acta Sci. Math. Szeged 42, 121-131. Matroid matching and some applications, J. Comb. Theory B28, 208-236. The matroid matching problem, in Algebraic Methods in Graph Theory, Coll. Math. Soc. J. Bolyai 25, 495-517.
318 Matroids and Sperner's Lemma, Europ. J. Combin. 1, 65-66. Efficient algorithms: an approach by formal logic, in Studies on Math. Programming (ed. A. Prekopa), Akademiai Kiado, 119-126. 1981 Mathematical structures underlying greedy algorithms (with B. Korte), in Fundamentals of Computation Theory (ed. F. Gecseg), Lecture Notes in Comp. Sci. 117, Springer, 205-209. On additive arithmetic functions satisfying a linear recursion (with A. Sarkozi and M. Simonovits), Ann. Univ. Ebtos 24, 205-215. The ellipsoid method and its consequences in combinatorial optimization (with M. Grotschel and A. Schrijver), Combinatorica 1, 169-197. Remarks on a theorem of Redei, Studia Sci. Math. Hung. 16, 449-454. Cycles through given vertices of a graph (with A. J. Bondy), Combinatorica 1, 117-140. Khachiyan's algorithm for linear programming (with P. Gacs), Math. Prog. Study 14, 61-68. 1982 Factoring polynomials with rational coefficients (with A. K. Lenstra and H. W. Lenstra) Math. Ann. 261, 515-534. Brick decompositions and the matching rank of graphs (with J. Edmonds and W. R. Pulleyblank), Combinatorica 2, 247-274. On generic rigidity in the plane (with Y. Yemini), SI AM J. Alg. Discr. Methods 1, 91-98. Some combinatorial applications of the new linear programming algorithms, in: Combinatorics and Graph Theory (ed. S. B. Rao), Lecture Notes in Math. 885, Springer, 33-41. Bounding the independence number of a graph, Ann. Discr. Math. 16, 213-223. Selected topics of matroid theory and its applications (with A. Recski), Suppl. Rendiconti del Circ. Mat. Palermo 2, 171-185. Borsuk's Theorem and the number of facets of centrally symmetric polytopes (I. Barany), Acta Math. Hung. 40, 323-329. 1983 Perfect graphs, in More Selected Topics in Graph Theory (eds. L. W. Beineke and R. L. Wilson), Academic Press, 55-67. Ear-decompositions of matching-covered graphs, Combinatorica 3 105-117. Structural properties of greedoids (with B. Korte), Combinatorica 3 359-374.
319 Submodular functions and convexity, in Mathematical Programming: The State of the Art (eds. A. Bachem, M. Grotschel and B. Korte), Springer, 235-257. Self-dual polytopes and the chromatic number of distance graphs on the sphere, Acta Sci. Math. Szeged 45, 317-323. Remarks on a theorem of Redei (with A. Schrijver), Studia Math. Hung. 16, 449-454. On the number of complete subgraphs of a graph II (with M. Simonovits), in Studies in Pure Math., to the memory of P. Turan (ed. P. Erdos), Akademiai Kiado, 459-495. 1984 Algorithmic aspects of combinatorics, geometry and number theory, in Proc. Int. Congress Warsaw 1982, Polish Sci. Publishers - NorthHolland, 1591-1595. Geometric methods in combinatorial optimization (with M. Grotschel and A. Schrijver), in Progress in Combinatorial Optimization (ed. W. R. Pulleyblank), Academic Press, 167-183. Greedoids - A structural framework for the greedy algorithm (B. Korte), in Progress in Combinatorial Optimization (ed. W. R. Pulleyblank), Academic Press, 221-243. Polynomial algorithms for perfect graphs (with M. Grotschel and A. Schrijver), Ann. Discrete Math. 2 1 , 325-256. A polynomial-time test for total dual integrality in fixed dimension (with W. Cook and A. Schrijver), Math. Programming Study 22, 64-69. Corrigendum to our paper "The ellipsoid method and its consequences in combinatorial optimization" (with M. Grotschel and A. Schrijver), Combinatorica 4, 291-295. Greedoids and linear objective functions (with B. Korte), SIAM J. Algebraic and Discrete Methods 5, 229-238. Polynomial factorization and the nonrandomness of bits of algebraic and some transcendental numbers (with R. Kannan and A. K. Lenstra), in Proc. 16th ACM Symp. on Theory of Computing, 191-200. Shelling structures, convexity, and a happy end (with B. Korte), in Graph Theory and Combinatorics (ed. B. Bollobas), Academic Press, 219-232. 1985 A note on selectors and greedoids (with B. Korte), Euro. J. Combinatorics 6, 59-67. Posets, matroids, and greedoids (with B. Korte), in Matroid Theory, Coll. Math. Soc. J. Bolyai 40 (eds. L. Lovasz and A. Recski), North-Holland, 239-265. Polymatroid greedoids (with B. Korte), J. Comb. Theory B 38, 41-72.
320
Basis graphs of greedoids and 2-connectivity (with B. Korte), Math. Programming Study 24, 158-165. Computing ears and branchings in parallel, 26th IEEE Annual Symp. on Found, of Corny. Sci., 464-467. Some algorithmic problems on lattices, in Theory of Algorithms (eds. L. Lovasz and E. Szemeredi), Coll. Math. Soc. J. Bolyai 44, NorthHolland, 323-337. Vertex packing algorithms, Proc. ICALP Conf, Springer. Homotopy properties of greedoids (with A. Bjorner and B. Korte), Adv. Appl. Math. 6, 447-494. Relations between subclasses of greedoids (with B. Korte), Zeitschr. Oper. Res. A: Theorie 29, 249-267. 1986 Algorithmic aspects of some notions in classical mathematics, in Mathematics and Computer Science (eds. J. W. de Bakker, M. Hazewinkel and J. K. Lenstra), CWI Monographs 1, North-Holland, 51-63. Relaxations of vertex packing (with M. Grotschel and A. Schrijver), J. Combin. Theory B40, 330-343. Discrepancy of set-systems and matrices (with J. Spencer and K. Vesztergombi), Euro. J. Combin. 7, 151-160. Non-interval greedoids and the transposition property (with B. Korte), Discrete Math. 59, 297-314. A note on perfect graphs (with K. Cameron and J. Edmonds), Periodica Math. Hung. 17, 173-175. A physical interpretation of graph connectivity (with N. Linial and A. Wigderson), Proc. 27th Annual IEEE Symp. on Found, of Comp. Sci., 39-48. Covering minima and lattice point free convex bodies (with R. Kannan), Proc. Conf. on Foundations of Software Technology and Theoretical Comp. Sci., Lecture Notes in Comp. Science 241, Springer, 193-201. Connectivity algorithms using rubber bands, Proc. Conf. on Foundations of Software Technology and Theoretical Comp. Sci., Lecture Notes in Comp. Science 241, Springer, 394-411. Searching in trees, series-parallel and interval orders (with U. Faigle, R. Schrader and G. Turan), SI AM J. Computing 15, 1075-1084. Homomorphisms and Ramsey properties of antimatroids (with B. Korte), Discrete Appl. Math. 15, 283-290. 1987 The chromatic number of Kneser hypergraphs (with N. Alon and P. Frankl), Trans. Amer. Math. Soc. 298, 359-370.
321 On some combinatorial properties of algebraic matroids (with A. Dress), Combinatorica 7, 39-48. Matching structure and the matching lattice, J. Comb. Theory B43, 187-222. Pseudomodular lattices and continuous matroids (with A. Bjorner), Acta Sci. Math. Szeged 51, 295-308. 1988 Rubber bands, convex embeddings, and graph connectivity (with N. Linial and A. Wigderson), Combinatorica 8, 91-102. Geometry of numbers: An algorithmic view, in ICIAM '87: Proc. 1st Internatl. Conf. on Industr. Appl. Math. (eds. J. McKenna and R. Teman), SIAM, 144-152. Lattices, Mobius functions and communication complexity (with M. Saks), 29th IEEE Annual Symp. Found. Comp. Sci., 81-90. The chromatic number of the graph of large distances (with P. Erdos and K. Vesztergombi), in Combinatorics, Proc. Coll. Eger 1987, Coll. Math. Soc. J. Bolyai 52, North-Holland, 547-551. How to tidy up your set-system? (with C. A. J. Hurkens, A. Schrijver and Eva Tardos), in Combinatorics, Proc. Coll. Eger 1987, Coll. Math. Soc. J. Bolyai 52, North-Holland, 309-314. The intersection of matroids and antimatroids (with B. Korte), Discrete Math. 73, 143-157. Covering minima and lattice point free convex bodies (with R. Kannan), Ann. Math. 128, 577-602. 1989 Extremal problems for discrepancy (with K. Vesztergombi), in Irregularities of Partitions (eds. G. Halasz and V. T. Sos), Algorithms and Combinatorics 8, Springer, 107-113. Orthogonal representations and connectivity of graphs (with M. Saks and A. Schrijver), Linear Alg. Appl. 114/115, 439-454. Examples and algorithmic properties of greedoids (with B. Korte and O. Goecke), in Combinatorial Optimization, Como 1986 (ed. B. Simeone), Lecture Notes in Math. 1403, Springer, 113-161. Disks, balls and walls: The analysis of a combinatorial game (with R. Anderson, P. Shor, J. Spencer, E. Tardos and S. Winograd), Amer. Math. Monthly. An on-line graph coloring algorithm with sublinear performance ratio (with M. Saks and W. T. Trotter), Discrete Math. 75, 319-325. Faster algorithms for hard problems, Information Processing '89 (ed. G. X. Ritter), Elsevier, 135-141.
322
Polyhedral results for antimatroids (with B. Korte), in Combinatorial Mathematics, Proc. 3rd Intern. Conf. (eds. G. S. Bloom, R. L. Graham and J. Malkevitch), Annals of the NY Academy of Sciences 555, 283-295. On the graph of large distances (with P. Erdos and K. Vesztergombi), Discrete Comput. Geometry 4, 541-549. Geometry of numbers and integer programming, in Mathematical Programming, Recent Developments and Applications, Kluwer, 177-201. Singular spaces of matrices and their application in combinatorics, Bol. Soc. Braz. Mat. 20, 87-99. Some recent results on graph matching (with M. D. Plummer), in Graph Theory and its Applications: East and West (eds. M. F. Capobianco, M. Guan, D. F. Hsu and F. Tian), Ann. NY Acad. Sci. 576, 389-398. On the number of halving planes (with I. Barany and Z. Fiiredi), Proc. 5th Symp. Comp. Geom., 140-144. 1990 Matrix cones, projection representations, and stable set polyhedra (with A. Schrijver), in Polyhedral Combinatorics, DIMACS Series in Discrete Mathematics and Theoretical Computer Science I, 1-17. The shapes of polyhedra (with R. Kannan and H. E. Scarf), Math. Oper Res. 15, 364-380. The mixing rate of Markov chains, an isoperimetric inequality, and computing the volume (with M. Simonovits), Proc. 31st IEEE Annual Symp. on Found, of Comp. Sci., 346-354. Communication complexity: A survey, in Paths, flows, and VLSI-Layout, (eds. B. Korte, L. Lovasz, H. J. Promel and A. Schrijver), Springer, 235-265. Entropy splitting for antiblocking pairs and perfect graphs (with I. Csiszar, J. Korner, K. Marton and G. Simonyi), Combinatorica 10, 27-40. On the number of halving planes (with I. Barany and Z. Fiiredi), Combinatorica 10, 175-183. 1991 Graphs with given automorphism group and few edge orbits (with L. Babai and A. J. Goodman), Euro. J. Combin. 12, 185-203. Chip-firing games on graphs (with A. Bjorner and P. Shor), Euro. J. Combin. 12, 283-291. Cones of matrices and set-functions, and 0-1 optimization (with A. Schrijver), SIAM J. Optim. 1, 166-190. Geometric algorithms and algorithmic geometry, Proc. Int. Congress of Math, Kyoto, 1990, Springer-Verlag, 139-154.
323
Approximating clique is almost NP-complete (with U. Feige, S. Goldwasser, S. Safra and M. Szegedy), Proc. 32nd IEEE Annual Symp. on Found, of Comp. Sci., 2-12. Search problems in the decision tree model (with M. Naor, I. Newman and A. Wigderson), Proc. 32nd IEEE Annual Symp. on Found, of Comp. Sci., 576-585. 1992 A matching algorithm for regular bipartite graphs (with J. Csima), Discrete Appl. Math. 35, 197-203. How to compute the volume? Jber. d. Dt. Math.-Verein, Jubildumstagung 1990, B. G. Teubner, Stuttgart, 138-151. Linear decision trees, hyperplane arrangements, and Mobius functions (with A. Yao and A. Bjorner), in Proc. 24th ACM Symp. on Theory of Computing, 170-177. Two-prover one-round proof systems: their power and their problems (with U. Feige), in Proc. 24th ACM Symp. on Theory of Computing, 733-744. On integer points in polyhedra: a lower bound (with I. Barany and R. Howe), Combinatorica 12, 135-142. On the randomized complexity of volume and diameter (with M. Simonovits), Proc. 33rd IEEE Annual Symp. on Found, of Comp. Sci., 482-491. The generalized basis reduction algorithm (with H. Scarf), Math, of OR 17, 751-764. Chip-firing games on directed graphs (with A. Bjorner), J. Algebraic Combinatorics 1, 305-328. 1993 Dating to marriage (with J. Csima), Discrete Appl. Math. 41, 269-270. The cocycle lattice of binary matroids (with A. Seress), Euro. J. Combin. 14, 241-250. Random walks in a convex body and an improved volume algorithm (with M. Simonovits), Random Structures and Alg. 4, 359-412. Communication complexity and combinatorial lattice theory (with M. Saks), J. Comp. Sys. Sci. 47, 322-349. Note: on the last new vertex visited by a random walk (with P. Winkler), J. Graph Theory 17, 593-596. A Monte-Carlo algorithm for estimating the permanent (with N. Karmarkar, R. M. Karp, R. Lipton and M. Luby), SIAM J. Comput. 22, 284-293. 1994 Stable sets and polynomials, Discrete Math. 124, 137-153.
324
Linear decision trees, subspace arrangements, and Mobius functions (with A. Bjorner), J. Amer. Math. Soc. 7, 677-706. Chessboard complexes and matching complexes (with A. Bjorner, S. Vrecica and R. Zivaljevic), J. London Math. Soc. 49, 25-39. 1995 Search problems in the decision tree model (with M. Naor, I. Newman and A. Wigderson), SI AM J. Disc. Math. 8, 119-132. Isoperimetric problems for convex bodies and a localization lemma (with R. Kannan and M. Simonovits), Disc. Comput. Geometry 13, 541-559. Randomized algorithms in combinatorial optimization, in Combinatorial Optimization, Papers from the DIMACS Special Year (eds. W. Cook, L. Lovasz and P. Seymour), DIMACS Series in Discrete Mathematics and Combinatorial Optimization 20, AMS, 153-179. Exact mixing in an unknown Markov chain (with P. Winkler), Electronic J. Combinatorics 2, paper R15, 1-14. Mixing of random walks and other diffusions on a graph (with P. Winkler), in Surveys in Combinatorics, 1995 (ed. P. Rowlinson), London Math. Soc. Lecture Notes Series 218, Cambridge Univ. Press, 119-154. Efficient stopping Rules for Markov Chains (with P. Winkler), Proceedings of the 1995 ACM Symposium on the Theory of Computing, 76-82. The cocycle lattice of binary matroids II (with A. Seress), Linear Algebra and its Applications 226-228, 553-566. On the invariance of Colin de Verdiere's graph parameter under clique sums, (with A. Schrijver and H. v. d. Hoist), Linear Algebra and its Applications 226-228, 509-518. 1996 The rank and size of graphs (with A. Kotlov), J. Graph Theory 23, 185-189. Random walks on graphs: A survey, in Combinatorics, Paul Erdos is Eighty, Vol. 2 (eds. D. Miklos, V. T. Sos and T. Szonyi), Janos Bolyai Mathematical Society, Budapest, 353-398. Approximating clique is almost NP-complete (with U. Feige, S. Goldwasser, S. Safra and M. Szegedy), J. ACM 43, 268-292. 1997 The membership problem in jump systems, J. Comb. Theory B70, 45-66. On Conway's thrackle conjecture (with J. Pach and M. Szegedy), Discrete Comput. Geom. 18, 369-376. Random walks and an 0*(n5) volume algorithm for convex bodies (with R. Kannan and M. Simonovits), Random Structures and Algorithms 11, 1-50.
325
The Colin de Verdiere number and sphere representations of a graph (with A. Kotlov and S. Vempala), Combinatorica 17, 483-521. Mixing times for uniformly ergodic Markov chains (with D. Aldous and P. Winkler), Stochastic Processes and their Applications 7 1 , 165-185. 1998 A Borsuk theorem for antipodal links and a spectral characterization of linklessly embeddable graphs (with A. Schrijver), Proceedings of the American Mathematical Society 126, 1275-1285. Reversal of Markov chains and the forget time (with P. Winkler), Combinatorics, Probability and Computing 7, 189-204. Mixing times (with P. Winkler), in Microsurveys in Discrete Probability (eds. D. Aldous and J. Propp), DIMACS Series in Discrete Math, and Theor. Comp. Sci., AMS, 85-133. Random walks and the regeneration time (with A. Beveridge), J. Graph Theory 29, 57-62. 1999 The Colin de Verdiere graph parameter (with A. Schrijver), Proc. Conf. Combinatorics, in Graph Theory and Combinatorial Biology, Bolyai Soc. Math. Stud. 7, Janos Bolyai Math. Soc, 29-85. Lifting Markov chains to speed up mixing (with F. Chen and I. Pak), in Proc. 31st Annual ACM Symp. on Theory of Computing, ACM, 275-281. A logarithmic Cheeger inequality and mixing in random walks (with R. Kannan), in Proc. 31st Annual ACM Symp. on Theory of Computing, 282-287. On the null space of a Colin de Verdiere matrix (with A. Schrijver), Ann. ITnstitute Fourier 49, 1017-1026. Hit-and-run mixes fast, Math. Programming, series A 86, 443-461. Expository Papers A matroidelmelet rovid attekintese (A short survey of matroid theory), Mat. Lapok 22 (1971), 249-267. A szitaformularol (On the sieve formula), Mat. Lapok 23 (1972), 53-69. Kombinatorikus optimalizacio (Combinatorial optimization), Magyar Tudomdny 25 (1980), 736-742. A new linear programming algorithm: better or worse than Simplex Method?, Math. Intelligencer 2 (1980), 141-146. Mit ad a matematikanak es mit kap a matematikatol a szamitogeptudomany? (What does computer science get from
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mathematics and what does it give to it?), Magyar Tudomdny 35 (1990), 1041-1047. The mathematical notion of complexity, Proc. IF AC Symposium, Budapest, 1984. Algorithmic mathematics: an old aspect with a new emphasis, in Proc. 6th ICME, Budapest, 1988, J. Bolyai Math. Soc, 67-78. The work of A. A. Razborov, Proc. of Int. Congress of Math, Kyoto, 1990, Springer-Verlag, 37-40. Features of computer language: communication of computers and its complexity, Acta Neurochirurgica 56 [Suppl.] 91-95. Random walks, eigenvalues, and resistance, Appendix to Chap. 31, Tools from linear algebra, in Handbook of Combinatorics (eds. R. Graham, M. Grotschel and L. Lovasz), Elsevier (1995) 1740-1748. Combinatorial optimization (M. Grotschel), Chap. 28 of Handbook of Combinatorics (eds. R. Graham, M. Grotschel and L. Lovasz), Elsevier (1995) 1541-1597. Combinatorics in computer science (with D. B. Shmoys and E. Tardos), Chap. 40 of Handbook of Combinatorics (eds. R. Graham, M. Grotschel and L. Lovasz), Elsevier (1995) 2003-2038. Combinatorics in pure mathematics (with L. Pyber, D. J. A. Welsh and G. M. Ziegler), Chap. 41 of Handbook of Combinatorics (eds. R. Graham, M. Grotschel and L. Lovasz), Elsevier (1995) 2039-2082. Information and complexity (how to measure them?), in The Emergence of Complexity in Mathematics, Physics, Chemistry and Biology (ed. B. Pullman), Pontifical Academy of Sciences, Vatican City 1996, Princeton University Press, 65-80. One mathematics, The Berlin Intelligencer, Mitteilungen der Deutschen Math.-Verein, Berlin (1998), 10-15.
JOURNAL OF COMBINATORIAL THEORY, Series A 25. 319-324 (1978)
Note Kneser's Conjecture, Chromatic Number, and Homotopy L. LOVASZ Bolyai Institute, Jozsef A ttila University, H-6720 Szeged, Aradi vertanuk tere 1, Hungary Communicated by the Editors Received March 4, 1977
If the simplicial complex formed by the neighborhoods of points of a graph is (k — 2)-connected then the graph is not ^-colorable. As a corollary Kneser's conjecture is proved, asserting that if all n-subsets of a (2n — A:)-element set are divided into k + 1 classes, one of the classes contains two disjoint n-subsets.
1. INTRODUCTION
Kneser [6] formulated the following conjecture in 1955, whose proof is the main objective of this note. THEOREM 1. If we split the n-subsets of a (2n + k)-element set into k + I classes, one of the classes will contain two disjoint n-subsets.
It is easy to split the n-subsets into k + 2 classes so that the assertion does not remain valid. For let 1,..., 2n + k be the given elements and let Kt contain those subsets whose first element is i. Then Kx, K2,..., Kk+1, Kk+2. u • • • u Kk+n+1 is a partition of the n-subsets into k + 2 classes such that any two n-subsets in the same class intersect. Let us construct a graph KGn%k as follows. The vertices of KGnik are the n-subsets of {1,..., 2n + k) and two of them are joined by an edge iff they are disjoint. These graphs are often called Kneser's graphs. Note that KG2il is the well-known Petersen graph. Now Theorem 1 can be rephrased as follows: THEOREM
1'. The chromatic number of Kneser's graph KG„ik is k + 2.
This conjecture, or special cases of it, have turned out to play many roles in various fields of graph theory. In particular, the case n < 3 has been proved and applied by Garey and Johnson [4] and Stahl [5]. Here we mention the following: Kneser's graph has the property that each odd circuit of it has 319 Reprinted by permission of the publisher Copyright © 1978 by Academic Press, Inc. All rights of reproduction in any form reserved.
L. LOVASZ
320
lengths at least 2njk + 1. So if we know that it has high chromatic number, we see that Kneser's graph is an example of triangle-free high-chromatic graphs (and of even more). Erdos and Hajnal [2, 3] have constructed several other classes of graphs with similar properties, among others the following graph, which is often called Borsuk's graph. 1 Let the vertices of graph Bk be the points of the ^-sphere Sk, two of them being adjacent iff their distance is at least 2 — e for some e > 0 (i.e., iff they are almost antipodal). It is easy to see that if e is small, this graph contains no short odd circuits. The fact that its chromatic number is k + 2 is equivalent to the following well-known theorem of Borsuk: BORSUK'S THEOREM.
If
Sk = Fx u ••• u f
w
,
where F1,...,Fk+1
are
k
closed subsets ofS , then one of the sets F{ contains two antipodal points. We shall prove Theorem 1 by using some techniques of algebraic topology and Borsuk's theorem. In fact we shall derive a more general lower bound for the chromatic number of certain graphs. To formulate this result we need some preparation. Let G be a graph. Define the neighborhood complex Jf(G) as the simplicial complex whose vertices are the vertices of G and whose simplices are those subsets of V(G) which have a common neighbor. For any complex Jf, let Jf denote the polyhedron determined by Jf. A topological space T is called n-connected if each continuous mapping of the surface Sr of the (r + I) dimensional ball into T extends continuously to the whole ball, for r = 0, l,...,n. THEOREM
2. If ^(G)
COROLLARY.
is (k -\- 2)-connected then G is not k-colorable.
^V(.G) is never homotopically trivial.
In the case k = 2 we obtain: If the neighborhood complex of G is connected then G is not bipartite. This is trivial since the color-classes of any 2-coloration of G are components of Jf(G). For connected graphs the converse is also true if A: = 2: If G is not bipartite then any two vertices x, y can be connected by a walk x = x0, x1,..., x2j) = y of even length and then x 0 , x2,..., x2»-2 > r X23J i s a walk in ^V(G) connecting x to y, thus J (G) is connected. For k ^= 3 the condition of Theorem 2 is not necessary, which is shown by any graph with large chromatic number and girth. It seems to be an interesting question whether any topological property of ^(G) is equivalent to the A:-colorability of G. On the other hand, could Theorem 2 be strengthened 1
1 am indepted to Miklos Simonovits for pointing out the analogy between Kneser's and Borsuk's graphs, which is the underlying idea of this paper.
KNESER'S CONJECTURE
321
by considering homology instead of homotopy, or as follows ? If the (k — 2)dimensional homotopy group of J/~(G) is trivial, then the chromatic number of G differs from k. The fact that Kneser's graph satisfies the conditions of Theorem 2 is not quite obvious; we shall prove the following, slightly more general result: THEOREM 3. Let S be a finite set and n, k natural numbers. Consider the simplicial complex Jf whose vertices are the n-subsets ofS and whose simplices are those sets A0 ,..., Am of n-subsets for which
IM
< n + k.
Then J f is (k — \)-connected. Since for | S | = 2n + k the complex X above is the neighborhood complex of KG„ik, Theorems 2 and 3 together imply Theorem 1.
2. PROOF OF THEOREM 2
Let J^G) denote the barycentric subdivision of JV{G). The vertices of Jf-^G) are those sets XC V(G) whose elements have a common neighbor, and some of them span a simplex iff they form a chain with respect to inclusion. It is trivial that ^(G) and ^\(G) are homeomorphic. Let X C V(G) and denote by v(X) the set of common neighbors of X. Then v maps the set of vertices of ^V[(G) into itself, and since X C Y implies v{X) 2 v{Y), it is simplicial, i.e., maps the vertices of any simplex onto vertices of a simplex of •^l(G). Let us extend it simplicially to a continuous mapping of -^[(G) into itself. We denote this extended mapping by v. Note that v3 = v
and
v3 — v.
(1)
We define mappings 9r:
ST -> A{G)
(r = 0, 1,..., k - 1)
by induction on r such that
(2)
for all x e Sr (here — x is the point antipodal to x). First let r = 0 and v an arbitrary point of A(G). Set
322
L. LOVASZ
Second, let r ^ 1 and assume that cpr-^. S r _ 1 ->• *A^(G) is defined so that (2) holds. Denote by 5+ and S~ the upper and lower hemisphere of Sr, so that S+ n S~ = S r _ 1 . Let us extend <pr_! to a continuous mapping ifi: S+ —*• *^(G). This is possible by the assumption that *^(G) is ^-connected. Define now
if x e 5+,
,„
On 5 r _ 1 = S+ n S" the two definitions coincide, and in fact both yield
=
t\<pr-i(x))
= v(9r-l(
— x)) =
<pr-l(x),
since (2) is valid for r — 1. Thus (3) defines a continuous mapping of 5 r into ^ ( G ) . Moreover, if x e 5+ then
= ?2(«A(-x)) = v( 9r (x)).
So (2) is inherited and the definition of <pr is complete for all r < k — 1. Suppose now that G admits a ^-coloration. Let ^ denote the subcomplex of ~W(G) formed by those simplices whose vertices have a common neighbor of color i (1 < / < £). Then trivially
jr(G) = J[\J ••• u Jfk. Moreover, ^
n v(^) =
0.
Assume indirectly that x e JV{ and v(x) e JTt. Then x belongs to the simplex. of Jf(G) spanned by the neighborhood of a vertex v e V(G) of color /. In the barycentric subdivision ~^(G), x belongs to the interior of a unique simplex spanned by vertices of ^[(G), i.e., subsets of V(G), say X1,..., Xm; and we have X j , X2,..., Xm C v(v). Then y, = v(A'?) a r and v(x) is contained in the interior of the simplex of ^ ( G ) spanned by some of Y1, Y2,..., Ym . Since v(x) eJfi,\X follows similarly that there is a vertex u e V{G) of color / such that some F,- C v(w). But then u, v are adjacent vertices of G of color /, which is a contradiction. Now let Fj = cp^xi^K). Then Ft is a closed subset of S1"1 and clearly fj u ••• u f t = 5,fc_1. Also Ft contains no two antipodal points; in fact, if x eFj and —xeFt then 9k-i(X>
6
^
323
KNESER'S CONJECTURE
and
3. PROOF OF THEOREM 3
Let A = {A0,..., Am} be a simplex in Jf. Put m
U(A)
=[]Ai, i=0
and denote by M(A) the simplex spanned by all ^-subsets of U(A). We call A crowded if | U(A)\ < n + k. The proof goes by induction on \S\. For \S\ < n + k the assertion is obvious, since J f is a simplex. So we may assume that \ S\ > n + k. Let Jf' denote the closed subcomplex of c€ whose simplices are the crowded simplices of Jf, and let Cf0 be the subcomplex whose simplices are the simplices of dimension < k — 1 (the (k — l)-dimensional skeleton of df). First we show that Cf0 can be deformed into JT' in ct. We do so by defining a continuous mapping >p: X~0 —• Jf' such that (*)
for each simplex A of df0,
I/J(A)
lies in M(A).
This condition clearly implies that tfi is homotopic in Jf to the injection of cta into jf. We define tp(A) by induction on the dimension of A. Tf dim A = 0 then 4 is automatically crowded and we may set >p(A) = A. Assume now that dim A > 0 and that I/J is defined on the boundary A of A such that (*) is fulfilled. Consider the subcomplex jf'A of Jf' induced by the vertices of M{A). By the induction hypothesis, I/J(A) lies in Cfc'A and hence, by the other induction hypothesis on | S |, from which we know that c£"A is (r — l)-connected, we know that ifi can be extended over the interior of A. This completes the definition of
ifueXbutv£X,
Then cpuv is simplicial, i.e., if A = {A0,..., Am} is a simplex in Jf' then so is
324
L. LOVASZ
Am then U(tpuv(A)) C U(A); if w e A0 u ••• u Am and r £ ^ 0 u ••• u 4TO then. U(
I U(
k-l.
Thus 9oUK can be considered as a continuous mapping of X' into itself. Also observe that (**)
9uv(A) u A is contained in the simplex of j f spanned by the n-subsets of U(A) u {v}.
Therefore
This maps each w-subset of S on {u±,..., un}. Since it is, by the remark above, homotopic to the injection of X' into X, it follows that X' can be contracted in X to a single point. Since we have shown that X0 can be deformed into X', it follows that X0 can be contracted in X to a single point. This completes the proof.
REFERENCES 1. K. BORSUK, Drei Satze iiber die H-dimensionale euklidische sphare, Fund. Math. 20 (1933), 177-190. 2. P. ERDOS AND A. HAJNAL, On chromatic number of graphs and set-systems, Acta Math. Acad. Sci. Hungar. 17 (1966), 61-99. 3. P. ERDOS AND A. HAJNAL, Kromatikus grafokrol, Mat. Lapok 18 (1967), 1-4. 4. M. GAREY AND D . S. JOHNSON, The complexity of near-optimal graph coloring, J. Assoc. Comput. Much. 23 (1976), 43-49. 5. S. STAHL, n-tuple colorings and associated graphs, / . Combinatorial Theory Ser. B 20 (1976), 185-203. 6. M. KNESER, Aufgabe 300, Jber. Deutsch. Math.-Verein. 58 (1955).
333 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. IT-25, NO. 1, JANUARY 1979
On the Shannon Capacity of a Graph LASZLO LOVASZ
Abstract—It is proved that the Shannon zero-error capacity of the pentagon is V5 . The method is then generalized to obtain upper bounds on the capacity of an arbitrary graph. A well-characterized, and in a sense easily computable, function is introduced which bounds the capacity from above and equals the capacity in a large number of cases. Several results are obtained on the capacity of special graphs; for example, the Petersen graph has capacity four and a self-complementary graph with n points and with a vertex-rransitive automorphism group has capacity Vn .
I.
INTRODUCTION
L
ET THERE BE a graph G, whose vertices are letters in an alphabet and in which adjacency means that the letters can be confused. Then the maximum number of one-letter messages which can be sent without danger of confusion is clearly a(G), the maximum number of independent points in the graph G. Denote by ot(Gk) the maximum number of ^-letter messages which can be sent without danger of confusion (two ^-letter words are confoundable if for each 1 < / < k, their rth letters are confoundable or equal). It is clear that there are at least a(Gf such words (formed from a maximum set of nonconfoundable letters), but one may be able to do better. For example, if C5 is a pentagon, then a(C|) = 5. In fact, if i;,,' • • ,c 5 are the vertices of the pentagon (in this cyclic order), then the words 0,0,, o2°3> v3v5, t)4u2, and t)5o4 are nonconfoundable. It is easily seen that 0(G)=sup^ct(G*) = lim yja(Gk) . k
*—*
A general upper bound on 0(G) was also given in [6] (this bound was discussed in detail by Rosenfeld [5]). We assign nonnegative weights w(x) to the vertices x of G such that 2
w(x)
for every complete subgraph C in G; such an assignment is called a fractional vertex packing. The maximum of l£xw(x), taken over all fractional vertex packings, is denoted by a*(G). It follows easily from the duality theorem of linear programming that a*(G) can be defined dually as follows: we assign nonnegative weights q(C) to the cliques C of G such that
2 9(C) > i CBx
for each point x of G and minimize ~Zcq(C). With this notation Shannon's theorem states 6(G)
This number was introduced by Shannon [6] and is called the Shannon capacity of the graph G. The previous considII. THE CAPACITY OF THE PENTAGON eration shows that e(G)>a(G) and that, in general, equality does not hold. Let G be a finite undirected graph without loops. We The determination of the Shannon capacity is a very say that two vertices of G are adjacent if they are either difficult problem even for very simple small graphs. connected by an edge or are equal. Shannon proved that a(G)-8(G) for those graphs which The set of points of the graph G is denoted by V(G). can be covered by a(G) cliques (the best known such The complementary graph of G is defined as the graph G graphs are the so-called perfect graphs; see [1]). However, with V(G)= V(G) and in which two points are connected even for the simplest graph not covered by this result— by an edge iff they are not connected in G. A k-coloration •he pentagon—the Shannon capacity was previously un- of G is a partition of V(G) into k sets independent in G. known. Note that this corresponds to a covering of the points of the complementary graph by k cliques. The least k for which G admits a A-coloration is called its chromatic number. lanuscript received February 23, 1978; revised June 20, 1978. A permutation of V(G) is an automorphism if it prehe author is with the Department of Combinatorics, University of serves adjacency of the points. The automorphisms of G erloo, Walerloo, ON, Canada, on leave from the Bolyai Institute, ef Attila University, H-6720 Szeged, Aradi vertaniik t. 1, Hungary.
01979 IEEE
334 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. IT-25, NO. 1, JANUARY 1979
form a permutation group called the automorphism group hence of G. If for each pair of points JC, y G V(G) there exists an automorphism mapping x onto_y, then the automorphism 9(G°H)< max group is called vertex transitive. Edge transitivity is defined in an analog manner. A graph is called regular of = 9(G)9(H). degree d if each point is incident with d edges. Note that graphs whose automorphism groups are vertex transitive Remark: We shall see later that equality holds in are regular. This does not necessarily hold for edge transi- Lemma 2. tivity (as, for example, in the case of a star). Lemma 3: a(G)<9(G). If G and H are two graphs, then their strong product Proof: Let («„••-,«„) be an optimal orthonormal G-H is defined as the graph with V{G-H)= V(G)X V{H), in which (x,y) is adjacent to (x',y') iff x is adjacent to x' representation of G with handle c. Let {1.••-,&}, for k in G a n d / is adjacent t o y in H. If we denote by G the example, be a maximum independent set in G. Then strong product of k copies of G, then a(Gk) is indeed the a,,- • • ,uk are pairwise orthogonal, and so maximum number of independent points in Gk. We shall use linear algebra extensively. For various I = <:-'> 2 {cTu,)2>a(G)/9(G). properties of (mostly semidefinite) matrices, see, for exami= \ ple, [4]. All vectors will be column vectors. We shall denote by / the identity matrix, by / the square matrix all Theorem 1: 9(G) <9(G). of whose entries are ones, and by j the vector whose Proof: By Lemmas 1 and 2, a(Gk)<9(Gk)<9(G)k. entries are ones (the dimension of these matrices and vectors will be clear from the context). Theorem 2: 0 ( C 5 ) = V 5 . Besides the inner product of vectors v, w (denoted by Proof: Consider an umbrella whose handle and five r t; w, where T denotes transpose), we shall use the tensor ribs have unit length. Open the umbrella to the point product, defined as follows. If v = (v],- • • ,vn) and w = where the maximum angle between the ribs is TT/2. Let u„ (M/|, • • • , wm), then we denote by v ° w the vector u2, «3. «4, "5 be the ribs and c be the handle, as vectors (OIMV* • • ,viwm,v2wl,- • • ,vnwm)T of length nm. A simple oriented away from their common point. Then «,,- • • ,a 5 computation shows that the two kinds of vector multi- is an orthonormal representation of C5. Moreover, it is plication are connected by easy to compute from the spherical cosine theorem that c r«1, = 5 ~' / 4 , and hence (x°y)T(u°w) = (xrv)(yTw). (1) 1 0(C 5 )„) of unit vectors in a Euclidean follows. space such that if i a n d / are nonadjacent vertices, then t), and Vj are orthogonal. Clearly, every graph has an orthonormal representation, for example, by pairwise orthogoIII. FORMULAS FOR 9(G) nal vectors. To be able to apply Theorem 1 to estimate or calculate Lemma 1: Let {«,,-••,«„) and (U],••-,»„) be orthonormal representations of .G and //, respectively. Then the the Shannon capacity of other graphs we must investigate vectors u, ° »• form an orthonormal representation of GH. the number 9(G) in greater detail. The proof is immediate from (1). Theorem 3: Let G be a graph on vertices {1, • • - , « } . Define the value of an orthonormal representation Then 9(G) is the minimum of the largest eigenvalue of ( a , , - • • ,«„) to be any symmetric matrix (a^)"^ [ such that 1 mm max a:y = l, if i = / o r if / a n d / a r e nonadjacent. (2) c "'<"(cr«,)2 Proof: where c ranges over all unit vectors. The vector c yielding 1) Let («,,••• ,«„) be an optimal orthonormal repthe minimum is called the handle of the representation. Let 9(G) denote the minimum value over all representa- resentation of G with handle c. Define tions of G. It is easy to see that this minimum is attained. Call a representation optimal if it achieves this minimum value. Lemma 2: 9(G-H) <9(G)»(H). Proof: Let («!,,••• .«„) and (v„-• • ,v„) be optimal and orthonormal representations of G and H, with handles c and d, respectively. Then c°d is a unit vector by (1), and
a„=l-
('Wl)'
",v=l. A
=(«iXJ-r
'*j.
335 LOVASZ: SHANNON CAPACITY OF A G R A P H
Then (2) is satisfied. Moreover,
matrix satisfying (3) and (4). Then using (2) and (3), Tr BJ = 2
>*j,
2 btJ= 2
i-ij-i
2
"ijb^TrAB,
,-ij-i
and so
and
#(C)-Trfl/ = Tr(#(G)/-^)£. + #(G)--
&(G)-a,
Here both d(G)I-A and B are positive semidefinite. Let ' - ( - * ) • «,,-•• ,«„ be a set of mutually orthogonal eigenvectors of These equations imply that d(G)l — A is positive semidefiB, with corresponding eigenvalues X„- • • , \ , > 0. Then nite, and hence the largest eigenvalue of A is at most 9(G). T r ( # ( G ) / - / 0 f l = 2 e/WG)/-/!)&>,. 2) Conversely, let A = (a,-,) be any matrix satisfying (2), and let X be its largest eigenvalue. Then XI —A is positive semidefinite, and hence there exist vectors = 2\-*,r(#(G)/-/Oe,>0. i= i xv- • • ,xn such that 2) We have to construct a matrix B which satisfies the previous inequality with equality. For this purpose let Let c be a unit vector perpendicular to xt,- • • ,xn, and set ('n/i)."'' '('mJm)('k<Jk) be the edges of G. Consider the (m + l)-dimensional vectors 1
W
VX «?=^(l + x?)-l,
(h,hJy,hlmh^(2hl)2)T
h=
Then
where h = (hl,- • • ,hn) ranges through all unit vectors and
i = l, • • • , « ,
z = (0,0,--,0,#(G))r and for nonadjacent / and_/',
Claim: z is in the convex hull of the vectors A. Suppose this is not the case. Since the vectors A form a compact set, there exists a hyperplane separating z from So (i^,-• • ,i/n) is an orthonormal representation of G. all the A, i.e., there exists a vector a and a real number a such that aTha. Moreover, Set 1 = \, i = l,---,n, a = (al,- • • ,am,y) . (c\f ^ . - i ( l + ^ ) - 0 .
and hence &(G)<\. theorem.
This completes the proof of the
Then in particular aTh <,a, for A = (1,0,-• • ,0); whence y
The next theorem gives a good characterization of the value »(G). Theorem 4: Let G be a graph on the set of vertices 0 ' " • " . " } , and let B — (biJ)1J_, range over all positive semidefinite symmetric matrices such that *, = 0
(3) (4)
Then t?(G)=max B
Since the largest eigenvalue of -4 = (fly) is equal to max [xTAx: \x\ = 1),
for every pair (ij) of distinct adjacent vertices and T r £ = l.
2 2 "„AA<«-
i-ij-i
TTBJ.
Note that Tr BJ is the sum of the entries in B. Proof; 1) Let A =(aj).)JJ._, be a matrix satisfying (2) with largest eigenvalue #(G\ and let B be any symmetric
this implies that the largest eigenvalue of (a ) is at most a. Since (atJ) satisfies (2), this implies &(G)*ia, a contradiction. This proves the claim. By the claim, there exist a finite number of unit vectors A,,- • • ,AW and nonnegative reals a,,- • • .a^ such that ", + ••• +
= ;.
(5) (6)
336 *
IEE
Set
TRANSACTIONS ON INFORMATION THEORY, VOL. I T - 2 5 . NO. 1, JANUARY 1 9 7 9
Set v = w
i
N b
a
i/\w\
d-\ 2
w
i\/\ 2
w
i •
h
ij= 2 pK> pj P-\
B'(bu). The matrix B is clearly symmetric and positive semidefinite. Further. (6) implies b,, =0,
Then the vectors vt form an orthonormal representation of G by (7) and (3). Moreover, using the Cauchy-Schwarz inequality we get
k=l,--,m
>(tH(dTv,)J=(±dTW)j2
and Tr A / - » ( G ) while (5) implies
= (f2>,) = (.£ •".•) =*(G)-
TrB=l. This completes the proof.
This completes the proof. Lemma 4: Let («,,•-•.«„) be an orthonormal repreNote that since we have equality in the Cauchysentation of G and (a,,- • • ,vn) be an orthonormal repre- Schwarz inequality, it also follows that sentation of the complementary graph G. Moreover, let c and d be any vectors. Then (
£ i-1
/Yoo/; By (1). the vectors «,•"<;,• satisfy ( V !*)(., • » , ) - ( n r « , ) ( t > r » > « , . Thus they form an orthonormal system, and we have (c>df>
£((c")r(«,-e,))2
which is just the inequality in Lemma 4. Corollary // If (C|,•••,«„) is an orthonormal representation of G and d is any unit vector, then
Theorem 6: Let A range over all matrices such that a,j = 0 if ij are adjacent in G, and let \,(A)> • • • >X„(A) denote the eigenvalues of A. Then
Proof: 1) Let A be any matrix such that a, ; =0 if i and 7 are adjacent. L e t / = ( / „ • • • ,/„) r be an eigenvector belonging to A,(/() such t h a t / 2 = - \/\n(A) (note that since Tr A = 0, the least eigenvalue of A is negative). Consider the matrices F=diag (/,,-•• ,/„) and B = F(A-\„(A)I)F.
9(G)> 2 (t^rf)2. Corollary 2: »(G)9(G)>n. We give now another minimax formula for the value 9(G), which shows a very surprising duality between G and its complementary graph G. Theorem 5: Let (i?,,- • • ,t>m) range over all orthonormal representations of G and d over all unit vectors. Then #(G) = max 2
T
(d v:f.
Obviously B is positive semidefinite. Moreover, bfJ = 0 if / and j are distinct adjacent points, and Tr£=-A„(/4)TrF2=l. So by Theorem 4,
#(G)>TrB/= £ £ V^-A„(,0 £ / 2 '" - 1 y - 1
2
1-1
2
= 2{A l (^)/ -A^)/ } =
l - ^ .
Proof: By Corollary 1 we already know that the 2) The fact that equality is attained here follows by inequality > holds. We construct now a representation of a more or less straightforward inversion of this argument G and a unit vector with equality. Let B = (b,) be a and is omitted. positive semidefinite symmetric matrix satisfying (3) and Corollary 3: (See Hoffman [3].) Let A, > • • • > \, be the (4) such that Tr BJ = ft(G). Since B is positive semidefieigenvalues of the adjacency matrix of a graph G. Then nite, we have vectors »,,••• , H>„ such that the chromatic number of G is at least Note that
i> 2 =i.
(iU)2=»(G).
Proof: The chromatic number of G is least #(G). In fact, if («,,• • • ,«„) is an orthonormal representation of G,
337 LOVAS2: SHANNON CAPACITY OF A GRAPH
c is any unit vector, and ./,,- • • ,Jk are the color classes in Then trivially, B also satisfies (3), and any A-coloration of G, then TTB=\
2(<-r«,f= 2 2 UT«,f< 2 i-* /-l
m - l IEJ„
»-l
from which the assertion follows by Theorem 5. Now the adjacency matrix of G satisfies the condition in the theorem (with G instead of G), which implies the inequality in the corollary. IV.
TrBJ =
9(G)
(using PJ — JP = J). Also trivially, Bis symmetric and positive semidefinite and satisfies P^'BP^B, for all PS T. Since T is transitive on the vertices, this implies btl — l / « , for all ;'. Constructing the orthonormal representation (f|,- • • ,vn) and the unit vector d as in the proof of Theorem 5, we have
SOME FURTHER PROPERTIES OF 9(G)
The results in the previous section make the value 9(G) by (8). So from the definition of 9(G), quite easy to handle. Let us derive some consequences. Theorem 7: »(G-H) =
9(G)9(H).
9(G) < max '<•'•''(jTv,f
Proof: We already know that
= -r-^-, 9 G ( )
and hence »(GH)i9(G)9(H).
9(G)9(G)
/= 1
Then u( ° H>- is an orthonormal representation of GH (this follows since it is an orthonormal representation of GH and GND G-H). Moreover, c°d is a unit vector. So
9(GH)> 2 £((«>, °M,)V)) 2 , - l y - l
= 2 2 W'fWf
Equality holds if the automorphism group of G is transitive on the edges. Corollary 5: For odd n. n cos (ir/n) »(C„) = 1 +cos {ir/n) '
i-ij-i
Proof: Consider the matrix J — xA, where x will be chosen later. This satisfies condition (2) in Theorem 3, and hence its largest eigenvalue is at least &(G). Let t> Theorem 8: If G has a vertex-transitive automorphism denote the eigenvector of A belonging to \ . Then since A is regular, vi =/', and therefore, y, «2, - • • ,vh are also eigengroup, then vectors of J. So the eigenvalues of J — xA are n — 9(G)9(G) = n. JCX], — xX2,- • •, — x \ , . The largest of these is either the first or the last, and the optimal choice of JC is x = n/(ki — Corollary 4: If G has a vertex-transitive automorphism X„) when they are both equal to - n\n/(\l — Xn). This group, then proves the first assertion. e(G)@(G)
-2(«,r*)22(»/<02-#(G)*(/o. , - i
S
,--i
U
-&H(&'-' '}
value is at most &(G). By Theorem 3, it is equal to &(G). Moreover, C is clearly of the form J — xA. Hence the second assertion follows.
338 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. IT-25, NO. 1, JANUARY 1979 V.
COMPARISON WITH OTHER BOUNDS ON
If G is self-complementary, then
CAPACITY
0(G-G) = ©(G 2 ) = 0(G) 2 . This proves the theorem. The proof also shows that in Proof: We use Theorem 4. Let («,) be an orthonor- these cases & = #. mal representation of G and c be a unit vector such that Theorem 13: Let n>2r, and let the graph K(n,r) be defined as the graph whose vertices are the r-subsets of an ^-element set S, two subsets being adjacent iff they are disjoint. Then Let C be any clique in G. Then { « , : / £ C} is an orthonormal set of vectors, and hence Q(K(n,r))-(»2\). Theorem 10: &(G)
Corollary 6: The Petersen graph, which is isomorphic with A"(5,2), has capacity four. Hence the weights (c */,) form a fractional vertex packCorollary 7: (See Erdos, Ko, and Rado [2].) ing, and so a(K(n,r)) = ("-_\). /EC 7
2
Note that
1= 1
A very simple upper bound on @(G) is the dimension of an orthonormal representation of G.
«*W«,r))-(?)/[i]
Theorem 11: Assume that G admits an orthonormal which is larger than " unless r is a divisor of n. representation in dimension d. Then Proof of Theorem 13: The r subsets containing a ${G)
(9)
2 *r = °1
V , 7- \2
1
UczT
There are ( 1 — I _ , ) linearly independent vectors (xT) of this type. For each such vector, define
Therefore 9(G)
*A = VI.
We can use our methods to calculate the Shannon capacity of graphs other than the pentagon. We of course deal only with graphs G such that a(G)a(G-G)>\V(G)\. On the other hand, we have by Theorems 1, 6, and 7 that e ( G - G ) < » ( G - G ) - » ( G ) * ( G ) = |K(G)|.
2
*T
TCA
APPLICATIONS
m-r for every AcS, \A\ = r. It is not difficult to see, and actually well-known, that the numbers xT can be calculated from the numbers xA, whence there are I
I—I . I linearly independent vectors of type (xA). Claim: Every (xA) is an eigenvector of the adja-
cency matrix of K(n,r) with eigenvalue (— l)'!
I-
In fact, for any A0c S such that \A0\ = r, we have /fn/<„-0
TnA„-2>\
r
To determine this value we set
A=
2 \TnA0\-
l I
\
r
i I
339 7
LOVASZ: SHANNON CAPACITY OF A GRAPH
Then summing (9) for every UcS and \U nA0\-i, we get
such that \U\ = t~\
(,+ l)/3, + l + ( / - 0 / 3 , = 0. This may be considered as a recurrence relation for the fj: and yields
A=(-iy(;.)/3 0 whence
A,=(-i)'/3, = (-i)'^„ which proves the claim. By this construction we have found
' + ,?,(CH,-, ))=(") linearly independent eigenvectors (there is no problem with the eigenvectors belonging to different values of 1 since they belong to different eigenvalues). Therefore, we have all eigenvectors, and it follows that the eigenvalues of K(n,r) are the numbers
(-!)'(«;:;'),
I, respectively, and Theorem 9 yields !n-r-\\ln\
VII.
0(G)0(G) = 0(GG)>a(GG)>|K(G)|. This, in turn, would imply an affirmative answer to the last question of Problem 1: n<0(C„)0(C„)<#(C„)O(C„) = «; hence 0(C„) = t)(C„) and 0(C„) = i»(C„)). Corollary 7 shows an example where the calculation of &(G) helps to determine a(G) in a nontrivial way. Are there any further examples? ACKNOWLEDGMENT
,=o,i,..,,
So the largest and smallest eigenvalues are ( " 7 r ) and - 1 " — r~
Various properties of ir(G) established in this paper suggest further problems which would be solved by an affirmative answer to Problem 1. Problem 2: Is 0(G- H) = 0(G)0(W)? (Note that @(GH)>@(G)@(H) is obvious.) Problem 3: Is it true that 0 ( G ) - 0 ( G ) > | K(G)|? Note that an affirmative answer to Problem 2 would imply an affirmative answer to Problem 3:
My sincere thanks are due to K. Vesztergombi and M. Rosenfeld for numerous conversations on the topic of this paper—also to T. Nemetz, A. Schrijver, and the referees of the paper for pointing out several errors in, and suggesting many improvements to, the original text. It is a pleasure to acknowledge that A. J. Hoffman and M. Rosenfeld have also found extensions of the original idea (in Section II) to other graphs. Among others Hoffman found Theorem 9. and Rosenfeld found #(C„).
CONCLUDING REMARKS
The purpose of introducing 9(G) has been to estimate 0(G). So the obvious question is as follows. Problem 1: Is # = 0? More modestly, find further graphs with #(G) = 0(G). In particular, do odd circuits satisfy #(G) = 0(G)? This last question pinpoints a difficulty which seems to be crucial. In all cases known to the author where 0 ( G ) is precisely determined, there is some k (k = \ or 2, in fact) such that a(Gk) = Q{G)k. But if 0(G) = t»(G) for the seven-circuit, for example, then no such k can exist, since no power of #(C 7 ) is an integer.
REFERENCES [1] C. Berge, Graphs and Hypergraphs. Amsterdam and London North-Holland; New York: American Elsevier, 1973. [2] P. Erdos, C. Ko, and R. Rado, "Intersection theorems for systems of finite sets," Quart. J. Malk. Oxford, vol. 12, pp. 313-320, 1961. [3] A. J. Hoffman, "On eigenvalues and colorings of graphs," in B. Harris, Ed., Graph Theory and its Applications. New York and London: Academic, \910,pp. 79-91. [4] P. Lancaster, Theory of Matrices. New York and London: Academic, 1969. [5] M. Rosenfeld, "On a problem of Shannon," Proc. Amer. Math. Soc, vol 18, pp. 315-319, 1967. [6] C. E. Shannon, "The zero-error capacity of a noisy channel," IRE Trans. Inform. Thoery, vol. IT-2, no. 3, pp. 8-19. Sept. 1956.
340 August 1988
Algorithmic mathematics: an old aspect with a new emphasis Ldszlo Lovdsz
0. Introduction The development of computers is perhaps the single most significant technological breakthrough in this century. It is natural that it has not left untouched closely related branches of science like mathematics and its education. It is also natural that whichever fields have come into contact with it, heated debates have started and very different views, extreme and moderate, progressive and conservative, have been put forth. Is algorithmic mathematics of higher value than classical, structureoriented, theorem-proof mathematics, or does it just hide the essence of things by making them more complicated then necessary? Does teaching of an algorithm lead to a better understanding of the underlying structure, or is it a more abstract, more elegant setting that does so? Is the algorithmic way of life best (Maurer 1985), or is applied mathematics just bad mathematics (Halmos 1981)? Should computers be introduced in elementary/secondary/college education of mathematics? I want to start with a disclaimer: I will not attempt to give an answer to all these questions. The point I will try to make is that algorithmic mathematics (put into focus by computers, but existent and important way before their development!) is not an antithesis of the "theorem-proof" type classical mathematics. Rather, it enriches several classical branches of mathematics with new insight, new kinds of problems, and new approaches to solve these. So: not algorithmic or structural mathematics, but algorithmic and structural mathematics! The interplay between the algorithmic and structural sides of mathematics is manyfold; I will only mention the two most important lines. The design and analysis of algorithms and the study of algorithmic solvability uses deeper and deeper tools from classical structural mathematics on the one hand; and an algorithmic perspective has a more and more profound effect on the whole framework of many fields of classical mathematics on the other hand. Let me examplify. Perhaps the first notion of an "algorithm" that was defined clearly enough so that the question of "algorithmic solvability" could be raised was the notion of a geometric construction by ruler and compass, formulated by the Greek geometers. The mathematical interest of this notion is that there are both solvable and unsolvable construction problems. The design of construction algo-
341 rithms has been stimulating in geometry for a long time, and has contributed to the development of important tools (very useful also independently of construction problems) like inversion or the golden ratio. But the proof of unsolvability of basic construction problems (trisecting an angle, squaring a circle, doubling a cube, constructing a regular heptagon etc.) illustrates this effect more dramatically. Such negative results were inaccessible for the Greek mathematics and for quite a while later on; it required the notion of real numbers and a substantial part of modern algebra to prove them, or even to formulate them with an exactness that made them accessible to mathematical methods. In fact, modern algebra was inspired by the desire to prove such negative results (besides the non-constructibility of certain configurations, the non-solvability of equations of degree at least 5 was of the same nature). As another example, let us consider the notion of primes. These numbers were also studied by the ancient Greeks, and they proved several basic properties of them. The beautiful but very hard theory of prime numbers has been a major branch of mathematics (and a source of inspiration for many other branches) throughout the history of modern mathematics too. But only the development of computers, and even more the establishment of computational complexity theory, raised the fundamental algorithmic problem: how to test whether a given number is prime? how to find the prime factorization if it is not? (Mathematicians in the 18-th and 19-th century, in particular Gauss, did make extensive computations regarding primes, and developed ingenious tricks to help their work. But appearantly they did not consider their algorithms as mathematical results.) These questions have profound applications in computing, and the simple elementary procedures to solve them are by far not satisfactory (practice requires the consideration of numbers up to several hundred digits). Over the last 10 years or so, more and more advanced methods from number theory have been applied to design more and more sophisticated and efficient algorithms to answer these questions. Note that the interaction of mathematics and algorithms in these two examples is different: in the first, we have a certain notion of an "algorithm" (a construction procedure), and we want to prove that it does not suffice to solve certain problems. Such questions may be notoriously hard, and, as our example shows, the proof may require deep mathematics or even the development of entirely new fields. The theory of computing today is full of unsolved problems of a similar nature; there are very few methods to prove negative results about the algorithmic solvability of problems, in particular if limits are imposed on the time (or other resources) that the algorithm can use. In such cases, computer science is an "external" user of
342 mathematics: it supplies hard problems, which need be modelled and solved just like hard problems in mechanics or astronomy. In the second example, computer science penetrates classical mathematics, putting old questions in a new perspective. Often, this is done by requiring constructions where "pure" existence proofs have been supplied by the classical theories. In other cases, one requires efficient procedures where "theoretically finite" case-analyses have been at hand (like in prime testing). In some branches of mathematics, e.g. in graph theory, this process has provided a whole new framework for the field (cf. Lovasz 1986). In my talk, I would like to focus on this second development. This new perspective on several issues in classical mathematics is, of course, a challenge to mathematics education. The introduction of computers (at whatever level) is only a very partial answer. I will give some remarks about how I think this challenge can be met; but I believe that it will take much more work — experimental and theoretical — before the contours of the answer will be clear.
1. Some old results from a new perspective The approximation of irrational numbers by rational ones has a long history. Suppose that we have a real number a, that we want to approximate by a rational number with a small denominator. The first, trivial approach is to round: take any positive integer q, and round the product qa to the nearest integer p. Then p/q is a reasonably good rational approximation of a; in fact, \a — E\ < j - . Can we do better, with some positive integer Q in place of the 2 here? More precisely, can we find, for a given real number a and positive integer Q, a rational number p/q such that \a — 2| < -X-l It is clear that in this case q cannot be chosen arbitrarily any more; but how large does it have to be? A theorem of Dirichlet states that for every given real number a and positive integer Q, we can find integers p and q such that 0 < q < Q and \a — 2| < ^ - . In other words, \qa — p\ < i . There are two basic proofs of this fundamental result, and I would like to discuss them here to illustrate the difference between the algorithmic and nonalgorithmic approaches. First proof: Consider the Q + 1 numbers Qa - [OaJ,
la - [la\,
,... ,
Qa - [Qa\
These numbers all lie in the interval [0,1), so some two of them, say ka — [ka\ and la — [la\ (0 < k < I < Q) are closer to each other than 1/Q. Let q = I — k and
343 p = [la\ — [ka\. Then q < Q and \qa -p\ = \{l - k)a - {{la\ - \ka\)\ = \(ka - [ka\) - {la - \la\)\ < - . So the rational number p/q proves Dirichlet's Theorem. Second proof (sketch): It is well known that every real number a can be expressed as a continued fraction 1
a = CLQ -\
.
ai +
1 02 +
a3 + ...
This expansion may be finite (if a is rational) or infinite (if a is irrational). For example, we have the expansions 11
,
3
,
1 2+
,
1 2
3
T
+ 1 +
2
and v/2 = l + ( v / 2 - l ) = l + ^ — = 1 + ~ - = 1+ v/2 + 1 2+ (V2-l)
2 +
_
1 1
2+ ^ Now if we stop in a continued fraction expansion in the k-th step, we get a rational number Pk , 1
— qk = a0 +
1
1
a2+ ... + —
a-k
There are a number of basic facts that can be proved about this rational number, called the A;-th convergent of a. What we need here is the fact that Pk/qk is a very good approximation of a: \a
< qk
qkqk+i
Hence, if we let k be the largest subscript for which qk < Q (it is known that qk tends to infinity, in fact exponentially fast), then
344
i.e. the rational number Pk/qk proves Dirichlet's Theorem. Which of these proofs is "better" ? There is little doubt that the first one is much simpler: not only is it shorter but it is self-contained, while the second uses quite a bit from the theory of continued fractions. But suppose that we also want to find the rational number in question. Before going into a discussion of this, we have to clarify one thing: how is a given? Since we want to have an algorithm now, we better assume that a is represented in some finite explicit form; let us assume that it is rational, say a = a/b. Of course, we have to assume then that Q < b, else the approximation problem is trivial. Which algorithm can we derive from the first proof? It tells us that we should form all the numbers hot — [ka\, then find two, which are close, then ... But if we want to form all these numbers anyway, we can check with the same amount of work which of them is less that 1/Q. So the first proof of the theorem does not provide us with any non-trivial algorithm to find the rational number whose existence it certifies; it is a "pure existence proof" in this sense. (It could be argued that the structural insight gained from the first proof can be developed further to obtain an efficient algorithm. This, however, takes further work and deeper insight.) Is the second proof better from this point of view? How much work does it take to compute the continued fraction expansion? We show that not only is it easy to obtain this expansion, but also that there is nothing mystical about its use in an approximation problem. Suppose that we want to find a good rational approximations of a number a. As a first approach, we could use the rational number ao/1, where ao = [^JTo obtain a better approximation, we have to approximate the error, i.e., x\ = a — ao. Since this number is less than 1, it is a natural idea to replace it with its reciprocal, and approximate this by its integer part a\ = [ l / ^ i j - Now this approximation again has an error X2 = l/x\ — a%. Take the reciprocal again etc. The positive integers a0,ai,... obtained this way are just the coefficients in the continued fraction expansion of a. If a = a/b then we obtain ao by dividing a by b with remainder; then ao is the quotient and if r is the remainder then the error x\ = r/b. So l / z i is the rational number b/r and we repeat the procedure with this in place of a/b. It is clear that this is a finite procedure, and so the continued fraction expansion of a can be obtained. The rest of the proof gives a simple recipe to obtain the approximating rational number. (Some of my readers may have observed at this point that to obtain the continued fraction expansion of a rational number a/b, we carry out exactly the same
345 arithmetic operations as in the euclidean algorithm used to compute the greatest common divisor of a and b.) So the second proof is just the analysis of a very natural iterative algorithm to find better and better approximations of a number. But is this algorithm any better than the trivial one derived from the first proof? The answer is: much better! The number of steps in the continued fraction expansion of a/b (equivalently, the number of steps in the euclidean algorithm to compute g.c.d.(a, b)), is proportional to the number of digits of a and b\ the number of steps in the first trivial algorithm is proportional to Q, which may be as large as b itself. If a, b and Q are 100 digit numbers, then the first algorithm takes 10 100 steps while the second, less than 500. The usual way to measure the running time of an algorithm is to compare the number of bit-operations with the number of bits necessary to write down the input. So a fc-bit integer (having k bits in its base 2 expansion) contributes k to the input; in the diophantine approximation problem, the size of the input is the number of bits in a, b and Q, which is essentially log2 a + log2 b + log2 Q. In this model of computation, the length of the numbers also influences the time spent on a single arithmetic operation. For example, multiplication of two fc-bit integers by the method taught at school takes about k2 bit-operations, so such an operation contributes k2 to the running time etc. An important distinction to make is whether the running time grows as a polynomial of the input length or faster. Polynomial algorithms tend to be mathematically interesting and usually — though not always — practically feasible. Brute force case-distinctions often lead to exponentially many cases to distinguish (all subsets of a set, etc.) and thereby to exponentially growing running times. The algorithm derived from the first proof needs exponential time; the algorithm derived from the second is polynomial. So we see that: - The first proof is an existence proof; it is elegant, short, but does not give an algorithm to find the approximation. It takes more work and further ideas to develop it into an efficient algorithm. - The second proof is just an analysis of an elegant, natural and efficient algorithm to construct good approximations. It takes work to analyse how good these approximations are. The first proof also generalizes easily to the problem of simultaneously approximating several numbers by rationals with a common denominator. The algorithmic proof (continued fractions) also has some generalizations to this case, but those are
346
substantially less elegant and do not yield as good approximations as the generalization of the first proof. To find a polynomial time algorithm for this simultaneous diophantine approximation problem, which would find the approximating rational numbers whose existence is guaranteed by the almost straightforward extension of the first proof is unsettled!
2. A glimpse of complexity theory Most of us have met problems sounding like "Characterize those sets (numbers,...) with the property that ***.", and also the student who gives, provocatively, the answer "A set has property *** if and only it has property ***." What is wrong with this answer? And what happens if the student hides the triviality by slightly re-phrasing the property * * *? When does the answer begin to be non-trivial and thereby acceptable? Or should we simply ban this kind of problems as meaningless? One of the great successes of the theory of computing is that it is able to define in a mathematically exact way which characterizations are "good" and which are more-or-less just rephasing the question, at least for a large class of structures and properties. There is no room in this talk, of course, to develop this theory. But I try to illustrate the idea on an important example. Consider a system of inequalities anxi + auX2 + • • • + ainxn a2\X\
< b\,
+ a22X2 + • • • + CL2nXn < b2, (1)
: a-mi^i + am2X2 H
h amnxn
< bm.
Since we are interested in algorithmic aspects, we assume that the inequalities have rational coefficients. When does (1) have a solution? Let us investigate the following two answers to this question: Theorem A. The system of inequalities (1) has a solution if and only if the system o n ( « i - vi) + ai2(u2 -V2)-\
h ain(un
- vn) < &i,
fl2l(«l - ^ l ) + 022("2 - v2) H
\- a2n(un ~ Vn) < b2, (2)
o m i(ui - vi) + am2(u2 -v2)-\ has a non-negative solution.
1- amn(un
- vn) < bm
347 T h e o r e m B . The system of inequalities (1) has a solution if and only if the system «n2/i + «2i^2 ~i
h amiym
= 0,
ai22/i + «222/2 -i
\- am2ym = 0, ;
O-lnVl + a2nV2 H hyi
+ b2V2 ^
(3)
h a-mnVm = 0, 1- bmym
= -1
has no non-negative solution. (Note that a solution of (3) can be viewed as a linear combination of the inequalities in (1) with non-negative multipliers — the j/j — that yields the trivially non-solvable inequality Ozi + . . . + 0xn < —1.) Both theorems give necessary and sufficient conditions for the solvability of (1). But Theorem A is an essentially trivial trick to show that solvability is easily transformed into non-negative solvability, while Theorem B is an important result (called Farkas' Lemma). What makes the difference? Assume that I want to use a concrete case of (1) in this talk as an illustration of a system of solvable linear inequalities. How can I convince you that it is indeed solvable? Easy: I just show you a solution. Suppose now that I want to use another concrete case of (1) to illustrate an unsolvable system. How can I convince you that it is indeed unsolvable? To try all possible values for the variables? There is no easy way at hand. Do the necessary and sufficient conditions formulated in the two theorems above help? If we apply Theorem A, I have to exhibit that (2) does not have a non-negative solution — this is not any easier than the original task. But if we apply Theorem B, it suffices to exhibit that (3) does have a non-negative solution — and this I can do by showing you one. So the condition in Theorem B does have an entirely different logical structure from the original property and from the condition given in Theorem A. It turns out that a large part of graph theory, optimization, number theory etc. has an analogous structure. Important properties of graphs, numbers etc. have the feature that if they are present, there is an easy way to exhibit this (e.g. composite numbers, 4-colorable graphs etc.) Basically, these properties are defined in terms of the existence of a certain object (a number is composite if it has a proper divisor; a graph is 4-colorable if it has a proper coloring with 4 colors etc.) If I want to convince you that the number 617532176011725742711064069114397791170668109866281
348 is composite, all I have to do is to show you the divisor 7858321551080267055879091 (Of course you still have to verify that'this is a divisor; but, as we know, this takes only polynomial time.) Such properties are called NP-properties (named after a technical definition involving Non-deterministic Polynomial-time Turing-machines, whose details I don't have to give here). Sometimes (but not always!) the negation of NP-properties is again an NP-property. Theorems establishing such equivalences are often among the most important results in the field (like the Farkas Lemma above). They are sometimes called good characterizations. This classification of properties is also closely related to algorithms. Most properties for which we would like to find a polynomial-time algorithm to decide them belong to NP in a natural way. If a property can be decided by a polynomialtime algorithm, then both the property and its negation are NP-properties, i.e., it has a good characterization. The converse may not be true: it is not known whether every well-characterized property can be decided in polynomial time (probably not, but as I remarked before, we do not have the means to prove such negative results in this area). But, usually, to find a good characterization is an important step towards the construction of a polynomial time algorithm. The example of the Farkas Lemma is very illustrative: almost a century after the proof of the lemma, a polynomial time algorithm to decide if (1) has a solution was given by Khachiyan in 1978 (the famous Ellipsoid Method). Of course, this article cannot discuss the theory of algorithms in any reasonable detail; we only sketched what was necessary to support our arguments on a possible new framework for various fields in mathematics. (For an introduction to the theory of algorithms, see e.g. Sedgewick 1983.)
3. What does this imply in math education? Whatever it implies, should be regarded with utmost caution and moderation. I feel that math education should follow what happens in math research, at least to a certain extent, in particular those (rare) developments there that fundamentally change the whole framework of the subject. Algorithmic mathematics is one of these. However, the range of the penetration of an algorithmic perspective in classical mathematics is not yet clear at all, and varies very much from subject to subject (as well as from lecturer to lecturer). Graph theory and optimization, for example, have been thoroughly re-worked from a computational complexity point
349 of view; number theory and parts of algebra are studied from such an aspect, but many basic questions are unresolved; in analysis and differential equations, such an approach may or may not be a great success; set theory does not appear to have much to do with algorithms at all. Our experience with "New Math", the adaptation of the set-theoretic foundations of mathematics in lower level mathematics education, warns us that drastic changes may be disastrous even if the new framework is well established in research and college mathematics. So let me just raise some ideas on the teaching of algorithms on various levels, emphasizing that they must be carefully discussed and tried out before any large scale implementation is attempted. Basically, some algorithms and their analysis could be taught about the same time when theorems and their proofs first occur, perhaps around the age of 14. Of course, certain algorithms (for multiplication and division etc.) occur quite early in the curriculum. But these are more recipes than algorithms; no correctness proofs are given (naturally), and the efficiency is not analyzed. The children have to learn (and practice) how to carry out these simple algorithms. This is like teaching theorems (axioms) without proofs, or teaching empirical facts in the sciences without experiments: necessary but not leading to really deep understanding. What I would consider as the beginning of learning "algorithmics" is to learn to design, rather than execute, algorithms. (For an elaboration of this idea, see e.g. Maurer 1984). The euclidean algorithm, for example, is one that can be "discovered" by students in class. In time, a collection of "algorithm design problems" will arise (just as there are large collections of problems and exercises in algebraic identities, geometric constructions or elementary proofs in geometry). Along with these concrete algorithms, the students should get familiar with basic notions of the theory of algorithms: input-output, correctness and its proof, analysis of running time and space, good characterizations read off from algorithms, algorithms motivated by good characterizations etc. Some possible types of algorithm-design problems, suitable probably already on the high school level: enumeration problems where no closed formula exists; elementary optimization problems in graph theory (e.g. maximum independent sets in trees, shortest paths, listing of cliques, circuits etc.); sorting and searching; simple (though inefficient) methods for primality testing, factorization, and many other problems in number theory; Gaussian elimination and other manipulations in linear algebra; convex hull and other elementary plane geometry constructions. In college, the shift to a more algorithmic presentation of the material should, and will, be easier and faster. Already now, some subjects like graph theory are taught in many colleges quite algorithmically: shortest spanning tree, maximum
350 flow and maximum matching algorithms are standard topics in most graph theory courses. This is quite natural since, as I have remarked, computational complexity theory provides a unifying framework for many of the basic graph-theoretic results. In other fields this is not quite so at the moment; but some topics like primality testing or cryptographic protocols provide nice applications for a large part of classical number theory.
4. Computers and algorithms At this point, I have to comment on the use of computers in the teaching of these topics. One should distinguish between an algorithm and its implementation as a computer program. The algorithm itself is a mathematical object; the program depends on the machine and/or on the programming language. It is of course necessary that the students see how an algorithm leads to a program that runs on a computer; but it is not necessary that every algorithm they learn about or they design be implemented. The situation is (again) analogous to that of geometric constructions with ruler and compass: some constructions have to be carried out on paper, but for some more, it may be enough to give the mathematical solution (since the point is not to learn to draw but to provide a field of applications for a variety of geometric notions and results). Let me insert a warning about the shortcomings of algorithmic language. There is no generally accepted form of presenting an algorithm, even in the research literature (and as far as I see, computer science text books for secondary schools are even less standardized and often even more extravagant in handling this problem.) The practice ranges from an entirely informal description to programs in specific programming languages. There are good arguments in favor of both solutions; I am leaning towards informality, since I feel that implementation details often cover up the mathematical essence. Let me illustrate this by two examples. An algorithm may contain a step "Select any element of set 5". In an implementation, we have to specify which element to choose, so this step necessarily becomes something like "Select the first element of set S". But there may be another algorithm, where it is important the we select the first element; turning both into programs hides this important detail. (Also, it may turn out that there is some advantage in selecting the last element of S. Giving an informal description leaves this option open, while turning the algorithm into a program forbids it.) To give my second example, recall that the Fibonacci numbers are defined by the recurrence -Pfc+i = -Pfe + jFfc-i
351 (and by F0 = 0 and Fi = 1). This recurrence provides a trivial algorithm to compute these numbers. Turning this recurrence into a program, we would get something like this: if F is the current Fibonacci number, and G is the previous, then (with an auxiliary variable H) H:=F,
F:=G
+ F,
G := H.
We see that the program contains a number of details that do not belong mathematically to the procedure of computing Fibonacci numbers: it stores Fk+i in the same register where Fk used to sit, but to do so, it has to "salvage" Fk since its value will be needed in the next step, and to do so, we need an "auxiliary variable" etc. 1 To show the other side of the coin, the main problem with the informal presentation of algorithms is that the "running time" or "number of steps" is difficult to define — as we have experienced above. Unfortunately, this depends on the details of implementation (down to a level below the programming language; it depends on the data representation and data structures used). Sometimes there is a way out by specifying which steps are counted (e.g. comparisons in a sorting algorithm, or arithmetic operations in an algebraic algorithm); but this is "cheating" in a sense since we disregard the time needed ' o handle the data, which may be as much as, or even more than, the time used by the "mathematical essential" steps. It should be mentioned that the polynomiality of an algorithm is "robust", i.e., it does not depend on implementation (although different implementations may have different polynomials in the bound on their running time). The route from the mathematical idea of an algorithm to a computer program is long. It takes the careful design of the algorithm; analysis and improvements of running time and space requirements; selection of (sometimes mathematically very involved) data structures; and programming. In college, to follow this route is very instructive for the students. But even in secondary school mathematics, at least the mathematics and implementation of an algorithm should be distinguished. An important task for mathematics educators of the near future (both in college and high school) is to develop a smooth and unified style of describing and analyzing algorithms. A style that shows the mathematical ideas behind the design; that facilitates analysis; that is concise and elegant would also be of great help in overcoming the contempt against algorithms that is felt nowadays both from the side of the teacher and of the student. X
I am grateful to Jack Edmonds for these examples and arguments — and regret not to have given a more detailed exposition of them.
352 References P. R. Halmos (1981), Applied mathematics is bad mathematics, in Mathematics Tomorrow (ed. L. A. Steen), Springer, 9-20. L. Lovasz (1986), An algorithmic Theory of Numbers, Graphs, and Convexity, CBMS-NSF Reg. Conf. Series in Appl. Math. 50; SIAM. S. Maurer (1984), Two meanings of algorithmic mathematics, Mathematics Teacher 430-435. S. Maurer (1985), The algorithmic way of life is best; reflexions by R. G. Douglas, B. Korte, P. Hilton, P. Renz, C. Smorynski, J. M. Hammersley and P. R. Halmos; College Math. Journal 16, 2-18. R. Sedgewick (1983): Algorithms, Addison-Wesley.
,ASZL6 LOVASZ
One Mathematics "here i s no n a t u r a l way ;o divide mathematics vlathematics is torn by many divi;ion lines. The most prominent of hese runs between Pure and applied Mathematics. The conroversy around Bourbaki focuses in Abstract vs. Concrete. The listinction between Structural vlathematics (whose main results ire theorems and proofs) and Ugorithmic Mathematics (whose •esults are algorithms and their inalysis) can be traced back to incient times.Thcre is a deep iivision (or at least so it appears) >etween Continuous and Discrete Vlathematics. ioinc of these divisions are consequences of different working environments: For example, applied nathematicians work under very lifferent financing conditions and ;uccess criteria from pure mathenaticians. Some others are cul:ural: Various branches of mathenatics have their own system of inferences, journals, prizes; a lifferent set of mathematical con:epts, basics and perhaps values hat are assumed in conversation. ^jid of course the appreciation of :he abstract varies very much with personality and individual taste. Should we accept this partition as i fact of life: "I am a bad pure abstract algorithmic discrete mathematician"? Some say that this is rood enough, and one should ac:ept the splitting of mathematics nto smaller and smaller independint branches as a fact of life. . feel that accepting this would •eally lead to tragic consequences,
ind that there is a deep unity of our sci;nce that gives it its strength and vitality. will argue that recent trends in our icience make these division lines more ;omplex than they appear; that we have to io our best to bridge these gaps; and that :hese same new trends may provide means :o do so.
Three new trends changing the tforld of mathematics The size of the community. It is a commonplace that the number of mathematical jublications (along with publications in 'ther sciences) has increased exponenially in the last 50 years. Mathematics has jutgrown the small and close-knit comnunity that it used to be. And with inreasing size, the profession is becoming nore diverse, more structured and more ;omplex. Vlathematicians are conservative people; '. don't mean that we are right-wing (as far ts I can see, we span the political spec:rurn just like every other profession), but ve don't push for changes: "We are reluc.ant to spend time on anything other than rying to prove that p * NP (or the Rienann Hypothesis, or whatever problem ceeps us infatuated at the moment). So pretend that mathematical research is ts it used to be. We believe that we find ill the information that might be relevant jy browsing through the new periodicals •n the table in the library, and that if we publish a paper in an established journal, hen it will reach all the people whose •esearch might utilize our results. V larger structure is never just a scaled-up version of the smaller. In larger and nore complex animals an increasingly arge fraction of the body is devoted to
Reprinted with permission. The Berlin Inteligencer, International Congress of Vlathematics, Berlin, August 1998, pp. 10-] 5. © Springer-Verlag.
354
overhead": The transportation ofmatediscrete: simple basic questions ial and the coordination of the function ike finding matching patterns, ot if various parts. In larger and more :racing consequences of flipping :omplex socieries an increasingly large aver substrings, sound more raction of the resources is devoted to ramiliar to the graph theorist than ion-productive activities like transporta- :o the researcher of differential :ion and information processing. We have equations. A question about the o realize and accept that a larger and nformation content, redundancy, larger part of our mathematical activity ir stability of the code may sound should, and will, be devoted to communi- :oo vague to a classical mathemacation. This is easy to observe: The numtician but a theoretical computer ler of professional visits, conferences, icientisc will immediately see at workshops, research institutes is increaseast some tools to formalize it ng fast, e-mail is used more and more. 'even if to find the answer may be The percentage of papers with multiple :oo difficult at the moment). luthors has also jumped. But probably we Even physics has its encounters will reach the point soon where mutual with unusual discrete mathematipersonal contact does not provide suffi;al structures: elementary parcient information flow. ades, quarks and the like are very There is another consequence of the Combinatorial; understanding increase in mass: the inevitable formation basic models in statistical mechanof smaller communities, one might say ics requires graph theoty and subcultures. These seem to arise on a probability. random basis, but then they persist and Economics is a heavy user of determine research directions for quite a mathematics - a n d much of its long time. One such subculture is Disneed is not part of the traditional crete Mathematics —Theory of Compuapplied mathematics toolbox. ting - Operations Research. The success of linear programming in economics and operations research depends on conditions I DON T SEE ANY REASON, OTHER THAN of convexity and unlimited divisiC U L T U R A L , WHY COMPUTATIONAL C O M P L E X I bility; taking indivisibilities into TY THEORY SHOULD BE EMBRACED BY THE account (for example, logical DESIGNERS OF DISCRETE A L G O R I T H M S , BUT decisions, or individuals) leads to VIEWED: WITH GRAVE S U S P I C I O N BY THE integer programming and other M A J O R I T Y OF DESIGNERS OF NUMERICAL combinatorial optimization ALGORITHMS . models, which are much more 'New areas of application. The traditional (difficult to handle. finally, there is a completely new area of application of mathematics is area of applied mathematics: comphysics; and no doubt this area involves J:he deepest mathematics and the greatest puter science. The development of success stories. The branch of mathemat- electronic computation provides a vast array of well-formulated, ics used in these applications is analysis, difficult, and important mathematthe real hatd core of mathematics. ical problems, raised by the study But in the boom of scientific research in 0f algorithms, data bases, format the second half of this century, many languages, ctyptography and comother sciences have come to the point puter security, VLSI layout, and whete they need serious mathematical much more. Most of these have to tools. Quite often the traditional tools of do with discrete mathematics, fotanalysis are not adequate. Tial logic, and ptobability. For example, biology tries to understand Which branches of mathematics :he genetic code: a gigantic task, which ,vill be applicable in the near future s the key to understanding life and, ultis utterly unpredictable. Just 25 Tiately, ourselves. The genetic code is
vears ago questions in number theory like j low many primes there are between j ; x 10 200 and 4 x io 2 0 0 seemed to belong ! :o the purest, most classical and comI iletely inapplicable mathematics; now j 'elated questions belong to the core of J piathematical cryptology and computer \ security. ! \t would seem that this diversity of appli- ! cations is anothet centrifugal force; but I :hink that, to the contrary, it should itrengthen the flow of information across . division lines. j
NO FIELD CAN R E T R E A T I N T O ITS I V O R Y
I
TOWER AND C L O S E ITS D O O R S TO APPL
j
C A T I O N S ; NOR CAN ANY FIELD CLAIf)l TO
I
BE THE A P P L I E D M A T H E M A T I C S .
New tools: computers. Computers, of course, are not only sources of interesting and novel mathematical problems. They also provide new tools for doing and organizing our research. There is obviously a large variation in the relationship between mathematicians and computers. Some avoid computers altogether; others are glued to theirs. I use them for e-mail and word processing like inost of us; less regularly, I use them for experimentation, and for getting information through the web. I have become iddicted to searching in the Mathematical Reviews database, and I •ind it mote and more convenient to get nformation by browsing through electronic journals and, perhaps even tiore significantly, through home pages if other mathematicians. Are these uses of computers just toys or at lest matters of convenience? I think not, md that each of these is going to have a profound impact on our science. Electronic journals and databases, home pages, and e-mail provide new ways of dissemination of results and ideas. In a sense, they reinforce the increase in the volume of research: not only are there increasingly more people doing research, but an increasingly large fraction of this information is available at our fingertips [and often increasingly loudly and aggressively: the etiquette of e-mail is far from feolid). But we can also use them as ways
if coping with the information explosion. At first sight, word processing just looks like a convenient way of writing papers. JThe final output of mathematical research [s still a printed paper, read by others in a journal or perhaps more and more from a manuscript printed out in their office. But slowly many features of electronic versions become available that are superior to the usual printed papers: Hyperlinks, colored figures and illustrations, animations and the like. A mathematical paper is almost never read in a strictly linear fashion: one jumps back to refresh a definition, jumps ahead to see how a certain lemma is applied, skips proofs at first reading, returns repeatedly to check how the arguments work on a certain example-this is more reminescent of browsing the internet than of reading a novel. And if a document is not read in a linear way, why wrirc it in a linear way? I will not discuss here the opportuniries fand traps) provided by these features of felectronic publication; but it is quite probable that they will gradually transorm the way we write papers; and jossibly through rhis also the way wc do *esearch. New forms of mathematical activity The traditional 2500 year old paradigm pf mathematical research is defining horions, stating theorems and proving them. Perhaps less recognized, but almost this old, is algorithm design (rhink of rhe Euclidean Algorithm or Newton's Method). While different, these two ways of doing mathematics are strongly interconnected. It is also obvious that computers have increased the visibility and respectability of algorithm design substantially.
Surveys. The most serious threat p the unity of mathematics is the sheer size of mathematical research. No one can read even a tiny fraction of new research papers. One solution to this problem is the creation of an activity that deals with the secondary processing of research results. For lack of (1 better word, I'll call rhis expository writing, although I'd like to consider it more as a form of mathematical research than as a
F
orm of writing: finding the ramiScations of a result, its connections with results in other fields, explaining, perhaps translaring it for people coming from a different subculture.
The community has invented this activity already: There is more and more demand for expositions, surveys, mini courses, handbooks and encyclopedias. Many conferences are mostly or exclusively devoted to survey-type talks; publishers much prefer volumes of survey articles to volumes of research papers. Wc organize the International Congress every 4 years (and many other regional meerings of the
Same kind). While some matheI inaticians feel that the Congress is worthless (and it is if you consider ir as a big research conference), people in other fields envy us for r. It is a great asset if used for naintaining the unity of our field, is a forum forgiving surveys, expositions of the most important new results and new areas and methods of applications. Yet we all feel reluctance towards accepting expository and survey However, as a consequence of the increase writing as scientific achievement. jn the size of the research community, There is often a reservation abour this paradigm must be enriched by new somebody's writing an exposition forms of scienrific achievemenr. These of somebody else's new result may include writing good expositions and (I personally feel that this activity surveys, formulating problems and conjecphould be encouraged instead). tures, compiling examples, experimenting If, as suggested, writing expositions ind reporting the results. Let me comshould become a highly regarded nent on the first two of these. research activity, one has to find
356
conjecture is g o o d , o n e expects that its
Whole euclidean plane). It is a
into o u r p i c t u r e of achievements, i n c l u d i n g jobs,
resolution should advance o u r k n o w l e d g e
patural a n d powerful m e t h o d to
P r o m o t i o n s , grants?
iubstantially. M a n y m a t h e m a t i c i a n s feel
s t u d y discrete structures by
{We k n o w little a b o u t the criteria for a good
:hat this is the case w h e n we can clearly
' e m b e d d i n g " t h e m in the c o n t i n -
m a t h e m a t i c a l survey. We don't have a good for-
;ee the place of the conjecture, a n d its
uous world.
hial criterion m a r k i n g a g o o d t h e o r e m , either.
?robable solution, in the b u i l d i n g of
T h e leading r h e m e in c o m b i n a t o -
However, t h e r e are s o m e reasonably exact neces-
Tiathematics; b u t there are conjectures so
al o p t i m i z a t i o n in the 6o's a n d
;ary c o n d i t i o n s (the t h e o r e m should be new, n o n -
iurprising, so utterly inaccessible by
o's was the application of tech-
:riviai, a n d correct), a n d the c o m m u n i t y tends
c u r r e n t m e t h o d s , t h a t their resolution
niques o f linear p r o g r a m m i n g to
:o agree o n o t h e r criteria t h a t are m o r e difficult
m u s t b r i n g s o m e t h i n g n e w - w e just don't
c o m b i n a t o r i c s . It is q u i t e easy to
to formalize like interest a n d significance.
k n o w where.
f o r m u l a t e the m o s t i m p o r t a n t
ays of evaluating it. H o w s h o u l d surveys fit
Let m e propose a radical idea: Let us evaluate Surveys in the way h u m a n i t i e s evaluate their Achievements. W e t e n d t o look d o w n u p o n these ;ireas as "soft", a n d believe t h a t (in c o n t r a s t to o u r own "hard a n d exact" science) success is a •natter of luck or, worse, good abilities in self-pro-
t
combinatorial optimization prob-
i s c r e t e and c o n t i n u o u s
ems as linear p r o g r a m s w i t h i n t e -
he m o s t intrinsic a m o n g the division
grality c o n d i t i o n s , and it is quite
lines is discrete vs. c o n t i n u o u s , because it
easy to solve these, if we disregard
nvolves basic structures a n d m e t h o d s of ur science. In this lasr section, which
the integrality c o n d i t i o n s ; the g a m e is to find ways to wrire u p
m o t i o n . Clearly this feeling is far from the t r u t h ,
necessarily gets a bit m o r e technical, I p u t
these linear p r o g r a m s in such a
ind. h u m a n i t i e s have their own ways of recog-
forth s o m e examples d e m o n s t r a t i n g h o w
way t h a t disregarding integrality
l i z i n g excellence in intellectual achievements.
m u c h we c o u l d lose if we let this c h a s m
c o n d i t i o n s is justified.
We c o u l d o n l y gain b y l e a r n i n g h o w t o d o this
g r o w wider, a n d h o w m u c h we can gain
\The power of tools from elsewhere,
without o u r m o r e direct criteria.
by b u i l d i n g bridges over it.
To s u p p o r t m y plea for the u n i t y of
fnftnite
Tiathematics, let me discuss o n e
OUR S C I E N C E C O U L D O N L Y C A I N 8V HtJRE OF THE M E T H O D S
FROM THE
T R E A S U R Y OF H U M A N T H O U G H T
ADAPTING
VAST
I N T O OUR
OWN
P U R S U I T OF K N O W L E D G E .
Problems and conjectures. In a small c o m m u n i t y , everybody knows w h a t the m a i n p r o b l e m s are. p u t in a c o m m u n i t y o f 1 0 0 , 0 0 0 people, p r o b l e m s have to be identified a n d stated in a precise way. 'oorly stated p r o b l e m s lead to b o r i n g , irrelevant ults. T h i s elevates the f o r m u l a t i o n of conjecures t o t h e r a n k of research results. C o n j e c t u r i n g
i
iecame an art in the h a n d s of the late Paul Erdos, / h o f o r m u l a t e d m o r e conjectures t h a n p e r h a p s
£ll m a t h e m a t i c i a n s before h i m p u t together. H e Considered his conjecrures as p a r t of his m a t h e matical oeuvre as m u c h as his t h e o r e m s . O n e of U y m o s t prized m e m o r i e s is the following c o m m e n t from h i m : "I never envied a t h e o r e m from Enybody; b u t I envy you for this conjecture." )f course, w i t h conjectures we run into the same ifficulty as with surveys: It is difficult to f o r m u late w h a t makes a good conjecture. A n d indeed, there is a lot of controversy a r o u n d the style of fcrdos conjectures. It is easy to agree that if a
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to finite.
It is perhaps unnecessary
:o argue t h a t discrete a n d c o n t i n u o u s
recent d e v e l o p m e n t in the t h e o r y
Tiathematics c o m p l e m e n t each o t h e r
of algorithms. M y starting example
m d that each utilizes m e t h o d s a n d tools
(s a simple a l g o r i t h m i c p r o b l e m in
from the other. W e use the finite to
g r a p h t h e o r y : Given a (finite)
a p p r o x i m a t e rhe infinite. T o discretize a
g r a p h , find a p a r t i t i o n of its n o d e
c o m p l i c a t e d c o n t i n u o u s s t r u c t u r e has
pet into two classes so t h a t the
ilways been a basic m e t h o d - from the
l u m b e r of edges c o n n e c t i n g the
definition o f the R i e m a n n integral
w o classes is as large as possible.
:hrough t r i a n g u l a t i n g a manifold in (say)
In spite of its simplicity, this is a
l o m o l o g y t h e o r y to numerically solving a
rather i m p o r t a n t p r o b l e m ; see the
martial differential e q u a t i o n o n a grid.
t n o n o g r a p h o f Deza and Laurent
'.n spite o f this, I feel t h a t t h e status o f
for a d e s c r i p t i o n o f its far-reaching
amplications of discrete m a t h e m a t i c a l
;onnections.)
m e t h o d s in c o n t i n u o u s m a t h e m a t i c s s less than satisfactory. Perhaps c o m b i n a rorics has nor yet reached the d e p t h a n d lower of analysis or algebra. Perhaps p a r t )f the reasons is cultural: A discrete n a t h e m a t i d a n is m o r e likely to have studied Galois T h e o r y or the B o r s u k - U l a m JTheorem than a "classical" m a t h e m a t i c i a n , say, R a m s e y T h e o r y or the M a x - F l o w Vlin-Cut T h e o r e m . Finite to infinite.
It is a slightly m o r e
mbtle t h o u g h t t h a t the infinite is often 'or p e r h a p s always?) a n a p p r o x i m a t i o n o f h e large finite. C o n t i n u o u s structures ire often cleaner, m o r e s y m m e t r i c , a n d icher t h a n their discrete c o u n t e r p a r t s Tor example, a p l a n a r grid has a m u c h mailer degree of s y m m e t r y t h a n the
Unfortunately, this p r o b l e m is N P iard. If you are n o t familiar w i t h :his basic n o t i o n of complexity :heory, it means roughly t h a t there js n o efficient (polynomial time) ilvay no find t h e best p a r t i t i o n (at least subject to the hypothesis t h a t p). W e m u s t settle for less: ay, to find an a p p r o x i m a t e l y o p t i m a l p a r t i t i o n . It is very easy to find a p a r t i t i o n w h e r e at least half Lf the edges go across; this was first observed by E r d o s in t h e 6o's in a different c o n t e x t . Since no p a r t i t i o n can pick u p m o r e t h a n all edges, this gives an a p p r o x i m a t e s o l u t i o n that achieves at feast 50% of the o p t i m u m .
Can we do better? This really innocent question remained unanswered until fairly recently, almost Simultaneously, two important results were obtained: Goemans and Williamson gave an efficient approximation algorithm that gets within 13% of the optimum; and building on a series of weaker results, Hastad proved that no efficient (polynomial time) approximation algorithm whatever can do better than 6%. This is a surprisingly small gap for this kind of problem, but from our point of view, it is more important that both results depend on tools that come from quite unexpected places. The negative result is one of several lower bounds an approximability of various optima, proved by iimilar means.The first of these proofs was an application of a result in the theory of interactive proof systems, a very interesting, but at first sight, quite specialized area in complexity theory. Later improvements revealed that the most important mathematical construction in the b roof is an error-correcting code obtained by algebraic methods.
Deeper u n i t y Probability theory is only one illustration of the unity in mathematics that goes deepet than just using tools from other branches. Many of the basic questions are not a priori discrete or continuous in nature - they can be modelled as discrete problems, or continous problems. In the last years, I have worked on sampling algorithms (algorithms generating a uniformly distributed random element From a set that is large and often only implicitly described). This question leads to estimating the mixing time of Markov chains (the number of steps before the chain becomes essentially stationary). From the point of view of this application, |t is natural to consider finite Markov chains - a computation in a computer is necessarily finite. But in the analysis, it depends on the particular application whether one prefers to use a finite, or a general measurable, state space. All the essential (and very interesting) connec(The algorithm itself depends on another line of tions that have been discovered hold in previous results based on connections between both models. In fact, the genetal mathedistant areas. The key step is the use of semimatical issue is dispersion: We might definite optimization, which is an extension of be intetested in dispersion of heat in linear programming, building heavily on the a material, or dispersion of probability spectral theory of symmetric matrices. Again, during a random walk, or many other this is not an isolated result; the use of semirelated questions. There is always a Lapladefinite optimization (combined with randomcian operator that describes one step of ized algorithms) has been very successful in the the dispersion. The speed of the disperdesign of approximation algorithms. sion is governed by the spectral gap of Probability. This brings me to a topic that seems the Laplacian; but if infotmation about to bridge most of the division lines in mathematthe spectral gap is not available, one can ics. The importance of probabilistic methods also relate the dispersion speed to isopein combinatorics, graph theory, and the theory of rimetric inequalities in the state space. {llgorithms is exploding. Beside their traditional To establish isoperimetric inequalities, use in Monte-Carlo methods in integration and pne most often constructs (explicitly or simulation, randomized algorithms are used for implicitly) muiticommodity flows. counting, exact and approximate optimization, My second example is more vague. I start primality testing, and one could go on. With probably the most important series In non-algorithmic graph theory, the probabiliof results in graph theory in the last stic method was first introduced in the 50's decade or two, the Graph Minor Theory by Erdos. As a method for showing the existence developed mainly by Robertson and bf objects (graphs, or colorings of a given graph, Seymour. Recall Kutatowski's classical etc.), it is now basic and extremely powerful. theorem: A graph can be embedded in the Probability enters proofs of theorems whose plane if and only if it does not contain statement has nothing to do with probability. pvo specific gtaphs. The notion of conThe role of probability is certainly not restricted tainment can be defined here in several to combinatorics and graph theory: Just let me different but equivalent ways; let us settle mention sieve methods in prime number theory, on "containment as a minor", which Or the explanation of tutbulence in terms of means the following: H is a minor oft; if it statistical mechanics.
can be obtained from G by deleting edges and nodes, and contracting some edges to single nodes. The class of planar graphs (just like the class of graphs embeddable in any other fixed surface) is closed under taking of minors. It is clear therefore that this class can be characterized by excluded minors (just list alt minor-minimal nonplanar graphs). The point in Kuratowski's Theorem is to show that this set of excluded minors consists of two graphs only. Wagner formulated the daring conjecture in the 30's that every class of graphs that is closed under taking of minors can be pharacterized by a finite list of excluded minors. The central result in rhe Robertfcon-Seymour theory is the proof of this conjectute. However, what 1 would like to comment on is an "auxiliary" result, Which describes, in a sense, large graphs not containing a fixed graph H as a minor. Informally, it says that every such graph can be constructed in the following way: Take an arbitrarily large graph embedded in a surface with bounded genus; add edges connecting nodes on the same face at bounded distance; add abounded
number of further nodes; and glue together such graphs along bounded sets of nodes in a tree-like fashion. Here,"bounded"means a bound depending on the graph G but nothing else. We can see the beginnings of a "global" theory of graphs emerging here: What does a huge graph look like? What hidden structures can be identified in this seemingly unstructured universe? Probably there is a more general theory, identifying 3-dimensional, 4-dimensional etc. structures in large graphs. But the formidable difficulties in the Robertson-Seymour theory (stretching over 19 papers now) warn us that such a theory ivill not be easy to establish. Recently I learned about the work of David and Semmes, and I could not help noticing some analogy with the Robertson-Seymour theory. They give a decomposition of a "reasonable"metric space into pieces of different dimension on different scales. Is there more to this analogy? put speaking of "global" graph theory brings other important results to mind. The Regularity Lemma of Szemeredi states that every huge graph can be "decomposed" into a bounded number of pieces that look "random" (the number of pieces depends on the error in approximating randomness; an exact statement would take too much
preparation again). Recently, a flow of exciting applications of this basic lemma emerged. Does Szemeredi's Lemma have a more general setting? An indication may be the recent work of Frieze and Kannan, showing a connection between Szemeredi's Lemma and low-rank approximation of matrices. There is no natural way to divide mathematics, but serious communication gaps can arise unless we realize that we have to pay for avoiding them: Pay not only with organizational effort but also with research time devoted to expository Writing and reading those expositions, to popularizing mathematics and to listening to mathematical problems from various areas of applications. Acknowledge!
am indebted to Tom Zaslavsky for
reulingrimu
refully and for suggesting many
aszld Lovasz is Profe 1 fJomputer Sc ence at Yale University, (view Haven.
Wolf Prize in Mathematics, Vol. 2 (pp. 359^15) eds. S. S. Chern and F. Hirzebruch © 2001 World Scientific Publishing Co.
Curriculum Vitae
Born Orange, New Jersey, USA, February 20, 1931 Positions Held 1954-57 Assistant Professor, Princeton University 1957-60 Associate Professor, Princeton University 1960-67 Professor, Princeton University 7/1/1963-7/1/73 Member, Institute for Advanced Study 1967-69 Professor, University of California, Los Angeles 1969-70 Professor, Massachusetts Institute of Technology 7/1/1970-7/88 Professor, Institute for Advanced Study 8/88- Professor and Director, Institute for Mathematical Sciences, SUNY at Stony Brook Honors and Societies American Mathematical Society Fields Medal, 1962 National Academy of Sciences, USA, 1963American Philosophical Society, 1965National Medal of Science, 1967 James Madison Medal, Princeton University, 1977 Wolf Prize, 1989
360 Topological Methods in Modern Mathematics
PUBLICATIONS OF J O H N MILNOR —THROUGH 1992
1.
On the total curvature of knots, Ann. of Math. (2) 52 (1950), 248-257.
2.
An axiomatic approach to measurable utility, (with I. N. Herstein), Econometrica 21 (1953), pp. 291-297.
3.
The characteristics of a vector field on the two-sphere, Ann. of Math. (2) 58 (1953), 253-257.
4.
On total curvatures of closed space curves, Math. Scand. 1 (1953), 2 8 9 296.
5.
Sums of positional games, in "Contributions to the Theory of Games, Vol. II," (H. W. Kuhn and A. W. Tucker, eds.), Ann. of Math. Stud. No. 28, Princeton University Press, Princeton, N. J., 1953, pp. 2 9 1 301.
6.
Games against nature, in "Decision Processes," (R. M. Thrall, C. H. Coombs and R. L. Davis, eds.), Wiley, New York; Chapman and Hall Ltd., London, 1954, p p . 49-59. (Reprinted in "Game Theory and Related Approaches to Social Behavior," (M. Shubik, ed.), Wiley, New York, 1964.)
7.
Some experimental n-person games, (with G. Kalish, J. Nash, and E. D. Nering), in "Decision Processes," (seeprevious reference), pp. 301-327.
8.
Link groups, Ann. of Math. (2) 59 (1954), 177-295.
9.
Construction of universal bundles, I, Ann. of Math. (2) 63 (1956), 2 7 2 284.
xv
PUBLICATIONS OF JOHN MILNOR
10.
Construction of universal bundles, II, Ann. of Math. (2) 63 (1956), 430436.
11.
On the immersion of n-manifolds in n + l-space, Comment. Math. Helv. 30 (1956), 275-284.
12.
On manifolds homeomorphic to the 1-sphere, Ann. of Math. (2) 64 (1956), 399-405.
13.
The geometric realization of a semi-simplicial complex, Ann. of Math. (2) 65 (1957), 357-362.
14.
Groups which act on Sn without fixed points, Amer. J. Math. 79 (1957), 623-630.
15.
Isotopy of links, in "Algebraic Geometry and Topology. A Symposium in Honor of S. Lefschetz," (R. H. Fox, D. C. Spencer and A. W. Tucker, eds.), Princeton University Press, Princeton, N.J., 1957, pp. 280-306.
16.
On games ofsurvival, (with L. S. Shapley), in "Contributions to the Theory of Games, Vol. Ill," (M. Dresher, A. W. Tucker and P. Wolfe, eds.), Ann. of Math. Stud. No. 39, Princeton University Press, Princeton, N. J., 1957, p p . 15-45.
17.
On the existence of a connection with curvature zero, Comment. Math. Helv. 32 (1958), 215-223.
18.
Some consequences of a theorem of Bott, Ann. of Math. (2) 68 (1958), 444-449.
19.
On the parallelizability of the spheres, (with R. Bott), Bull. Amer. Math. Soc. (N.S.) 64 (1958), 87-89. .
20.
The Steenrod algebra and its dual, Ann. of Math. (2) 67 (1958), 150-171.
21.
On simply connected A-manifolds, in "Symposium Internacional de Topologia Algebraica," Univ. Nac. Autonoma de Mexico and UNESCO, Mexico City, 1958, pp. 122-128.
22.
On the Whitehead homomorphism J, Bull. Amer. Math. Soc. (N.S.) 64 (1958), 79-82.
23.
Differentiable structures on spheres, Amer. J. Math. 81 (1959), 962-972.
24.
Sommes de varietes differentiables et structures differentiables des spheres, Bull. Soc. Math. France 87 (1959), 439-444.
25.
On spaces having the homotopy type of a CW-complex, Trans. Amer. Math. Soc. 90 (1959), 272-280.
26.
Bernoulli numbers, homotopy groups, and a theorem of Rohlin, (with M. A. Kervaire), in "Proc. Internat. Congress Math. 1958," Cambridge University Press, 1960, pp. 454-458.
362 PUBLICATIONS OF JOHN MILNOR
27.
Two remarks on fibre homotopy type, (with E. Spanier), Pacific J. Math. 10 (1960), 585-590.
28.
On the cobordism ring Q* and a complex analogue, I, Amer. J. Math. 82 (1960), 505-521.
29.
A procedure for killing homotopy groups of differentiable manifolds, in "Differential Geometry. Proc. Symposia Pure Math. Vol. Ill," Amer. Math. S o c , Providence, R. I., 1961, pp. 39-55.
30.
Two complexes which are homeomorphic but combinatorially distinct, Ann. of Math. (2) 74 (1961), 575-590.
31.
On 2-spheres in 4-manifolds, (with M. A. Kervaire), Proc. Nat. Acad. Sci. U.S.A. 4 7 ( 1 9 6 1 ) , 1651-1657.
32.
Variedades lisas confrontera, An. Inst. Mat. Univ. Nac. Autonoma Mexico 1 (1961), 82-116.
33.
A unique decomposition theoremfor 3-manifolds, Amer. J. Math. 84 (1962), 1-7.
34.
A duality theorem for Reidemeister torsion, Ann. of Math. (2) 76 (1962), 137-147.
35.
A survey of cobordism theory, Enseign. Math. (2) 8 (1962), pp. 16-23.
36.
On axiomatic homology theory, Pacific J. Math 12 (1962), 337-341. (Reprinted in "Algebraic Topology: A Student's Guide," by J. F. Adams, Cambridge University Press, 1972.)
37.
An example of anomalous singular homology, (with M. G. Barratt), Proc. Amer. Math. Soc. 13 (1962), 293-297.
38.
The work of J. H. C. Whitehead, in "The Mathemadcal Works of J. H. C. Whitehead, Vol. 1," Pergamon Press, Oxford, New York, 1962, pp. xxi-xxxiii.
39.
Topological manifolds and smooth manifolds, in "Proc. Internat. Congress Math. Stockholm 1962," Inst. Mittag-Leffler, Djursholm, 1963, pp. 132-138.
40.
Spin structures on manifolds, Enseign. Math. (2) 9 (1963), 198-203.
41.
Groups of homotopy spheres, (with M. A. Kervaire), Ann. of Math. (2) 77 (1963), 504-537.
42.
"Morse Theory. Based on lecture notes by M. Spivak and R. Wells," Ann. of Math. Stud. No. 51, Princeton University Press, Princeton, N. J., 1963. (Russian translation 1965, Japanese translation 1968.)
43.
On the Betti numbers of real varieties, Proc. Amer. Math. Soc. 15 (1964), 275-280.
xvii
PUBLICATIONS OF JOHN MILNOR
44.
Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci. U.S.A. 51 (1964), 542.
45.
Microbundles, I, Topology 3 (1964) Suppl. 1, 53-80.
46.
Most knots are xvild, Fund. Math. 54 (1964), 335-338.
47.
Differential topology, in "Lectures on Modern Mathematics, Vol. II," T L. Saaty, ed., Wiley, New York, 1964, pp. 165-183. (Russian translation: Uspehi Mat. Nauk 20 (1965), pp. 41-54.)
48.
Some curious involutions of spheres, (with M. W. Hirsch), Bull. Amer. Math. Soc. (N.S.) 70 (1964), 372-377.
49.
Some free actions of cyclic groups on spheres, in "Differential Analysis. Bombay Colloquium," Oxford University Press, London, 1964, pp. 3 7 42.
50.
On the structure of Hopf algebras, (with J. C. Moore), Ann. of Math. (2) 81 (1965), 211-264.
51.
On the Stiefel-Whitney numbers of complex manifolds and of spin manifolds, Topology 3 (1965), 223-230.
52.
Remarks concerning spin manifolds, in "Differential and Combinatorial Topology. A Symposium in Honor ofMarston Morse,'" Princeton University Press, Princeton, N. J., 1965, pp. 55-62.
53.
"Lectures on the /i-Cobordism Theorem. Notes by L. Siebenmann and J. Sondow," Princeton University Press, Princeton, N.J., 1965. (Russian translation 1969.)
54.
"Topology from the Differential Viewpoint," University Press of Virginia, 1965. (Polish translation 1969.)
55.
Whitehead torsion, Bull. Amer. Math. Soc. (N.S.) 72 (1966), 358-426.
56.
Singularities of 2-spheres in 4-space and cobordism of knots, (with R. H. Fox), Osaka J. Math. 3 (1966), 257-267.
57.
Solution of the congruence subgroup problem for SL„(/i > 3) and Sp2„(fl > 2), (widi H. Bass and J. P. Serre), IHES Publ. Math. No. 33 (1967), 59-137.
58.
On characteristic classes for spherical fiber spaces, Comment. Math. Helv. 43 (1968), 51-77.
59.
A note on curvature and fundamental (1968), 1-7.
60.
"Singular Points of Complex Hypersurfaces," Ann. of Math. Stud. No. 6 1 , Princeton University Press, Princeton, N. J., 1968.
group, J. Differential Geom. 2
364 PUBLICATIONS OF JOHN MILNOR
61.
Infinite cyclic covering, in "Conference on the Topology of Manifolds," (J. G. Hocking, ed.), Prindle, Weber and Schmidt, Boston, 1968, pp.115-113.
62.
Growth offinitelygenerated solvable groups,]. Differential Geom. 2 (1968), 447-449. On isometries of inner product spaces, Invent. Math. 8 (1969), 83-97. Semi-characteristics and cobordism, (with G. Lusztig and F. P. Peterson), Topology 8 (1969), 357-359. A problem of cartography, Amer. Math. Monthly 76 (1969), 1102-1112. Torsion et type simple d'homotopie, (with O. Burlet), in "Essays on Topology and Related Topics. Memoires dedies a George de Rham," SpringerVerlag, Berlin, New York, 1970, pp. 12-17. Algebraic K -theory and quadratic forms, Invent. Math. 9 (1970), 318-344.
63. 64. 65. 66.
67. 68.
Isolated singularities defined by weighted homogeneous polynomials, (with P. Orlik), Topology 9 (1970), 385-393. 69. Symmetric inner product spaces in characteristic 2, in "Prospects in Mathematics," Ann. of Math. Stud. No. 70, Princeton University Press, Princeton, N. J., 1971, pp. 59-75. 70. "Introduction to Algebraic A'-Theory," Ann. of Math. Stud. No. 72, Princeton University Press, Princeton, N. J., 1971. (Russian translation 1974.) 71. On the construction FK, in "Algebraic Topology: A Student's Guide," by J. F. Adams, Cambridge University Press, 1972, pp. 119-136. 72. "Symmetric Bilinear Forms," (with D. Husemoller), Springer-Verlag, Berlin, New York, 1973. 73. 74.
75.
76.
"Characteristic Classes," (with J. Stasheff), Ann. of Math. Stud. No. 76, Princeton University Press, Princeton, N. J., 1974. Isolated critical points of complex functions, in "Differential Geometry. Proc. Symposia Pure Math. Vol. XXVII, Part I," Amer. Math. S o c , Providence, R. I., 1975, pp. 381-382. On the 3-dimensional Brieskorn manifolds M(p, q, r), in "Knots, Groups and 3-manifolds. Papers dedicated to the memory of R. H. Fox," (L. P. Neuwirth, ed.), Ann. of Math. Stud. No. 84, Princeton University Press, Princeton, N.J., 1975, p p . 175-225. Hilbert's problem 18: on crystallographic groups, fundamental domains, and on sphere packing, in "Mathematical Developments Arising from Hilbert Problems. Proc. Symposia Pure Math. Vol. XXVIII," Amer. Math. S o c , Providence, R. I., 1976, pp. 491-506.
xix
PUBLICATIONS OF JOHN MILNOR
77.
Differential Geometry (problem list), ibid., 54-57.
78.
Curvatures of left invariant metrics on Lie groups, Adv. in Math. 21 (1976), 293-329.
79.
On deciding whether a surface is parabolic or hyperbolic, Amer. Math. Monthly 84 (1977), 43-46.
80.
On fundamental groups of complete affinely flat manifolds, Adv. in Math. 25 (1977), 178-187.
81.
Characteristic numbers of 3-manifolds, (with W. Thurston), Enseign. Math. (2) 23 (1977), 249-254.
82.
Analytic proofs of the "hairy ball theorem " and the Brouwerfixed point theorem, Amer. Math. Monthly 85 (1978), 521-524.
83.
Hyperbolic geometry: the first 150 years, Bull. Amer. Math. Soc. (N.S.) 6 (1982), 9-24.
84.
On the homology of Lie groups made discrete, Comment. Math. Helv. 58 (1983), 72-85.
85.
On the geometry of the Kepler problem, Amer. Math. Monthly 90 (1983), 353-365.
86.
Remarks on infinite-dimensional Lie groups, in "Relativity, Groups and Topology II," Les Houches Session XL, 1983, B. S. DeWitt and R. Stora, eds., Elsevier Science Publishers B.V., North Holland, 1986, p p . 1007-1058.
87.
Values of large games II: oceanic games, (with L. S. Shapley, written in 1961), Math. Oper. Res. 3 (1978), 290-307.
88.
On polylogarithms, Hurwitz zeta functions, and the Kubert identities, Enseign. Math. (2) 29 (1983), 281-322.
89.
On the concept ofattractor, Comm. Math. Phys. 99 (1985), 177-195; 102 (1985), 517-519.
90.
Directional entropies of cellular automaton-maps, in "Disordered Systems and Biological Organization," (Bienenstock et al., eds.), SpringerVerlag, Berlin, New York, 1986, pp. 113-115.
91.
The work of M. H. Freedman, in "Proc. Internat. Congress Math. Berkeley, 1986," Amer. Math. S o c , Providence, R. I., 1987, 13-15.
92.
Non-expansive Henon maps, Adv. in Math. 69 (1988), 109-114.
93.
On the entropy geometry of cellular automata, Complex Systems 2 (1988), 357-386.
PUBLICATIONS OF JOHN MILNOR
94.
On iterated maps of the interval, (with W. Thurston, preprint 1977), in "Dynamical Systems" (J. C. Alexander, ed.), Lect. Notes Math. 1342, Springer-Verlag, Berlin, New York, 1988, pp. 465-563. 95. Dynamical properties of plane polynomial automorphisms, (with S. Friedland), Ergodic Theory Dynamical Systems 9 (1989), 667-699. 96. Self-similarity and hairiness in the Mandelbrot set, in "Computers in Geometry and Topologoy," (M. C. Tangora, ed.), Marcel Dekker, New York, Basel, 1989, pp. 211-257. 97. Dynamics in one complex variable, Stony Brook Institute for Mathematical Sciences Preprint # 1 9 9 0 / 5 . 98. Remarks on iterated cubic maps, Stony Brook Institute for Mathematical Sciences Preprint #1990/6. Experiment. Math. 1 (1992), 5-24. 99. Fixed points of polynomial maps, II. Fixed point portraits, (with L. R. Goldberg), Stony Brook Institute for Mathematical Sciences Preprint #1990/14. To appear in Ann. Sci. Ecole Norm. Sup. 100. The Fibonacci unimodal map, (with M. Lyubich), Stony Brook Institute for Mathematical Sciences Preprint #1991/15. To appear in J. Amer. Math. Soc. 101. Hyperbolic components in spaces of polynomial maps, Stony Brook Institute for Mathematical Sciences Preprint # 1 9 9 2 / 3 . 102. Local connectivity of Julia sets: expository lectures, Stony Brook Institute for Mathematical Sciences Preprint # 1 9 9 2 / 1 1 . 103. Remarks on quadratic rational maps, Stony Brook Institute for Mathematical Sciences Preprint #1992/14. 104. On cubic polynomials with periodic critical point, in preparation.
xxi
367 ANNALS OF MATHEMATICS
Vol. 64, No. 2, September, 1956 © 1956 The Johns Hopkins University Press. Reprinted with permission.
ON MANIFOLDS HOMEOMORPHIC TO THE 7-SPHERE BY JOHN MILNOR1
(Received June 14, 1956)
The object of this note will be to show that the 7-sphere possesses several distinct differentiable structures. In §1 an invariant X is constructed for oriented, differentiable 7-manifolds M satisfying the hypothesis (*) H3(M7) = H*(M7) = 0. (Integer coefficients are to be understood.) In §2 a general criterion is given for proving that an n-manifold is homeomorphic to the sphere S". Some examples of 7-manifolds are studied in §3 (namely 3-sphere bundles over the 4-sphere). The results of the preceding two sections are used to show that certain of these manifolds are topological 7-spheres, but not differentiable 7-spheres. Several related problems are studied in §4. All manifolds considered, with or without boundary, are to be differentiable, orientable and compact. The word differentiable will mean differentiable of class C. A closed manifold M" is oriented if one generator n tHn(Mn) is distinguished. §1. The invariant X(Af7) For every closed, oriented 7-manifold satisfying (*) we will define a residue class X(Af7) modulo 7. According to Thorn [5] every closed 7-manifold M7 is the boundary of an 8-manifold B*. The invariant X(M') will be defined as a function of the index T and the Pontrjagin class pi of B*. An orientation v t Hs(Bg, M7) is determined by the relation 6V = ft. Define a quadratic form over the group H*(Bg, M 7 )/(torsion) by the formula a —> (v, a 2 ). Let T(B ) be the index of this form (the number of positive terms minus the number of negative terms, when the form is diagonalized over the real numbers). Let pi tH (Bg) be the first Pontrjagin class of the tangent bundle of B*. (For the definition of Pontrjagin classes see [2] or [6].) The hypothesis (*) implies that the inclusion homomorphism i:H\B\
M7) -»H*(B g )
is an isomorphism. Therefore we can define a "Pontrjagin number"
q(B*) = <,, (f W ) THEOREM 1. The residue class of 2q(B*) — r(B*) modulo 7 does not depend on the choice of the manifold B8. Define \(M ) as this residue class.2 As an immediate consequence we have: 7 COROLLARY 1. If \(M ) j£ 0 then M is not the boundary of any 8-manifold having fourth Betti number zero. 1
The author holds an Alfred P. Sloan fellowship. 399
400
JOHN MILNOR
Let B\, B\ be two manifolds with boundary M7. (We may assume they are disjoint.) Then C8 = B\ U B\ is a closed 8-manifold which possesses a differentiable structure compatible with that of Bi and B\. Choose that orientation v for C8 which is consistent with the orientation in of B\ (and therefore consistent with — v2). Let q(Cs) denote the Pontrjagin number {v, p\((?)). According to Thom [5] or Hirzebruch [2] we have r(C8) = < y , A ( 7 p 2 ( C 8 ) - p ? ( C 8 ) ) ; and therefore 45r(C 8 ) + q(C8) = 7(v, p2(C*)) = 0
(mod 7).
2q((?) - r(C 8 ) = 0
(mod 7).
This implies (1) LEMMA
1. Under the above conditions we have
(2)
r(Cf) = r(B\) -
r(B\)
(C8) = q(B\) -
q{B\).
and (3)
Formulas 1, 2, 3 clearly imply that 2q{B\) - r{B\) = 2q(Bl) - r{B%)
(mod 7);
which is just the assertion of Theorem 1. PROOF OF LEMMA 1. Consider the diagram
# n ( B i , M) © Hn(B2, M) —
IT(C, M)
t'i © it
Hn(Bx) © Hn(B2)
k
Hn(C)
£Jote that for n = 4, these homomorphisms are all isomorphisms. If a = jh~\ai © «2) tH\C), then (4)
(v, a) = (v;jh~\al
© at)) = (Vl © ( - * ) , a\ © a\) = (vu a\) - <m , a\).
Thus the quadratic form of C8 is the "direct sum" of the quadratic form of By and the negative of the quadratic form of B 2 . This clearly implies formula (2). Define ax = i7lpi(-Bi) and a2 = iT'piC^). Then the relation *(pi(C)) = pi(Bi) © pi(B»)
implies that 4
Similarly for n = 4fc — 1 a residue class X(M") modulo snn(Lt) could be defined. (See [2] page 14.) For k = 1, 2, 3, 4 we have stti(Lk) = 1,7, 62, 381 respectively.
369 MANIFOLDS HOMEOMORPHIC TO THE 7-SPHERE
jh~\ax
-101
0 a2) = pi(C).
The computation (4) now shows that (y, pl(0)
= <*, , al) - (* , a\),
which is just formula (3). This completes the proof of Theorem 1. The following property of the invariant X is clear. LEMMA 2. If the orientation of M is reversed then X(M ) is multiplied by — 1. As a consequence we have 7 1 COROLLARY 2. If \(M ) ^ 0 then M possesses no orientation reversing diffeo3 morphism onto itself. §2. A partial characterization of the n-sphere Consider the following hypothesis concerning a closed manifold M" (where R denotes the real numbers). (H) There exists a differentiable function f:M" —> R having only two critical points Xo, Xi. Furthermore these critical points are non-degenerate. (That is if Wi, • • • , un are local coordinates in a neighborhood of Xo (or xi) then the matrix (d^f/dUidUj) is non-singular at xo (or xi).) n THEOREM 2. If M satisfies the hypothesis (H) then these exists a homeomorphism n of M onto S" which is a diffeomorphism except possibly at a single point. Added in proof. This result is essentially due to Reeb [7]. The proof will be based on the orthogonal trajectories of the manifolds / = constant. Normalize the function / so that /(xo) = 0, f(xi) = 1. According to Morse ([3] Lemma 4) there exist local coordinates Vi, • • • , vn in a neighborhood V of xt so that /(x) = v\ + • • • + i)!„ for i e F. (Morse assumes that / is of class C3, and constructs coordinates of class C1; but the same proof works in the CT case.) The expression ds — dv\ + • • • + dv2n defines a Riemannian metric in the neighborhood V. Choose a differentiable Riemannian metric for Mn which coincides with this one in some neighborhood4 V of x 0 . Now the gradien of / can be considered as a contravariant vector field. Following Morse we consider the differential equation J=grad//||grad/||2In the neighborhood V this equation has solutions
(»i(0, • • • , ».(0) = (oi(0*, • • • , aM) for 0 ±S t < E, where a = (ai, • • • , o„) is any n-tuple with ^fa\ = 1. These can be extended uniquely to solutions xa(t) ior 0 ^ t ^ 1. Note that these solutions satisfy the identity 3
A diffeomorphism /is a homeomorphism onto, such that both/ and/ -1 are differentiable. * This is possible by [4] 6.7 and 12.2.
370 402
JOHN MILNOR
/(*«(<)) = IMap the interior of the unit sphere of Rn into Mn by the map MO*,'
• • • , an(ty) -» x„(t).
It is easily verified that this defines a diffeomorphism of the open n-cell onto Mn — (xi). The assertion of Theorem 2 now follows. Given any diffeomorphism g'.S"^ —> S"~x, an n-manifold can be obtained as follows. n CONSTRUCTION (C). Let M (g) be the manifold obtained from two copies of R" by matching the subsets R" — (0) under the diffeomorphism 1 u —+ v ~
T.
/ n
u
\
q I ,,——- I.
II « II VII « 1 1 / (Such a manifold is clearly homeomorphic to <Sn. If g is the identity map then n M (g) is diffeomorphic to Sn.) n COROLLARY 3. A manifold M can be obtained by the construction (C) if and only if it satisfies the hypothesis (H). PROOF. If M"(g) is obtained by the construction (C) then the function
/(x) = || w j|7(i+'IMI 2 ) = i / ( i + || Ml2) will satisfy the hypothesis (H). The converse can be established by a slight modification of the proof of Theorem 2. §3. Examples of 7-manifolds Consider 3-sphere bundles over the 4-sphere with the rotation group SO(4) as structural group. The equivalence classes of such bundles are in one-one correspondence6 with elements of the group ir3(SO(4)) ^ Z + Z. A specific isomorphism between these groups is obtained as follows. For each (h, j) e Z + Z letfhj:B3 — SO(4) be defined by fkj(u) • v = u vu', for v « R . Quaternion multiplication is understood on the right. Let i be the standard generator for ^(S4). Let &,• denote the sphere bundle corresponding to (ft,) t ir3(SO(4)). LEMMA 3. The Pontrjagin class pi(£»y) equals ± 2{h — j)t. (The proof will be given later. One can show that the characteristic class c(hj) (see [4]) is equal to (h + j)t.) For each odd integer k let Mk be the total space of the bundle &, where h and j are determined by the equations h + j ' = 1, h — j = k. This manifold M\ has a natural differentiable structure and orientation, which will be described later. LEMMA 4. The invariant \(Mk) is the residue class modulo 7 of k — 1. LEMMA 5. The manifold Ml satisfies the hypothesis (H). Combining these we have: 8
See [4] §18.
371 MANIFOLDS HOMEOMORPHIC TO THE 7-SPHERE
403
THEOREM 3. For k ^ 1 mod 7 the manifold Mk is homeomorphic to S but not diffeomorphic to S . (For k = ± 1 the manifold Ml is diffeomorphic to S 7 ; but it is not known whether this is true for any other k.) Clearly any differentiable structure on S7 can be extended through R* — (0). However: COROLLARY 4. There exists a differentiable structure on S which cannot be extended throughout R . This follows immediately from the preceding assertions, together with Corollary 1. PROOF OF LEMMA 3. It is clear that the Pontrjagin class pife.,) is a linear function of h and.?. Furthermore it is known that it is independent of the orientation of the fibre. But if the orientation of S is reversed, then &y is replaced by £_;•_/• . This shows that Pi(^y) is given by an expression of the form c(h — j)i. Here c is a constant which will be evaluated later. A PROOF OF LEMMA 4. Associated with each 3-sphere bundle Mk —> S there is a 4-eell bundle Pk:Bl -* S\ The total space Bk of this bundle is a differentiable manifold with boundary Ml . The cohomology group H (Bk) is generated by the element a = pk (i). Choose orientations n, v for Mk and Bk so that
{v, or1*)2) = +i. Then the index r(Bl) will be + 1 . The tangent bundle of Bk is the "Whitney sum" of (1) the bundle of vectors tangent to the fibre, and (2) the bundle of vectors normal to the fibre. The first bundle (1) is induced (under pk) from the bundle £hj, and therefore has Pontrjagin class pi = pk {c(h — j)i) = cka. The second is induced from the tangent bundle of S4, and therefore has first Pontrjagin class zero. Now by the Whitney product theorem ([2] or [6]) Pi(Bt) = cka + 0. For the special case A: = 1 it is easily verified that B\ is the quaternion projective plane Pz{K) with an 8-cell removed. But the Pontrjagin class pi(P2(K)) is known to be twice a generator of H (P^K)). (See Hirzebruch [1].) Therefore the constant c must be ± 2 , which completes the proof of Lemma 3. Now q(B\) = <*, (r 1 (±2fca)) : ) = 4fc2; and 2q - r = 8fc2 - 1 = k2 - 1 (mod 7). This completes the proof of Lemma 4. 4 PROOF OF LEMMA 5. As coordinate neighborhoods in the base space <S take the complement of the north pole, and the complement of the south pole. These can be identified with euclidean space R under stereographic projection. Then a point which corresponds to u t R under one projection will correspond to u' = u/\\ u ||2 under the other. The total space Mk can now be obtained as follows. Take two copies of R X S3 and identify the subsets (724 — (0)) X S under the diffeomorphism (u, v) -
«i/)
=
(M/||U||2,MW/||M||)
372 404
JOHN MILNOR
(using quaternion multiplication). This makes the differentiable structure of Ml precise. Replace the coordinates («', v') by («", v') where u" = w'(t/) _1 . Consider the function f:MI —* R defined by /(*) = »(p)/(l + || « ||*)* = « ( « » ) / ( l + || u" ||V; where 9?(t>) denotes the real part of the quaternion v. I t is easily verified that / has only two critical points (namely (u, v) = (0, ± 1 ) ) and that these are nondegenerate. This completes the proof. §4. Miscellaneous results THEOREM 4. Either (a) there exists a closed topological 8-manifold which does not possess any differentiable structure; or (b) the Pontrjagin class pi of an open 8-manifold is not a topological invariant. (The author has no idea which alternative holds.) PROOF. Let Xk be the topological 8-manifold obtained from Bl by collapsing its boundary (a topological 7-sphere) to a point x0. Let or « H*(Xl) correspond to the generator a e H*(Bk). Suppose that Xl, possesses a differentiable structure, and that pi(Xk — (x0)) is a topological invariant. Then pi(Xl) must equal ±2fca, hence
2q(Xl) - r(Xl) = 8A;2 - 1 = k2 - 1
(mod 7).
2
But for fc ^k 1 (mod 7) this is impossible. Two diffeomorphisms /, g'-Mi —> Mt will be called differentiably isotopic if there exists a diffeomorphism M" X R —* M% X R of the form (x, t) —» (h(x, I), t) such that f{x)
{t
0)
HX t) = ! ^ h{x l) ' \g(x) (t i> l ) . LEMMA 6. If the diffeomorphisms f, g'.S""1 —* S"~l are differentiably isotopic, then the manifolds Mn(j), Mn(g) obtained by the construction (C) are diffeomorphic. The proof is straightforward. 6 THEOREM 5. There exists a diffeomorphism f:S* —* <S of degree +1 which is not differentiably isotopic to the identity. Proof. By Lemma 5 and Corollary 3 the manifold Ml is diffeomorphic to M7(f) for some / . If / were differentiably isotopic to the identity then Lemma 6 would imply that M\ was diffeomorphic to <S7. But this is false by Lemma 4. PRINCETON UNIVERSITY REFERENCES
1. F. HIRZEBRTJCH, UeberdiequaternionalenprojektivenR&ume, S.-Ber. math.- naturw. Kl. Bayer. Akad. Wiss. Munchen (1953), pp. 301-312. 2. , Neue topologische Methoden in der algebraischen Geometrie, Berlin, 1956.
MANIFOLDS
HO.MEO-MOHPHIC TO T H E
/-SPHERE
405
3. M. M O R S E , Relations between the numbers of critical points of a real function of n independent variables, T r a n s . Amcr. M a t h . S o c , 27 (1925), p p . 345-396. 4. X. STEENHOD, T h e topology of fibre bundles, Princeton, 1951. 5. R. T H O M , Qitelgues proprieles qlohale des variUCs dijferentiables, Comment. Math. Helv., 28 (1954), p p . 17-80. 6. Wc W E X - T S U N , Sur les classes caractcristiques des structures fibrics sphcriques, Actual. sci. industr. 1183, Paris, 1952, pp. 5-89. 7. O. K E E B , Sur certain proprieles topoloiiiques des varictes feuilletecs. Actual, sci. industr. 1183, Paris, 1952, p p . 91-154.
374 ANNALS OF MATHEMATICS
Vol. 68, No. 2, September, 1958 © 1958 The Johns Hopkins University Press. Reprinted with permission.
SOME CONSEQUENCES OF A THEOREM OF BOTT BY JOHN MILNOR1
(Received February 11, 1958)
It will be shown that the following theorem, due to R. Bott [3], can be used to solve several well known problems ; including the problem of the existence of division algebras, and the parallelizability of spheres2. (Independent solutions of these problems, also based on Bott's work, have been given by Kervaire and Hirzebruch.) THEOREM OF BOTT. For any Om-bundle ? over the sphere S4k, the Pontrjagin class py(£) e Hik(Sik ; Z) is divisible by (2k — 1)!. (This result was conjectured, and proved up to powers of 2, by Borel and Hirzebruch [I])The following result, which follows from Wu Wen-Tsiin [19], will also be needed. Since Wu's paper is in Chinese a proof is included in the appendix. The epimorphism Z -> Z4 induces a homomorphism H*(K; Z) -> H*(K; Zt) which will be denoted by a -*• (a)t. Let i : Zz-^ Zt denote the inclusion homomorphism. For any Om-bundle f over a complex K, the class (Pk(£))i 6 H (K\ Z4) is determined by the Stiefel-Whitney classes w,(6) e Hl(K; Zz). In particular if the Stiefel-Whitney classes w^), •• • , wik-i(E) are zero then (pk(£))i = i*wilc(£). Combining these two results, the following is obtained. THEOREM OF W U . ik
1. There exists an Om-bundle £ over the sphere Sn with w„(?) ^t 0 only for n = 1, 2, 4 or 8. (Examples of such bundles can be given as follows : for n — 1 the 2fold covering of the circle, and for n = 2, 4 or 8 the On-bundle over Sn associated with the Hopf fibering S2""1 ->• S*.) THEOREM
PROOF. According to Wu [16] such a bundle can exists only if n is a power of 2. Hence it is certainly sufficient to consider the case n = 4k, k > 2. The identity
(ft(O)* = **«>„(£) e ff"0S"; zt) = zt is valid, since the lower Stiefel-Whitney classes must be zero. In other words the class w4Jt(?) is zero if and only if pk(£) is divisible by 4. But 1 J
The author holds a Sloan fellowship. A preliminary account of this work has been given in [20]. 444
375 A THEOREM OF BOTT
445
pt(f) is known to be divisible by (2k — 1)!. For k > 2 this proves that w«(f) = 0. r THEOREM 2. The sphere S is parallelizable only for r = 1,3, 7. (Compare Steenrod and Whitehead [10].) / PROOF. The fibering SOr > SOr+1 > Sr associated with the tangent r bundle of S has the following homotopy sequence : » 7Tr(SOr +l) - ^ nr(Sn —
TTr-ASO,)
>
ffr.,(SOrtl)
>0.
r
The group 7Tr(S ) will be identified with the integers. Then 9(1) e nr~i(SOr) is the element which corresponds to the tangent bundle of Sr. (See Steenrod [9, § 18]). For each X e 7rr(SOr+l) let £ denote the corresponding SO r+r bundle over Sr+1, and let X(£) denote its Euler class ( = top Stief el-Whitney class with integer coefficients). Let ^ be the standard generator of Hr+1(Sr+1 ; Z). Then f*(X) is equal to the negative of the " Euler number " <X(f), fi). [Proof. Let o(f) e Hr+i(Sr+'; 7tr(SOr+l)) denote the obstruction to the existence of a cross-section of $. Then X($), the obstruction to the existence of a cross-section in the associated sphere bundle, is equal to /*(o(f)). According to Steenrod [9, p. 180] the identity
0 Z+Z
1 %%%!£
f
2
3
4
5
6
7
Z + Zi
Z%
Z +Z
Z2
Z
Z-2
446
JOHN MILNOR
PROOF. This follows from Bott's computation [2] of the stable groups ffr_1(SOr+1), together with the exact homotopy sequence used to prove Theorem 2.
3. Let Min be a simply-connected differentidble manifold such that the cohomology group H\M!n ; Z) is infinite cyclic for i — 0, n, 2n, and zero otherwise. Then n must be 2, 4, or 8. (Examples are provided by the complex, quaternion, and Cayley projective planes. It will be shown in a later paper [7] that the condition of simple-connectivity can be eliminated. This will give an answer to Problem 5 of [5]). THEOREM
If a generates H"(Min ; Z), then the Poincare duality theorem implies that a w a generates Hin(Mln ; Z). Hence PROOF.
Sq" : Hn(M2n ; Z2)
> Hin{Mln ; Z,)
in non-zero. The formulas of Wu [15] now imply that the Stiefel-Whitney class w„ of the tangent bundle 6 is non-zero. Choose a map g : Sn ->• Min which, under the Hurewicz homomorphism, corresponds to a generator of H„(MM ; Z). Then the bundle 0' over Sn induced from 6 by g will satisfy w„(^') =£ 0. Therefore n must be 1, 2, 4 or 8. Since the case n = 1 is easily excluded, this completes the proof. Bott's theorem is related to the question of the existence of maps with Hopf invariant 1 as follows. Let J : nn^SOJ) -*• n-m+n_1(Sm) be the homomorphism of G. W. Whitehead [13], and let rn : 7Tm+n.1{Sm) -> Z3 be the generalized Hopf invariant of Steenrod [8], which is defined using the functional Sqn operation. For each odd prime q let Tq.i '• ^m + 2i(,-l)-l('S' m )
> Zt
denote the corresponding homomorphism based on the reduced gth power ^'. THEOREM 4a. The image J^n^SOn), m^n, contains an element JX with generalized Hopf invariant Tn(Jfy different from zero only if n equals 2, 4, or 8. THEOREM 4b. The image «7^24(,_1)_1(iSOro), m 2; 2i(q — 1), contains an element JX with r 9 ,«(^) different from zero only ifi = 1. PROOF OF 4b. Let $ be the SOm-bundle over S" associated with J, where n = 2i(q — 1). Let E be the total space of the associated bundle having the unit ball Bm as fibre, so that the boundary E is the total space of the associated sphere bundle. According to [7, Theorem 3, Corollary 1],
A THEOREM OF BOTT
447
the collapsed space E\E can be obtained from the sphere Sm by attaching an (m + w)-cell, using on attaching map in the homotopy class JX. Thus the generalized Hopf invariant rg,4(«//l) is non-zero if and only if the homomorphism ^ " : Hm(E, E ; Zq)
> Hm+n(E, E; Zq)
is non-zero. Let 0 : H3{Sn ; Zq) - • H1+m(E, E ; Z„) denote the isomorphism of Thorn [12]. According to Wu [18, § IV] the class tfr^W) e H"(Sn ; Zt) can be expressed as a polynomial in the Pontrjagin classes of £, reduced modulo q. But these Pontrjagin classes are zero, except for p((,-i)/2(f) which is divisible by (i(q — 1) — 1)!. For i > 1, since the number (i(q — 1) — 1)! is divisible by q, it follows that the operation ^ " must be zero. Theorem 4a is proved in a similar way, using Theorem 1 together with Thorn's definition of the Stiefel-Whitney classes. (See [12]). Appendix Following Hirzebruch [6] define the Pontrjagin class pk of an Om-bundle as (— 1)* times the Chern class c2t of the Z7m-bundle induced by the inclusion Om-*Um. This is slightly different from the Pontrjagin class as defined by Pontrjagin and Wu. (Compare [17, Theorem 4]). Consider the exact sequence of cohomology group corresponding to the coefficient sequence PROOF OF THE THEOREM OF W U .
0
> Z2 —^-+ Z, - ^ Z2
•0;
as well as the Pontrjagin squaring operation 5P : H*(K; Zt)
> H4*(K; Z<) .
(See for example Whitehead [14]). LEMMA 1. The Pontrjagin class pk of any Om-bundle is related to the Stiefel-Whitney classes wu ••• , wik by an identity
(P*)t — ¥>(wiK) + i+Mw^, ••• , wik) where fk is a polynomial with coefficients in Z2. It is clearly sufficient to consider the case of the universal bundle over the Grassmann space Gm(R), with m large. The identity j^(w) = w^jw holds for any cohomology class w. Comparing this with the relation3 PROOF.
~3~See Wu [Tvfheorem 3}.
448
JOHN MILNOR 3*{(Pk\)
= (P*)i =
W2k^lV.lk
it follows that (p*)4 - ^w.ik e (kernel j j = i*Hlk{Gn{R); Z2) . Since the cohomology ring H*{Gm{R); J?,) is generated by the StiefelWhitney classes, this proves Lemma 1. To prove the theorem it is only necessary to show that the coefficient of wik in fk is non-zero. Let T denote the universal E7m-bundle over the complex Grassmann space GJC). Recall that the cohomology ring H*(Gm(C) ; Z) is a polynomial ring4 generated by the Chern classes of T. The inclusion Um -> 02m induces an 02m-bundle over Gm(C) which will be denoted by TB. Applying Lemma 1 to this bundle TJJ, the relations5 P*(T*) = ck(ry - 2c*_1(T)ct+1(T) +
± 2c0(T)c«(T)
and «w(Tfl) = 0 ,
w.2r(rR) = (cr(T))2
show that the polynomial fk must satisfy fk(0, w2, 0,wt,---,
wik) = w2k-2w.ik+z + w2k.iW2k+i + • • • + wawlk .
Therefore/^(O, 0, • • • , 0, ivik) — wik ; which completes the proof. PRINCETON
UNIVERSITY REFERENCES
1. A. BOREL and F . HlRZEBRUCH, Characteristic classes and homogeneous spaces, to appear (Amer. J. Math.). 2. R. BOTT, On the stable homotopy of the classical groups, Proc. Nat. Acad. Sci. U.S.A. 43 (1957), 933-935. 3. , The space of loops on a Lie-group, to appear (Mich. J. Math.). 4. S. S. CHERN, Characteristic classes of Hermitian manifolds, Ann. of Math. 47 (1946), 85-121. 5. F . HlRZEBRUCH, Some problems on differentiate and complex manifolds, Ann. of Math. 60 (1954), 213-236. 6. , Neue topologische Methoden in der algebraischen Geometrie, Springer, 1956. 7. J. MILNOR, On spaces with a gap in cohomology, to appear. 8. N. E. STEENROD, Coholomogy invariants of mappings, Ann. of Math. 50(1949), 954-988. 9. , T h e topology of fibre bundles, Princeton, 1951. 10. and J.H.C. WHITEHEAD, Vector fields on the n-sphere, Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 58-63. 11. E. STIEFEL, Uber Richtungsfelder in den projektiven Rmimen und, einen Satz aus der reelen Algebra, Comm. Math. Helv. 13 (1940), 201-218. 12. R. THOM, Espaces fibres en spheres et carres de Steenrod, Ann. sci. Ecole norm, sup 69 (1952), 109-182. * Chern [4]. 5 See Hirzebruch [6, p. 68] and Steenrod [9, p. 212].
A THEOREM OP BOTT
449
13. G. W. WHITEHEAD, On the homotopy groups of spheres and rotation groups, Ann. of Math. 43 (1942), 634-640. 14. J.H.C. WHITEHEAD, On simply connected 4-dimensional polyhedra, Comm. Math. Helv. 22 (1949), 48-92. 15. WU WEN-TSUN, Classes caracteristigues et i-carres d'une variete, C. R. Acad. Sci. Paris 230 (1950), 508-511. 16. , Les i-carres dans une variete grassmannienne, C. R. Acad. Sci. Paris 230 (1950), 918-920. 17. , On Pontrjagin classes I, Scientia Sinica 3 (1954), 353-367. 18. , On Pontrjagin classes II, Scientia Sinica 4 (1955), 455-490. 19. , On Pontrjagin classes III, Acta Math. Sinica 4 (1954), 323-346. 20. R. BOTT and J. MILNOK, On the parallelizability of the spheres, Bull. Amer. Math. Soc. 64 (1958), 87-89.
Inventiones math. 9, 318-344 (1970)
Algebraic X-Theory and Quadratic Forms JOHN MILNOR (Cambridge,
Massachusetts)
The first section of this paper defines and studies a graded ring K^F associated to any field F. By definition, KnF is the target group of the universal n-linear function from F* x • • • x F' to an additive group, satisfying the condition that ax x • • • x a„ should map to zero whenever a i + a i+i = l- Here F' denotes the multiplicative group F - 0 ^ Section2 constructs a homomorphism d: KnF—>.£„_! F associated with a discrete valuation on F with residue class field F. These homomorphisms 8 are used to compute the ring K^Fff) of a rational function field, using a technique due to John Tate. Section 3 relates K^ F to the theory of quadratic modules by defining certain " Stiefel-Whitney invariants " of a quadratic module over a field F of characteristic 4=2. The definition is closely related to Delzant [5]. Let W be the Witt ring of anisotropic quadratic modules over F, and let IcW be the maximal ideal, consisting of modules of even rank. Section 4 studies the conjecture that the associated graded ring {W/I,I/I2,I2/I\...) is canonically isomorphic to K^F/IK^F. Section 5 computes the Witt ring of a field F(t) of rational functions. Section 6 describes the conjecture that K^F/IK^F is canonically isomorphic to the cohomology ring H*(GF;Z/2Z); where GF denotes the Galois group of the separable closure of F. An appendix, due to Tate, computes K^F/HC^F for a global field. Throughout the exposition I have made free use of unpublished theorems and ideas due to Bass and Tate. I want particularly to thank Tate for his generous help. §1. The Ring K^F To any field F we associate a graded ring K.F^KoF.J^F^F,...) as follows. By definition, KXF is just the multiplicative group F* written additively. To keep notation straight, we introduce the canonical isomorphism
Algebraic K-Theory and Quadratic Forms
319
where l(ab)=l(a) + l(b). Then K^F is defined to be the quotient of the tensor algebra (Z, KtF, KlF®KlF, KlF®K1F®K1F,...) by the ideal generated by all I {a}® I (I — a), with a 4=0,1. In other words each KnF,n^.2,is the quotient of the w-fold tensor product Kx F® • • • ® Kx F by the subgroup generated by all l(at)®•••®l{a„) such that a ; + a i + i = l for some i. In terms of generators and relations, K^ F can be described as the associative ring with unit which is generated by symbols 1(a), aeF', subject only to the defining relations l(a b)=1(a) + 1(b) and 1(a) 1(1 — a)=0. Explanation. This definition of the group K2 F is motivated by work of R. Steinberg, C. Moore, and H. Matsumoto on algebraic groups; and has already been the object of much study. (Compare references [2 — 4, 7 — 9, 17].) For n ^ 3 , the definition is purely ad hoc. Quite different definition of K„ for n2:3 have been proposed by Swan [18] and by Nobile and Villamayor [11]; but no relationship between the various definitions is known. First let us describe some fundamental properties of the ring K^ F. (Examples will be given in §§1.5 — 1.8.) Lemma 1.1. For every £eKmF and every neKnF, the identity
r,i = (-ir^n is valid in Km+nF. Proof (following Steinberg). Clearly it suffices to consider the case m = n = l. Since — a = (l — a)/(l — a - 1 ) for a4=1, we have
l(a)l(-a)=l(a)l(l-a)-l(a)l(l-a-1) = l(a)l(l-a) + l(a~l)l(l-a-1)
= 0.
Hence the sum 1(a) 1(b)+1(b) 1(a) is equal to 1(a) l(-a) + 1(a) 1(b) + 1(b) 1(a) + 1(b) l(-b) =
l(a)l(-ab)+l(b)l(-ab)
= l(ab)l(-ab)
= 0;
which completes the proof. Here are two further consequences of this argument: Lemma 1.2. The identity l(a)2 = l(a) l( — 1) is valid for every 2
l(a)eK1F.
For the equation 1(a) l( — a) = 0 implies that 1(a) = 1(a) (/(—1) + /(— a)) must be equal to 1(a) l( — l).
320
J. Milnor:
Lemma 1.3. / / the sum a1 + ---+a„of non-zero field elements is equal to either 0 or 1, then l(a^) ... l(a„) = 0. Proof by Induction on n. The statement is certainly true for n = 1,2; so we may assume that nS:3. If a1 + a2-0, then the product l{a^l{a2) is already zero. But if a t + a2 4= 0, then the equation «i/(«i + a2) + a 2 /(ai + a2) = l implies that
{lia^-li^
+ a^^ia^-lia.
+
a^Q.
Multiplying by l(a3)... l(a„), and using 1.1 and the inductive hypothesis l(a1 + a2)l(ai)...l(a„)
= 0,
the conclusion follows. Here is an application. Theorem 1.4. The element —I is a sum of squares in F if and only if every positive dimensional element of K^ F is nilpotent. Proof. If — 1 is not a sum of squares, then F can be embedded in a real closed field, and hence can be ordered. Choosing some fixed ordering, define an n-linear mapping from Kt F x • • • x Kj F to the integers modulo 2 by the correspondence it w ~n \^ l - s g n f o ) l-sgn(a„) /(a 1 )x---x/(a„)i-^ ^ • Evidently the right hand side is zero whenever a ; + aj + 1 = l. Hence this correspondence induces a homomorphism K„F—Z/2Z; which carries /( —1)" to 1. This proves that the element /( — 1) is not nilpotent. Conversely, if say — l = aj-\ \-af, then it follows from 1.3 that /(-a 1 2 ).../(-a r 2 ) = 0; hence
/(-l)r = 0mod2XrF.
Since 2J(-1) = 0, it follows immediately that i ( - l ) r + 1 = 0 . For any generator y = l(ax)... l(an) of the group K„F, it follows from 1.2 that ys is equal to a multiple of f(—1)"<S-1). Hence y s =0 whenever n ( s - l ) > r . Similarly, for any sum y1H \-yk of generators, the power {y1 H h yk)s can be expressed as a linear combination of monomials y'i •••y'k w i t n h H 1- k = s- Choosing s > k, note that each such monomial is a multiple of/(-l)" ( s - f c ) . If s>k + r/n, it follows that (V! + --- + yfc)s = 0; which completes the proof.
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To conclude this section, the ring K^F will be described in four interesting special cases. Example 1.5 (Steinberg). If the field is finite, then iC 2 F=0. In fact K^ F is cyclic, say of order q — 1; so § 1.1 implies that K2 F is either trivial or of order ^ 2, according as q is even or odd. But, if q is odd, then an easy counting argument shows that 1 is the sum of two quadratic nonresidues in F; from which it follows that X 2 F = 0. This implies, of course, thatK„F=0forn>2also. Example 1.6. Let R be the field of real numbers. Then every KnR, n ^ l , splits as the direct sum of a cyclic group of order 2 generated by 1{~Y)", and a divisible group generated by all products 1(0^... l{a„) with at,..., a„ > 0. This is easily proved by induction on n, using the argument of §1.4 to show that /( —1)" is not divisible. Example 1.7. Let F be a local field (i.e. complete under a discrete valuation with finite residue class field), and let m be the number of roots of unity in F. Calvin Moore [10] proves that K2 F is the direct sum of a cyclic group of order m and a divisible group. We will show that K„F is divisible for n ^ 3 . Consider the algebra K^F/pK^F over Z/pZ; where p is a fixed prime. If p does not divide m, then Moore's theorem clearly implies that K2 F/p K2F = 0. Suppose that p does divide m. We claim then that: (1) the vector space K1F/pK1F has dimension 2^2 over Z/pZ; (2) the vector space K2F/pK2F has dimension 1; and (3) for each a=t=0 in K^/pK^F there exists /? in K1F/pK1F so that
a 0*0.
In fact (1) is clear; (2) follows from Moore's theorem; and (3) is an immediate consequence of the classical theorem which asserts that, for each aeF' which is not a p-th power, there exists b so that the p-th power norm residue symbol (a, b)Fis non-trivial. (See for example [20, p. 260].) The correspondence l(a)l(b)^(a,b)F clearly extends to a homomorphism from K2 F to the group of p-th roots of unity. So, taking a=1(a), /? = /(&), the conclusion (3) follows. Proof that every generator a P y of K3F/pK3F is zero. Given a, /?, y one can first choose /?'=(= 0 so that oe/?' = 0 (using (1) and (2)), and then choose y' so that /?' y' = /? y (using (2) and (3)). The required equation
Thus K3F/pK3F = 0 for every prime p; which proves that K3F is divisible.
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Example 1.8. Let F be a global field (that is a finite extension of the field Q of rational numbers, or of the field of rational functions in one indeterminate over a finite field). Let Fv range over all local or real completions of F. The complex completions (if any) can be ignored for our purposes. The inclusions F —> Fv induce a homomorphism K2 F —> ©„ K2 F„/(max. divis. subgr.), where each summand on the right isfinitecyclic by 1.6 and 1.7. Bass and Tate [3] have shown that the kernel of this homomorphism is finitely generated, but the precise structure of the kernel is not known. Moore has shown that the cokerael is isomorphic to the group of roots of unity inf. The structure of KnF is not known for n ^ 3 , but Tate has proved the following partial result: The quotient K„F/2K„F maps isomorphically to the direct sum, over all real completions Fv, of KmFJ2K„Fv*Z/2Z. Thus the dimension of K„F/2K„F as a mod 2 vector space is equal to the number of real completions. Tate's proof of this result is presented in the Appendix. It may be conjectured that the subgroup 2K„F is actually zero for w^3, so that K„F itself is a vector space over Z/2Z. As an example, for the field Q of rational numbers the isomorphism KnQ^Z/2Z for ng;3 can be established by methods similar to those of §2.3. § 2. Discrete Valuations and the Computation of K*F(i) Suppose_that a field F has a discrete valuation v with residue class field F (=FV). The group of units (elements u with ord„u=0) will be denoted by U, and the natural homomorphism U—>Fm by ui->u. An element n of F' is prime if ord„ n — 1. Lemma 2.1. There exists one and only one homomorphism d = dv from KnF to K„_iF which carries the product l(n)l(u2)...l(u„) to l(u2)...l(iin) for every prime element n and for all units u2, ...,«„. This homomorphism d annihilates every product of the form /(u t )... l(u„). (For n = 1 the defining property is to be that dl{n) = 1.) Remarks. Evidently d is always surjective. For n = l the homomorphism d can essentially be identified with the homomorphism ord„: F'-^Z;
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and for n = 2 it is closely related to the classical "tame symbol" n'u1,niu2h->(—l)iJ M'2/U{ which is utilized for example in [3]. To begin the proof, note that any unit ut can be expressed as the quotient % ujn of two prime elements. So the property
follows immediately from the defining equation. Proof of Uniqueness. Choose a prime element n. Since F' is generated by 7t and U, it follows that K„ F is generated by products of the form l(n)rl(ur+1)...l(un). If r = l , then the image of any such product under d has been specified; and if r > l then using the identity l{n)r = l(n) /( — l) r _ 1 it is also specified. But if r=0, then any such product maps to zero. This proves that d is unique, if it exists. Proofof Existence1. It will be convenient to introduce an indeterminate symbol x which is to anticommute with all elements of Kx F. Given any n-tuple of elements l(nilu1),...,l(ni''un)eKlF, construct a sequence of elements cpjeKjF by the formula (xi1 + l(u1))...(xi„ + l(un)) = xn(p0 + xn-1(p1 + --- + (p„. Evidently each
Thus (peK„_1F, and evidently
/(nS).
If two successive n'JUj add up to 1, we will prove that
extends uniquely to a ring homomorphism 0„ from KtF Setting
to this enlarged ring. Now,
0 » = ,Ha) + {3(a) with i^(a) and d(o) in K^F, we obtain the required homomorphism d.
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is well defined and extends to a homomorphism
K^^K^F. Since it is clear that l{nu)l(u2)...l{u^ maps to l(u2)...l(u„), this will complete the proof. To avoid complicated notation, we will carry out details only for the case n" «i + n'2 u2 = 1. There are four possibilities to consider. If i*! > 0, then it follows easily that i2 = 0,
u 2 = l.
Hence the factor x i2 + l(u2) is zero and it certainly follows that cp = 0. The case it = 0, i2 > 0 is disposed of similarly. If i-! = i2 = 0, then ul+u2=T, hence {xii + l(ui))(xi2 + l(u2))=0, so again cp = 0. Finally suppose that ix <0. Then clearly i'i = i 2 and u2 = — uv In this case the product (x iy + /("i)) (x i 2 + l(u2)) evidently simplifies to x2i\ + xhl(P\.) x i(
Hence the expression Yj " Pj
can
x{xil + i1l(^l)){xii
^
+ 0.
e wr tten as
i
+ l(u3))...(xin + l(un)).
Cancelling the initial x, and then substituting /(— 1) for the remaining x's, we evidently obtain an expression for cp. But this substitution carries xif -Mi/( —1) to l(—l)if + i1l(—l) = 0. So (p = 0 in this case also; which completes the proof of 2.1. A similar argument proves the following. Lemma 2.2. Choosing some fixed prime element n, there is one and only one ring homomorphism iP: K*F-+K„F which carries l(n'u) to l(u)for every unit u. In fact ij/ is defined by the rule /(«'»Il1).../(7l'"liIJ ^/(SJ.../(S,,). Details will be left to the reader. Evidently this homomorphism \j/ is less natural than d, since it depends on a particular choice of n.
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Now let F be an arbitrary field. We will use 2.1 and 2.2 to study the field F(i) of rational functions in one indeterminate over F. Each monic irreducible polynomial 7tsF[t] gives rise to a (7i)-adic valuation on F(t) with residue class field F [t]/(7i). Here (n) denotes the prime ideal spanned by n. Hence there is an associated surjection 5.: K . F W - K . ^ F M A n ) . Theorem 2.3. These homomorphisms 8n give rise to a split exact sequence
0-K B F-K B F(t)->©K,,_iFM/(jO->0, where the direct sum extends over all non-zero prime ideals (n). This theorem is essentially due to Tate. In fact the proof below is an immediate generalization of Tate's proof for the special case n = 2. Proof.Keeping n fixed, let LdcKnF(t) be the subgroup generated by those products /(/]).../(/„) such that fu ...,f„eF[i] are polynomials of degree S d. Thus L 0 <=. L j c L 2 <= • • •
with union K„ F(t). Using the homomorphism
^:
KnF(t)^KnF
of 2.2, where n is any monic (irreducible) polynomial of degree 1, we see easily that L 0 is a direct summand of K„ F(t), naturally isomorphic to K„ F. Let n be a monic irreducible polynomial of degree d. Then each element g of the quotient F [t]l{ri) is represented by a unique polynomial geF\f\ of degree
K^Fiqi^^LJL^
which carries each product I (g2) •.. l(g„) to the residue class ofl(n) l(g2)... l(g„) modulo Ld_x. Proof. First consider the correspondence ' ( g 2 ) x '' • x l(gn)>->• I(n)l(g2)
• • • l(gn) m o d L d _ t
from K j F I T I / ^ x ••• x Kt F[f]/(n) to Ld/Ld_1. We will show that this correspondence is linear, for example as a function of g 2 . Suppose that g2=«2g2mod(7t),
where g 2 , g'2, g 2 are polynomials of degree < d. Then g2 = *:./'+g 2 g 2
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where/is also a polynomial of degree
Since the case/=0 is straight forward, this proves that our correspondence is (n — l)-linear. To prove that this correspondence gives rise to a homomorphism /(g2)-'(g n )^'(*)/(g 2 ).../(g n ) from Kn_lF to LJLd_u it isnow only necessary to note that the image is zero whenever g,-+gj+i = 1 and hence gj + g J+1 = 1. This proves 2.4. Lemma 2.5. The homomorphisms d% give rise to an isomorphism between LJLd_1 and the direct sum of Kn_1F[f]/(n) as % ranges over monic irreducible polynomials of degree d. Proof. Inspection shows that each 8„ induces a homomorphism
LJL^^K^FiqUn). Furthermore it is clear that the composition
is either the identity or zero, according as n = n' or n =# n'. So to complete the argument we need only to show that LJLd_^ is generated by the images of the/i„. Consider any generator of Ld, expressed as a product 1(f)... '(/s)'(gs+i)•••'(&.) where f, ...,/, have degree d and g J+1 , ...,g„ have degree < d. If s ^ 2 then we can set
f2=-af+g with aeF' and degree g < d. If g =t= 0 it follows that
afjg+fi/g = l hence (/(fl) + /(/l)-Ife))('(/2)-*fe)) = 0. Thus the product /(/,) l(f2) can be expressed as a sum of terms
1(f) l(g) + l(g) l(fi)-Ha) HfiHH") 1(g)-l(g)\
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each of which involves at most one polynomial of degree d. A similar situation obtains when g=0. It follows, by induction on 5, that every element of Ld can be expressed, modulo Li_1, in terms of products '(/iH(g2)---'(gn) where only f has degree d. If f is irreducible, then setting f—an this product evidently belongs to the image of hn. But if f is reducible then the product is congruent to zero modulo Ld_v Thus LJLd_1 is generated by the images of the homomorphisms h„, which completes the proof of 2.5. An easy induction on d now shows that the homomorphisms dK induce an isomorphism from Ld/L0 to the direct sum of Kn_1F[t']/(n), taken over all monic irreducible n of degree g d. Passing to the direct limit as d —>• 00, this completes the proof of Theorem 2.3. To conclude this section, let us record a similar, but easier statement. Lemma 2.6. Suppose that afield E is complete under a discrete valuation with residue class^ field E=F. Then for any prime p distinct from the characteristic ofE there is a natural split exact sequence O^KnF/pK„F-+KnE/pK„E^UK„_1F/pKn_lF^O. Proof. If a unit of E maps to 1 in F, then it has a p-th root. Hence the correspondence l(u)h^l(u) vaoApK^E is well defined. This correspondence extends to a ring homomorphism
K.F^K„E/pK.E. Further details will be left to the reader. § 3. The Stiefel-Whitney Invariants of a Quadratic Module For the rest of this paper we will only be interested in the quotient of the ring K^F by the ideal IK^F. To simplify the notation, let us set knF=KnF/2KnF. Thus k„F is a graded algebra over Z/2Z, with k^F^F'/F"2. We will always assume that F has characteristic 4= 2. The symbol knF will stand for the algebra consisting of all formal series £0 + £1 + £2 H— with £i€ktF. Thus knF is additively isomorphic to the cartesian product i 0 F x ^ f xk2Fx ••-. Let M be a quadratic module over F. That is M is a finite dimensional vector space with a non-degenerate symmetric bilinear inner product. Then M is isomorphic to an orthogonal direct sum e-© of one dimensional modules. Here denotes the one dimensional quadratic module such that the inner product of a suitable basis vector with itself is a.
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Define the Stiefel-Whitney invariant w(M)eknF of a quadratic module Ms©---© by the formula w(Af) = (l + /( fll ))(l + /(a 2 ))...(l + /(flr)). Thus w(M) can be written as l+w 1 (M)+--- + wr(M) where w{{M), the i-th Stiefel-Whitney invariant, is equal to the i-th elementary symmetric function of /(a^,..., l(ar) considered as an element of/c.F. Evidently wl is just the classical "discriminant" of M, and w2 is closely related to the classical Hasse-Witt invariant. Remark. This definition is very similar to the definition proposed by Delzant [5]. However Delzant's Stiefel-Whitney classes belong to the cohomology H*(GF; Z/2Z) of the maximal Galois extension of F. They are precisely the images of our w, under a canonical homomorphism k*F^H*{GF;Z/2Z) which is described in § 6. Lemma 3.1. The invariant w(M) is a well defined unit in the ring knF and satisfies the Whitney sum formula w(M®N) = w(M)w(N). Proof Just as in the classical proof that the Hasse-Witt invariant is well defined, it suffices to consider the rank 2 case. (Compare O'Meara [12, p. 150].) Suppose then that e <&>£<«>©<£>. Then the discriminant ah must be equal to a/? multiplied by a square; or in other words (4)
l(a) + l(b) = l(a.) + l(p) 2
mod2K,F.
2
Furthermore, the equation a = ax + by must have a solution x,yeF. Since the case x=0 or y = 0 is easily disposed of, we may assume that x + 0, y + 0 . Then the equation 1 = a x 2 /a + b y2la. implies that 0 = (Z(fl) + 2I(x)-/(a))(Z(fc) + 2Z(y)-Z(a)) s(Z(a)-Z(a))(Z(6)-Z(a))
mod2K2F.
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Rearranging terms, and then substituting (4) this implies that l(a)l(b)=l(a)(l(a) + l(b)-l(ct)) = l{oi)l{P)
mod2K2F;
which completes the proof. Remark. Delzant shows that a quadratic module over a number field is determined up to isomorphism by its rank and Stiefel-Whitney cohomology classes. But Scharlau points out that the corresponding statement for an arbitrary field is false. The same statements, proofs, and examples apply to our Stiefel-Whitney invariants. Now let us introduce the Witt-Grothendieck ring WF, consisting of all formal differences M — N of quadratic modules over F; where M — N equals M' — N' if and only if the orthogonal direct sum M&N' is isomorphic to M'®N. (Compare [5, 14].) The product operation in WF is characterized by the identity
w(M-N) by definition. Next consider a generator
(5)
= w(M)/w(N)
£= «ai>-
of the ideal/"F. Lett = 2" _ 1 . Lemma 3.2. The Stiefel-Whitney invariant w of such a product £, is equal to either . ,, , ., . ,, „„ „ \+l{al)...l{an)l(-l)t-n or (l + l(al)...l(an)l(-l)'->)-1 according as n is odd or even. Proof. Multiplying out the formula (5), we obtain
£ = E±, to be summed as s 1; ...,£„ range over 0 and 1. Here and subsequently, + stands for the sign (— l)£l + ' ' + £ " + n . Therefore w(£) = n ( l + ei/(
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Consider the corresponding product (6)
r](i+£lXl+...+£jiXn)±i
in the ring of formal power series with mod 2 coefficients in n indeterminates. If we substitute 0 for some xh then evidently this product becomes 1. Hence the product (6) must be equal to l + X l - - - X „ / ( X l , •••,*„)
for some formal power series / Therefore w(Q = l +
l(ai)...l(an)f{l(al),...,l(a„))
= l + /(a1).../(an)/(/(-l),...,/(-l)); using §1.2. To compute the power series /(/( — l),...,/( —1)) it suffices to substitute *! = • • • = x„ = x in (6), so as to compute f(x, ...,x). Evidently the product reduces to either (1 + x)' or (1 + x ) - ' according as n is odd or even; where r=2"~ 1 . For n odd it follows that l + xnf(x,...,x) so that
= (l + x)' = l + x',
/(x,...,x) = x'- n ;
and a similar computation can be carried out for n even . This completes the proof. Corollary 3.3 If t = 2n~1, then the invariants wlt... w, _! annihilate the ideal I"F, while w, induces a homomorphism wt:
I"F/Itt+1F-+ktF
which carries the product (<<*!>-
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it is not difficult to show that wr ws = Y (U
r-i,s-i)wr+s_il{-1)',
to be summed over 0 ^ i ^ Min {r, s). Here (i,j, k) stands for the trinomial coefficient (i+j+k)\/i\j\k\. But if r is a power of 2, and if s=0mod2r, then this identity takes the simple form vvrws = w r + s which completes the outlined proof. §4. The Surjection
KJ2KU-^PIP+1
Let F be a field of characteristic 4= 2. The Witt ring W= WF can be defined as the quotient ffi/H, where l^is the Witt-Grothendieck ring of §3, and H is the free cyclic additive group spanned by <1>©< —1>. Clearly H is an ideal, so that W is a ring. Note that the augmentation ideal I in W maps bijectively to a maximal ideal in W. This image ideal will be denoted by / = IF. (Remark. The utility of working with W, rather than W, will become apparent only in § 5.) As in § 3, we set k„F=K„F/2K„F. This will sometimes be abbreviated aSkn = KJ2Kn. Theorem 4.1. There is one and only one homomorphism sn:
KF-^rF/r^F
which carries each product l{a^)... l(a„) in k„F to the product
«a1>-
modulo I F. The homomorphisms st and s 2 are bijective (compare [13]); and every s„ is surjective. Proof. The correspondence
l(ai)x...xl(an)^Y\ from Kt x ••• x Kx to I"/r+1
«a,.>-
modr+1
is n-linear since
-
mod/2.
Furthermore, if ai + a i + 1 = l then an easy computation shows that « a , . > - < l » « a i + 1 > - < l » = 0,
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so the image is zero. Thus this correspondence gives rise to a homomorphism Kn->In/In+1. This homomorphism annihilates 2K„ since 2/(a 1 ).../(a„) = /(a?)/(a 2 ).../(a„) with -
r/In+1^In/I"+1-+k,
of §3.3. Evidently the composition wt°sn is just multiplication by /(-l)'-". But if n equal 1 or 2, then t = n, and the appropriate statement is that w„ ° s„ is the identity. This shows that sx and s 2 are bijective; which completes the proof of 4.1. Remark 4.2. For n>2, this argument proves the following: If multiplication by l{ —1)'_" carries k„F injectively into ktF, then the homomorphism s . k p^/y/n+i is necessarily bijective. Evidently there are two key questions in relating k^ to the Witt ring W. Let F be any field of characteristic 4= 2. Question 4.3. Is the homomorphism s„:fe„F—>In/In+1 bijective for all values ofn? Question 4.4. Is the intersection of the ideals I" equal to zero? (Compare [13, 14].) This section will conclude by proving two preliminary results. (See also §§5.2 and 5.8.) Lemma 43. IfF is a global field, or a direct limit of global fields, then both questions have affirmative answers. Proof. Using Tate's explicit computation of k^ F for a global field (§1.8 or the Appendix), we see that multiplication by /( —1) induces isomorphisms k3 F-+ k4F—• ks F—> • ••.
Together with §4.2, this proves that s„ is bijective in the case of a global field. The corresponding statement for a direct limit follows immediately.
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As to the intersection of the ideals I", first note that each embedding of F in the real field gives rise to a ring homomorphism
WF^WR^Z called the signature. Note that an element of 7 3 F is zero if and only if its signature at every embedding F—> R is zero. In the case of a global field, this statement follows immediately from the Hasse-Minkowski theorem; and for the direct limit of a sequence F1cF2<=F3c:--of global fields it follows easily using the isomorphisms WljmF^timWFt and Emb (lim Fx, R) = hm Emb (Fa, R). But each such signature carries the ideal IF to 2Z, and hence carries the intersection of the ideals I"F to f] 2"Z = 0. This completes the proof. Lemma 4.6. Now suppose that F is a field such that k2 F has at most two distinct elements. Then again the sn are bijective and f] I"=0. Notice that this includes the case of a finite, or local, or real closed, or quadratically closed field; as well as any direct limit of such fields. Proof. If /cj, modulo the null-space of the pairing k1®k1-+k2, has dimension 4=1, then Kaplansky and Shaker [6] show that a quadratic module is completely determined by its rank, discriminant, and HasseWitt invariant. It follows that 7 3 = 0. But just as in §1.7 one sees that /c3 = 0. Since sx and s 2 are already known to be bijective, it certainly follows that every s„ is bijective. On the other hand if k% modulo this null-space has dimension 1, then it is easy to define the "signature" of a quadratic module, and to show that the rank, discriminant, and signature form a complete invariant. (Compare [6, Lemma 1].) Since the signature of an element in I" is divisible by 2", it follows that f] I"=0. Futhermore, techniques similar to those of §1.4 show that k„ is cyclic of order 2, generated by /( — 1)", for every n ^ 2; hence §4.2 implies that every s„ is bijective. This completes the proof. §5. The Witt Ring of a Rational Function Field This section will study the Witt ring, using constructions very similar to those of §2. First consider a field £ which is complete under a discrete valuation v, with residue class field E of characteristic 4= 2. Let n be a prime element.
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Theorem of Springer. The Witt ring WE contains a subring W0 canonically isomorphic to WE. Furthermore WE, splits additively as the direct sum of W0 and <7t> W0. In fact W0 can be defined as the subring generated by as u ranges over units of E, and the isomorphism W0 —> WE is defined by the correspondence h-> . For the proof, see T. A. Springer [16]. Since _<7r>2 =
5<7tU> = <M>.
Note however that d depends on the particular choice of the prime element n. The proof is straightforward. Corollary 5.2. / / the questions 4.3 and 4.4 have affirmative answers for the residue class field E, then they also have affirmative answers for E. Proof. It will be convenient to identify WE with the sub-ring W0 a WE. Note that the ideal IE then splits as a direct sum JE = J E e ( < 7 r > - < l » W E . It follows inductively that r £ = /"£©«7!:>-
(7„)
Consider the diagram knE j
>/c„_1£
l _ E-> PE/F^E-*
n+1
FE/I
> knE
l
_ r^E/rE,
where the top sequence comes from §2.6, the vertical arrows from §4.1, and the bottom sequence is the quotient of (7„) by (7„+1). Checking that this diagram is commutative, and then applying the Five Lemma, the conclusion follows.
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Now consider a field E = F(t) of rational functions. For each monic irreducible n€F\t] we can form the 7t-adic completion E„, with residue class field _ En^F[t]l{n). Let d,: WE->WE„ denote the composition of the natural map WE —> WEn with the homomorphism d of 5.1. Evidently d„(u}=0 and dn(jiu) = (ju). Theorem 5.3. These homomorphisms dn give rise to a split exact sequence 0->WF->WE-»eWE«->0, where E = F(t\ and where the summation extends over all monic irreducible polynomials n in F\_t~\. The proof will be based on the Tate technique already utilized in §2.3. Let LdcWE denote the subring generated by all > such that feF[t~\ is a polynomial of degree ^d. Thus L 0 <= L j c L 2 <= • • •
with union WE. Additively, Ld is generated by all products <_/i.. .X> where the f{ are polynomials of degree ^ d. Note that L 0 is just the image of the natural homomorphism WT— WE. Lemma 5.4. In fact WF maps bijectively to L 0 . Furthermore L0 is a retract of WE under a ring homomorphism p:
WE^WFsL0.
Proof. Choose some monic polynomial n of degree 1, and define p by the conditions /><"> = <">, p<7tu> = . Here u denotes any unit with respect to the (Tr)-adic valuation. It follows from Springer's theorem, applied to the (7t)-adic completion, that p is a well defined ring homomorphism. Since the composition
WF^WE-^WF is the identity, this proves 5.4. Now suppose that
d^l.
Lemma5.5. The additive group Ld is generated, modulo LA_U by expressions <7t gj... gs> where n is an irreducible polynomial of degree d, and g j , . . . , gs are polynomials of degree < d. Furthermore if f is the poly-
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nomial of degree
then
<7t/> = < t g i - - - g s >
mod fa), modLj.!.
Proof. First note that the identity (8)
= + -
holds in the Witt ring of any field. Consider a generator of L„, where the polynomials fx,..., fr are distinct, monic of degree d, and where gj,..., gs have degree < d. If r ^ 2, then defining a polynomial ft of degree < d by the identity (8) becomes
)l", where each exponent ev is 0 or 1. These exponents are well defined except that we may simultaneously replace each E„ by 1 — £„. Now suppose that n=3. For fixed c the correspondence a, b\-+q>(a, b, c) can be described as above. Thus there exist exponents ev(c) so that (p(a,b,c) = Y[v(a,bY/c). Fixing b and c, consider the idele (dv) whose r-th component is Using the symmetry relation (p(a,b,c)=
) are the groups GL(/i) itself and GL(n)XGL(m), n>/w, with H as above. L It is well known that for G=GL(n)XGL(m), G=GL(n,C)XGL(m,Q. It is possible to show that the JL-functions constructed using the subgroup (4) correspond to the representation of LG which is the tensor product of the standard representations of GL (n, C) and GL (m, C). How to construct L-functions for other representations of LG is almost unknown. We now describe the local L and e factors, following the pattern given by J. Tate. We consider a set of integrals and we claim that all these integrals converge in some half-plane and can be meromorphically continued to the whole complex plane. We define L(n, s) in such a way that L(n, s) times any of these integrals is an entire function. For the definition of e we should consider another set of integrals. There is a natural one-one correspondence between these sets such that the corresponding integrals are proportional, and the coefficient of proportionality is the £-factor.
515 Tate Theory for Reductive Groups and Distinguished Representations
587
For GL(2), for instance, we put
L(W,s) =
fwi£x§\x\W-'d*x,
where W belongs to the Whittaker model. (Recall that a Whittaker model for an irreducible representation % of GL (n, kp) is a right invariant space W(n, \jf) of smooth functions on GL(n,kJ satisfying W(zg)=\lr(ztt+...+zn-1JW(g), Vz£Z„, such that the representation of GL(n, k^ on W{n, ^) is equivalent to n.) Then we have the following theorem: (1) Both integrals converge absolutely in some half-plane Rej>c 0 . (2) They are rational functions of q'. (3) There exist two Euler factors Ujt, s) and L(it, s) such that L~\n, s)L{W, s) and L~Hn,s)L(W,s) are polynomials in q',q~*. (4) There is a function e(n, s, ij/) which is of the form aq"' and such that L-l(n, 1 -s)lQV, 1 - s ) = e(7t, s, lftL-Hx, s)L{W, s) for all »P€W(*,^). (5) If Jt and ^r are unramified, W^0 is invariant under the corresponding maximal compact group and fV0Q J)=l, then L(W0, s) = Z,(TT, s).
For GL(n)XGL(m) we have to consider the following integrals. Let re and x be irreducible representations of GL (n, &,) and GL (m, fcp). We put for WtW(n,i(r) and P P e ^ r , ^ " 1 ) (x 0 0) Z,(FK «",«) = fW\y 1, 0hf 1 (Jc)|detx| , - 1/8 d(x, v), ^ 10 0 lJ l(W, W\s)=fwl(l
1, +1 1 s+,+1/a x d ;c. 0 V (*)|detx|-
Then for the integrals L and L the same theorem as in the case GL (2) holds. (II) Rankin-Selberg construction. R. Rankin and A. Selberg gave a method of analytic continuation for the convolution of Dirichlet series relating to classical modular forms. The reformulation of their construction in the language of representation theory was considered by H. Jacquet ([10] for GL(2)XGL(2) with explicit computation) and Piatetski-Shapiro [7] in generality, but without explicit computation). We will outline [7]. Assume that H is a reductive subgroup of G and P a maximal parabolic subgroup of H. For simplicity we assume that the center of G is trivial. We put P=MNt where AT is a reductive group and N its unipotent radical. Let
,s), \fr£RA, and therefore /(>, s)=
/
(
,s)dg,
where ty(g),0(g))=
f
= II
and hence that J{q>, s) has the Euler product K
p (k) 03 3 such that pi Of. = E and lim |T(0i) = 0. + 0, i.e., v is the ground state for L (see also [5]). Some results on the existence of a ground state for the Schro'dinger operator with almost periodic potential were obtained by Kozlov [8], In this paper we study the eigenfunctions Lv = ev for small e. By analogy with the Bloch functions of the Schrodinger operators with periodic potential, we give the following definition. Definition 1. )). » (T) r ( ) ' 1 + wt ((p + (0) 1 ) o r , on l e t t i n g ip = nto and m u l t i p l y i n g b o t h s i d e s by e ^ n P , - q n x ( ( n + 1) co) e***1* + (2 - 2nX cos 2n (nco + k (no))) • e inp ?11 (raw) - qn ((n - i ) ©) e ^ D * = e g u ( n co)e in " Sequence v = W n ) , v n = e i n P q n ( n w ) i s both a Bloch and an e i g e n s e q u e n c e , i . e . , Lv = e v . Another Bloch s e q u e n c e , with quasimomentum —p, i s c o n s t r u c t e d u s i n g glt ( / \ _ ( r " 1(,p) '•",( )\ V m— ^ r - i ( 9 _ ( D j u,L(«p)r-i ( )( where W^2' ^ SE~. Then w ' 2 ' must s a t i s f y t h e e q u a t i o n C)WM (cp) — WW (tp + co) C<*> + V eAW — 0. This y i e l d s , f o r t h e i n d i v i d u a l m a t r i x e n t r i e s , uff ( ) + / ? (1 + iftp,)-1 j* ( t. LEMMA 1. Suppose that at the n-th step there is given the system i("((p + (o) = [(o" °)+Aw(9)]dl"( ) CO = A$ (9). a] < c n / a | | / | | i , with cn = 0(n\ogn), andcn otherwise independent of B. The case p > 2 in (a) above was proved by Bourgain [3] and was based on his key lemma (below). Using this lemma both Bourgain [6] and Carbery [10] independently extended the result to p > §. (The role of the special exponent | will be clarified momentarily.) Part (b) of the theorem, which is proved by very different methods, is in Stein and Stromberg [104]. The main idea of the proof of (a) is the following remarkable universal decay estimate for the Fourier transform of the characteristic function XB, of a convex symmetric body B in R n of unit volume. Let X B ( £ ) = fB e~2ntx'^ dx. LEMMA. XB(£(0)> , et C°, W°, W°, D° les projections de C, Wi, Wj, D dans V/Vi- On voit aisement que u° est la representation naturelle de G°, que C° est une chambre de cette representation dont W°, W° sont les murs, et que C° — D° est l'intersection des demi-espaces radiciels de w° limites par W° et W° et contenant C°. D'autre part, Wj et Wj sont les seuls murs de C contenant Vj, par consequent, si Wi = Wkg, k = i ou j . De ces remarques, il resulte immediatement que la proposition a demontrer sera vraie pour G si elle Test pour G°, c'est-a-dire pour n = 2, mais dans ce cas, c'est une proposition de geometrie euclidienne plane, facile a demontrer. 3.14. L e m m e . Pour tout g G G et tout i € S(A), Wi n Cg est I'adherence d'une face de Cg. Si Wi = Wkg (k e S(A)) (cas oil Wi est un mur de Cg), ri = g~xrkg-
Lp(a(gp)dp))f(gpt
In many situations Weil representations are distinguished. For instance, this is the case when G is the split three-dimensional unitary group. Here we can also prove the local assumption made above (uniqueness of jrp—T,). Details were given by the author in lectures in Yale University (1977-1978)11 1
During the congress I have learned that T. Shintani has considered independently afterwards (1978) a similar construction in the special case of the holomorphic modular forms in the two dimensional ball. He did not give a definition for local L and e factors. He found remarkable necessary and suflScient conditions'for holomorphic modular form in the two-dimensional ball to be an eigen function of Hecke operators in terms of its Fourier—Jacobi series.
518 590 I. Piatetski-Shapiro: Tate Theory for Reductive Groups and Distinguished Representations We can also apply this method to a group G which is a central extension of G, for instance the metaplectic covering of GL (2). Here again Weil representations are distinguished (Gelbart) and the local uniqueness assumption is a theorem (joint work with S. Gelbart and R. Howe). The main result which was obtained in this case is the construction of /.-functions; it will be published in a forthcoming paper of S. Gelbart and the author (see also [11]). The main application is the proof of the existence of a Shimura correspondence. This says that there exists an injective map Sp from the set of irreducible admissible representations of Gp into the set of irreducible admissible representations of Gp which preserves L and £ and has the following property: if n = ® np is an irreducible automorphic cuspidal representation, then S(«) = (g> Sf(np) is an irreducible automorphic representation and is cuspidal iff n does not come from a Weil representation. Unfortunately, the image of S or Sr is unknown. Despite lack of evidence I dare to conjecture that the analogous map exists for any central extension of any reductive group. I am very grateful to D. Zagier for reading this talk and for his help in its preparation. References 1. H. Jacquet and R. P. Langlands, Automorphic forms on GL(2), Lecture Notes in Math, vo 114, Springer-Veriag, Berlin and New York 1970. 2. R. P. Langlands, Problems in the theory of automorphic forms. Lectures in Modern Analysis and Applications III, Lecture Notes in Math., Vol. 170, Springer-Veriag, Berlin and New York, 1970, pp. 18—87. 3. R. Godement and H. Jacquet, Zeta-functions of simple algebras. Lecture Notes in Math., Vol. 260, Springer-Veriag, Berlin and New York, 1972. 4. R. Langlands, Euler products. Yale Math. Monographs, Yale Univ. Press, New Haven and London, 1971. 5. On the functional equations satisfied by Eisenstein series. 6. H. Jacquet, G. Shalika and J. Piatetski-Shapiro, Automorphic forms on GL (3), Ann. of Math. (to appear). 7. J. Piatetski-Shapiro, Euler subgroups. Lie groups, Budapest, 1975. 8. I. Piatetski-Shapiro, Zeta-functions o/GL(n), Univ. of Maryland, Nov. 1976. 9. M. Novodvorsky, Foncttons J pour des groupes orthogonaux, Acad. Sci. Paris, 280 (1975); Fonction J pour GSp (4), C. R. Acad. Sci. Paris. 280 (1974). 10. H. Jacquet, Automorphic forms on GL (2), Lecture Notes in Math. Vol. 114, Springer-Veriag, Berlin and New York, 1970. 11. S. Gelbart and I. Piatetski-Shapiro, Automorphic L-functions of half-integral weight Proc. Nat. Acad. Sci USA, Vol 75, 1978, pp. 1620—1623. TEL-AVIV UNIVERSITY RAMAT-AVTV, TEL-AVIV, ISRAEL
YALE UNIVERSITY NEW HAVEN. CONNECTICUT,
USA
519
TWO C O N J E C T U R E S O N L-FUNCTIONS
I.I.
PIATETSKI-SHAPIRO
We would like to present two conjectures on L-functions of representations. The first deals with the possible poles of the L-functions of cuspidal automorphic representations. The second deals with using the L-function to determine whether a representation is automorphic. In both we deal simply with the L-function of the representation twisted only by characters. 1. Poles of L-functions. Our first conjecture concerns the question of when the L-function of a cuspidal automorphic form can have poles. As is well known, for the general linear group GLn, the L-functions of cuspidal automorphic representations are entire (at least for n > 1). For other groups, the situation should not be so simple. Recall that we say that an automorphic representation n of a group G(A) is CAP (cuspidal associated to parabolic) if it is cuspidal, but there is an induced representation, induced from automorphic representations on the Levi of a proper parabolic subgroup, with which 7r is locally isomorphic for almost all places v [3J. Our conjecture essentially says that all poles of Lfunctions cuspidal representations for split groups are caused by the CAP phenomenon. Conjecture 1. Let G be a split reductive algebraic group defined over a global field k, G ^ GL\. Let IT be a unitary cuspidal representation of G(A). Let x be a unitary character of kx\Ax. s (a) L(s,7r x x) * entire unless TX is CAP. (b) If G is not isogenous to a product of factors of the form GLmi(k) or SLmi(k), then there exist IT and x as above such that L(S,TV X X) has a pole, i.e., G(A) has CAP representations. (c) Suppose the derived group of G is simple. If -K is unitary CAP, then the poles of L(s,ir x x) *n the domain Re(s) > | ; if they exist, should be simple and lie in a finite progression in | Z of step size 1 (even type) or | (odd type). As we mentioned, if G = GLn with n > 2 and IT any unitary cuspidal representation, then L(s,n x %) is entire. GLn(A) does not have CAP representations. If G — GSpi and IT is a cuspidal representation, then the phenomenon we are interested in is related to the Saito-Kurokawa lift [4]. In this case we
520
know that the condition that L(s,ir x \) has a pole for some character x is equivalent to IT being CAP and associated to either the Borel subgroup or the Siegel parabolic [3]. These conditions in turn are equivalent to IT being (essentially) a theta lift (or Saito-Kurokawa lift) from a cusp form on the twofold cover SL2 [3]. If for example we take IT — 0\Q [5], which is the theta lift of the Weil representation, then IT is CAP and L{s,ir) has simple poles at s = I and s = | . So #10 is of even type. We should remark that in some sense it is more natural to analyze the GSp4 phenomenon in terms of PGSp4 ~ S05, since the dual pair involved in the theta correspondence is really (SO5, SL2). Then the Siegel parabolic of PGSpn becomes the Klingen parabolic of SO5 and we see that the cuspidal representations whose L functions exhibit poles for GLi-twists are precisely those which are CAP and associated to parabolics which fix a line. Note that we are not claiming that all CAP representations have poles that can be detected only by character twists. The situation when twisting with higher rank general linear groups could be even more interesting. 2. Converse t h e o r e m s for GLn. Our second conjecture deals with the question of determining whether an irreducible admissible representation of GLn(A) is automorphic or not based on the analytic properties of its L-function. A converse theorem is a way of telling whether an irreducible admissible representation II of GLn(A) is automorphic based on the analytic properties of its Z-function. II has a decomposition II = ®'II„, where Tlv is an irreducible admissible representation of GLn(kv). To each 11^ is associated a local L-function L(s,Hv) and a local e-factor e(s,Hv,ipv). Hence formally we can form L(s,U) = JjL(s,II 1 ) ) V
and
e(s,U,ip) = Jj£(s,IL J ,i/>„)V
We will always assume the following two things about IT: (1) L(s,H) converges in some half-plane Re(s) 2> 0, (2) the central character wn of II is automorphic, that is, invariant under kx. Under these assumptions, e(s, 11,-0) = e(s,n) is independent of our choice of ip.
Converse theorems will involve twists but now by cuspidal automorphic representations of GLm(A) for certain m. Let r = ®'TV be a cuspidal automorphic representation of GLm(A) with m < n. Then again we can formally define L(s,U x r ) and £-(s,II x r ) since again the local factors make sense whether II is automorphic or not. A consequence of (1) and (2) above and the cuspidality of r is that both L(s,H x T) and L(s,Il x f) converge absolutely for Re(s) S> 0, where n and f are the contragredient representations, and that £(s,II x r ) is independent of the choice of ip.
521
We say that L(s,U x r ) is nice if (1) L(s,H x T) and L(s,Tl x f) have analytic continuations to entire functions of s, (2) these entire continuations are bounded in vertical strips of finite width, (3) they satisfy the standard functional equation L(a,U x T) = e ( s , n x T)L(1 - s,ft X f). The basic converse theorems for GLn which are known are of the following types. The first type of theorems is those in [1]. The basic theorem states that if L(s,U. x r ) is nice for all cuspidal representations r of GLm(A) for 1 < m < n — 1, then II is cuspidal automorphic. There are variants, where we greatly restrict the ramification of the twisting representations r, but at the cost of only being able to conclude that II is quasi-automorphic. For example, in the case of class number one fields we can restrict to twists by r which are unramified at all finite places, i.e. only level one twists. The second type of theorems is those in [2]. Here we restrict the rank of our twists and require that L(s,H x r ) is nice only for cuspidal automorphic r on GLm (A) for 1 < m < n — 2 and may still conclude that IT is cuspidal automorphic. Here there are also versions where we allow mild restrictions on the ramification of our twisting r by only requiring twists that are unramified in a fixed finite set of places. In this case we again can only conclude that n is quasi-automorphic. What we conjecture is that in fact it is enough to require only twists by GL\. Such a simple test for automorphicity would have many applications to the problem of lifting automorphic representations to GLn. Conjecture 2. Let n be an irreducible admissible generic representation of GLn(A) whose central character uu is trivial on kx and whose L-function L(s,TV) is convergent in some half-plane. Assume that L(s,U
Publ. Math.
522 2. J.W. Cogdell and I.I. Piatetski-Shapiro, Converse theorems for GLn, II, J. Reine Angew. Math. 507 (1999), 165-188. 3. LI. Piatetski-Shapiro, Cuspidal automorphic representations associated to parabolic subgroups and Ramanujan conjectures, Number Theory Related to Fermat's Last Theorem (Neil Koblitz, ed.), Progress in Mathematics No. 26, (Birkhauser, 1982), pp. 143-152. 4. LI. Piatetski-Shapiro, On the Saito-Kuorkawa lifting, Invent. Math. 71 (1983), 309-338. 5. LI. Piatetski-Shapiro, L-functions for GSp(4), Pacific J. Math., Olga Taussky-Todd Memorial Issue (1997), 259-275. DEPARTMENT OF MATHEMATICS, YALE UNIVERSITY, N E W HAVEN, C T
06520
Wolf Prize in Mathematics, Vol. 2 (pp. 523-551) eds. S. S. Chern and F. Hirzebruch © 2001 World Scientific Publishing Co.
Jean-Pierre SERRE College de France, 3, rue d'Ulm, 75005 PARIS
Outre les traditionnels Curriculum Vitae et Liste des Publications, on trouvera ciapres des textes, en grande partie inedits, qui commentent et eclairent (du moins je l'espere) certaines de mes publications. Ce sont: - Une lettre a John McCleary sur les groupes d'homotopie, leur role, et l'apport de ma these (1950-1951). - Une lettre a David Goss expliquant la genese de Faisceaux Algebriques Coherents (1953-1954). - Une lettre a Pierre Deligne (Juillet 1967) sur la variation des elements de Frobenius et sur les representations ^-adiques. - Une lettre a David Goss sur I'historique de la conjecture de Taniyama-Weil (1955-1967). - Un extrait d'une allocution prononcee lors de la remise du prix Balzan, a Berne, en 1985. - Une interview par Marian Schmidt (Janvier 1986), parue dans Hommes de Science, Hermann, Paris, 1990. - Une lettre a Jacques Tits (Mars 1993), resumant mes principaux sujets d'interet entre 1987 et 1993.
524 Curriculum Vitae
Ne a Bages (Pyrenees Orientales, Prance) le 15 Septembre 1926, de Jean Serre et Adele Serre (nee Diet), pharmaciens. Marie le 10 aout 1948 a Josiane Heulot. Un enfant: Claudine Serre, nee le 29 Novembre 1949. Eleve a I'ecole primaire de Vauvert (1932-1937), puis au Lycee de gargons de Nimes (1937-1945). Bachelier es sciences et es lettres (1944). Eleve a l'Ecole Normale Superieure, 45 rue d'Ulm (1945-1948). Agrege des sciences mathematiques (1948). Attache, puis charge, puis maitre de recherches au C.N.R.S. (1948-1954). Docteur es sciences (1951). Maitre de conferences a la Faculte des Sciences de Nancy (1954-1956). Charge du cours Peccot au College de France (1955). Professeur au College de France, chaire d'Algebre et Geometrie (1956-1994); professeur honoraire depuis Octobre 1994. Cours dans des universites etrangeres: Alger (1965, 1966), Bonn (1976), CalTech (1997), Eugene (1998), Geneve (1999), Gottingen (1970), Harvard (1957, 1964, 1974, 1976, 1979, 1981, 1985, 1988, 1990, 1992, 1994, 1995, 1996), McGill (1967), Mexico (1956), Moscou (1961, 1984), Princeton (1952, 1999), Singapour (1985), U.C.L.A. (2001), Utrecht (1974). Sejours a I'l.A.S. (Princeton): 1955, 1957, 1959, 1961, 1963, 1967, 1970, 1972, 1978, 1983, 1999. Sejour a 1'I.H.E.S. (Bures sur Yvette): 1963-1964.
525 Cours au College de France (1955-1994)
1955-1956: 1956-1957: 1957-1958: 1958-1959: 1959-1960: 1960-1961: 1961-1962: 1962-1963: 1964-1965: 1965-1966: 1966-1967: 1967-1968: 1968-1969: 1970-1971: 1971-1972: 1972-1973: 1973-1974: 1974-1975: 1975-1976: 1976-1977: 1977-1978: 1979-1980: 1980-1981: 1981-1982: 1982-1983: 1983-1984: 1984-1985: 1985-1986: 1987-1988: 1988-1989: 1989-1990: 1990-1991: 1991-1992: 1992-1993: 1993-1994:
Cohomologie et geometrie algebrique (cours Peccot) Groupes algebriques et theorie du corps de classes Algebre locale - theorie des intersections Homologie des groupes - applications arithmetiques Corps locaux et isogenies Fonctions zeta et series L Analyse p-adique Questions arithmetiques relatives aux groupes algebriques Courbes elliptiques Groupes formels et courbes elliptiques Fonctions modulaires - applications arithmetiques Groupes p-divisibles et varietes abeliennes Groupes discrets Representations abeliennes des groupes de Galois - applications aux courbes elliptiques Fonctions L Fonctions L et formes modulaires (suite) Cohomologie des groupes discrets Formes modulaires - applications arithmetiques Representations lineaires de groupes de Galois - exemples - problemes Quelques questions de theorie analytique des nombres Quelques questions de theorie analytique des nombres (suite) Autour du theoreme de Mordell-Weil Autour du theoreme de Mordell-Weil (suite) Nombres de classes, formules de masse, nombres de Tamagawa Formes quadratiques et nombres de Tamagawa Courbes algebriques et varietes abeliennes sur un corps fini Groupes de Galois et representations ^-adiques Groupes lineaires modulo p et points d'ordre fini des varietes abeliennes Groupes de Galois et formes modulaires Problemes galoisiens Problemes galoisiens (suite) Quelques problemes de cohomologie galoisienne Corps de fonctions et cohomologie galoisienne Quelques proprietes des representations galoisiennes associees aux motifs Quelques exemples d'invariants cohomologiques
[Je n'ai pas donne de cours en 1963-1964, 1969-1970, 1978-1979 et 1986-1987.]
526
Publications Articles et seminaires La plupart sont reproduits dans: Oeuvres - Collected Papers (1949-1998), 4 volumes, Springer-Verlag, Heidelberg, 1986 et 1999.
Ouvrages Groupes algebriques et corps de classes, Hermann, Paris, 1959; 2° edit. 1975, 204 p. (traduit en anglais et en russe). Corps Locaux, Hermann, Paris, 1962; 3° edit. 1980, 245 p. (traduit en anglais). Cohomologie Galoisienne, Lecture Notes in Math. 5, Springer-Verlag, 1964; 5° edition revisee et completee, 1994, 181 p. (traduit en anglais et en russe). Lie Algebras and Lie Groups, Benjamin, New York, 1965; 2° edit., Lecture Notes in Math. 1500, Springer-Verlag, 1992, 168 p. (traduit en russe). Algebre Locale-Multiplicites, Lecture Notes in Math. 11, redige avec la collaboration de P.Gabriel, 1965; 3° edit., 1975, 160 p. (traduit en anglais et en russe). Algebres de Lie Semi-simples Complexes, Benjamin, New York, 1966, 135 p. (traduit en anglais et en russe). Representations lineaires des groupes finis, Hermann, Paris, 1968; 5° edit., 1998, 182 p. (traduit en allemand, anglais, espagnol, japonais, polonais et russe). Abelian l-adic Representations and Elliptic Curves, redige avec la collaboration de W.Kuyk et J.Labute, Benjamin, New York, 1968, 195 p.; 2° edit., AK Peters, Wellesley, 1998 (traduit en japonais et en russe). Cours d'Arithmetique, Presses Universitaires de France, Paris, 1970, 188 p.; 4° edit., 1995 (traduit en anglais, chinois, japonais et russe). Arbres, Amalgames, SL2, redige avec la collaboration de H.Bass, Asterisque 46, S.M.F., 1977, 189 p.; 3° edit., 1983 (traduit en anglais et en russe). Lectures on the Mordell-Weil Theorem, traduit et edite par M.Brown, d'apres des notes de M.Waldschmidt, Vieweg, 1989, 218 p.; 3° edit., 1997. Topics in Galois Theory, notes written by H.Darmon, Jones & Bartlett Boston. 1992, 117 p. Les traductions anglaises ont ete publiees par Springer-Verlag (Heidelberg) et les traductions russes par les editions Mir (Moscou).
527 Lettre a John McCleary, 11 Mars 1997 Dear Professor McCleary, Let me try to answer the questions you ask in your letter: First, about the role of homotopy groups in algebraic topology. Nowadays one can safely say that, if a topological problem depends on the knowledge of some homotopy group, that group can be computed (more likely: has already been computed). In the 1940s it was the other way around: every interesting question (on fiber spaces, on obstructions, on homotopy type, whatever) depended on the computation of homotopy groups which were usually unknown. And, by "unknown", I mean completely unknown: before the fall of 1950, one did not even know that 7Tj(Sn) is a finitely generated group; even worse, it was not even conjectured to be so: there was no evidence at all. The specialists, such as Eilenberg or the Whiteheads, knew that the 7Tj of a finite simplicial complex X are not always finitely generated: take for X a bouquet of Si and S2 for instance; then ^ ( X ) is free of infinite rank. But they had no idea that this was just due to the fundamental group being infinite. Hence, for the topologists of that period, computing homotopy groups (of spheres, especially) was a first priority. Now let me explain a bit about my thesis, and how it came out. I had learned about the spectral sequence of a fiber space through Leray's lectures and (much more importantly) through conversations with Borel. In June 1950, we managed to use spectral sequences to prove a rather curious looking result: R n cannot be fibered with compact fibers — except if the fibers have only one point each, see C.R.A.S. 230 (1950). A few weeks later, I noticed a paper where somebody computed the cohomology of the Eilenberg-MacLane space K(Z, 2) by using the fact that the complex projective space Poo(C) is a model of K(Z, 2). Clear enough! But I was used to see Poo(C) as the base of a fibered space with trivial homotopy (the sphere SQO), the fiber being Si, which is a K(Z, l)-space. Looking at the homotopy exact sequence, this made me guess that there might be, for every (TT,TI), a fiber space with trivial homotopy, base K(n,n) and fiber K{ir,n — 1). If such a space existed, I could play with its Leray spectral sequence and try to get, for instance, the cohomology mod p of K(Z, 3), starting with that of K(Z, 2), which was already known. And then the cohomology of K(Z, 4), etc. Of course there was no guarantee that Leray's theory would be strong enough to make the computation. But it was worth trying, and I spent a few weeks doing so. One neat result being that (if the method was right) the Q-cohomology of K(Z, n) was easy: an exterior algebra on one generator if n is odd, and a polynomial algebra on one generator if n is even. Another interesting point was that the computations mod p were different for each p. I could "localize at p"; this was the first germ of C-theory. There was still a puzzling question: why should there exist a fiber space with trivial homotopy, base K(n, n) and fiber K(TT, n — 1)? I soon realized that the general question should be: for any space X, construct a fiber space E with trivial
528 homotopy and with base X. How to find such a space? A week or two later, I saw that the space of paths on X with given origin (and free extremity) had to be the answer. Of course that space was not a "fiber space" for any of the standard definitions. But I convinced myself that, at least, it had the "homotopy lifting property" and I was a priori sure that this was enough. There was still a serious technical difficulty: Why did Leray theory apply to such a space? It was clear that Leray was working in a "Cech" context, while I had to work in a "singular" one. Obviously I could make no use of the existing Leray theory, tailored for locally compact spaces. It was necessary to transpose it to the singular theory. Here again, I was stuck. However, in October 1950, I took part in a Bourbaki meeting, north of Paris, and one fine day Cartan and Koszul asked me what I was doing with Eilenberg-MacLane cohomology, homotopy groups, etc. I told them I had plenty of new things, but they all depended on a would-be extension of Leray theory to the singular context. Then I think it was Koszul who told me he had already toyed with the idea of filtering the singular complex of the fiber space, and that it looked encouraging. Back home, I immediately started the computations. I had a hard time proving the necessary lemmas (lemmas 3, 4, 5 of Chap. II of my thesis). But once this was done a lot of consequences followed, both on Eilenberg-MacLane complexes and on homotopy groups. A month later (December 1950) I announced them in the Comptes rendus, in a series of three notes, and I started writing my thesis. This went very smoothly. By March or April 1951 the text was ready. On the advice of Eilenberg, it was sent to the Annals, where Steenrod gave it priority so that it appeared at the end of 1951. Copies of my thesis were sent to a few people, such as J. H. C. Whitehead at Oxford and G. Whitehead at M.I.T., and their students started working on it (especially John Moore). From that point on, homotopy groups could be attacked in a systematic way (but not always entirely computed, of course) and their basic properties could be proved: for instance that 7Tj(X) is finitely generated if X is a finite simply connected complex, and is finite if X = S„ (except for i = n, and for i = 2n — 1 if n is even). One of the main applications of these being the computations of cobordism groups by Thom the next year (which could not have been possible two years before), which themselves led to Hirzebruch's proof of Riemann-Roch, and to so many other things. I hope this answers more or less your questions on homotopy groups. (You also ask about Adams spectral sequence; this is something which came several years later, at a time I was not doing any more Topology. I don't have any personal comment on that.) About spectral sequences in algebra: I believe that the first one is the one on the cohomology of group extensions. I was working on it at the same time (fall 1950) as on the loop spaces. I deduced it from the Cartan-Leray spectral sequence for Galois coverings; it was a very easy proof, published in the C.R. (October 1950). I knew of course of Lyndon's paper, where one can see that he had been very close to getting himself a spectral sequence.
529 It was a near miss. Probably he had trouble first with the very notion of spectral
sequence (which he clearly did not know, even though Leray had already published his first accounts of it), and second with his topologist viewpoint of working mainly with constant coefficients. A few months after the C.R. note, I received a letter from Hochschild, who had read it, and constructed a proof based on an explicit filtration of cochains. We agreed to publish jointly our two proofs; the paper appeared in T.A.M.S., 1953. (Incidentally, Hochschild's method, while much more complicated to justify, has the great advantage of giving explicit maps, which can actually be computed in special cases; this is not the case for my original method, based on "general nonsense". There would be a lot to say about explicit computations or the lack of them - in Homological Algebra. But I will not go into this painful subject here.) Well, I guess that the above answers at least a part of your questions . . . With best regards, Yours J.-P. Serre
530 Lettre a David Goss, 12 Decembre 1991 Dear Goss, Since you ask about the genesis of FAC, I'll try to reconstruct for you what the sequence of ideas was. In 1953, I had started working on applying coherent analytic sheaves to complex manifolds and algebraic varieties, following the trend of Cartan with Stein manifolds. The letter to Borel of April 1953, reproduced in my CP, n°20, tells clearly where I stood then: I had the duality theorem, and I had the idea that Riemann-Roch is an "Euler-Poincare formula" (an idea that Kodaira-Spencer also had at about the same time). This was purely analytic, but I find a surprising sentence in my letter (p. 248, lines 4-5), hinting at the "siirement possible" existence of "faisceaux algebriques". Still, I don't think I had any precise idea at that time that it was indeed not only possible but easy. This had to wait for a few months. Meanwhile, I investigated more closely the coherent analytic sheaves on the projective space Pjv- If M is such a sheaf, I knew (fall 1953) that, by the twisting operation il M(n)", I would get a sheaf having lots of sections (for n large), and hence I would get a graded module which would conversely reconstruct M. This correspondence between sheaves and graded modules over the ring C[Xo,... ,XN\ was very striking, and it was natural to expect that it would also work on any algebraically closed field once a good definition of "faisceaux algebriques coherents" be found. I had also a kind of philosophical reason to search for such a thing: when one works with analytic sheaves, the cohomology is defined by Cech cochains which are systems of holomorphic sections on some open sets; these sections are allowed to have terrible singularities on the boundaries of these sets. Still, one recovers in the end finite dimensional spaces; one can't help thinking that all these terrible singularities could be avoided and that just nice algebraic poles would be enough. Another reason for looking towards an algebraic theory was that Weil had developed (in his Chicago notes) a very satisfying theory of fiber spaces in algebraic geometry, based on the Zariski topology. All I had to do was to stop being afraid of the non-Hausdorff character of the Zariski topology (especially when I noticed that some involved axiom of Weil's "Foundations" just means that the diagonal is closed, so that the Zariski topology has some claims to being Hausdorff, after all . . . ) . I must have taken the plunge around Christmas 1953, because I find in my correspondence with Borel a letter, dated February 10, 1954, where I explain to him what the Zariski topology is, what the coherent sheaves are, and what the main cohomological theorems on them are (with the exception of the duality theorem, which I did not know how to prove at the time). I started writing FAC a few months later. It went very smoothly, as far as I can remember, except for: 1 - I was maybe halfway through it when I got a letter from Igusa telling me: we are having a seminar on your stuff (a short note was going around) and we don't know how to prove the cohomology exact sequence for sheaves; how do you prove it? The sad truth was that I had assumed it would be trivial, and I had been
531 using it happily with excellent results. But, when I tried to mimic the standard proof for paracompact spaces, I realized that Igusa was right: that proof did not extend. Fortunately I already had so many results on the cohomology of sheaves that I knew they could not be wrong. After a day or two, I managed to salvage the situation by a technical trick: show the cohomology exact sequence is OK up to H1 included (FAC, n°24); do the theory of affine varieties using H1 only; then show that the cohomology exact sequence is OK whenever the sheaves involved are coherent (FAC, n°47). Ouf, as we say in French. A close shave. If I had noticed that difficulty earlier I might not have had the courage to start this game; a cohomologist can't live without exact sequences. 2 - During the writing up, I saw that I could use the Ext functors of CartanEilenberg to express the cohomology, and I got in this way results substantially equivalent to the duality theorem I knew in the analytic case (FAC, n°72). 3 - At the time, I had the (wrong) idea that a paper had to be written several times in succession before being sent to a journal. My manuscript was about 110 pages long, and in reasonably good shape. Still, I started retyping it (in the style of the time: with five or six carbon copies, so that Borel, Cartan, Kodaira could each have one - don't forget that Xerox machines did not exist). It took me an entire month, after which: a) I was exhausted; b) the only changes I had made were to interchange two sections and to suppress Remark 3 of n°81 (that remark being the germ of my Lecture-Notes-to-be on intersection multiplicities). I swore never to do that again. Well, that's about it. I am afraid it is not a very interesting story: there was no real "flash" of discovery. The whole process was rather logical, and technically very easy, even for somebody who knew little algebra and not much geometry. The writing itself was not hard; I was following the style I had learned from Bourbaki and the Cartan seminar: taking my time and giving almost complete details. The FAC setting, over an algebraically closed field, was very satisfactory for a former topologist: points, continuous functions and the like. And the points were honest closed points, not big overwhelming points as in Weil — and later in Grothendieck. It was a kind of Paradise. Still, I knew I could not dwell very long there: I was already attracted (under Lang's influence, mainly) by non-algebraically closed fields such as Q or F , . Maybe it was not really Paradise; just a little oasis. Best regards
J.-P. Serre
PS. You might be interested in reading the report that Zariski wrote on a seminar that he and others had on the FAC theory (even before FAC was entirely written): see Bull.A.M.S. 62 (1956), 117-141.
532 L e t t r e a P i e r r e Deligne, 24 Juillet 1967 Cher Deligne, Je t'avais dit que je t'enverrais des commentaires sur les representations Z-adiques et la fonction de Ramanujan. Avant de le faire, il est necessaire de se rendre compte du cadre ou se place ce genre de question. C'est ce que je vais essayer d'expliquer: Un des problemes centraux (j'aurais tendance a dire: le plus important — en tout cas pour moi) de l'arithmetique consiste a comprendre "comment Frobenius varie", ou, ce qui revient au meme, a comprendre les fonctions zeta et L des schemas. Ceci est l'enonce vague du probleme. Mais, grace a Weil, nous sommes a peu pres surs que la reponse est celle-ci: ces fonctions zeta sont des cas particuliers des fonctions, a priori toutes differentes, associees aux formes modulaires sur les groupes semi-simples. Bien sur, cela n'est pas demontre; on ne sait meme pas quelle forme precise doit prendre cet enonce, a part le cas des courbes elliptiques sur Q, ou Weil + Swinnerton-Dyer rendent pratiquement certaine la correspondance entre: classes de courbes elliptiques/Q (a Q-isogenie pres), de conducteur N et facteurs/Q de la jacobienne de la courbe modulaire XQ(N), stables par les operateurs de Hecke (et aussi par une certaine operation supplementaire WQ(N) peu importe 1 ). Je vais quand meme essayer d'etre plus precis, et de decrire divers types d'objets dont on espere qu'ils se correspondent: 1. Systemes de representations l-adiques d'un corps de nombres Tu sais ce que c'est. Soient K un corps de nombres et S un ensemble fini de places de K. Pour tout I premier, soit Gitx le groupe de Galois de la plus grande extension de K non ramifiee en dehors des places de S et de celles divisant I. On se donne pour chaque I une representation continue fi : GilK -»• G L n ( Q j ) . On suppose ceci: si v £ S est une place de K, l'image par fi du Frobenius de v (qui est definie si (u,l) = 1) a un polynome caracteristique a coefficients dans Q et independant de I. II semble que ceci soit une notion assez raisonnable; en tout cas, je ne connais aucun exemple de tel systeme qui soit pathologique (i.e. ne provienne pas de fagon simple d'un schema sur Q). D'autre part, meme dans le cas ou Taction est abelienne Bien sur, j'aurais du ajouter que le facteur considere est de dimension 1, et aussi qu'il est "new", i.e. de multiplicite 1.
533 (cf. le cours au College que tu avais suivi 2 ), je suis oblige de faire des hypotheses supplementaires pour pouvoir classer ces representations. 2. Representations l-adiques dans un groupe algebrique sur Q C'est une variante de la notion precedente, le groupe G L n etant remplace par un groupe G defini sur Q. Plus precisement, on se donne deux choses: a) Pour tout v £ S, une classe de conjugaison Fv de G definie sur Q (attention, cela veut dire une classe de conjugaison de G(Q) qui est invariante par Galois; je me fous de savoir si elle est realisable sur Q, et comment). (Pour les applications interessantes, G est reductif, pas necessairement connexe, et les Fv sont semi-simples.) b) Pour chaque I une representation continue fi '• GitK -> G(Qi), groupe des Qj-points de G. Axiome: ce que tu penses, i.e. que // transforme le Frobenius attache a v en une classe de conjugaison qui est tout juste (sur une extension de corps convenable) celle Fv donnee en a). L'amusant est que la donnee b) entraine la donnee a); mais j'ai tenu a formuler a) independamment, car a) determine fort probablement b) et de plus on verra plus tard des cas ou on dispose de a) et pas (pour l'instant) de b). II me faut un nom pour une donnee du type a) + b); disons un G-systeme de representations Z-adiques, ou simplement un G-systeme. Si p : G —>• GL„ est une representation lineaire definie sur Q, et si je dispose d'un G-systeme, p le transforme en un systeme de representations i-adiques. C'est clair - et c'est l'un des avantages du G-point de vue. II y aurait une notion a definir, celle de G-systeme essentiel (ou minimal). Definition possible: le groupe G est le plus petit groupe algebrique (sur Q - ou sur Q;? je n'en sais rien) contenant l'image de /;. II faudrait que ce soit independant del. 3. Fonction L attachee a un systeme de representations l-adiques Si v $L S, soit Fv l'image de son Frobenius par un //; il devrait etre vrai que les valeurs propres de Fv ont pour valeurs absolues des nombres de la forme (Nv)1/2, pour certains entiers i ne dependant que de (/;) mais pas de v. On peut alors former le produit eulerien
l[l/det(l-Fv(Nv)-s)
Lf(s) = v<£S
et il converge pour R(s) assez grand. C'est la fonction L cherchee (a des termes elementaires pres, voir plus loin). (Note que la connaissance de L n'est pas equivalente — sauf si K = Q — a celle de ses facteurs locaux
Lf,v =
l/det{l-Fv(Nv)-s);
II s'agit du cours de 1965/1966, resume dans Oe.71; voir aussi le resume du cours de 1966/1967, Oe.78.
534 en effet, des v ayant meme caracteristique residuelle peuvent se melanger. II m'arrivera pourtant, dans la suite, d'appeler "fonction L" la collection des termes locaux; je m'en excuse d'avance. A vrai dire, a ce degre de generality, c'est sans importance; par restriction des scalaires je pourrais me ramener au cas ou K — Q. II n'y a que pour les corps de fonctions sur un corps fini qu'on est irremediablement baise.) Ce qui precede s'applique bien sur aussi quand on a un G-systeme. Tout caractere \ de G, realisable sur Q, donne lieu a une fonction Lx correspondante; en fait, il est raisonnable de ne pas se borner aux x definis sur Q, et de prendre toutes les representations de G (sur un corps de nombres quelconque); on obtient des fonctions L dont les coefficients ne sont plus rationnels, mais algebriques. On sait bien qu'on peut s'en contenter. Avantage: si l'on connait toutes ces fonctions L, on connait le G-systeme, ou du moins sa partie a). En effet, une classe de conjugaison semi-simple est caracterisee par les valeurs que prennent dessus les caracteres. 4. Exemple: le cas elliptique On se donne une courbe elliptique E sur K. On lui associe un G-systeme ainsi: (i) Si elle n'a pas de mult.complexe, on prend G = GL2, et la representation £-adique connue (Weil). Ca a les proprietes a) et b). (ii) Si elle a de la mult.complexe definie sur K, soit L le corps quadratique imaginaire de la mult.complexe, et soit G le tore de rang 2 sur Q egal a i?z,/Q(G m ). II opere sur la representation Z-adique de la courbe, et Ton constate que Frob. est dedans; on a un G-systeme. (iii) Si elle a de la mult.complexe, mais pas definie sur K (i.e. si K ne contient pas L), on prend pour G le normalisateur du tore maximal defini en (ii). On verifie qu'on a un G-systeme. De plus, dans chaque cas, je me suis arrange pour avoir un systeme essentiel. Le G-systeme ainsi construit a une representation canonique de degre 2; la fonction L associee est la fonction zeta de E (a des termes triviaux pres). Mais on peut s'amuser davantage: si l'on fait le produit tensoriel i-ieme de cette representation, la fonction L associee permet de calculer (avec celles relatives aux j < i) la fonction zeta de E x • • • x E (i fois). 5. Un super-exemple: les motifs Je vais faire semblant de savoir ce que c'est. II parait que c'est equivalent a la categorie des modules sur un groupe proalgebrique 3 defini sur Q. Soit G un quotient de ce groupe qui soit, lui, algebrique sur Q. II doit etre vrai qu'il lui est associe un G-systeme essentiel (vu que toute la theorie des motifs est conjecturale, j'ignore si cela peut etre demontre — meme si l'on admet ce que Grothendieck appelle les "conjectures standard"). Les fonctions L associees sont celles des motifs. Ce sont vraiment a elles qu'on en a. II s'agit des groupes de Galois motiviques introduits par Grothendieck; leurs proprietes (conjecturales) sont resumees dans Oe.161.
535 Bien entendu, l'exemple 4 est le cas particulier de 5 ou Ton prend pour motif une courbe elliptique; dans ce cas, je suis sur du groupe G! 6. Un sous-exemple: cohomologie a supports compacts C'est au fond la meme chose que l'exemple precedent, a ceci pres que ce dernier s'occupait de schemas projectifs lisses et de leur cohomologie, alors que maintenant on prend n'importe quoi, et la cohomologie a supports compacts. On n'est pas tres sur d'avoir un G-systeme (en tout cas le G ne serait plus reductif); mais en tout cas on a (si les conjectures de Weil sont vraies) un systeme de representations Z-adiques (on ne sait pas tres bien decrire le S exceptionnel). Changeons maintenant de point de vue. semi-simples:
Passons du cote des groupes
7. Construction d'un demi-G-systeme a partir de formes modulaires Helas, je suis bien incompetent, et je dois me borner a ceci: soit Go un groupe semi-simple (reductif devrait suffire) sur Q, deploye (?). II a un systeme de racines R, et on sait ce qu'est le systeme dual Rv. Soit G le groupe semi-simple ayant ce systeme dual Rv pour systeme de racines. Eh bien, si h est une "forme modulaire" sur Go (en quel sens?), fonction propre des "operateurs de Hecke" (?), Langlands (notes de Yale) lui associe un demi-G-systeme, i.e. une donnee du type a) de 2. Et, ce qui est bien plus beau, il peut montrer dans certains cas que les fonctions L associees (par le procede du n° 3) ont une equation fonctionnelle explicite, et sont holomorphes ou il faut. Pour SL2 sa theorie doit etre plus ou moins equivalente a la correspondance de Hecke—Weil entre series de Dirichlet et formes modulaires. Voila a peu pres ou on en est; pour etre plus complet, il faudrait que j'ajoute des precisions a 3: d'une part l a determination des "bons" facteurs locaux aux mauvaises places (y compris l'infini); je crois savoir ce qu'il faut mettre 4 ; d'autre part, une conjecture precise sur les zeros et poles des fonctions L (j'en avais parle au College, quand tu y etais), et le lien avec certaines equidistributions. 5 Ce sont la des details. Le vrai probleme (pour la resolution duquel je n'ai aucune idee) est: ameliorer Langlands pour trouver une correspondance bijective entre "formes modulaires" de type convenable, et G-systemes (ou du moins ceux associes aux motifs, a supposer qu'il en existe d'autres). Ce grand probleme contient des quantites de sous-problemes, fort interessants, et dont certains sont abordables. Celui sur r (j'y reviens enfin) en est un: on est dans un cas ou Ton a une donnee du type 2/a), dont on a toutes raisons de croire (a cause par exemple du fait que la fonction L associee est OK) qu'elle est une moitie de G-systeme. 'On demande de trouver l'autre mditie. Si l'on etait extremement. optimiste, on demanderait un motif sur Q, de poids 11, ayant bonne reduction partout, et correspondant a V; au langage pres, c'etait une question que m'avait posee Weil il y a deja longtemps. On ne voit malheureusement pas comment S
J'ai publie deux ans plus tard (Oe.87) une recette donnant ces facteurs. Voir la-dessus Oe.161, §13.
536 faire. C'est pourquoi je demande seulement un systeme de representations Z-adiques (obtenu par de la cohomologie a supports compacts). Bien a toi
J.-P. Serve
PS. Je me rends bien compte que je t'ai resume la situation de facon bien peu symetrique: je suis familier avec les fonctions zeta des schemas (ou motifs), mais pas avec celles des groupes semi-simples. Du cote de ces derniers il y a des ingredients qui m'echappent completement (representations de dimension infinie, formule de Selberg, . . . ) . II faut que tu en tiennes compte. J'aurais du egalement dire que 1'equation fonctionnelle des fonctions zeta ne semble demontrable que dans les cas oil le "pont" avec le semi-simple est fait (semisimple, ou reductif, bien sur).
537 Lettre a David Goss, 30 Mars 2000 Dear Goss, You are right: the history of the "modularity conjecture" has been somewhat distorted recently, and it would be good to put the record straight. Let me try. As you well know, the main actors have been Taniyama, Shimura and Weil. What they have published on it is as follows: 1. Taniyama. At the Tokyo-Nikko conference (1955), the organizers asked for a list of open questions. This list was typed and distributed to all participants (but it was not included in the Conference volume). Taniyama contributed several such questions. One of them (problem n°12) is about elliptic curves; you may find it (in Japanese) in his Collected Papers, p. 167, and (in English) in mine, Vol. Ill, p. 399 (see below). It is clear that Taniyama had been influenced by the results of Eichler of 1954 (which had also made a deep impression on Weil). His conjecture was, more or less, that Eichler's construction gives the zeta functions of all the elliptic curves over Q. Unfortunately, he chose to state it over an arbitrary algebraic number field; this made him invoke a "field of automorphic functions" which does not make sense in such a general setting. Still, it was a brilliant insight. 2. Shimura. He clarified Eichler's theory by using the action of the Hecke algebra on the Jacobian of the modular curve (1958); this allowed him to split the Jacobian, up to isogeny. He obtained the "Eichler-Shimura" relation (for the reduction mod p of Tp) for large enough (but unspecified) primes p. It was Igusa (1959) who proved the important fact that this holds for every p not dividing the level. As for the modularity conjecture, Shimura published nothing on it. He did not mention it (not even as a "problem") in his 1971 book, nor in any of the many papers on modular forms he wrote between 1955 and 1985. The most he did was to ask a few people (verbally, only) whether they believed in it or not. An explicit statement in print would have been more useful; maybe he felt he did not have enough evidence to do so. 3. Weil. In his paper on "Funktionalgleichungen" (Coll. Papers, [1967a]), he mentions the conjecture, tongue-in-cheek, as an "exercise for the interested reader", without quoting Eichler or Taniyama (as he could have). He adds two decisive ingredients: a) A characterization of modular forms by functional equations of Hecke type for the corresponding L functions, and their twists by Dirichlet characters. A remarkable aspect of his theorem is the way the constant of the functional equation depends on the twisting character. This has been the starting point of what is now called "converse theorems" in Langlands theory. b) He suggests that, not only every elliptic curve over Q should be modular, but its "level" (in the modular sense) should coincide with its "conductor" (defined in terms of the local Neron models, say).
538 Part b) was a beautiful new idea; it was not in Taniyama, nor in Shimura (as Shimura himself wrote to me after Weil's paper had appeared). Its importance comes from the fact that it made the conjecture checkable numerically (while Taniyama's statement was not). I remember vividly when Weil explained it to me, in the summer of 1966, in some Quartier Latin coffee house. Now things really began to make sense. Why no elliptic curve with conductor 1 (i.e. good reduction everywhere)? Because the modular curve X0(l) of level 1 has genus 0, that's why! I went home and checked a few examples of curves with low conductor: I did not know any with conductor < 11, nor with conductor 16? No surprise, since XQ(N) has genus 0 for such values of N, etc. Within a few hours, I was convinced that the conjecture was true. I was not the only one to be convinced: people such as Birch, Tate, SwinnertonDyer, Mazur, . . . felt the same way; moreover, a lot of numerical evidence was soon collected by Swinnerton-Dyer and others. Of course, there were several loose ends which needed tying up, but this was done within a few years: - the Galois-representation definition of the conductor, and of the gamma factors of the functional equation (Ogg, Tate, myself); - the newform theory of Atkin-Lehner (1970); - the fact that an elliptic curve is determined, up to isogeny, by its Z-adic representation (for any given I). This was harder. I did it in my McGill lectures (1967) when the j-invariant is not an algebraic integer (a case which turned out later to be sufficient for Wiles), but I could not do it in general; it had to wait until 1983, when Faltings proved the general Tate conjecture. This period (end of the '60s and beginning of the '70s) was a very exciting one for people working on modular forms, elliptic curves and the like. To wit: - Langlands's theory (especially his 1967 Yale notes), with its relations with motives; - Deligne's construction (1968) of the Z-adic representations associated with modular forms of weight > 2, confirming a conjecture I had made the year before; - the theory of modular forms mod p (Swinnerton-Dyer, 1970), which I applied to define p-adic modular forms and to construct the p-adic zeta function of an arbitrary totally real number field (Antwerp, 1972); - Shimura's correspondence between modular forms of half-integral weight and those of integral weight (1972): a surprising, and beautiful, application of Weil's "converse theorem"; - the crowning part (1973): Deligne's proof of Weil's conjecture for varieties over finite fields, and, as a consequence, the proof of the Ramanujan-Petersson conjecture. Quite a list, don't you think? Note that, during the ten years following Weil's paper, the modularity conjecture was called "Weil's conjecture", and Taniyama's original insight was all but forgotten. Around 1976, I bought a copy of Taniyama's Collected Papers, and I
539 noticed that "problem n°12" was included there in Japanese, but not in English. To make it more widely available, I reproduced its 1955 English version in my 1977 paper on /-adic representations; a fitting place, since the notion of system of ^-adic representations is due to Taniyama! From then on, I started saying "Taniyama-Weil conjecture" instead of "Weil conjecture": it seemed natural to me that the credit be divided between the two of them. Little did I know that I was thus starting a bitter controversy. In the '90s, Lang took the matter to heart (as he often does) and launched a big campaign in order to have Weil's name removed and the conjecture called "Taniyama-Shimura", which I find strange in view of Shimura's record (or absence of record, see above). I still feel that "Taniyama-Weil" is more accurate. Maybe your suggestion of "modularity conjecture" is even better? Anyway, one should not take such terminology quarrels too seriously. As Weil was fond to say, "Pell's equation" is not due to Pell, and Klein did not do much on the "Kleinian functions" of Poincare . . . Best wishes
J.-P. Serre
PS. Lang's paper in the 1995 Notices describes a would-be discussion between Shimura and myself, at the Institute, in 1962-64 (sic). You ask whether this discussion actually happened. The answer will surprise you: I don't know! I have no memory of it. However, it is perfectly possible that Shimura said once "... don't you believe that every elliptic curve is modular?" and that I replied something like "... why should it be so?". I know very well that memory erases what is not important. If he had given me even one little piece of evidence, I would have been impressed and I would not have forgotten. (The discussion with Weil, on the other hand, was memorable; the evidence was there.) PPS. You would probably be interested by the letters I exchanged with Tate, Swinnerton-Dyer, Shimura, . . . , between 1966 and 1968, on the modularity conjecture. No controversy then: just mathematics.
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Extrait du discours de reception du prix Balzan, Berne, 1985 . . . J'aimerais vous dire quelques mots du travail des mathematiciens, de ce qu'ils cherchent et des methodes qu'ils emploient. Ce n'est pas facile: il y a tant d'idees regues — et tant d'idees fausses — sur les mathematiques! Essayons pourtant: D'abord, que cherche le mathematicien? Comme tout scientifique, il cherche a comprendre. Mais comprendre quoi? Quel est ce materiau, a la fois impalpable et infiniment resistant, sur lequel il travaille? Prenons un exemple, tire d'une branche des mathematiques qui m'est chere, la theorie des nombres. Cette theorie etudie l'un des phenomenes naturels les plus fondamentaux, a savoir la suite 0, 1 , 2 , 3 , . . . des nombres entiers. L'homme de la rue n'y voit rien que de tres banal: ce n'est pas plus mysterieux que ne le sont l'eau ou la lumiere, pense-t-il. Le mathematicien, lui, sait que cet objet d'apparence innocente renferme tout autant de mysteres que l'eau pour le chimiste et la lumiere pour le physicien. Comment explorer ces mysteres? Le physicien se sert d'instruments toujours plus perfectionnes: microscopes, accelerateurs, etc. Nous faisons de meme. Mais nos instruments ne sont pas materiels (mis a part batons de craie, machines a ecrire et ordinateurs), ils sont purement intellectuels, ce sont ce que nous appelons des "theories": calcul differentiel, algebre, topologie, geometrie algebrique . . . Ainsi, si nous desirons etudier les decompositions des entiers en sommes de nombres premiers (problemes du type Goldbach), nous utiliserons la theorie dite "du crible", inventee il y a pres de 2000 ans et considerablement perfectionnee dans les dernieres 50 annees (notamment par mon ami Enrico Bombieri, qui a regu le prix Balzan en 1980). Pour d'autres questions, nous utiliserons des methodes algebriques, ou les nombres premiers apparaissent comme les analogues des points d'une courbe (cette curieuse analogie, qui date du debut du siecle, s'est revelee d'une extraordinaire recondite). II serait facile de multiplier les exemples. Ceux que je viens de donner permettent deja d'entrevoir les deux aspects du travail du mathematicien: - la construction de methodes generates, ou de theories; - leur application a la resolution de problemes concrets. Bien sur, c'est la une vue tres schematique, d'autant plus que la notion de "probleme concret" est eminemment subjective (certains reseaux d'un espace a 24 dimensions sont tres "concrets" pour mon collegue Jacques Tits, ils le sont un peu moins pour moi et ne le seraient sans doute pas du tout pour beaucoup de gens). En outre, toute theorie donne naissance a de nouveaux problemes; ceux-ci sont vite considered comme "concrets"; leur solution demande la creation de nouvelles theories . . . Processus sans fin! Ce developpement, un peu effrayant au premier abord par sa complexity, est heureusement compense par les nombreux "ponts" que Ton construit entre differentes theories (par exemple, on applique en arithmetique des idees provenant de la topologie). Ces ponts facilitent le passage d'un domaine a
541 un autre; ils permettent a ceux qui le desirent d'eviter une trop grande specialisation. Ce sont eux qui mettent en evidence l'unite des mathematiques — unite si forte que mon maitre Nicolas Bourbaki n'hesite pas a parler de "la mathematique" plutot que "des mathematiques". Mesdames et Messieurs, j'ai sans doute depasse le temps qui m'etait imparti, et j'ai peur de ne vous avoir donne qu'une idee tres superficielle de ce qu'est mon metier. Avez-vous devine, par exemple, que c'est un metier passionnant? Et qu'il donne de grandes joies? J'espere que oui. Je vous remercie.
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Entretien avec Jean-Pierre Serre (M. Schmidt, Hommes de Science, 218-227, Hermann, Paris, 1990)
JE SUIS NE le 15 septembre 1926 et j'ai ete eleve par mes parents, tous deux pharmaciens, qui habitaient une petite ville du Midi de la France, Vauvert, pres de Nimes. Jusqu'a onze ans, j'ai frequente l'ecole communale de Vauvert; ensuite, je suis alle au lycee de Nimes oil j'ai fait toutes mes etudes secondaires, de la sixieme a l'hypotaupe.
l'annee, le premier prix, ce qui m'a encourage a essayer d'entrer a l'Ecole normale superieure. Apres un an de preparation, a l'hypotaupe de Nimes, j'ai ete recu second au concours de 1945 et je suis monte a Paris, comme on dit dans le Midi.
Tres jeune, j'ai aime les mathematiques, ou plutot le "calcul", comme Ton disait. Je me souviens encore du plaisir que j'ai eprouve, a l'age de six ou sept ans, quand j'ai compris comment on pouvait retrouver la table de multiplication par 9 sans avoir a l'apprendre par cceur. Plus tard, a Nimes ou j'habitais une pension de famille tout en suivant les cours du lycee, j'avais des camarades plus ages et je me plongeais dans leurs livres de mathematiques ; il m'arrivait aussi de resoudre leurs problemes a leur place, a la fois par gout et par interet: cela m'evitait (pas toujours...) les brimades des "grands". Plus tard encore, vers quinze ans, je me suis mis a feuilleter puis a lire les livres de calcul difrerentiel qui appartenaient a ma mere. Celle-ci, en effet, aimait beaucoup les mathematiques quand elle etait jeune (un gout qui lui venait, m'a-t-on dit, de sa propre mere) et, lorsqu'elle etait etudiante en pharmacie a Montpellier, elle avait suivi les cours de Mathematiques generates et passe avec succes l'examen correspondant. C'est ainsi que je me suis familiarise avec derivees, integrales, determinants, etc., bien avant que ce ne soit au programme du lycee.
Non : j'etais heureusement trop jeune pour etre envoye dans un camp de travail par le gouvernement de Vichy. Une seule alerte, mais pas tres serieuse : en mai 1944, les troupes allemandes ont requisitionne les eleves des classes terminales du lycee et les ont envoyes construire des fortifications en Camargue, aux Saintes-Maries-dela-Mer. Un certificat medical m'a permis d'echapper a cette corvee, qui m'aurait empeche de me presenter au Concours general: notre medecin de famille m'a trouve une "albuminuric orthostatique" qui me rendait "inapte a la station deboutprolongee". Cette providentielle maladie ne s'est plus manifestee depuis.
A dix-sept ans, je suis entre en Mathematiques elementaires (la terminale C actuelle), ou j'ai eu la chance d'avoir un excellent professeur ; il m'a fait preparer le Concours general oil j'ai obtenu, a ''. fin de
21S
La guerre a-t-elle modifie le cours de votre vie ?
Quel souvenir gardez-vous de votre sejour a l'Ecole normale ? Mon arrivee a l'Ecole, en 1945, a ete pour moi un changement extraordinaire. Pour la premiere fois, j'avais des camarades avec lesquels je pouvais discuter de mathematiques et travailler. Je pense en particulier a Daniel Lacombe, qui s'occupe maintenant de logique et de pedagogie, et a Georges Poitou, l'actuel directeur de l'Ecole. Nous logions tous trois dans la meme turne et nous nous posions quantite de questions : comment demontrer le theoreme de Jordan sur les courbes planes (nous ne l'avons jamais vraiment compris...), comment resoudre des equations integrales d'apres
543 .1 I". A N - P I F. R R E S E R R E Goursat et F. Riesz, etc. C'etait passionnant. Nous suivions aussi - avec moins d'enthousiasme - les cours de Sorbonne donnes par G. Valiron, P. Montel, G. Julia, P. Garnier. Nous avions egalement des cours reserves aux normaliens, par G. Bouligand, dont j'ai garde un vif souvenir : sans etre tres clairs, ils etaient toujours stimulants. A la fin de 1'annee, nous avions obtenu tous les certificats de licence exiges par l'Ecole, de sorte que, 1'annee suivante, nous etions debarrasses de toute obligation universitaire. Cette seconde annee a ete pour moi un veritable regal: j'etais enfin libre de faire ce que je voulais, sans souci d'aucune sorte. J'en ai notamment profite pour faire beaucoup de sport, sous l'influence de Lacombe et aussi du professeur d'education physique, Ruffin, dont 1'enthousiasme etait irresistible : athletisme (que j'aimais beaucoup et que j'ai continue a pratiquer chez Ruffin pendant plus de vingt ans), boxe et judo (vite abandonnes), natation (avec un succes tres limite), ping-pong et ski (que je pratique encore). En mathematiques, je suivais quelques cours en Sorbonne, sans beaucoup m'y interesser, et je lisais, un peu au hasard, quantite d'ouvrages de la bibliotheque de l'Ecole. Je me suis fait envoyer de New York, oil j'avais de la famille par alliance, trois livres introuvables en France a cette epoque, qui m'ont fait une forte impression : Theorie des operations line'aires de S. Banach, Moderne Algebra de B.L. van der Waerden et Theory of Lie Groups de C. Chevalley. Je les ai lus avec passion, ligne a ligne, du debut a la fin, comme on peut le faire a vingt ans - et comme on n'a plus l'occasion de le faire ensuite. La troisieme et derniere annee a ete moins agreable. II me fallait preparer l'agregarion, qui etait extremement scolaire a l'epoque. Les problemes de geometrie elementaire et de mecanique etaient specialement rebutants. J'etais pourtant contraint d'y travailler : l'agregerepetiteur, Frenkel, m'avait affirme que je ne serais engage au C.N.R.S., a ma sortie de l'Ecole, que si j'etais recu dans un tres bon rang. J'ai done pris l'agregarion au serieux ; en fait, trop au serieux : j'ai finale-
ment ete recu premier, ce qui etait bien inutile ! Pendant toute cette annee, je n'ai guere fait de mathematiques interessantes : la preparation a l'agregation ne m'en laissait pas le temps. Deux exceptions pourtant: d'abord le seminaire Cartan, qui venait de commencer et portait alors sur l'analyse harmonique, la transformation de Fourier et la these de Godement; on y demontrait des theoremes difficiles par des methodes parfaitement claires et "bourbachiques", ce qui me piaisait beaucoup. Je suivais egalement les cours de Leray au College de France, ou il exposait ses nouvelles theories sur la topologie algebrique, avec tout plein d'anneaux filtres et de "couvertures" ; je n'y comprenais rien, mais cela me fascinait. Qu'avez-vous fait ensuite ? Je me suis marie et je suis entre au C.N.R.S. Pendant un an, j'ai habite Auxerre, ou ma femme etait professeur ; puis nous sommes revenus a Paris, que nous n'avons plus quitte depuis. Mon patron au C.N.R.S. etait Cartan, et je participais activement a son seminaire, frappe des exposes comprise. Le sujet en etait la topologie : complexes, formule de Kiinneth, homologie singuliere (j'avais copie a la main les quelque quarante pages de Singular Homology Theory de S. Eilenberg, n'ayant pas ose ecrire pour demander un tire a part), faisceaux, espaces fibres et groupes d'homotopie. J'avais garde un pied a l'Ecole normale. J'y faisais un cours a demi officiel aux eleves de premiere annee et, un peu plus tard, j'y tenais un seminaire encore moins officiel, que nous appelions "le seminaire de la turne 100", ou Blanchard, Cerf, Lions, Malgrange puis Berger et Bruhat ont fait leurs premieres armes. Souvent, leurs exposes prefiguraient leurs travaux ulterieurs : Lions et Malgrange sur les distributions, Cerf sur la theorie de Morse, Bruhat sur les representations de groupes localement compacts et Berger sur les espaces de Riemann. Ce petit seminaire nous a ete bien utile.
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C'est a cette epoque que j'ai ete recrute par Bourbaki. Cela s'est passe de la facon suivante : en juin 1948, a la fin de ma troisieme annee a l'Ecole normale, j'avais eu l'occasion d'assister a une seance du congres Bourbaki, qui se tenait en salle E ou F et au cours de laquelle on lisait et discutait les redactions destinees aux prochains livres. Cela m'avait beaucoup interesse. Quelques mois plus tard, alors que j'etais a Auxerre, j'appris que le prochain congres allait se tenir a Nancy, rue de la Craffe. Sans hesiter, j'ai pris le train pour Nancy; Bourbaki a accepte que je participe aux discussions et, a la fin du congres, m'a invite a venir au suivant comme "cobaye" puis comme membre a part entiere. C'est ainsi qu'a debute une collaboration qui a dure pres de vingt-cinq ans et m'a beaucoup marque. Mes debuts au C.N.R.S. se presentaient done assez bien : cours a l'Ecole, seminaire Cartan, Bourbaki... II y avait pourtant un gros point noir : je n'avais pas de sujet de these, je ne trouvais rien et je ne savais meme pas dans quelle direction je devais chercher : topologie avec Cartan, representations de groupes avec Godement, ou autre chose ? Situation banale, sans doute, mais qui me demoralisait; si cela continuait, j'etais decide a abandonner le C.N.R.S. pour devenir professeur de lycee ! Cela n'a pas continue. J'ai fini, deux ans apres ma sortie de l'Ecole, par "demarrer". Ce demarrage, je le dois en grande partie au mathematicien Suisse Armand Borel; venu de Zurich, ou il avait ete l'eleve de H. Hopf, il etait arrive a Paris a l'automne 1949 et il y est reste deux ans. Nous avons fait connaissance au seminaire Cartan et nous avons immediatement sympathise. Un peu plus age que moi, il avait deja plusieurs publications a son actif et il m'a beaucoup appris sur la technique de la recherche. De plus, ce qui a ete capital, il avait reussi a comprendre la mysterieuse theorie de Leray, suites spectrales incluses, et il me l'a expliquee. Un certain dimanche de juin 1950 (le jour ou a commence la guerre de Coree), nous avons trouve une application de cette theorie qui, sans etre difficile, etait d'un genre nouveau : nous avons demontre qu'il n'existe pas de fibration d'un espace euclidien dont lesfibressoient compactes et non
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reduites a un point. Nous avons redige la-dessus une note pour les Campus rendus de VAcademie des sciences; presentee le lendemain a l'Academie par Elie Cartan, elle est parue deux semaines plus tard : les delais de publication etaient courts en ce temps la ! Mis en confiance par ce succes, j'ai entrepris d'explorer la theorie de Leray pour voir quelles autres applications on pouvait en tirer. Je me suis rapidement apercu que Ton pouvait l'utihser pour l'etude des groupes d'homotopie, et en particulier des groupes d'homotopie des spheres : un sujet qui etait au coeur de la topologie algebrique de l'epoque et sur lequel on savait fort peu de chose (on ignorait meme si les groupes en question etaient finis ou infinis). A vrai dire, il y avait une difficulte technique serieuse : la theorie de Leray, sous sa forme originale, ne s'appliquait pas aux espaces fibres que j'etais amene a introduire, les espaces de lacets. II fallait proceder autrement. C'etait surement possible, j'en etais persuade ; mais comment faire ? La encore, j'ai eu de la chance : au congres Bourbaki suivant, a Royaumont, Cartan et Koszul m'ont suggere d'employer une certaine filtration du complexe singulier a laquelle je n'avais pas pense. De retour a Paris, je n'ai pas eu grand mal a montrer que cette filtration avait toutes les proprietes revees. La difficulte etait vaincue. II n'y avait plus qu'a recolter les resultats. J'en ai fait une these, soutenue en mai 1951 et parue a la fin de l'annee aux Annals ofMathematics. Mon apprentdssage etait termine. Qu'avez-vous fait apres voire these ? Au debut de 1952, j'ai ete invite par l'universite de Princeton, ce qui m'a permis de faire connaissance avec S. Lefschetz, E. Artin, S. Lang, J. Tate et bien d'autres. Je suivais avec assiduite le seminaire Artin-Tate sur le corps de classes : utiliser des groupes de cohomologie dans des questions arithmetiques, quel plaisir pour le topologue que j'etais ! Je donnais moi-meme des cours, sur ma these ainsi que sur sa suite, la "C-theorie". Malheureusement, il m'a fallu ecourter mon sejour, ma femme ayant du subir une serie d'intervenrions chirur-
545 JEAN-PIERRE gicales tres douloureuses consecutives a un accident de laboratoire oii elle avait failli perdre une main. Je suis done revenu a Paris. Peu apres, j'ai ete nomme a Nancy, ou enseignaient Delsarte, Dieudonne, Godement et Schwartz (et, plus tard, Lions): une bonne equipe ! Mes nouveaux collegues auraient aime que je vienne m'installer comme eux a Nancy, mais je m'y suis refuse : je ne voulais pas quitter Paris a cause du seminaire Cartan, et aussi a cause de ma femme qui travaillait a sa these, sous la direction de A. Pullman, a l'lnstitut du radium. J'ai done "commute" : chaque semaine, je venais passer deux jours a Nancy, pendant lesquels je participais au seminaire et je faisais mes cours. Ceux-ci portaient sur la mecanique dite rationnelle (par opposition a la mecanique reelle ?), sujet qui me rebutait lorsque je preparais la licence et l'agregation et que je m'etais empresse d'oubher; il m'a fallu l'apprendre a nouveau, pour le re-oublier ensuite, definitivement cette fois (du moins, je l'espere); quel gachis ! Cetteperiodenanceenne n'apas durelongtemps. En 1954, j'ai recu l'une des deux medailles Fields du Congres international d'Amsterdam, et peu de temps apres Leray m'a propose d'etre candidat a une chaire au College de France. J'ai commence par refuser : j'ai une facheuse tendance a dire "non" quand on me propose queique chose, quitte a changer d'avis ensuite... Heureusement, Leray et Lichnerowicz sont revenus a la charge quelques mois plus tard et m'ont finalement convaincu de poser ma candidature. J'ai redige une notice et fait les visites d'usage. A la fin de 1955, le College a vote la creation d'une chaire d'Algebre et geometrie et j'en suis devenu titulaire en 1956. II y a maintenant trente ans que je l'occupe : plus de la moitie de ma vie ! Que repre'sente pour vous Venseignement au College de Frame ? Enseigner au College est un privilege merveilleux et redoutable. Merveilleux a cause de la liberte dans le choix des sujets et du haut niveau de l'auditoire : cher-
SERRE
cheurs au C.N.R.S., visiteurs etrangers, collegues de Paris et d'Orsay - beaucoup sont des habitues qui viennent regulierement depuis cinq, dix ou meme vingt ans. Redoutable aussi: il feut chaque annee un sujet de cours nouveau, soit sur ses propres recherches (ce que je prefere) soit sur celles des autres; comme un cours annuel dure environ vingt heures, cela fait beaucoup ! Et vos voyages ? Je vais regulierement aux Etats-Unis, ou je passe a peu pres trois mois tous les deux ans, soit a 1'Institute for Advanced Study de Princeton soit a Harvard. II m'arrive aussi d'aller en Allemagne (a Bonn, a Oberwolfach), en Grande-Bretagne (a Londres, a Cambridge) et dans d'autres pays: en Suisse, en Suede, au Japon, etc. Etl'U.R.S.S.
?
J'ai passe trois semaines a Moscou en 1961; je m'y suis fait des amis et j'aurais aime y retourner. Malheureusement, en 1970, a propos du Congres international de Nice, j'ai ete amene a crftiquer de facon assez vive la conduite des autorites mathematiques sovietiques, qui avaient refuse des visas a certains conferenciers (cette pratique n'a, helas, pas changeJ.Mes critiques ont eu au moins un resultat tangible : j'ai ete mis sur une liste noire et, pendant de nombreuses annees, il ne m'a plus ete possible d'aller en U.R.S.S., malgre les efforts de mes amis russes. Cela a dure jusqu'en 1983. A cette date, j'ai eu le plaisir de recevoir une lettre du grand maitre des mathematiques sovietiques, l'academicien I.M. Vinogradov (directeur depuis cinquante ans de l'lnstitut Steklov et cinq fois prix Lenine), qui m'informait que la punition etait levee et que j'etais de nouveau le bienvenu en U.R.S.S. Je lui ai repondu que j'etais enchante de l'apprendre et j'y suis alle l'annee suivante pour un sejour de deux semaines a Leningrad et a Moscou, qui s'est fort bien passe.
Peut-etre irai-je une troisieme fois, qui sait ? J'ai une grande estime pour mes collegues sovietiques : malgre des conditions materielles et morales difficiles (pour ne
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pas dire plus), ils ont reussi a creer une Ecole mathemarique qui est devenue l'une des premieres du monde.
seule. On peut aussi se fixer de facon consciente un programme et le realiser pas a pas. C'est morns amusant, mais tout aussi efficace.)
Quelles sont les principals influences que vous avez subies ?
Pourriez-vous decrire vos travaux les plus importants de facon accessible pour un non-spe'cialiste ?
Celles d'Henri Cartan et d'Armand Borel ont ete decisives pour ma formation. Ensuite, c'est celle d'Andre Weil, a la fois par les sujets memes de ses travaux et par sa facon de les aborder ; ainsi, pendant une dizaine d'annees, j'ai lu et relu regulierement son Avenir des Mathematiques, un texte extraordinaire ou sont predits en quelques pages la plupart des progres de la theorie des nombres durant les vingt annees suivantes.
J'ai bien peur que ce ne soit pas possible. Disons seulement que j'ai travaille sur des sujets ou geometrie et algebre se melent en proportions variables : nomotopic, fonctions de variables complexes, geometrie algebrique, theorie des groupes, arithmetique. Ces sujets ne sont pas independants : il y a entre eux quantites de relations, les unes evidentes, les autres plus ou moins cachees. C'est ce qui explique que Ton puisse passer de l'un a l'autre : au fond, ce sont les memes idees qui interviennent sous des deguisements differents.
Quels sont, pour vous, lesproblemes importants ou interessants en mathematiques ? Que j'aimerais pouvoir repondre a cette question ! Helas, je n'en suis pas capable. Cela fait bien des annees que, comme editeur ou "referee", je suis amene a proposer l'acceptation ou le refus de nombreux articles dans des revues aussi differentes que les Comptes rendus de I'Academie des sciences et Inventiones Mathematicae. Je me decide en general tres vite. Pourtant, il m'est impossible de dire sur quel critere precis je me base pour juger de la qualite d'un texte : ce n'est pas programmable ! En ce qui concerne mon propre travail, j'essaie de ne pas me poser la question de son interet, ni de son importance. Un probleme m'intrigue ? J'y reflechis, meme si d'autres jugent qu'il n'est qu'un simple exercice. Parfois, d'ailleurs, la methode que Ton est amene a fabriquer se revele avoir quantiti d'applications que Ton n'avait pas entrevues au debut; parti d'un exercice, on se retrouve avec une theorie ! C'est l'une des surprises agreables du metier. (La technique de travail que j'evoque ici est celle que je prefere ; elle a l'avantage de faire une grande part a l'imagination et a l'inconscient. Mais ce n'est pas la
224
Depuis une vingtaine d'annees, je m'interesse particulierement aux formes modulaires, un sujet plein de mysteres attirants et de rencontres delicieuses. Dans le texte sur YAvenir des Mathematiques que je citais tout a l'heure, Weil parlait deja de cesproduits euleriens, dont les recbercbes de Hecke viennent seulement de nous reveler Vextreme importance en theorie des nombres et en theorie des fonctions. II ajoutait: "Ce domaine est encore pour nous si mysterieux , les questions qui s'y posent sont si nombreuses et si fascinantes que toute tentative pour i les classer par ordre d'importance serait prematuree." Quarante ans plus tard, les questions sont encore plus nombreuses et tout aussi fascinantes. Vous avez public une dizaine de livres. Comment sont-ils ne's ? Aucun de ces livres n'a ete ecrit de faijon systematique. Je n'ai jamais eu le courage de m'installer devant ma machine a ecrire et de me mettre a taper "chapitre I" ; meme pour la redaction de ma these, j'ai commence par le chapitre III. Je me suis borne a regrouper et completer des notes de cours ou de seminaires deja redigees. C'est ainsi, par exemple, que sont nes Corps Locaux, Cours d'Arithmetique et les Arbres.
547 J E A N - P IK R R E S [•'. R R E Vous avez public' recemment vos CEuvres completes ? Completes ? J'espere bien que non : il y a encore beaucoup de choses que j'aimerais mettre au point et publier. Quant aux trois volumes auxquels vous faites allusion, ils reproduisent l'essentiel de ce que j'ai fait jusqu'en 1984, avec des notes indiquant les questions resolues et donnant des references a des publications plus recentes. Rediger ces notes m'a amene a me replonger dans des textes vieux de vingt ou trente ans, parfois davantage : un exercice instructif, plus facile que je ne l'aurais cru a priori. Quel est pour vous le rapport entre les mathematiques etla realite ? Vous posez la question comme si les deux choses etaient separees. Pour moi, les mathematiques font partie de la realite : quoi de plus concret qu'un nombre premier ou une sphere ? Pour vous donner un exemple, j'ai en ce moment des conjectures (c'est-a-dire des enonces que je ne sais pas demontrer, mais que je crois vrais) sur les formes modulaires. Ces conjectures, quand on les exprime dans des cas particuliers, font intervenir des nombres tout a fait explicites : 2 , 3 , 1 . . . et il est possible de les tester par des calculs numeriques, soit a la main, soit sur ordinateur. Un seul contre-exemple suffirait a les demolir ; jusqu'a present, on n'en a pas trouve. N'estce pas aussi concret qu'un travail experimental ? D'ailleurs, l'une des choses que j'aime le plus en mathematiques, c'est de voir de grandes theories abstraites, comme celles de Grothendieck, qui, appliquees a des cas particuliers, ont des consequences numeriques tres fines que Ton ne sait pas obtenir par d'autres moyens. Et la physique ? Franchement, je n'y connais rien ! Je sais, bien sur, que certains mathematiciens rirent leur inspiration de pro-
blemes poses par la physique, mais cela ne m'est jamais arrive ; il faut croire que je n'ai pas Tesprit physicien". Ya-t-il des differences de style mathematique d'un pays a I 'autre, et d'une epoque a Vautre ? II faut d'abord insister sur le fait que les mathematiciens du monde entier forment une grande famille, independamment des frontieres et des langues. Lorsque l'un de nous trouve un resultat important, les autres en sont presque aussitot informes et, peu de temps apres, peuvent verifier la demonstration et en utiliser les principes. II n'y a ni methode secrete ni chasse gardee. En voici un exemple. Au debut des annees 1950, on parlait aux Etats-Unis de French Topology pour designer l'ensemble des methodes (faisceaux, suites spectrales, espaces de lacets) creees par Leray, Cartan et leurs eleves. Cela n'a pas dure. Deux ou trois ans plus tard, ces methodes avaient ete assimilees et les topologues americains, russes, anglais, allemands... s'en servaient avec encore plus de succes que leurs collegues francais : la French Topology avait ete absorbee par la Topology tout court. II faut s'en rejouir : Andre Weil a tres bien explique pourquoi la "Science francaise", si chere aux journalistes et aux politiciens, est une notion contradictoire : si elle n'est que francaise, ce n'est plus de la science ! Cela dit, il y a pourtant des pays qui possedent une ecole particulierement forte sur tel ou tel sujet. Cela peut etre du soit a l'influence d'un homme (Courant a New York pour l'analyse, Grothendieck a Bures-surYvette pour la geometrie algebrique), soit a l'existence d'une longue tradition (Cambridge pour la theorie des nombres, Princeton pour la topologie). Quant aux differences de style entre les epoques, elles existent, mais elles ne sont pas tres importantes. Ainsi, les mathematiques des annees 1980 me paraissent plus concretes que celles des annees 1960 ; mais peut-etre est-ce une illusion ? En tout cas, il n'y a a aucun moment de rupture avec le passe, encore moins de revolution ; nous
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548 nous sentons solidaires de nos collegues d'il y a un siecle et bien des problemes qu'ils se posaient nous interessent encore, meme si nous les voyons sous un angle different. Cette "longue duree" est l'un des charmes des mathematiques. La politique vous interesse-t-elle, etde quelle facon ? Elle m'interesse comme n'importe qui, mais pas davantage. La seule chose que je puisse dire a ce sujet, c'est que je semble avoir un talent particulier pour me tromper dans mes previsions et pour voter pour des candidats qui ne seront pas elus. Mon vote ressemble au kiss of death de la Maffia : celui qui le recoit n'en rechappe pas. Avez-vous une philosophic personnelle ? Essayer d'etre heureux ! (C'est agreable, non seulement pour soi, mais aussi pour les autres : peu de gens s'en rendent compte.) Jusqu'a present, j'ai eu la chance d'y parvenir. Quant a la vraie philosophic, je ne m'y suis jamais vraiment interesse - ou plutot je n'ai jamais trouve de philosophe qui ecrive quelque chose qui m'interesse, a de rares exceptions pres : je pense notamment a des remarques de Sartre et de Jeanson sur la morale, dont j'ai effectivement tire profit. Meme deception avec la philosophic des sciences : ce que j'en ai lu ne me donne pas envie d'en parler. L'histoire des sciences, en revanche, est un sujet passionnant quand il est traite par des scienufiques de bon niveau ; mais cela n'a rien a voir avec la philosophic.
Quelle est votre sensihilite a Part ? Mon education artistique a ete plutot ratee. Je n'ai pas ecoute de morceau de musique digne de ce nom avant l'age de vingt ans, alors que j'etais eleve a l'Ecole normale (il s'agissait des Concertos brandebourgeois de Bach, qui ont ete pour moi une revelation). Depuis, j'en suis reste a la musique ancienne : Bach, Telemann, Vivaldi, Monteverdi... Le seul compositeur plus recent que j'apprecie est Debussy: je ne me lasse pas d'ecouter Pelle'as et Me'lisande. Meme deficience en peinture : les musees de Nimes n'etaient pas riches. La liste des peintres que j'aime vraiment est tres reduite. II y a Van Gogh et Rembrandt (pour ses vieillards), et surtout le Greco avec ses personnages dont on a le sentiment qu'ils voient Dieu directement. Mais vous venez de dire que vous e'tiez athee ? Je n'en admire que davantage les gens qui voient Dieu : il n'y a la aucune contradiction. D'ailleurs on ne peut pas reprocher a Dieu de ne pas exister, ce serait mesquin. (Pour clarifier ceci, laissez-moi vous raconter un reve que j'ai fait il y a quatre ou cinq ans. J'avais une conversation avec Dieu, en tete a tete, je ne sais plus sur quel sujet. Tout a coup, II me dit: II y a quelque chose queje dois t'avouer: Je n 'existe pas. Je Lui ai repondu : Je le savais depuis longtemps. Cela n 'a pas d'importance. Et nous nous sommes quittes avec un bon sourire.) Et la lecture ?
Avez-vous une religion, ou une conception personnelle de la religion ? Mon pere, fils de pasteur, etait protestant et ma mere catholique. J'ai recu une education religieuse protestante mais, peu apres ma premiere communion, j'ai constate que la religion m'etait indifferente et je suis devenu athee ; je le suis reste.
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Depuis mon enfance, c'est l'une de mes occupations favorites et il en est de meme pour ma femme. Nous nous sommes consume une assez bonne bibliotheque, aussi bien en langue anglaise que francaise. Parmi mes auteurs favoris, je citerais Saint-Simon, Stendhal, Proust, Giono, Queneau, Yourcenar, d'Ormesson et, pour les poetes, Verlaine, Rimbaud, Saint-John Perse ;
549 JEAN-PIERRE
aux Etats.-Unis, Faulkner, O'Connor, Nabokov, LB. Singer, Updike ; en Grande-Bretagne, Kipbng et Forster; en Allemagne, T. Mann et H. Boll; en Scandinavie, Lagerlof, Lagerkvist, Hamsun, Dinesen ; en Italie, Buzatti et Calvino ; au Japon, Kawabata, Tanizaki et Mishima ; en Argentine, Borges. II faudrait ajouter a ces listes celle des auteurs que j'ai aimes mais que je n'aime plus, comme Dostoi'evski, Gide, Sartre.
SERRE
Un autre de mes sports favoris, moins dangereux celuila, est le ping-pong. J'y joue depuis ma jeunesse, mais ce n'est que depuis une dizaine d'annees que je fais partie d'un club, celui de l'Assemblee nationale, l'ASCAN. J'y vais regulierement; l'entrainement a lieu le lundi en fin d'apres-midi, juste apres la seance de l'Academie des sciences ; du quai de Conri a l'Assemblee, le chemin n'est pas long: c'est bien commode.
Quelles stmt vos activites ban des matbematiques ? Je pratique plusieurs sports. D'abord le ski, a Val d'Isere et Zermatt le plus souvent. Je suis alle une fois passer une semaine au Canada, dans les Montagnes Rocheuses, pour faire de l'heliski: un helicoptere vous depose sur un sommet, avec un moniteur et d'autres slrieurs, et vous descendez dans la neige fraiche ; l'helicoptere vous attend en bas, il vous remonte ailleurs et ainsi de suite. C'est beau mais impressionnant: droit dans des pentes raides, en neige profonde, a travers les arbres... Brr ! Je ne sais pas si j'aurai le courage de recommencer.
Mais le sport que je prefere est sans doute l'escalade a Fontainebleau. Les rochers ne sont pas hauts : trois ou quatre metres en moyenne ; on peut en tomber sans se blesser serieusement. La difficulte n'est pas la; elle est dans les mouvements, souvent subtils, qu'il faut trouver pour arriver a passer en utilisant les minuscules prises de la voie choisie. C'est un exercice a la fois physique et intellectuel. On peut meme y travailler la nuit: on connait si bien le rocher que Ton se dit: tiens, si je mettais le pied la, et la main sur telle prise, ca passerait peut-etre ? Au fond, ce n'est pas tellement different des mathematiques ! 14 Janvier 1986
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Lettre a Jacques Tits, 13 Mars 1993 Cher Tits, Tu me demandes dans quelles directions j'ai travaille ces dernieres annees. La reponse est bien simple: en theorie de Galois. Et cela sous divers aspects: Conjecture de Duke Math. J. (1987) sur les representations modulaires de degre 2, a determinant impair, de Gal(Q/Q), et leurs relations avec les formes modulaires (mod p). Cette conjecture aurait (si elle est vraie, ce qui parait probable) quantite de consequences mirifiques, la moindre d'entre elles etant le theoreme de Fermat. Elle a deja suscite quantite de travaux (Ribet, Edixhoven, Coleman, . . . ) qui en ont nettoye les abords. Mais personne n'a d'idee sur la partie centrale. II y a une interpretation adelique quaternionienne qui est fort interessante, et que j'ai exposee dans le cours 1987/1988. Helas je n'ai rien redige la-dessus. 1 Extensions de Q, ou de Q(T), a groupe de Galois donne. Cela a ete le sujet des cours 1988/1989 et 1989/1990. La methode "de rigidite" est celle qui donne les meilleurs resultats, mais ceux-ci sont encore bien incomplets. J'ai publie la-dessus des notes de cours: Topics in Galois Theory, Jones & Bartlett, Boston, 1992. Probleme d'Abhyankar. II s'agit de determiner les groupes finis qui sont groupes de Galois d'une extension de k(T) non ramifiee sauf a l'infini, le corps k etant algebriquement clos de caracteristique p > 0. J'ai remis a la mode cette vieille question, et resolu le cas ou le groupe de Galois est . . . resoluble! Tout recemment, la conjecture vient d'etre demontree par Raynaud et generalised par Harbater. De plus, Abhyankar lui-meme a fabrique quantite d'exemples ou le groupe de Galois est un groupe interessant, e.g. un groupe de Mathieu. Formes trace et G-algebres galoisiennes. J'ai travaille la-dessus avec Eva Bayer (article a paraitre 2 a VAmer. J. Math.). Les resultats ne sont vraiment satisfaisants que lorsque le 2-groupe de Sylow du groupe considere est abelien element aire. Cohomologie negligeable. Je me suis rendu compte a l'occasion du travail precedent que certaines classes de cohomologie d'un groupe fini G sont "negligeables" au sens de la cohomologie galoisienne: elles donnent 0 dans la cohomologie de tout corps. Comment determiner ces classes? C'est la un probleme non trivial, sur lequel j'ai un certain nombre de resultats partiels. 3 L'un des plus surprenants concerne le groupe symetrique Sn: tout element de Hl(Sn, Z/mZ), i > 0, m impair, est negligeable; ce fait est lie a la determination des invariants cohomologiques des algebres etales, determination que j'ai exposee a Harvard en 1992, mais pas encore au College. Voir: Two letters on quaternions and modular forms (mod p), Oe.169. Torsions quadratiques et bases normales autoduales, Amer. J. Math. 116 (1994), ( = Oe.163). Voir Cohomologie Galoisienne, 5° edition, Chap. Ill, Annexe, §7, p. 170.
1-63
551 Cohomologie galoisienne des corps de fonctions. C'etait le sujet du cours de 1991/1992, ou j'ai essaye de clarifier des resultats (en principe) connus sur les residus, avec applications notamment au groupe de Brauer (specialisation, obstruction de Manin, etc.). Problemes de cohomologie galoisienne. Je suis revenu a cette vieille question dans le cours de 1990/1991, ainsi que dans un cours a Harvard. 4 Le cas du groupe orthogonal, etudie en profondeur par Pfister, Merkurjev-Suslin, Rost et autres, suggere I'existence d'invariants cohomologiques superieurs (demarrant en dimension 3) d'un type nouveau. J'ai cherche a construire ces invariants; je n'y suis parvenu que tres imparfaitement. Heureusement, Rost a eu plus de reussite; d'apres les lettres qu'il m'a envoyees,5 il a effectivement obtenu les proprietes que Ton voulait. Bien sur, ce n'est qu'un premier pas. Meme pour F4, on ne sait pas que ces invariants "separent" des elements distincts, i.e. donnent une description complete du H1 galoisien etudie. II est d'ailleurs evident qu'une theorie plus achevee devra utiliser Bruhat-Tits . . . nous n'en sommes pas encore laRepresentations f-adiques attachees aux motifs. C'est le sujet du cours de 1992/1993, et c'est aussi l'un de mes themes favoris; j'ai commence a faire des cours la-dessus il y a 20 ou 25 ans! Je dois dire (pour m'excuser, au cas ou ce serait necessaire) qu'il est fascinant de voir des groupes qui sont a la fois des groupes de Galois et des groupes algebriques (par exemple semi-simples). Je me souviens encore de mon enthousiasme, en 1962, lorsque je me suis rendu compte du fait suivant: le groupe de Galois 6 du ^module de Tate d'une courbe elliptique est un sous-groupe ferme de GL2(Z;), done est un groupe de Lie Z-adique, done a une algebre de Lie . . . Quelle est cette algebre de Lie? Une bonne partie de ce que j'ai fait depuis provient de cette question. Voila. S.et F.
C'est surement plus long que tu ne l'aurais desire.
Excuse-moi.
J.-P. Serve
Voir aussi Cohomologie galoisienne: progres et problemes, Sem. Bourbaki 1993/94, n°783 ( = Oe.166). 5 Voir M.-A. Knus, A. Merkurjev, M. Rost et J.-P. Tignol, The Book of Involutions, A.M.S. Colloquium n° 44 (1998), §31 et §40. C'est Tate qui a attire mon attention sur ce groupe (dans une lettre datee du 28 Juillet 1962), et m'a incite a determiner sa cohomologie, ce que j ' a i fait peu apres, cf. Oe.62.
Wolf Prize in Mathematics, Vol. 2 (pp. 553-597) eds. S. S. Chern and F. Hirzebruch © 2001 World Scientific Publishing Co.
Curriculum Vitae (by H. Klingen)
1896, December 31
Born in Berlin, Germany
1915-1917
Student of mathematics, astronomy and physics, Friedrich-Wilhelms-Universitat Berlin, teachers F. G. Frobenius, I. Schur, M. Planck
1917-1919
Short military service, dismissed due to illness
1919-1920
Student at the University of Gottingen, supervisor E. Landau
1920,June 9
Doctorate in mathematics, Dr. phil., University of Gottingen
1920-1921
Visiting lecturer, University of Hamburg, Assistant of R. Courant, University of Gottingen
1921, December 12
Habilitation, University of Gottingen
1922
Lecturer (Privatdozent), University of Gottingen
1922-1937
Professor of Mathematics, University of Frankfurt, successor on the chair of A. Schoenflies
1935
Visiting Professor, Princeton University and Institute for Advanced Study
1937-1940 1940 1940-1951
Professor of Mathematics, University of Gottingen Emigration to the United States Member of the Institute for Advanced Study, Princeton, Permanent Member since 1945
from 1951
Professor of Mathematics, University of Gottingen
1953
Visiting Professor, Johns Hopkins University, Baltimore
1955/56, 1959/60 1962/63, 1 9 6 6 / 6 7
Visiting Professor, Tata Institute of Fundamental Research, Bombay
1958
Visiting Professor, different universities in Japan
1959
Retired
1981, April 4
Died in his home at Gottingen
554
Honours and Awards (compiled by H. Klingen)
1947
Det Kongelige Danske Videnskabernes Selskap, Copenhagen, Member
1949
Akademie der Wissenschaften in Gottingen, Corresponding Member
1951
Akademie der Wissenschaften in Gottingen, Ordinary Member
1952
Det Norske Videnskaps-Akademi, Oslo, Member
1953
The University of Chicago, Doctor of Science honoris causa Indian Mathematical Society, New Delhi, Honorary Member
1954
Universite de Nancy, Docteur honoris causa
1956
The Royal Swedish Academy of Sciences, Stockholm, Foreign Member Academie des Sciences de l'Institut de France, Correspondant pour la section de geometrie Academia del Lincei, Rome, Foreign Member London Mathematical Society, Honorary Member
1958
Bayerische Akademie der Wissenschaften, Miinchen, Corresponding Member Deutsche Akademie der Naturforscher Leopoldina, Halle, Member
1959
Tata Institute of Fundamental Research, Bombay, Honorary Fellow
1960
Universitat Basel, Doctor honoris causa
1963
Order "Pour le merite fur Wissenschaften und Kiinste", Bonn, Member
1964
Universitat Frankfurt, Doctor honoris causa Decorated "Das Grosse Verdienstkreuz mit Stern", Germany
1965
Universitat Wien, Doctor honoris causa
1967
New York University, Doctor of Science honoris causa ETH Zurich, Doctor honoris causa National Academy of Sciences of the USA, Washington, Foreign Associate Member
1968
Academie Internationale d'Histoire des Sciences, Paris, Membre d'honneur
1973
Academie des Sciences de l'Institut de France, Associe etranger
1974
Osterreichische Akademie der Wissenschaften, Vienna, Honorary Member
1978
The Wolf Prize, Israel
1979
American Academy of Arts and Sciences, Boston, Foreign Honorary Member
Obituary (by Th. Schneider, reprinted from Jahrbuch der Akademie der Wissenschaften in Gottingen, 1982) Nachruf auf Carl Ludwig Siegel 31.Dezember 1 8 9 6 - 4 . April 1981 Das hohe Ansehen, das Carl Ludwig Siegel in der wissenschaftlichen und besonders in der mathematischen Welt genoS, fand seinen auSeren Niederschlag in den zahlreichen Ehrungen, die ihm zuteil geworden sind. Er war Ehrendoktor der Universitaten von Chicago, Nancy, Basel, Frankfurt, Wien, New York und der ETH Zurich, ferner Ehrenmitglied oder Mitglied zahlreicher angesehener Akademien des In- und Auslands, sowie des Ordens Pour le merite. Der Akademie der Wissenschaften in Gottingen gehorte er seit 1949 als korrespondierendes und seit 1951 als ordentliches Mitglied an. Die Fiille der Ehrungen war nur. ein Zeichen dafur, daE er einer der ganz groften Mathematiker seiner Zeit gewesen ist. Dabei war sein Auftreten von besonderer Zuriickhaltung und Bescheidenheit gepragt, und so hat er es moglichst vermieden, sich auf Kongressen zu zeigen. Seine gesammelten Werke sind schon zu seinen Lebzeiten veroffentlicht worden, und nachdem zu seinem 70. Geburtstag die ersten drei Bande erschienen waren, schrieb LeVeque in den Mathematical Reviews: „The collection stands as a monument to the genius of the author." Carl Ludwig Siegel wurde am 31.Dezember 1896 in Berlin geboren. Seine Eltern stammten aus dem Rheinland, er hatte keine Geschwister. In Berlin durchlief er auch die Schulen von der Gemeindeschule iiber die Realschule bis zur Oberrealschule. Von seiner Jugend ist auSer seinem Interesse fur Mathematik und seiner besonderen Neigung zum Zeichnen nicht viel bekannt. Uber seine Schulzeit auftert er sich in einem Nebensatz seines Aufsatzes „Erinnerungen an Frobenius" (1968), doch lassen sie mich hier ausfiihrlicher zitieren. Er hatte sich im Herbst 1915 in Berlin fiir das Studium der Astronomie eingeschrieben, und er schreibt dazu in den eben genannten Erinnerungen: „Als ich Herbst 1915 an der Berliner Universitat immatrikuliert wurde, war gerade ein Krieg in vollem Gange. Obwohl ich die politischen Ereignisse nicht durchschaute, so fafite ich in instinktiver Abneigung gegen das gewalttatige Treiben der Menschen den Vorsatz, mein Studium einer der irdischen Angelegenheiten moglichst fernliegenden Wissenschaft zu widmen, als welche mir die Astronomie erschien. DaE ich trotzdem zur Zahlentheorie kam, beruhte auf folgendem Zufall. Der Vertreter der Astronomie an der Universitat hatte angekiin-
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Nachruf auf Carl Ludwig Siegel
digt, er wiirde sein Kolleg erst 14 Tage nach Semesterbeginn anfangen, was ubrigens in der damaligen Zeit weniger als heutzutage iiblich war. Zu den Wochenstunden, Mittwoch und Sonnabend 9—11 Uhr, war aber auch eine Vorlesung von Frobenius iiber Zahlentheorie angezeigt. Da ich nicht die geringste Ahnung davon hatte, was Zahlentheorie sein konnte, so besuchte ich aus purer Neugier zwei Wochen lang dieses Kolleg, und das entschied iiber meine wissenschaftliche Richtung, sogar fiir das ganze weitere Leben. Ich verzichtete dann auf Teilnahme an der astronomischen Vorlesung, als sie schlieElich anfing, und blieb bei Frobenius in der Zahlentheorie." Die Vorlesungen von Frobenius haben ihn sehr beeindruckt. Er schildert diese in dem schon genannten Artikel so: „Frobenius sprach vollig frei, ohne jemals eine Notiz zu benutzen, und dabei irrte oder verrechnete er sich kein einziges Mai wahrend des ganzen Semesters. Als er zu Anfang die Kettenbriiche einfuhrte, machte es ihm offensichtlich Freude, die dabei auftretenden verschiedenen algebraischen Identitaten und Rekursionsformeln mit grofSter Sicherheit und erstaunlicher Schnelligkeit der Reihe nach anzugeben, und dabei warf er zuweilen einen leicht ironischen Blick ins Auditorium, wo die eifrigen Horer kaum noch bei der Menge des Vorgetragenen mit der Niederschrift folgen konnten." Und an spaterer Stelle des gleichen Artikels sagt Siegel: „ich habe bereits erwahnt, daS ich nicht erklaren kann, wodurch die starke Wirkung der Vorlesungen von Frobenius hervorgerufen wurde. Nach meiner Schilderung der Art seines Auftretens hatte die Wirkung eher abschreckend sein konnen. Ohne da£ es mir klar wurde, beeinfluSte mich wahrscheinlich die gesamte schopferische Personlichkeit des groEen Gelehrten, die eben auch durch die Art seines Vortrags in gewisser Weise zur Geltung kam. Nach bedriickenden Schuljahren unter mittelmaSigen oder sogar bosartigen Lehrern war dies fiir mich ein neuartiges und befreiendes Erlebnis." Am Ende dieses seines ersten Semesters erhielt Siegel dann den Eisensteinpreis, der in Berlin einmal jahrlich einem begabten Studenten der Mathematik verliehen wurde, und den Frobenius beantragt hatte. Die Losung des Problems, das Siegel zum Gegenstand seiner spateren Dissertation gemacht hat, hatte er bereits in seinem dritten Semester gefunden. Es handelt sich dabei um die Verscharfung eines Satzes von A. Thue aus dem Jahre 1908 iiber die Approximation algebraischer Zahlen durch rationale, auf die ich spater noch kurz zu sprechen kommen werde. Siegel schreibt dazu in einer Einfuhrung zu den im Jahre 1977 veroffentlichten Selected Mathematical Papers von A. Thue, nachdem er die weitere Entwicklung aus dem Thueschen Satz geschildert hat, folgendes: „Da ich selber am Beginn dieser weiteren Entwicklung beteiligt gewesen bin, so mag es vielleicht in diesem Zusammenhang gestattet sein, dafi
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ich dariiber eine personliche Episode erzahle. Wahrend meines dritten Studiensemesters erwahnte Professor I. Schur nach einem Vortrag uber die Pellsche Gleichung, dafi die entsprechend gebildete diesbezugliche Gleichung x n — dy n = 1, fiir jeden festen Exponenten n > 2 , stets nur endlich viele ganzzahlige Losungen besafie, und nannte dazu die Thuesche Arbeit. Als ich diese dann zu lesen versuchte, kam ich bald in Verwirrung durch die vielen Buchstaben c, k, 0 , co, m, n, a, s, deren tiefere Bedeutung mir ratselhaft schien. Um nun doch etwas mehr verstehen zu konnen, anderte ich die Anordnung der Hilfssatze, fiihrte auch neue Symbole ein, und unter diesen war dann, weniger durch geordnetes Denken als durch Zufall, noch ein bei Thue nicht aufgetretener Parameter, der zu meiner Verwunderung eine Verscharfung des Approximationssatzes ergab. Da ich in meiner Oberlegung keinen Fehler finden konnte, so schrieb ich sie auf, etwa 4 Seiten lang, und gab es Schur bei der nachsten Gelegenheit zur Beurteilung. Ich wurde aber in meiner Erwartung tief enttauscht, denn Schur reichte mir, ein paar Wochen sparer, das durch Lagern an der Sonne auf seinem Schreibtisch schon vergilbte Manuskript mit der kurzen Bemerkung zuriick, ich hatte blo6 mit Identitaten gerechnet, und aus diesen liefie sich nichts schliefien. Nachdem ich kurz darauf zu militarischer Verwendung eingezogen worden war, mag wohl auch jene Enttauschung zu meinem nervosen Zusammenbruch beigetragen haben. Meine Erbitterung uber den MiSerfolg bei Schur loste sich erst zwei Jahre sparer, als ich durch gliickliche Umstande Edmund Landau in Berlin sprechen konnte und dieser fiir mein Manuskript Interesse zeigte. Zunachst hatte ich in erneuter Fassung 6 Seiten dafiir verwendet, und Landau meinte nach Durchsicht, die Wahrscheinlichkeit fiir Richtigkeit sei 15 Prozent. Im Sommer 1919 konnte ich mein Studium in Gottingen wieder aufnehmen und schrieb unter Landaus standiger Kontrolle mehrere verbesserte Fassungen des Beweises, wobei die Seitenzahl allmahlich auf 40 anstieg und entsprechend jene Wahrscheinlichkeit auf 90 Prozent. Darauf erklarte schlieSlich Landau, er wolle meine Arbeit als Dissertation annehmen. Nach der am 2.6.1920 erfolgten Promotion war Siegel im WS 20/21 Lehrbeauftragter in Hamburg und danach drei Semester lang Assistent bei R. Courant in Gottingen. Seine auEergewohnliche wissenschaftliche Tatigkeit zeitigte rasche Erfolge. Allein in den beiden Jahren 1921, 1922 erschienen 13 Publikationen. In Gottingen drangte man ihn, sich schnellstens zu habilitieren, um ihn dadurch vielleicht am Orte noch etwas halten zu konnen. Doch die Ernennung zum Privatdozenten am 10.12.1921 konnte nicht verhindern, dafi er bereits am 1.8.1922 als Nachfolger von A. Schoenflies auf eine ordentliche Professur an die Universitat Frankfurt berufen wurde.
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Hier beginnt nun jene Zeit, die er in seinem Bericht zur Geschichte des Frankfurter Mathematischen Seminars anlafslich der 50-Jahr-Feier der Johann-Wolfgang-Goethe-Universitat ausfiihrlich geschildert hat, und in die nach seinen Worten die schonsten Jahre seines Lebens gefallen sind. Besonders weist er auf das harmonische Verhaltnis zu seinen damaligen Kollegen und auf das durch Max Dehn, dem er in lebenslanger Freundschaft verbunden war, initiierte einzigartige mathematisch-historische Seminar, an dem sich alle Kollegen beteiligten, hin. Bereits im Sommer 1930 war er fur ein Semester einer Einladung als Gastprofessor nach Gottingen gefolgt, und nachdem sich seit 1933 die politischen Verhaltnisse auf seine Frankfurter Kollegen besonders stark auswirkten, verlieE er zum 1.1.1938 Frankfurt endgultig, um in Gottingen eine Professur anzunehmen. Seit Kriegsbeginn 1939 trug er sich mit dem Gedanken der Emigration, und im Friihjahr 1940 ging er in die Vereinigten Staaten, wo er schon 1935 einen voriibergehenden Gastaufenthalt genommen hatte. Er fand am Institute for Advanced Study in Princeton zunachst bis 1945 ein Forschungsstipendium und danach eine feste Stellung, in der er bis zum Friihjahr 1951 verblieb. Den Winter 1946/47 hatte er wahrenddessen wieder in Gottingen als Gastprofessor verbracht, und 1951 nahm er erneut einen Ruf auf ein freigewordenes Ordinariat in Gottingen an. Zum 1.4.1959 liefi er sich in Gottingen emeritieren. Aber auch nach seiner Emeritierung fiihrte er die Vorlesungstatigkeit noch fiir mehrere Jahre weiter. Ferner gab er zwischen 1955 und 1967 insgesamt viermal, jedesmal fiir mehrere Monate, Gastvorlesungen am Tata-Institute for Fundamental Research in Bombay. In zunehmender Vereinsamung, aber volliger geistiger Frische, vollendete sich sein Leben in Gottingen am 4. April 1981. Sein Leben war erfiillt von seiner mathematischen Arbeit. Ein Ausgleich war fiir ihn die Natur, vor allem die Berge, die er nicht nur erwanderte, sondern deren Schonheit er auch in zahlreichen Olgemalden, Aquarellen und Pastellzeichnungen festgehalten hat. Aber auch in der Mathematik spielte fiir ihn das asthetische Moment, etwa in der Eleganz eines Beweises, der Perfektion und Ausgefeiltheit einer Vorlesung oder einer Veroffentlichung eine bedeutende Rolle. Gegen so manche neuere Stromungen in seiner Wissenschaft hegte er groEe Bedenken, da er dieselben als ungiinstig fiir die kiinftige Entwicklung der Mathematik ansah. Ich mochte an dieser Stelle den Anfang eines Satzes aus seinem Vorwort zur Reduktionstheorie quadratischer Formen zitieren, wo es heiSt: „Im Hinblick auf das Prokrustesbett, in das manche Neuerer den herrlichen Leib der Algebra gezwangt haben", usw.
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An der Entwicklung seiner Schiiler nahm er stets regen Anteil, und er blieb mit ihnen wissenschaftlich und menschlich auch weiterhin verbunden. Die meisten Schiiler hatte Siegel in den Jahren nach seiner Riickkehr aus den Vereinigten Staaten. In Frankfurt waren es nur wenige, die eine Doktorarbeit bei ihm geschrieben haben. Einem seiner damaligen Schiiler sagte er, sie sind der elfte, der eine Dissertation bei mir begonnen, und der funfte, der eine solche vollendet hat. Anspruchsvolle Seminare und hervorragende, aber iiberwaltigende Vorlesungen sorgten fur eine Auswahl. Dabei waren diese Vorlesungen von einer seltenen Klarheit. Jedes Wort war iiberlegt. Es drangt sich auf, an das zu denken, was er iiber die Vorlesungen von Frobenius geschrieben hat. Fast die Halfte der Publikationen Siegels bezieht sich auf Zahlentheorie, und er hat sich stets auch in erster Linie als Zahlentheoretiker gefuhlt. Sein schopferisches Wirken reicht jedoch weit dariiber hinaus in andere Disziplinen, so in die Funktionentheorie mehrerer Veranderlicher, verbunden mit der Theorie der quadratischen Formen, in die mathematische Astronomie, genauer die Himmelsmechanik, um nur diese jetzt zu nennen. Es iibersteigt bei weitem den vorgegebenen Rahmen, das Lebenswerk Siegels hier im einzelnen darzustellen. Ich muE mich daher auf einige wenige Themen seines Schaffens beschranken. Schon die Dissertation gibt innerhalb der Zahlentheorie die engere Richtung an, in die eine ganze Reihe von Arbeiten einzuordnen sind, namlich in das Gebiet der diophantischen Approximationen. Wie bereits angedeutet, behandelt die Dissertation die Frage, den Betrag der Differenz a — ~-, wobei a eine algebraische Zahl «-ten Grades, — eine rationale Zahl ist, in der Form g'^nach oben so abzuschatzen, dafi die Ungleichung nur endlich viele Losungen in rationalen-^ besitzt, d.h., den ft
*
giinstigsten Wert von k anzugeben. Thue hatte k = y + 1 + E gezeigt, und Siegel verbesserte dies auf ein k < 2n. Kurz darauf gelang es Siegel (1921), durch einen verallgemeinerten Beweisansatz, zwar nicht zu zeigen, daf? die Ungleichung fur den Exponenten k = e(log n + ~n n) nur endlich viele Losungen hat, aber daft, falls es doch unendlich viele gibt, fur die Nenner derselben eine GroSenaussage gemacht werden kann. Durch mehrere geeignete Anderungen der Beweisdurchfiihrung konnte erst 1955 durch K.F. Roth das bestmogliche Ergebnis mit endlich vielen Losungen fur k = 2 + E gezeigt werden, denn man weifi aus der Kettenbruchentwicklung, daE k = 2 nicht moglich ist. Eine effektive Schranke fur die Approximation algebraischer Zahlen wurde sogar erst 1970 durch Baker gegeben.
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Zu den Fragen aus dem Thueschen Gedankenkreis hat sich Siegel 50 Jahre nach seiner Dissertation noch einmal in einer weiteren Arbeit geaufiert. Bereits mit der Habilitationsschrift zeigt Siegel, daS er auch in anderen Bereichen der Zahlentheorie produktiv ist. Er schreibt in seinem Habilitationsgesuch (1921), „zuerst interessierte mich mehr die Richtung der Zahlentheorie, sowie der Gruppentheorie. Als ich dann durch eingehende Beschaftigung mit Funktionentheorie die machtigen Hilfsmittel kennengelernt hatte, mit denen sie insbesondere die Theorie der algebraischen Zahlkorper fordert, wandte ich mich der analytischen Zahlentheorie zu. Momentan begriinde ich die Anfange einer additiven Theorie der Zahlkorper, deren erste Satze in meiner Habilitationsschrift entwickelt sind." Es handelt sich dabei um die Verallgemeinerung des Lagrangeschen Satzes iiber die Darstellbarkeit natiirlicher Zahlen als Summe von 4 Quadraten auf algebraische Zahlkorper. Nachdem er bereits zuvor einen analogen Satz fur total positive Zahlen gezeigt hatte, behandelt er in der Habilitationsschrift und einer unmittelbar darauf erschienenen Arbeit unter Benutzung der analytischen Kreismethode von Hardy-Littlewood den allgemeinen Fall. Nach Vereinfachung dieser Methode durch Vinogradow greift er 1944 die Frage noch einmal auf und verallgemeinert den Waringschen Satz auf algebraische Zahlkorper. Mit Obertragungen beschaftigt er sich auch in den beiden Arbeiten iiber die Funktionalgleichung der Dedekindschen Zetafunktion (1922), namlich mit der Obertragung der entsprechenden Satze der Riemannschen Zetafunktion. Im Jahre 1929 erscheint die grofie zweiteilige Publikation mit dem bescheidenen Titel — „Uber einige Anwendungen diophantischer Approximazionen" —, die Siegels Genialitat voll erahnen lafit. Im ersten Teil, der sich mit Transzendenzuntersuchungen der Besselfunktionen und allgemeiner der von Siegel eingefiihrten E-Funktionen, sofern sie einer linearen homogenen Differentialgleichung hochsrens zweiter Ordnung geniigen, befafSt, wird nicht nur die Transzendenz von J0(^) fur algebraisches £ =t= 0 bewiesen, sondern dariiber hinaus eine effektive untere Schranke des Betrags eines Polynoms in J0(^) und J'0(E) angegeben. Die Beweismethode konnte sich kaum an Vorbilder anlehnen, denn sie hat mit den bis dahin bekannten Transzendenzuntersuchungen so gut wie nichts gemein. Hingegen befruchtete sie die Theorie der transzendenten Zahlen, die seit den Entdekkungen von Hermite und Lindemann aus dem 19.Jahrhundert stagnierte, aufs auSerste, die dann auch einen gewaltigen Aufschwung genommen hat. So konnte 1934 das Hilbertproblem iiber die Transzendenz von ab gelost werden, und Shidlovski dehnte 1954 die Siegelsche Methode auf E-
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Funktionen, die Differentialgleichungen hoherer Ordnung geniigen, aus. Der zweite Teil behandelt die diophantische Gleichung f(x,y) = 0, wobei f(x,y) ein Polynom mit Koeffizienten aus einem Zahlkorper sei, und gefragt wird, wann diese Gleichung nur endlich viele ganze Losungen hat. Siegel zeigt, daE dies sicher dann der Fall ist, wenn das Geschlecht des durch die Gleichung dargestellten algebraischen Gebildes grower als 0 ist, wobei der Approximationssatz aus seiner Dissertation und das Resultat aus der These von A. Weil beim Beweis benutzt werden. Nach der Bemerkung, daS die Gleichung f(x,y) = 0 selbst in rationalen Zahlen, wenn nur die Nenner beschrankt sind, und das Geschlecht grower 0 ist, nur endlich viele Losungen besitzt, schreibt Siegel: „Durch den Satz von Weil wird nahegelegt, das Theorem von Fermat und allgemeiner die Theorie der algebraischen diophantischen Gleichungen mit zwei Unbekannten von einer neuen Seite anzugreifen. Doch diirfte wohl der Beweis der Vermutung, daS jede solche Gleichung, wenn ihr Geschlecht grower 1 ist, nur endlich viele Losungen in rationalen Zahlen besitzt, noch die Uberwindung erheblicher Schwierigkeiten erfordern." Mit einer Arbeit iiber die Perioden elliptischer Integrale erster Gattung (1932) und einem Buch iiber transzendente Zahlen (1949) schliefst Siegel seine Transzendenzuntersuchungen ab. Interessant ist auch eine mathematisch-historische Untersuchung aus dem Jahre 1932 iiber Riemanns NachlaS zur analytischen Zahlentheorie, in der nicht nur der Frage nachgegangen wird, ob Riemann durch eine zu belegende heuristische Oberlegung auf seine beriihmte Vermutung hatte gekommen sein konnen, sondern auch gezeigt wird, wie stark Riemann die analytischen Hilfsmittel beherrscht hat. Eine vollig neue Entwicklung stofit Siegel dann mit seinen drei grundlegenden umfangreichen Arbeiten iiber die analytische Theorie der quadratischen Formen (1935/36/37) an. Der Ausgangspunkt ist eine zahlentheoretische Fragestellung, namlich der Satz von Legendre, dalS die diophantische Gleichung ax2 + bxy + cy2 = d dann und nur dann in rationalen Zahlen x,y losbar ist, wenn die entsprechende diophantische Kongruenz ax2 + bxy + cy2 = d (mod q) fur jeden Modul q eine rationale Losung besitzt. Hiervon hat Hasse die folgende Verallgemeinerung gegeben: Sei S die Matrix einer quadratischen Form Q von m Variablen, T die Matrix der quadratischen Form R von n Variablen und X die Matrix der homogenen linearen Substitution mit rationalen Koeffizienten, die Q in R iiberfiihrt, also X'SX = T, so folgt auch hier aus der rationalen Losbarkeit der Kongruenz X'SX = T (mod q) fiir jedes q die rationale Losbarkeit der Matrizengleichung. Hierzu wird, zunachst fiir definites S, eine quantitative Verscharfung angestrebt, also eine Aussage iiber Losungsanzahl, und diese
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wird in dem Siegelschen Hauptsatz gegeben. Er besagt, daf? die mittlere Losungsanzahl der Gleichung in unmittelbarem Zusammenhang zur mittleren Losungsanzahl der entsprechenden Kongruenzen steht. Die Bedeutung dieses arithmetischen Resultats besteht darin, daE hiervon eine analytische Interpretation gegeben wird. Siegel bemerkt dazu: „Dies ist wieder ein Beispiel dafiir, daf? die Funktionentheorie, der die Arithmetik so machtige Hilfsmittel verdankt, auch ihrerseits durch zahlentheoretische Probleme gefordert werden kann." Die genannte analytische Interpretation lauft auf eine Identitat zwischen Summen von Thetareihen und Eisensteinreihen hinaus. Sie verkniipft damit analytisch verschiedenartige Bildungen miteinander und stellt insofern eine tiefliegende Beziehung von Funktionen mehrerer Veranderlicher zueinander dar. Die Thetareihe hangt nun mit einer Siegelschen Modulform zusammen, so daE Siegels Theorie der definiten quadratischen Formen zu zwei verschiedenen Konstruktionsprinzipien fur Modulformen fiihrt, namlich der Bildung von Eisensteinreihen und Thetareihen. Im indefiniten Fall, den Siegel ebenfalls in mehreren Abhandlungen untersucht hat, sind einige Abanderungen zu treffen, aber auch hier besteht ein Zusammenhang zu Modulformen. Die Untersuchung der Modulfunktionen, die aus den Modulformen durch geeignete Quotientenbildung hervorgehen und eine Verallgemeinerung der elliptischen Modulfunktion einer Variablen sind, sowie der damit zusammenhangenden quadratischen Formen und der Eisensteinreihen war fur Siegel ein Anliegen, das er immer wieder aufgegriffen hat, aber selbst auf die Aufreihung der hierzu gehorenden Titel seiner Veroffentlichungen muE hier verzichtet werden. Ich mochte nur die beiden folgenden Arbeiten herausgreifen. Die „Einfiihrung in die Modulfunktionen n-ten Grades" (1939) im AnschluS an die Abhandlung „Uber die analytische Theorie der quadratischen Formen" enthalt einen systematischen Aufbau von der Modulgruppe n-ten Grades zu den Modulformen n-ten Grades und damit auch zu Modulfunktionen n-ten Grades, die man iibrigens heute Siegelsche Modulfunktionen nennt, und es wird gezeigt, da£ dieselben rational durch Eisensteinreihen ausgedriickt werden konnen. In einer mehr abstrakten, umfangreichen Abhandlung „Symplectic geometry" (1943) wird fur die Theorie der automorphen Funktionen von m Veranderlichen das geometrische und gruppentheoretische Fundament entwickelt. Eine solche Theorie basiert auf einem geeigneten Raum von m komplexen Veranderlichen, der fur m = 1 in die obere Halbebene iibergeht, auf der Theorie der Gruppe der analytischen Funktionen, die diesen Raum invariant lassen, der Theorie der diskontinuierlichen Untergruppen dieser Gruppe, und gipfelt schliefslich in der Diskussion der Fundamentalbereiche dieser Untergruppen.
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Die Siegelschen Modulfunktionen fanden rasch ein internationales Interesse, handelt es sich doch um eine Klasse analytischer Funktionen mehrerer Veranderlicher mit funktionentheoretisch interessanten Eigenschaften. Siegel hat mit seinen Arbeiten zahlreiche Autoren angeregt, die Theorie weiter zu entwickeln, und ein sehr produktives Forschungsgebiet geschaffen. Ferner hat er des ofteren Vorlesungen dariiber gehalten, die zum Teil auch in Buchform erschienen sind. Aber auch die Astronomie, fur die er sich ja im Jahre 1915 immatrikuliert hatte, um sich dann sogleich wieder von ihr abzuwenden, genauer die Himmelsmechanik, fand sein lebenslanges Interesse. In mehreren Vorlesungen und Seminaren, wie auch in einer Monographic, hat er sich hiermit beschaftigt. Ich vermag nicht zu sagen, ob es ein Zufall war, dal? seine erste grofie Arbeit zur Himmelsmechanik, namlich iiber das Dreikorperproblem mit dem Titel „Der Dreierstof?" (1941) zu einem Zeitpunkt erschien, als wieder ein Krieg ausgebrochen war. Allerdings hatte er bereits 1936 „Uber die algebraischen Integrale des restringierten Dreikorperproblems" geschrieben. Im Kriegsjahr 1941 lieS er einen Vortrag „On the modern development of celestial mechanics" und zwei weitere astronomische Arbeiten folgen, namlich „On the integrals of canonical systems" und „Some remarks concerning the stability of analytic mappings". SchlieElich veroffentlichte er 1951 „Ober eine periodische Losung im ebenen Dreikorperproblem" und „Beitrag zum Problem der Stabilitat". Die Titel mogen fiir sich sprechen und einen Eindruck vermitteln. Uber weitaus die meisten der 100 Veroffentlichungen habe ich nicht gesprochen. Es ware zu berichten gewesen von den Arbeiten aus den spateren Jahren zur Zahlentheorie, zu der Funktionstheorie und den quadratischen Formen, von Arbeiten zur Analysis, zu den L-Reihen, der Geometrie der Zahlen und noch manches mehr. Das gewaltige Schaffen Siegels lalSt sich nur durch das Studium seiner Schriften erfassen. Er lebte in seiner Wissenschaft, unterhielt sich gern iiber Mathematik, und interessierte sich bis zuletzt fiir die neuesten Fortschritte. Eine von ihm aufgegriffene Frage lieE ihn nicht ruhen, bis er sie in einer endgiiltigen Form gelost hatte. Unfertige Arbeiten schatzte er nicht. So hat er auch, wie er zuvor immer wieder gesagt hatte, keinen wissenschaftlichen Nachlal? hinterlassen. Es freute ihn, daS ihm im Jahre 1978 der Wolf-Preis verliehen wurde; fiir den Balzan-Preis war er vorgeschlagen worden. Siegels groSe Leistungen sind begriindet in seiner Vielseitigkeit, in den gewaltigen Schritten, mit denen er die Teile, die er behandelt hat, fordern
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konnte, in der Zusammenfiihrung von Arithmetik, Algebra und Funktionentheorie und der meisterhaften Beherrschung des analytischen Apparats. Dabei halfen ihm eine geniale mathematische Intuition, die Fahigkeit, die mathematischen Zusammenhange mit auSergewohnlicher Scharfe zu durchschauen und zu analysieren, eine ausgezeichnete Kenntnis der Literatur und ein exzellentes Gedachtnis. DaS auch eine ungeheuere Arbeitskraft dazu gehorte, sei nur am Rande erwahnt. Er wirkte auf diejenigen, die mit ihm in Beriihrung kamen, durch die Ausstrahlung seines iiberragenden Geistes und seiner Personlichkeit, und doch, oder vielleicht gerade deswegen, blieb er innerlich einsam. Es geniigte ihm, zu wissen, daf> er ein Werk hinterlassen wiirde, aus dem die Nachwelt ermessen konnte, wer er war. T h e o d o r S c h n e i d e r , Freiburg
565 Bibliography (reprinted from C.L. Siegel, Gesammelte Abhandlungen, Vol. IV)
Research Papers 1. Approximation algebraischer Zahlen Jahrbuch der Philosophischen Fakultat Gottingen Teil II. Ausziige aus den Dissertationen der Mathematisch-Naturwissenschaftlichen Abteilung 1921, 291-296 2. Approximation algebraischer Zahlen Mathematische Zeitschrift 10 (1921), 173-213 3. Darstellung total positiver Zahlen durch Quadrate Mathematische Zeitschrift 11 (1921), 246-275 4. Uber Naherungswerte algebraischer Zahlen Mathematische Annalen 84 (1921), 80-99 5. Ueber die Coefncienten in der Taylorschen Entwicklung rationaler Funktionen The Tohoku Mathematical Journal 20 (1921), 26-31 6. Ueber den Thueschen Satz Skrifter utgit av Videnskapsselskapet i Kristiania 1921, I. videnskabelig Klasse, 2. Bind, Nr. 16
Matematisk-Natur-
7. Neuer Beweis fiir die Funktionalgleichung der Dedekindschen Zetafunktion Mathematische Annalen 85 (1922), 123-128 8. Additive Theorie der Zahlkorper I Mathematische Annalen 87 (1922), 1-35 9. Bemerkungen zu einem Satz von Hamburger iiber die Funktionalgleichung der Riemannschen Zetafunktion Mathematische Annalen 86 (1922), 276-279 10. Uber die Diskriminanten total reeller Korper Nachrichten von der K. Gesellschaft der Wissenschaften zu Mathematisch-physikalische Klasse aus dem Jahre 1922, 17-24
Gottingen,
11. Neuer Beweis des Satzes von Minkowski iiber lineare Formen Mathematische Annalen 87 (1922), 36-38 12. Additive Zahlentheorie in Zahlkorpern Jahresbericht der Deutschen Mathematiker-Vereinigung
31 (1922), 22-26
13. Neuer Beweis fiir die Funktionalgleichung der Dedekindschen Zetafunktion II Nachrichten von der K. Gesellschaft der Wissenschaften zu Gottingen, Mathematisch-physikalische Klasse aus dem Jahre 1922, 25-31 14. Additive Theorie der Zahlkorper II Mathematische Annalen 88 (1923), 184-210
566
15. The integer solutions of the equation y2 = axn + bxn x + • • • + k The Journal of the London Mathematical Society 1 (1926), 66-68 16. Uber einige Anwendungen diophantischer Approximationen Abhandlungen der Preuftischen Akademie der Wissenschaften. mathematische Klasse 1929, Nr. 1
Physikalisch-
17. Uber die Perioden elliptischer Funktionen Journal fur die reine und angewandte Mathematik 167 (1932), 62-69 18. Uber Riemanns Nachlafi zur analytischen Zahlentheorie Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik 2 (1932), 45-80 19. Uber Gitterpunkte in convexen Korpern und ein damit zusammenhangendes Extremalproblem Acta Mathematica 65 (1935), 307-323 20. Uber die analytische Theorie der quadratischen Formen Annals of Mathematics 36 (1935), 527-606 21. Uber die Classenzahl quadratischer Zahlkorper Acta Arithmetics 1 (1935), 83-86 22. Uber die analytische Theorie der quadratischen Formen II Annals of Mathematics 37 (1936), 230-263 23. Uber die algebraischen Integrate des restringierten Dreikorperproblems Transactions of the American Mathematical Society 39 (1936), 225-233 24. Mittelwerte arithmetischer Funktionen in Zahlkorpern Transactions of the American Mathematical Society 39 (1936), 219-224 25. The volume of the fundamental domain for some infinite groups Transactions of the American Mathematical Society 39 (1936), 209-218 26. Uber die analytische Theorie der quadratischen Formen III Annals of Mathematics 38 (1937), 212-291 27. Analytische Theorie der quadratischen Formen Comptes Rendus du Congres international des Mathematiciens (Oslo) 1937, 104-110 28. Die Gleichung ax11 — byn = c Mathematische Annalen 114 (1937), 57-68 29. Formes quadratiques et modules des courbes algebriques Bulletin des Sciences Mathematiques, 2. serie 61 (1937), 331-352 30. Uber die Zetafunktionen indefiniter quadratischer Formen Mathematische Zeitschrift 43 (1938), 682-708 31. Uber die Zetafunktionen indefiniter quadratischer Formen II Mathematische Zeitschrift 44 (1939), 398-426 32. Einfiihrung in die Theorie der Modulfunktionen n-ten Grades Mathematische Annalen 116 (1939), 617-657
567 33. Einheiten quadratischer Formen Abhandlungen aus dem Mathematischen Seminar der Hansischen Universitat 13 (1940), 209-239 34. Der Dreierstofi Annals of Mathematics 42 (1941), 127-168 35. On the modern development of celestial mechanics American Mathematical Monthly 48 (1941), 4^0-435 36. Equivalence of quadratic forms American Journal of Mathematics 63 (1941), 658-680 37. On the integrals of canonical systems Annals of Mathematics 42 (1941), 806-822 38. Some remarks concerning the stability of analytic mappings Revista de la Universidad Nacional de Tucumdn A 2 (1941), 151-157 39. Iteration of analytic functions Annals of Mathematics 43 (1942), 607-612 40. Note on automorphic functions of several variables Annals of Mathematics 43 (1942), 613-616 41. Symplectic geometry American Journal of Mathematics 65 (1943), 1-86 42. Contribution to the theory of the Dirichlet i-series and the Epstein zetafunctions Annals of Mathematics 44 (1943), 143-172 43. Discontinuous groups Annals of Mathematics 44 (1943), 674-689 44. Generalization of Waring's problem to algebraic number fields American Journal of Mathematics 66 (1944), 122-136 45. On the theory of indefinite quadratic forms Annals of Mathematics 45 (1944), 577-622 46. Algebraic integers whose conjugates lie in the unit circle Duke Mathematical Journal 11 (1944), 597-602 47. The average measure of quadratic forms with given determinant and signature Annals of Mathematics 45 (1944), 667-685 48. The trace of totally positive and real algebraic integers Annals of Mathematics 46 (1945), 302-312 49. Sums of mth powers of algebraic integers Annals of Mathematics 46 (1945), 313-339 50. A mean value theorem in geometry of numbers Annals of Mathematics 46 (1945), 340-347 51. On the zeros of the Dirichlet L-functions Annals of Mathematics 46 (1945), 409-422
568 52. Note on differential equations on the torus Annals of Mathematics 46 (1945), 423-428 53. Some remarks on discontinuous groups Annals of Mathematics 46 (1945), 708-718 54. En brevveksling om et polynom som er i slekt med Riemanns zetafunksjon Norsk Matematisk Tidsskrift 28 (1946), 65-71 55. Indefinite quadratische Formen und Modulfunktionen Courant Anniversary Volume 1948, 395-406 56. Bemerkung zu einem Satze von Jakob Nielsen Matematisk Tidsskrift, B, 1950, 66-70 57. Uber eine periodische Losung im ebenen Dreikorperproblem Mathematische Nachrichten 4 (1951), 28-35 58. Indefinite quadratische Formen und Funktionentheorie I Mathematische Annalen 124 (1951), 17-54 59. Die Modulgruppe in einer einfachen involutorischen Algebra Festschrift zur Feier des 200jdhrigen Bestehens der Akademie der Wissenschaften in Gottingen, 1951, 157-167 60. Indefinite quadratische Formen und Funktionentheorie II Mathematische Annalen 124 (1952), 364-387 61. Uber die Normalform analytischer Differentialgleichungen in der Nahe einer Gleichgewichtslosung Nachrichten der Akademie der Wissenschaften in Gottingen, Mathematischphysikalische Klasse, 1952, Nr. 5, 21-30 62. A simple proof of rj{—1/r) = 77(r)y / r/i Mathematika 1 (1954), &• 4 63. Uber die Existenz einer Normalform analytischer Hamiltonscher Differentialgleichungen in der Nahe einer Gleichgewichtslosung Mathematische Annalen 128 (1954), 144~170 64. Meromorphe Funktionen auf kompakten analytischen Mannigfaltigkeiten Nachrichten der Akademie der Wissenschaften in Gottingen, Mathematischphysikalische Klasse, 1955, Nr. 4> 71-77 65. Zur Theorie der Modulfunktionen n-ten Grades Communications on Pure and Applied Mathematics 8 (1955), 677-681 66. Die Funktionalgleichungen einiger Dirichletscher Reihen Mathematische Zeitschrift 63 (1956), 363-373 67. A generalization of the Epstein zeta function Journal of the Indian Mathematical Society 20 (1956), 1-10 68. Zur Vorgeschichte des Eulerschen Additionstheorems Sammelband Leonhard Euler, Akademie-Verlag, Berlin 1959, 315-317
569 69. Uber einige Ungleichungen bei Bewegungsgruppen in der nichteuklidischen Ebene Mathematische Annalen 133 (1957), 127-138 70. Integralfreie Variationsrechnung Nachrichten der Akademie der Wissenschaften in Gottingen, physikalische Klasse, 1957, Nr. 4, 81-86
Mathematisch-
71. Vereinfachter Beweis eines Satzes von J. Moser Communications on Pure and Applied Mathematics 10 (1957), 305-309 72. Zur Reduktionstheorie quadratischer Formen Publications of the Mathematical Society of Japan 1959, Nr. 5 73. Zur Bestimmung des Volumens des Fundamentalbereichs der unimodularen Gruppe Mathematische Annalen 137 (1959), 427-432 74. Uber das quadratische Reziprozitatsgesetz in algebraischen Zahlkorpern Nachrichten der Akademie der Wissenschaften in Gottingen, Mathematischphysikalische Klasse, 1960, Nr. 1, 1-16 75. Uber die algebraische Abhangigkeit von Modulfunktionen n-ten Grades Nachrichten der Akademie der Wissenschaften in Gottingen, Mathematischphysikalische Klasse, 1960, Nr. 12, 257-272 76. Bestimmung der elliptischen Modulfunktion durch eine Transformationsgleichung Abhandlungen aus dem Mathematischen Seminar der Universitdt Hamburg 27 (1964), 32-38 77. Moduln Abelscher Funktionen Nachrichten der Akademie der Wissenschaften in Gottingen, physikalische Klasse, 1963, Nr. 25, 365-427
Mathematisch-
78. Zu zwei Bemerkungen Kummers Nachrichten der Akademie der Wissenschaften in Gottingen, physikalische Klasse, 1964, Nr. 6, 51-57
Mathematisch-
79. Uber die Fourierschen Koefnzienten der Eisensteinschen Reihen Det Kongelige Danske Videnskabernes Selskab. Matematisk-fysiske lelser 34 (1964), Nr. 6
Medde-
80. Beweis einer Formel fur die Riemannsche Zetafunktion Mathematica Scandinavica 14 (1964), 193-196 81. Zur Geschichte des Frankfurter Mathematischen Seminars Frankfurter Universitdtsreden 1965, Heft 36, Klostermann-Verlag 82. Faksimile eines Briefes an W.
GROBNER
83. Zu den Beweisen des Vorbereitungssatzes von Weierstrafi Abhandlungen aus Zahlentheorie und Analysis, Zur Erinnerung an Edmund Landau (1877-1938), 299-306, VEB Deutscher Verlag der Wissenschaften, Berlin 1968
570 84. Bernoullische Polynome und quadratische Zahlkorper Nachrichten der Akademie der Wissenschaften in Gottingen, physikalische Klasse, 1968, Nr. 2, 7-38
Mathematisch-
85. Zum Beweise des Starkschen Satzes Inventiones mathematicae 5 (1968), 180-191 86. Uber die Fourierschen KoefRzienten von Eisensteinschen Reihen der Stufe T Mathematische Zeitschrift 105 (1968), 257-266 87. Erinnerungen an Frobenius F. G. Frobenius, Gesammelte Ahhandlungen, Bd. I (1968), Addendum S. IV-VI 88. Abschatzung von Einheiten Nachrichten der Akademie der Wissenschaften in Gottingen, physikalische Klasse, 1969, Nr. 9, 71-86
Mathematisch-
89. Berechnung von Zetafunktionen an ganzzahligen Stellen Nachrichten der Akademie der Wissenschaften in Gottingen, physikalische Klasse, 1969, Nr. 10, 87-102
Mathematisch-
90. Uber die Fourierschen KoefRzienten von Modulformen Nachrichten der Akademie der Wissenschaften in Gottingen, physikalische Klasse, 1970, Nr. 3, 15-56
Mathematisch-
91. Einige Erlauterungen zu Thues Untersuchungen iiber Annaherungswerte algebraischer Zahlen und diophantische Gleichungen Nachrichten der Akademie der Wissenschaften in Gottingen, Mathematischphysikalische Klasse, 1970, Nr. 8, 169-195 92. Algebraische Abhangigkeit von Wurzeln Acta Arithmetica 21 (1972), 59-64 93. Uber Moduln Abelscher Funktionen Nachrichten der Akademie der Wissenschaften in Gottingen, physikalische Klasse, 1971, Nr. 4, 79-96
Mathematisch-
94. Periodische Losungen von Differentialgleichungen Nachrichten der Akademie der Wissenschaften in Gottingen, physikalische Klasse, 1971, Nr. 13, 261-283
Mathematisch-
95. Wurzeln Heckescher Zetafunktionen Nachrichten der Akademie der Wissenschaften in Gottingen, physikalische Klasse, 1972, Nr. 2, 11-20
Mathematisch-
96. Zur Theorie der quadratischen Formen Nachrichten der Akademie der Wissenschaften in Gottingen, physikalische Klasse, 1972, Nr. 3, 21-46
Mathematisch-
97. Normen algebraischer Zahlen Nachrichten der Akademie der Wissenschaften in Gottingen, physikalische Klasse, 1973, Nr. 11, 197-215
Mathematisch-
571 98. Beitrag zum Problem der Stabilitat Nachrichten der Akademie der Wissenschaften in Gottingen, physikalische Klasse, 1974, Nr. 3, 23-58
Mathematisch-
99. Zur Summation von L-Reihen Nachrichten der Akademie der Wissenschaften in Gottingen, physikalische Klasse, 1975, Nr. 18, 269-292
Mathematisch-
Books and Monographs Transcendental Numbers, Ann. of Math. Studies 16, Princeton University Press, 1949 Transzendente Zahlen, Bibl. Inst., Mannheim, 1967 (translation) Symplectic Geometry, Academic Press Inc., 1964 Vorlesungen iiber Himmelsmechanik, Grundlagen d. mathem. Springer, 1956
Wiss., Vol. 85,
Lectures on Celestial Mechanics (with J.K. Moser), Grundlagen d. mathem. Wiss., Vol. 187, Springer, 1971, and Classics in Mathematics, Springer, 1995 Zur Reduktionstheorie quadratischer Formen, Publ. of the Math. Soc. of Japan, No. 5, 1959 Topics in Complex Function Theory, Intersc. Tracts in Pure and Applied Mathematics, No. 25, Wiley-Interscience Vol. I Vol. II Vol. Ill
Elliptic Functions and Uniformization Theory, 1969 Automorphic Functions and Abelian Integrals, 1971 Abelian Functions and Modular Functions of Several Variables, 1973
Gesammelte Abhandlungen, Vol. I-IV, Springer, 1966 and 1979 Lectures on the Geometry of Numbers (rewritten by K. Chandrasekharan and R. Suter), Springer, 1989
Lecture Notes Baltimore, Johns Hopkins University Topics in Celestial Mechanics, 1953 (E.K. Haviland and D.C. Lewis, Jr.) Bombay, Tata Institute of Fundamental Research Lectures on Quadratic Forms, 1957 (K.G. Ramanathan) Lectures on Advanced Analytic Number Theory, 1961 and 1965 (S. Raghavan) Lectures on Riemann Matrices, 1963 (S. Raghavan, S.S. Rangachari) Lectures on the Singularities of the Three-Body Problem, 1967 (K. Balagangadharan, M.K. Venkatesha Murthy)
572 Gottingen, Mathematisches
Institut
Analytische Zahlentheorie, 1951 Himmelsmechanik, 1951/52 (W. Fischer, J. Moser, A. Stohr) Ausgewahlte Fragen der Funktionentheorie, Part I, 1953/54, Part II, 1954 (E. Gottschling) Automorphe Funktionen in mehreren Variablen, H. Klingen)
1954/55 (E.
Gottschling,
Quadratische Formen, 1955 (H. Klingen) Analytische Zahlentheorie, Part I, 1963 (K.F. Kiirten), Part II, 1963/64 (K.F. Kiirten, G. Kohler) Vorlesungen iiber ausgewahlte Kapitel der Funktionentheorie, Parts I-III, 1964-66 New York University Lectures on Analytic Number Theory, 1945 (B. Friedman) Lectures on Geometry of Numbers, 1945/46 (B. Friedman) Princeton, The Institute for Advanced Study and Princeton
University
Lectures on Analytic Theory of Quadratic Forms, 1935 (M. Ward), 2nd edition, 1949, 3rd edition, 1963 (U. Christian), 4th edition, 1995 Analytic Functions of Several Complex Variables, 1948/49 (P.T. Bateman)
Approximation algebraischer Zahlen Jahrbuch der Philosophischen Fakultat Gottingen Teil II Ausziige aus den Dissertationen der Matheraatisch-Naturwissenschaftlichen Abteilung 1921, 291- -296
1) Die Frage nach der Ordnung des Restgliedes bei der Approximation algebraischer Zahlen durch rationale Briiche ist erst in den letzten zwolf'Jahren wesentlich getordert worden. Im Jahre 1908 machte namlich Thue J ) die wichtige Entdeckung, daB bei jeder reellen algebraischen Zahl | vom Grade n > 2 und jedem s > 0 die Ungleichung (1)
-r^—
y
+ i+«
y
(y > o)
nur endlich viele Losungen in ganzen rationalen x, y besitzt. Vor Thue war nur durch Liouville 2 ) die Existenz einer positiven Zahl c = c(|) bekannt, fur welche die Ungleichung V
(y>0)
unlbsbar ist. Ich habe nun gefunden, dafi auf der rechten Seite von (1) der Exponent von y unter die Grofienordnung n herabgedrtickt werden kann; es ist namlich nur fur endlich viele ganze rationale x, y
6-
y
y
1 2\Jn
1) Bemerkungen iiber gewisse Naherungsbriiche algebraischer Zahlen. Uber rationale Annaberungswerte der reellen Wurzel der ganzen Funktion dritten Grades x3 — ax — 6. Om en generel i store hele tal ulasbar ligning. Skrifter udgivne af Videnskabs-Selskabet i Christiania, Jahrgang 1908. Uber Annaherungswerte algebraischer Zahlen. Journal fur die reine und angewandte Mathematik, Bd. 135 (1909), S. 284—305. 2) Sur des classes tres-(;tendues de quantitcs, dont la valeur n'est ni algebrique, ni meme reductible a des irrationnelles algebriques. Journal de Mathematiques pures et appliquees, Ser. 1, Bd. 16 (1851), S. 133—142. Siegel, Gesammelte Abhandlungen I
574
Dies gilt auch noch, wenn der Exponent 2 \Jn durch die Zahl 11
i i + * I + £ m^ festem s > 0 ersetzt wird. Dieser i X=l,...»\* + > Satz ist mit dem Thue'schen nur fiir n < 7 identisch; fiir alle n^.7 ist er scharfer als dieser. 2) Die Beweismethode ist einer Verallgemeinerung fahig, welche gestattet, einen Satz iiber die Approximation einer algebraischen Zahl durch andere algebraische Zahlen (also nicht nur durch rationale Briiche) herzuleiten; es ergibt sich I. Ist g vom Grade cl^.2 in Bezug auf einen algebraischen Zahlkbrper K und wird fiir jede Zahl a aus K die Funktion H{a) („Hohe von «") als das Maximum unter den absoluten Betragen der teilerfremden ganzen Koeffizienten in der im Korper der rationalen Zahlen irreduzibeln Gleichung fiir u erklart, so hat die Ungleichang
|g-g|<_A_ nur endlich viele Losungen in primitiven Zahlen t, aus K. Dies gilt auch noch, wenn der Exponent 2 \jd durch
min
f-m"i + 11 + £
i=l,...dV^ +
/
mit festem s > 0 ersetzt wird. I I . Man beschranke sich bei der Approximation von % nicht auf primitive Zahlen eines festen Korpers, sondern lasse fiir £ beliebige algebraische Zahlen eines festen Grades h < n zu. Dann hat die Ungleichung
|j--g|<
---^
nur endlich viele Losungen; und hierin kann 2 h \Jn noch durch den besseren W e r t
min (T"""""I + ^ r ' + A=l,...wU+1 /
£
ersetzt werden.
3) Die bedeutendste Anwendung, welche Thue') von seiner Abschatzung machte, ist der Beweis seines bekannten Satzes: 1st U{x, y) ein rationalzahliges homogenes irreduzibles Polynom vom Grade r^ 3, so hat die Diophantische Gleichung U(x,y)
= /.•
1) Vergl. die oben genannten Arbeiten.
fur jedes rationale Jc nur endlich viele Losungen in ganzen rationalen x, y '). Dieser Satz, sowie eine naheliegende Verallgemeinerung desselben von Maillet 2 ), lassen sich bei Benutzung meiner unter 2) zitierten Resultate folgendermafien verallgemeinern: Es sei U(x, y) eine homogene binare Form d ten Grades mit einfachen Linearfaktoren. Ihre Koeffizienten mogen einem Korper K0 vom Grade h0 angehoren. I) x und y seien ganzzahlige Variable eines festen Oberkbrpers K von K0, dessen Grad in Bezug auf den Korper der rationalen Zahlen mit h bezeich.net werde. Es sei d>2h(2h — Y) und V(x,y) ein zu U (x, y) teilerfremdes Polynom mit Koeffizienten aus K0, dessen Dimension
min
(- —-_- + ^l
ist. Dann hat die Diophantische Gleichung U(x. //) = V(x, y) nur endlich viele Losungen. II) x und y seien irgend welche ganze algebraische Zablen vom Grade ^ h. Es sei d >- 2 It4 (2 /<" li0 — 1) und V(x, y) ein zu U(x, y) teilerfremdes Polynom mit Koeffizienten aus Kg, dessen Dimension <.d-hl
min (-—° T + A) 1 = 1, ...dh0\* + 1 I
ist. Dann hat die Diophantische Gleichung U(x, y) = V(x, y) nur endlich viele Losungen. Ist also z. B. U(x, y) eine Form vom Grade > 13 mit Koeffizienten aus einem quadratischen Zahlkorper und durchlaufen x, y die ganzen Zahlen dieses Korpers, so stellt U(x, y) nur endlich oft ein und dieselbe Zahl dar. Fiir h — 1 besagt der Satz im wesentlichen: Haben V und V rationale Koeffizienten und ist die Dimension von V nicht grcifier als d — 2 \]d (d > 4), so hat die Gleichung U = V nur endlich viele Losungen in ganzen rationalen x, y. 1) Einen speziellen Fall des Thue'schen Satzes verscharft B. Delaunay in seiner Arbeit: La solution generale de l'e'quation X 3 g + Y3 = 1. Comptes rendus hebdomadaires des seances de l'Academie des Sciences, Paris, Bd. 162 (1916), S. 150—151. 2) Sur un theoreme de M. Axel Thue. Nouvelles Annales de Mathematiques.. Ser. 4, Bd. 16 (1916), S. 338—345. Determination des points entiers des courbes alge'briques unicursales a coefficients entiers. Comptes rendus hebdomadaires des seances de l'Academie des Sciences, Paris, Bd. 168 (1919), S. 217—220.
4) Aus seinem unter 3) geuannten Satze schloB Thue weiter: Sind zwei Linearformen ax + b, cx + d (« 4 0, c =J= 0) mit ganzen rationalen Koeffizienten voneinander verschieden (d. h. ad —be ^ 0), so gehoren nur fur endlich viele ganze rationale x ihre samtlichen Primteiler einem gegebenen endlichen Wertevorrat an; anders ausgedriickt, der groBte Primteiler von (a x-\-h) (c x + d) wird mit x unendlich. Diesen Satz iiber die spezielle Form acx*~ + (adJrb(-)x + bd iibertrug P o l y a ' ) auf alle ganzzahligen quadratischen Polynome px^ + qx + r, deren Diskriminante q2 — ipr =^ 0 ist. Der Fall p = 1, q = 0, r = 1 war schon friiher von Stormer 2 ) erledigt worden. DaB Gi-aufi den Polyaschen Satz zum mindesten fur den Fall p = 1, q = 0, r = Quadratzahl vermutet haben wird, geht aus seinen Tafeln zur Zyklotechnie 3 ) hervor. Mit Benutzung meiner scharferen Abschatzungen kann ich nun die analoge Fragestellung ftir beliebige Polynome einer Variabeln in beliebigen algebraischen Zahlkbrpern beantworten: 1st f(x) — u0x"l-\ + am ein Polynom mit ganzen Koeffizienten ao(=j; 0), ... am eines Korpers K, das mindestens zwei verschiedene Nullstellen h a t ' ) , so gehoren nur fiir endlich viele ganzzahlige x aus K die Primidealteiler von f(x) samtlich einem festen endlichen Vorrat von Idealen an. Also ist z. B. fiir alle hinreichend grofle ganze rationale x mindestens ein Primfaktor von x3 + 2 grofier als 1010. Insbesondere stellt ein Polynom (mit zwei verschiedenen Nullstellen) nur endlich viele Einheiten dar, wenn die Variable die ganzen Zahlen eines Korpers durchlauft. Das zu Beginn dieses Abschnitts zitierte Resultat von Thue besagt in anderer Form: Fiir jedes ganze rationale h 4= 0 strebt der groBte Primteiler von x + 1c gegen 00, wenn x eine Menge pTP"2 • • • IK' («, = 0, 1, 2, . . . , a2 = 0, 1, 2, . . . ; /;,, p2! ...p,. feste Primzahlen) durchlauft. Es laBt sich zeigen, daB sogar der groBte Primteiler, welcher in x + h zu u n g e r a d e r Potenz aufgeht, mit x iiber alle Grrenzen wachst. 5) Mit Hilfe meines unter 4) genannten Satzes laBt sich folgendes beweisen: 1) Zur arithmetischen Untersuchung der Polynome. Mathematische Zeitschrift, Bd. 1 (1918), S. 143—148. 2) Qaelques theoremes sur l'e'quation de Pell x2 — By'1 = + 1 et leurs applications. Skrifter udgivne af Videnskabs-Selskabet i Christiania, Jahrgang 1897. Sur une equation indeterminee. Comptes rendus hebdomadaires des seances de PAcademifc des Sciences, Paris, lid. 127 (1898), S. 752—754. 3) Werke, zweiter Band, herausgegeben von der Kbniglichen Gesellschaft der Wissenschaften in Gottingen, S. 477—496, 1876. 4) Diese Bedingung ist zugleich notwendig.
Es sei f(x)
ein rationalzahhges Polynom, a und b rationale
Zahlen, a =f= b, f(a) + 0.
Die Entwicklung von j ^ ~
(n j> 3)
nach Potenzen von x — b laute (x — a;
„_
0
gv
wo
y'v> ?» ganz rational und teilerfremd sind. Dann wachst der grofite Primfaktor von pv mit v iiber alle Grenzen. Eine Ausnahme bildet nur das Polynom
t\x) = /!^|x|o(-ir!t(3(s+p-^r,j(-^|J'+(^-«)^(^ wo r und s rationale Zahlen und g(x) ein rationalzahliges Polynom bedeuten. Derselbe Satz gilt mutatis mutandis auch fur Zahlkorper. Ferner folgt aus 4) leicht: P(x) Es sei f[x) = -yvy-r- eine rationale rationalzahlige Funktion P(x) und Q(x) relativ prim und nicht konstant oder Potenzen linearer Funktionen. Es durchlaufe x die natiirlichen Zahlen 1, 2, ... a, ..., und man setze')
{P (h)
III = t "^ ~ =
L
Dann wachsen der grofite Primteiler von pn und der grofite Primteiler von qn mit n iiber alle Grenzen. Auch dieser Satz lafit sich auf Zahlkorper ausdehnen. 6) Der Satz aus 4) laBt sich auch nach einer anderen Richtung hin verallgemeinern: Es sei W(x, y) ein Polynom wten Grades 2 ) (n 2^ 1) mit ganzen algebraischen Koeffizienten, das keinen homogenen Faktor (vom Grade > 0) enthalt. Die Koeffizienten von xn und yn seien 4" 0. Dann hat die Diophantische Gleicbung W(x, y) = 0 nur endlich viele Losungen in solchen ganzen Zahlen eines festen Korpers, deren Primteiler samtlich einer gegebenen endlichen Menge von Idealen angehoren. Der Spezialfall W(x,y) = x—y + x (x 4= 0) hiervon liefert den verallgemeinerten Tbue-Polya'schen Satz.
1) Die endlich vielen FaJle Q(n) = 0 sind auszuschliefien. 2) Grad bedeutet hochste in W(x, y) vorkommende Dimension.
Analytische Theorie der quadratischen Formen Comptes Rendus du Congres international des Mathtaiaticiens (Oslo) 1937, S. 104—110
Es ist bekannt, dafe zwischen der Arithmetik und der Theorie der analytischen Funktionen gewisse Analogien bestehen; insbesondere spiegeln sich manche Eigenschaften algebraischer Zahlkorper wider in ahnlichen Aussagen iiber algebraische Funktionen. Die funktionentheoretischen Satze liefern einen Zusammenhang zwischen den lokalen und den integralen Eigenschaften der analytischen Funktionen. Die Entwickelbarkeit in eine Reihe nach Potenzen der Ortsuniformisierenden auf der Riemannschen Flache ist eine lokale Eigenschaft der analytischen Funktion. Ihr entspricht in der Arithmetik der rationalen Zahlen die Entwicklung in eine />-adische Potenzreihe. Will man also in der Zahlentheorie Analogien finden zu den Integralsatzen der Analysis, so wird man danach fragen, inwieweit arithmetische Funktionen im Grofeen, also im Korper der rationalen Zahlen, bestimmt sind durch die entsprechenden arithmetischen Funktionen im Kleinen, also in den />-adischen KOrpern, oder, was auf dasselbe hinauskommt, durch das entsprechende arithmetische Problem modulo q, wobei q eine beliebige naturliche Zahl bedeutet. Einen Satz dieser Art verdankt man Legendre. Es seien a, b, c, d ganze Zahlen. Der Satz von Legendre besagt, dafe die diophantische Gleichung a x2 + b x y + cy 2 = d dann und nur dann in rationalen Zahlen x,y phantische Kongruenz ax* + bxy + cy2^d
losbar ist, wenn die dio-
(mod q)
fur jeden naturlichen Modul q eine rationale Losung besitzt. Es ist trivial, dafe diese Bedingung notwendig ist; die Bedeutung des Satzes liegt darin, dafe die Bedingung zugleich auch hinreichend ist, dafe also aus der lokalen LSsbarkeit die Losbarkeit im Grofeen folgt. Eine schone Verallgemeinerung des Satzes von Legendre hat Hasse gegeben. Sie bezieht sich auf die rationale Darstellbarkeit einer quadratischen Form R von n Variablen durch eine quadratische Form (J von m Variablen, also auf das Problem, Q in R Siegel, Gesammelte Abhandlungen II
durch eine homogene lineare Substitution mit rationalen Koefhzienten zu transformieren. Wir wollen der Einfachheit halber in diesem Vortrage nur den Fall behandeln, dafe die quadratischen Formen Q und R positiv-definit sind. Bedeutet S die Matrix der quadratischen Form Q von m Variabeln, T die Matrix der quadratischen Form R von n Variabeln und X die Matrix der linearen Transformation, welche Q in R uberfuhrt, so besteht die Matrizengleichung (i)
X'SX=T.
Hierin sind 5 und T gegeben, wahrend eine Matrix X mit m Zeilen und n Spalten von rationalen Elementen gesucht wird und X' die Transponierte zu X bedeutet. Es ist wieder trivial, dafe aus der rationalen Losbarkeit von (i) die rationale Losbarkeit der Kongruenz (2)
X'SX=T
(mod q)
fur jeden Modul q folgt; dabei soil eine solche Matrizenkongruenz bedeuten, daf3 entsprechende Elemente rechts und links miteinander kongruent sind. Der wichtige Satz von Hasse lautet nun, dafe umgekehrt aus der rationalen Losbarkeit von (2) fur jedes q auch wieder die rationale Losbarkeit von (ij folgt. Der Spezialfall m = 2, « = 1 ergibt insbesondere den Satz von Legendre, und andere Spezialfalle sind bereits von Smith und Minkowski behandelt worden. Es liegt jetzt nahe, folgende Frage zu stellen: Lafet sich die qualitative Aussage des Hasseschen Satzes zu einer quantitativen verscharfen, also zu einer Aussage iiber Ldsungsanzahl statt Ldsungsexistenz. Diese Fragestellung mufe aber noch modifiziert werden, damit sie zu einer befriedigenden Ldsung fiihren kann. Zunachst ist namlich leicht ersichtlich, dafe aus der Existenz einer einzigen rationalen Losung von (1) zugleich unendlich viele solche Losungen folgen. Um unter diesen eine endliche Anzahl auszuscheiden, wird man nur ganzzahlige LSsungen in Betracht ziehen. Ohne Beschrankung der Allgemeinheit kann noch vorausgesetzt werden, dafe auch die Elemente von 5 und T ganz sind. Es sei nun A(S, T) die Anzahl der Losungen von (1) in ganzen Matrizen X, d. h. Matrizen mit ganzzahligen Elementen, und Aq(S, T) die Anzahl der modulo q inkongruenten ganzen Losungen von (2). Das Problem lautet jetzt: Welcher Zusammenhang besteht zwischen A{S, T) und den Aq{S, 7")? Um zu erkennen, ob dieses Problem losbar ist, stellen wir folgende Betrachtung an. Man nennt zwei quadratische Formen Q und Qx mit den Matrizen S und S x aquivalent oder zur gleichen Klasse gehorig, wenn so-
wohl Q in Qx als auch Qx in Q durch eine lineare Substitution mit ganzen Koeffizienten transformiert werden kann, wenn also die Matrizengleichungen (3)
X'SX=SX,
X1'S1Xl
= S
beide ganze Losungen haben. Es ist klar, dafe sich die Anzahl A{S, T) nicht andert, wenn die quadratische Form mit der Matrix S durch eine aquivalente ersetzt wird. Daher ist A{S, T) eine Klasseninvariante. Nennt man ferner Q und Qx zum gleichen Geschlecht gehorig, wenn anstelle der Gleichungen (3) die entsprechenden Kongruenzen X'S
X=Sx(modq),
XX'SXXX^S
{mod q)
fur jedes q ganzzahlig losbar sind, so sind die Zahlen Aq{S, T) offenbar Geschlechtsinvarianten. Liefee sich nun A(S, T) aus den Aq{S, T) eindeutig berechnen, so ware diese Zahl ebenfalls eine Geschlechtsinvariante. Dies lafet sich nun aber durch ein Beispiel widerlegen. Wie man leicht einsehen kann, gehoren ^ = J C 2 + 55JV2 und Q1=-5xi + 1 Xy1 zum gleichen Geschlecht. Folglich haben bei beliebigem q die Kongruenzen Q= 1 (mod q) und Qx=\ (mod q) die gleiche Losungsanzahl. Andererseits ist aber £>= 1 ganzzahlig losbar und Qx = \ unlosbar. Dieses Beispiel zeigt zugleich, dafe Q und Qx nicht aquivalent sind, dafe also die Klasseneinteilung scharfer ist als die Geschlechtseinteilung. Aus einem Satze von Hermite ergibt sich, dafe jedes Geschlecht aus nur endlich vielen Klassen besteht. Liegen nun im Geschlechte von Q genau h Klassen, so wahle man aus jeder einen Reprasentanten und bilde mit den zugehOrigen Matrizen Sx, • • •, Sh die Anzahlen A(SX, T), • • •, A(Sh, T). Die h analogen Zahlen ^ , ( 5 ^ 7"), • • • •, Aq(Sh, T) haben alle denselben Wert Aq{S, T). Die ursprungliche Frage nach dem Zusammenhang zwischen A(S, T) und Aq(S, T) kann jetzt verniinftiger folgendermafeen gestellt werden: Besteht ein Zusammenhang zwischen den Aq(S, T) und den Zahlen A{SX, T), • • •, A{Sh, T)? Der Hauptsatz der Theorie besagt nun, dafe in der Tat diese Grofeen durch eine sehr einfache Beziehung miteinander verknupft sind. Ehe wir zur Formulierung dieses Hauptsatzes iibergehen, wollen wir noch die mittleren Werte von A,,{S, T) und A(S, T) definieren. In der Kongruenz (2) ist X eine ganzzahlige Matrix mit m Zeilen und n Spalten, also mit mn Elementen. Lafet man nun X samtliche modulo q inkongruenten Matrizen durchlaufen und nicht nur die Losungen jener Kongruenz, so erhalt man insgesamt qmn Matrizen X, da fur jedes Element von X genau q MSglichkeiten bestehen. Jedesmal ist dann X' S X=Y eine ganzzahlige symmetrische Matrix mit n Reihen. Da eine «-reihige symmetrische n(n+\) Matrix nur —
n(n+1)
unabhangige Elemente besitzt, so hat man q
Moglichkeiten fur Y. Daher ist n(n+1) n
2At(S,Y)=
2
2i=q
Y(moAq)
Y{moiq) n ( n + l)
und folglich kann man die Zahl qmn Aq{S, T) bezeichnen.
2
als den mittleren Wert von
Ganz entsprechend wird der mittlere Wert von A(S, T) erklart. Man « ( « + 1) deute die —— unabhangigen Elemente von Y als rechtwinklige cartesische Koordinaten eines Punktes im Raume von
Dimensionen. Vermoge
der Gleichung X'SX=Y wird dann ein beliebiges Gebiet^ dieses Raumes abgebildet auf ein Gebiet x im X-Raume, dessen Koordinaten die tnn Elemente von X sind. Man lasse nun y auf den Punkt T zusammenschrumpfen und bezeichne den Grenzwert des Volumenquotienten \dX Hm -. y^T}dY
= A„ (S, T)
y
als den mittleren Wert von A{S, T). In der Tat gelten fur ein beliebiges Gebiet y die Gleichungen $Ax(S,Y)dY=$dX y
*
ZA{S,Y)
= 2\.
Yiny
Xiax
Man setze noch A(S,S) = E(S); dies ist also die Anzahl der ganzzahligen Transformationen der quadratischen Form mit der Matrix S in sich selbst. Der Hauptsatz besagt dann: Es ist
,,
4(Slt T) A(Sh, T) + + EjSJ "" E(Sh) A„KSX, T) , A„{Sh, T) E(SX) "*" "*" E(Sh)
,. ,!£'.
Aq{S,T) mn-*Jz±H' q
wenn q eine geeignete Folge natiirlicher Zahlen durchlauft, z. B. die Folge 1!, 2!, 3 ! , • • •. Im Falle m<^n+l ist auf der rechten Seite noch der Faktor \ hinzuzufiigen, im Falle m = n aufeerdem noch im Nenner rechts der Faktor 2m(), wo m(q) die Anzahl der Primteiler von q bedeutet. Es sei noch bemerkt, da6 dieser Satz sich sinngemafe auf indefinite quadratische
Formen und auf quadratische Formen in beliebigen algebraischen Zahlkorpern iibertragen lafet. Der Ausdruck auf der rechten Seite von (4) kann als arithmetischer Ersatz fur die Integralbildungen der Funktionentheorie angesehen werden. Dies wird deutlicher, wenn man die rechte Seite von (4) als unendliches Produkt schreibt. Fur teilerfremde q, r ist namlich Aqr{S, T)=Aq{S, T)Ar(S, T); ferner ist fur die Potenzen q=pa jeder festen Primzahl p der Quotient n(n+l)
Aq(S, T): qm" 2 bei hinreichend grofeem a konstant, kann also gleich ap(S,T) gesetzt werden. Da auch die Zahlen Ax (Sx, T), • • •, Ax (Sh, T) alle den Wert Am (S, T) besitzen, so lafet sich die Formel des Hauptsatzes in der Gestalt A(S1,T) (5)
|
,
f (5h)
~ ^ ^ E(S,)^
A(Sh,T)
^
=AK(S,T)IIap(S,T)
E(Sh)
schreiben, wo p alle Primzahlen wachsend durchlauft. Die Faktoren auf der rechten Seite sind erklart durch die Arithmetik im Kleinen, namlich an den einzelnen Primstellen, wahrend sich der Ausdruck auf der linken Seite auf die Arithmetik im Grofeen bezieht. Der Hauptsatz ist seinem Wesen nach transcendenter Natur. Dementsprechend enthalt sein Beweis auch einen transcendenten Teil, namlich die Dirichletsche Methode der Mittelwertbildung. Aufeerdem sind eingehende arithmetische Uberlegungen notig. Da es nicht moglich ist, den Beweis in der noch zur Verfugung stehenden Zeit zu skizzieren, so ziehe ich es vor, nur noch einiges uber die Bedeutung des Satzes zu sagen. Ein Spezialfall der Formel (5), namlich S= T, ist bereits durch Minkowski entdeckt worden. Fur diesen Fall hat namlich der Zahler auf der linken Seite von (5) den Wert 1, und man erhalt einen Ausdruck fur die Grofee
h • • • + _ . „ . , die man seit Eisenstein als Geschlechtsmafe E (Si) E (Sh) bezeichnet. Die so entstehende Minkowskische Formel ist eine Verallgemeinerung der Eisensteinschen Formel fur das Geschlechtsmafe bei definiten ternaren quadratischen Formen und der Dirichletschen Klassenzahlformel bei binaren quadratischen Formen. Ein anderer Spezialfall entsteht, wenn man voraussetzt, dafe die Klassenzahl h des Geschlechtes von 5 den Wert 1 hat. Dann liefert namlich der Hauptsatz offenbar eine Aussage iiber A{S, T) selbst. Dieser Fall liegt z. B. dann vor, wenn S die Einheitsmatrix und m<^8 ist, also Q eine Summe von hochstens 8 Quadraten bedeutet. Nimmt man noch « = 1 , also fur die
Matrix T eine Zahl, so erhalt man aus (5) die Satze von Lagrange, Gauss, Jacobi, Eisenstein und Liouville iiber die Zerlegungen naturlicher Zahlen in Quadrate. Es ist bekannt, dafe Jacobi seinen Satz uber die Anzahl der Zerlegungen von natiirlichen Zahlen in 4 Quadrate aus der Theorie der elliptischen Funktionen entnahm; genauer gesagt, entsteht dieser Satz durch Koefnzientenvergleich aus einer gewissen Identitat zwischen Modulfunktionen. Es ist nun bemerkenswert, dafe auch fiir den Fall eines beliebigen -S und « = 1 der Hauptsatz in eine Beziehung zwischen elliptischen Modulfunktionen tibertragen werden kann. Hat man die dazu notigen Umformungen durchgefuhrt, so erhalt man zugleich einen Ansatz fiir eine funktionentheoretische Formulierung des Hauptsatzes im allgemeinen Fall, namlich fiir beliebiges tt. Auf diesem Wege gewinnt man dann ein iiberraschendes Ergebnis: Der Hauptsatz geht uber in eine sehr einfache Relation zwischen denjenigen n {tt + 1) analytischen Funktionen von Veranderlichen, die zu den 2«-fach periodischen meromorphen Funktionen von n Variablen in derselben Beziehung stehen wie die Modulfunktionen zu den elliptischen Funktionen. Es soil nun zum Schlufi noch diese Relation angegeben werden. Es sei Z eine M-reihige symmetrische Matrix mit komplexen Elementen, und zwar sei der Imaginarteil von Z die Matrix einer positiv-definiten quadratischen Form. Man lasse C alle ganzen Matrizen mit m Zeilen und n Spalten durchlaufen und bilde die unendliche Reihe
2^" 7(C ' SCZ) =/(5,Z), c wobei das Zeichen a die Spur der dahinterstehenden Matrix bedeutet, also die Summe der Diagonalelemente. Ferner setze man noch
ns^z) E(St)
•
+
/{sh,z)_ 1 E(Sh)
'EiS1)'t'
1 E(Sh)
=r(c.7) ^ ' ^
Fafet man nun in f(S,Z) alle Summanden zusammen, fiir welche CSC denselben Wert T hat, so wird in F(S,Z) der Koeffizient von e3lial-TZ:) genau die linke Seite von (5). Der Hauptsatz ergibt dann nach einer langeren Umformung schlie&lich eine Entwicklung der Gestalt m
(6)
F(S,Z)
=
s
£lyKL\KZ+L\"*.
Dabei ist der Koeffizient yKL von Z unabhangig und K,L durchlaufen ein voiles System von Paaren ganzzahliger w-reihiger Matrizen mit folgenden 3 Eigenschaften: 1) das Paar ist symmetrisch, d. h. es ist KL' = LK\
584
2) das Paar ist primitiv, d. h. es gibt keine gebrochene Matrix M, so dafe MK und ML beide ganz sind, 3) zwei verschiedene Paare KX,LX und KitL2 sind stets nicht-assoziiert, d. h. es gibt keine Matrix M, so dafe MKl = Ki und ML1 = Li ist. Nun lafet sich beweisen, dafe eine Partialbruchzerlegung der Form (6) fiir eine feste Funktion hochstens auf eine Art mOglich ist; die Koeffizienten yK L gewinnt man namlich eindeutig aus dem Verhalten der Funktion an ihren singularen Stellen. Auf diese Weise geht der Hauptsatz tiber in die einfache Aussage, dafe die Funktion F(S,Z) iiberhaupt in eine Partialbruchreihe der Form (6) entwickelbar ist. Um die funktionentheoretische Bedeutung von (6) zu verstehen, mufe man sich iiber die anaiytische Natur solcher Partialbruchreihen klar werden. Diese sind aber nun nichts anderes als Verallgemeinerungen der Eisensteinschen Reihen, welche explizite Ausdriicke fiir die elliptischen Modulfunktionen liefern. Die einfachsten unter diesen Reihen, namlich 2\ KZ-\-L\~e, sind bis auf einen trivialen Faktor invariant bei den Substitutionen Z=(AZ1 fur I
+ B){CZ1 + D)~l, welche entstehen, wenn
] eine beliebige 2«-reihige Matrix gesetzt wird, die dem Ubergang
von einer kanonischen Zerschneidung einer Riemannschen Flache vom Geschlecht n zu einer andern zugeordnet ist. Man kann beweisen, dafe sich aus diesen Reihen rational alie meromorphen Funktionen
Meromorphe Funktionen auf kompakten analytischen Mannigfaltigkeiten Naehrichten der Akademie der Wissenschaften in Gottingen. Mathematisch-physikalische Klasse, 1955, Nr. 4, 71—77 Vorgelegt in der Sitzung vom 18. F e b r u a r 1955
1. Es sei $ eine kompakte analytische Mannigfaltigkeit von n komplexen Dimensionen und K der Korper der auf *$ eindeutigen meromorphen Funktionen. Es handelt sich um den Beweis der folgenden beiden Satze. S a t z l : J e w + l Funktionen aus K sind algebraisch abhangig. S a t z 2: Oibt es n algebraisch unabhangige Funktionen flt . . . , / „ aus K, so ist K eine endliche algebraische Erweiterung des Korpers der rationalen Funktionen von f1, . . . , / „ . Ein wesentliches Hilfsmittel beim Beweise wird durch eine Verallgemeinerung des Schwarzschen Lemmas gegeben, die zuerst hergeleitet werden soil. Man fasse n komplexe Variable zk (k = 1, . . . , » ) zu einem Vektor z zusammen und bezeichne mit \z\ das Maximum der n absoluten Betrage \zk\. Fur jeden skalaren Faktor A ist dann offenbar \zX\ = |z| |A|. L e m m a : Es sei cp(z) eine im Gebiete \z\ < r konvergente Potenzreihe der Variabeln zlt ..., zn, welche dort absolut <; M ist. Treten in der Reihe keine Glieder der Ordnungen 0, 1, ..., h— 1 auf, so gilt W{z)\
(|2|
B e w e i s : Fur z = 0 ist die Behauptung trivial; es sei also 0 < \z\ <,r. Bedeutet A eine weitere komplexe Variable, so ist bei festem z die Funktion
im Kreise jA| < r—, regular und verscbiwindet bei X = 0 mindestens von h-ter Ordnung. Im gleichen Kreis ist dann auch die Funktion
A-W)
= xW
regular. Da auf dem Kreisrande die Abschatzung
\XW\=
\^hW{zX)\<Mi^)h
gilt und
2. Dem Beweise der beiden Satze seien einige historische Angaben vorausgescbickt. 1st 9J die additive Gruppe aller komplexen Vektoren z und G eine diskrete Untergruppe mit 2w unabhangigen Elementen cot (I = l,...,2n), so ist *$ = 9J/G kompakt und K der Korper der Abelschen Funktionen mit der Periodengruppe G. Fiir diesen Spezialfall haben die Satze in der geschichtlichen Entwicklung schon seit ungefahr 100 Jahren eine Rolle gespielt. Im Jahre 1860 hat R i e m a n n wahrend eines Besuches bei H e r m i t e zuerst von den Relationen zwischen den wl und dem sog. Thetasatze Mitteilung gemacht. Obwohl er Tiber diese Entdeckung nichts Schriftliches hinterlassen hat, so laBt sich vermuten, daB sie im Zusammenhang mit seiner Losung des Jacobischen Umkehrproblems durch Thetafunktionen entstanden ist, wobei er gerade die entsprechenden Relationen fiir die Perioden der Abelschen Integrale 1. Gattung bewiesen hatte. Dies macht es wahrscheinlich, daB er von einer ahnlichen Uberlegung geleitet war wie dann W e i e r s t r a B 1 bei seinen langjahrigen Bemiihungen um den Beweis der Periodenrelationen. WeierstraB erkannte namlich, daB man bei Benutzung von Satz 2 durch Umkehrung eines Systems von n Abelschen Funktionen zu Abelschen Integralen 1. Gattung kommen kann, deren Periodenrelationen dann das gewiinschte Resultat enthalten. Obwohl WeierstraB durch seine wichtigen Ergebnisse zur Algebra der Potenzreihen die Grundlagen fiir die allgemeine Theorie der meromorphen Funktionen mehrerer Variabeln geschaffen hat, so kam er doch nicht zu einem liiekenlosen Beweise von Satz 2 fiir den vorliegenden Fall. Die Ansatze von WeierstraB 2 wurden vollstandig erst nach seinem Tode veroffentlicht, nachdem die Beweisidee bereits durch eine unabhangig von WeierstraB entstandene Arbeit von P o i n c a r e und Pi c a r d 3 bekannt geworden war. Darin ist aber auch dieselbe Lticke vorhanden, und der gleiche Einwand betrifft die Untersuchung von W i r t i n g e r 4 . Die wesentliche Schwierigkeit fur die exakte Durchfuhrung im Sinne von WeierstraB riihrt davon her, daB man die Umkehrung einer durch n Abelsche Funktionen vermittelten Abbildung zu studieren hat. Im Lehrbuche von O s g o o d 5 wurde ein entsprechender Versuch gemacht, der aber auch nicht vollig befriedigend ausgefallen ist. Erst in neuester Zeit wurde durch T h i m m 6 eine einwandfreie Begriindung der Theorie meromorpher Abbildungen und damit speziell ein Beweis der Satze 1 und 2 auf dem von WeierstraB vorgesehenen Wege gegeben. 3. Im Jahre 1902 veroffentlichte P o i n c a r e 7 den ersten vollstandigen Beweis des Thetasatzes mit Hilfe der Potentialtheorie, und hieraus ergab sich unter Benutzung eines von F r o b e n i u s 8 bewiesenen Resultates zugleich ein Beweis von Satz 2 fiir den Korper der Abelschen Funktionen. Vorher war der Thetasatz bereits auf anderem Wege fiir n = 2 von A p p e l l 9 bewiesen worden, und hieran hat dann neuerdings C o n f o r t o 1 0 seinen allgemeinen Beweis angeschlossen. In seiner Arbeit gibt nun aber Poincare auch einen Ansatz zu einem direkten Beweise von Satz 1, welcher fiir die weitere Entwicklung von Bedeutung wurde. Er bildet aus w + 1 gegebenen Abelschen Funktionen
587 fk(z) (k = 0, ...,n)
das aUgemeine Polynom P vom Gesamtgrade m mit
-{-tit1) unbestimmten Koeffizienten und fordert, daB an einer Regularitatsstelle z = £ die Funktion P nebst alien partiellen Ableitungen der Ordnungen 1, 2, ..., h—-1 verschwindet. Dies ergibt insgesamt
homogene lineare Bedingungen fiir die Koeffizienten von P. Bei gegebenem m werde h so gewahlt, daB die Anzahl b < a und h moglichst groB ist. Es sei eine nicht-triviale Losung der linearen Gleichungen gewahlt. Nunmehr handelt es sich darum, aus dem analytischen Verhalten der fk den SchluB zu ziehen, daB fiir geniigend groBes m aus dem Verschwinden A-ter Ordnung an der einen Stelle £ das identische Verschwinden von P als Funktion von z folgt. Dafiir benotigt allerdings Poincare wieder Eigenschaften meromorpher Abbildungen, die nicht mit geniigender Strenge begriindet werden. Der Poincaresche Ansatz wurde vom Verfasser11-12 zur Untersuchung der homogenen algebraischen Abhangigkeit von gewissen automorphen Formen herangezogen. Hierbei treten geringere Schwierigkeiten auf, da die betrachteten Funktionen keine Polstellen haben, und der wesentliche SchluB wird durch Anwendung des oben formulierten Lemmas ermoglicht. Diese Methode wurde von H e r v e 1 3 weiter ausgebaut, indem statt der einen Interpolationsstelle £ beliebig viele benutzt werden. Das hat den Vorteil, daB die urspriinglich im groBen durchgefiihrte SchluBweise nur noch lokal anzuwenden ist. Eine weitere Verallgemeinerung verdankt man B o c h n e r 1 4 und M a r t i n 1 5 . Auch in der wichtigen Arbeit von Chow 1 6 tritt der Poincaresche Ansatz entscheidend auf. SchlieBlich erkannte Serre 1 7 , daB auf der so gewonnenen Basis auch die Satze 1 und 2 iiber meromorphe Funktionen sehr einfach zu beweisen sind, wenn man fiir Satz 2 ein bekanntes Resultat von Cousin 1 8 heranzieht. I n der vorliegenden Arbeit wird endlich noch dieses Hilfsmittel ehminiert und durch eine naheliegende algebraische Uberlegung ersetzt. Dabei wird Satz 1 benotigt, dessen Beweis nach dem Serreschen Vorbild dargestellt wird. Der vollstandige Beweis der beiden Satze ist nunmehr so kurz und elementar geworden, daB er bequem in einer Kollegstunde vorgetragen werden kann, wenn die Eindeutigkeit der lokalen Primfaktorzerlegung analytischer Funktionen als bekannt vorausgesetzt wird. 4. Der variable Punkt von ^S sei mit } bezeichnet. Fiir die Umgebung jedes Punktes a auf $ hat man n komplexe Ortskoordinaten zx, ...,zn von J, welche ein Gebiet \z\ < r auf eine abgeschlossene Umgebung S„ von o topologisch abbilden, wobei der Punkt z = 0 in a ubergeht. Ist der Durchschnitt zweier Umgebungen ^ n und $ 6 nicht leer, so sind dort lokal die beiden Koordinaten-
systeme za, zb durch eine analytische Transformation miteinander verkniipft. Eine auf % meromorphe Funktion /(g) hat in der Umgebung jedes Punktes a eine Quotientendarstellung
/(3)=
(!)
(jeff ;
^S
»)
dabei sind pa, qa Potenzreihen in zlt ..., z„, welche fiir \z\
(2)
<7„(z„) = jai(b)qa(za)
d e ft. ^ « s ),
wo die analytische Funktion jai($) lokal eine Einheit ist. Jetzt seien n -f- 1 Funktionen / und fk (k = 1, ..., w) des Korpers -ST gegeben. Auch fiir /*(j) gilt eine lokale Darstellung (1), worin zur Unterscheidung Pka> Qka staAt pa! qa und in (2) entsprechend j k a i gesetzt werde. Es sei femer S a die durch \z\ < Q — e _1 r definierte Umgebung von a, und es seien endlich viele Punkte a = alt ..., am nach dem Uberdeckungssatz so bestimmt, daB ganz ?j5 durch die entsprechenden Umgebungen £ a bedeckt wird. Man wahle zwei positive Zahlen fi, a>, so daB die Abschatzungen (3)
\j<*(i)\ < *"> I J 7 i f c * ( i ) l < e "
(aefa^£6;
a,b = a 1 , . . . , c t j
*= 1
gelten, und fuhre die natiirhche Zahl (4)
« == [conm] + 1
ein. Ferner sei t = th fiir h — 1,2, ... die groBte nicht-negative ganze Zahl, welche der Bedingung st" < mhn
(5) geniigt. Es folgt (6)
mhn < s(t + 1)" < (s + 1)(< + 1)»,
also tk -> oo fiir h -> oo. Man kann daher h so groB bestimmen, daB mit (4) sogar
gilt, und dann ist zufolge (5) auch (7)
fis + a>t < h.
5. Man bilde jetzt mit unbestimmten komplexen Koeffizienten das allgemeine Polynom F(x, xx, ..., xn) in n + 1 Variabeln x und xk (k = 1, ..., n), das in x den Grad s und in alien xk den Grad t besitzt. Die Anzahl der Koeffizienten ist (8) A = (s+l)(t + 1)«. Setzt man (9)
Qa = qsaf[qia,
Pa =
QaF{Uh,...,U),
4=1
so ist die Funktion Pa = Pa (z) auf \z\ <, ra regular. Man fordere, dafi bei z = 0 samtliche partiellen Ableitungen dieser Funktion von den Ordnungen 0, 1, ..., h— 1 verschwinden, und zwar soil dies fiir jedes a = a l s ..., <Xm erfullt sein. Da bei jedem a die Anzahl der zu annullierenden Ableitungen den Wert
hat und jedesmal fiir die unbestimmten Koeffizienten von F eine homogene lineare Gleichung entsteht, so erhalt man insgesamt (10)
B = mb< mh"
homogene lineare Gleichungen mit A Unbekannten. Zufolge (6), (8), (10) ist B < A, so dafi eine nicht-triviale Losung existiert. Es sei M das Maximum aller absoluten Betrage |P a (z)| fiir \z\
Jtf> — Jai llJkai *=1
nach (2), (9) die Beziehung Pi (zs) = Jot (h) Pa ( O erfullt und nach (3) die Abschatzung \Ja*W\<e"s+mt
( j e t , n tb) (jeft e ^ft,).
Es folgt M = | P 6 (z*)\ ^
Me"s+at-h,
wobei der Exponent nach (7) negativ ist, also M <, 0, M = 0, und damit das identische Verschwinden der Funktionen Pa(z) undF(f, flt ..., /„). Dies ergibt Satz 1.
6. Die gefundene algebraische Gleichung F(f,f1,...,fn) =0 hat in / hochstens den Grad s. Um Satz 2 zu beweisen, geniigt es, zu zeigen, daB die in (4) auftretenden Zahlen co, m von / unabhangig gewahlt werden konnen, falls zwischen f1, . . . , / „ keine algebraische Gleichung besteht. Nach dem bereits Bewiesenen gilt auf $ eine Gleichung (11) Gof + G i f - 1 + - - - + ( ? , = 0, worin G0,G1, ...,(?„ Polynome iaf1,...,fn sind und G0 nicht identisch auf $ verschwindet. Es moge die algebraische Gleichung (11) in bezug auf jede der einzelnen Funktionen /,. (k = 1, ..., n) vom Grade < g sein. Man betrachte jetzt die Quotientendarstellung (1) nur fiir die n Funktionen fk und lege dann die Umgebungen S 0 ohne Rucksicht auf / fest. Indem man ra, ,fa in dieser neuen Bedeutung verwendet und zur Abkiirzung (12)
IJql
=
Ra(z),
i =l
(13) (14)
Ri&^G^H^iz)
(l =
l,...,v),
RaG0f = Sa(z)
setzt, bekommt man aus (11) durch Multiplikation mit dem Faktor -fi^G^1 die Gleichung (15)
S-a + £H<,S?
= 0
(5eS„).
Nach (12), (13) sind die Produkte BaGt(l = 0, ..., v) und die Funktionen H^ (I = 1, . . . , v) fiir \z\
pa=Sa,qa=BaG0,
so gilt fiir /(j) die Zerlegung (1), wobei also die Umgebung ft„ nicht von / abhangt. Allerdings laBt sich nunmehi' nicht behaupten, daB pa, qa teilerfremd sind. Trotzdem bleibt die Beziehung (2) erhalten; man hat namlich dazu wegen (12), (16) nur n
Jab
=
11 Jiah
zu setzen. In dem fruher fiir Satz 1 durchgefuhrten Beweis konnen also jetzt die Umgebungen S£0, £ 0 und die Punkte a = alt ..., am von / unabhangig gewahlt werden, und aus (3) ersieht man das gleiche fiir die GroBe co. Damit ist gezeigt, daB / im Korper der rationalen Funktionen von j 1 , . . . , / „ auch einer irreduziblen Gleichung von beschranktem Grade v geniigt. Wird dann / = /„ so bestimmt, daB v moglichst groB ist, so liefert die Adjunktion von /„ den gesamten Korper K. Damit ist auch Satz 2 bewiesen.
Literatur 1
K. W e i e r s t r a B , Untersuchungen iiber die 2r-fach periodischen Functionen von r Veranderlichen. J. reine angew. Math. 89. 1880, 1—8. 2 K. W e i e r s t r a B , Allgemeine Untersuchungen iiber 2n-fach periodische Funktionen von n Veranderlichen. Math. Werke III, Berlin 1903, 53—114. 3 H. Poincare^ und E. P i c a r d , Sur un theoreme de Riemann relatif aux fonctions de n variables independantes admettant 2n systemes de periodes. C. r. Acad. Sci., Paris 97. 1883, 1284—1287. 4 W. W i r t i n g e r , Zur Theorie der 2n-fach periodischen Functionen (1. Abhandlung). Monatsh. Math. Phys. 6. 1895, 69—98. 5 W. F. Osgood, Lehrbuch der Funktionentheorie 112, Leipzig und Berlin 1932. ' W. T h i m m , Meromorphe Abbildungen von Riemannschen Bereichen. Math. Z. 60. 1954, 435—457. 7 H. P o i n c a r e , Sur les fonctions abeliennes. Acta math. 26. 1902, 43—98. 8 G. F r o b e n i u s , Grundlagen der Theorie der Jacobischen Functionen. J. reine angew. Math. 97. 1884, 16—48. ' P. A p p e l l , Sur les fonctions periodiques de deux variables. J. math, pures appl. (4) 7. 1891, 157—219. 10 F. C o n f o r t o , Funzioni abeliani e matrici di Riemann I, Roma 1942. 11 C. L. Siegel, Emfiihrung in die Theorie der Modulfunktionen n-ten Grades. Math. Ann. 116. 1939, 617—657. 12 C. L. Siegel, Note on automorphic functions of several variables. Ann. of Math. 43. 1942, 613—616. 13 M. H e r v e , Sur les fonctions automorphes de n variables complexes. C. r. Acad. Sci., Paris 226. 1948, 462—464, 11 S. B o c h n e r , Algebraic and linear dependence of automorphic functions in several variables. J. Indian Math. Soc. 16. 1952, 1—6. 15 S. B o c h n e r und W. T. M a r t i n , Complex spaces with singularities. Ann. of Math. 67. 1953, 490—516. 14 W. L. Chow, On compact complex analytic varieties. Amer. J. Math. 71. 1949, 893—914. 17 J- P. S e r r e , Fonctions automorphes. S6minaire Ecole norm, sup., autographierte Vortrage II, Paris 1953/54. 18 P. Cousin, Sur les fonctions de n variables complexes. Acta math. 19. 1895, 1—61.
Iteration of analytic functions Annals of Mathematics 43 (1942), 607—612
Let (1)
f{z) = JLakzk k-l
be a power series without constant term and denote by R > 0 its radius of convergence. The fixed point z = 0 of the mapping z —»/(z) is called stable, if there exist two positive finite numbers r0 ^ R and r ^ R, such that for all points z of the circle \z\ < r0 the set of image points zx = f(z), z n+ i = /(z n ) (n = 1, 2, • • • ) lies in the circle \z\ < r. It is easy to prove the stability in the case | ai | < 1, for then a positive number r0 < R exists, such that the inequality | f(z) | ±s | z | holds for | z | < r0 , and r = ro has the required property. Henceforth, the inequality | ai | ^ 1 is assumed. If z = 0 is stable, then the images zn(n = 1,2, • • •) of the points z of the circle \ z \ < ra under the mapping z —>/(z) and its iterations cover a domain D which is connected and contains the point z = 0. For all z in D, the image point f{z) again lies in D. Let
(2)
* = *>(r) = r + I > r 4 fc=2
be the power series mapping a certain circle | f | < p of the f plane conformally onto the universal covering surface of D. Then the formula v(f) = z —>/(z) = zi =
*»(axf) = /(?(r))
has a convergent solution p(f) = J- + • • • • On the other hand, it is obvious that z = 0 is stable, if | ai | = 1 and the functional equation (3) has a convergent solution. If ai is an n th root of unity, then z = 0 is stable, if and only if the (n — l) t h iteration of the mapping z —* f(z) is the identity. This is also easily proved by direct calculation. We assume now that | ai | = 1 and ai* ^ 1 for n = 1, 2, • • • .
By (1), (2) and (3), 00
CO
/
00
\
J
2>*(af- a,)f* = X > i K + Ec r f r
(4)
t=2
i-2
r=2
\
; /
hence Ck(k = 2, 3, • • •) is a polynomial in c 2 , • • • , cfc_i whose coefficients depend upon a,\, • • • , o A , and there exists exactly one formal (convergent or divergent) solution tp(£) = f + • • • of (3). The first example of a convergent series /(z) = aiz + • • • with divergent Schroder series
These ai are very closely approximated by certain roots of unity, and their linear Lebesgue measure on the unit circle | Oi | = 1 is 0. Until now, however, it was not known if there exists a number ai of absolute value 1, such that every convergent power series /(z) = a\Z + • • • has a convergent Schroder series. Julia3 tried to prove the erroneous hypothesis that the Schroder series is always divergent, if /(z) — atz is a rational function and not identically 0. We shall demonstrate the following THEOEEM :
Let
(5)
log | a? - 1 | = O(logn)
(n — « ) ;
then the Schroder series is convergent. Write a\ — e "*"; then the condition (5) may be expressed in the form TO !
-
_„
- I > Xn ",
n |
for arbitrary integers m and n, n ^ 1, where X and n denote positive numbers depending only upon co. I t is easily seen that (5) holds for all points of the unit circle | ai | = 1 with the exception of a set of measure 0. LEMMA 1: Let xp (p = 1, • • • , r) and yq (q = 1, • • • , s) be positive integers, r ^ 0, s ^ 2, r + s = t,
Z x, + Z y* = K P=l
Z 2/5 > 2o> '
9-1
4-1
y* = = 2o (? = "«
!. • • • > 5 );
t/ien
(6)
IlspIIj/Ufc'S1p-l
8
-l
' G . A. Pfeiffer, On the conformal mapping of curvilinear angles. The functional equation
PROOF: Denote the left-hand side of (6) by L and consider first the case k < It - 2. Then
k~"L ^ k~3 > (2t - 2)~3.
(7)
Assume now k ^ It — 2 and let
EH '+g*-* Then t^g
+ l^g
+ l + r^r,^k,
£
xP = fc - TJ + r,
whence '
«
*-i
S-i
fij-i+1,
if77 ^ ^ — 1 + (
((„ -
xin^g-l
9
- t + 2)g,
+ t
In the interval g + 1 S JJ ^ gr — 1 + £, (fc - v + l)d» ~ < + I) 2 ^ min {(A; - g)(g - t + 2)\ (k - g - t + 2)g2}; in the interval g — I -\- t ^ rj ^ k, (k -
V
+ 1)(„ - g - t
+ 2)V ^ (* -
9
- t + 2)g2;
in the interval 0 ^ £ ^ g, (k - g){g - i f - { k - g -
$g2 = {(k - g)t - (2k - 3g)gU ^ g(2g -
k)Z£0;
consequently g)(g - t + 2)2
L*(k-
(8)
Af3L >
k
~ 9 (9 ~~ lk+ 2 Y ^ 1(2« - 2)- 2 ^ ( a - 2)" 3 .
Now f - U
2' - 2
(i = 2, 3, •
• • • ) ,
and the lemma follows from (7) and (8). We use the abbreviation €» = | a? - 1 I"1
(n = 1, 2, • • •)•
On account of (5), the inequalities c < (2n)'
(» = 1,2, •••)
are fulfilled for a certain constant positive value v. We define W = 2 2 ' +1 ,
Ni = 8Wi = 2 6 ' +1 .
LEMMA 2: Let mt (I = 0, - • • , r) be integral, r ^ 0 and wio > wu >
••• >
m,
> 0; then
I I c , < A^+1 <»*> I I (jm-i - mi)
(9)
i-o
t
i-i
PROOF: The assertion is true in the case r = 0; assume r > 0 and apply induction. We have the identity
a\{al^ - 1) = (a? - 1) - (a? - 1)
(0 < q < p),
whence -l ^p—q
< =
-i ^p
I -i ~T" Cg
min (e, , €,) ^ 2ep_g < 2"+1(p - g)'. This simple remark is the main argument of the whole proof. Let emi {I = 0, • • • , r) have its minimum value for I = h. Then eOT„ < 2"+1 min {(mh-i — mh)", (mh — m* + i)'},
(10)
if we define moreover m_i = oo and mr+i = — °°. On the other hand, the lemma being true for r — 1 instead of r, we have
(ID
^ n«., < JVI / ^ - " ^
n
(WM _ mi)
Since mA_i — wu+i _ (nth-i — mh)(mh — «i+i)
m
1 * - i — w>>
1 < »U — ?WA+I
min (»u_i — WIA,m& — mh+\)'
the inequality (9) follows from (10) and (11). Consider now the sequence of positive numbers 5i = 1, fa, fa , • • • recurrently defined in the following way: For every k > 1, let ju* denote the maximum of all products 5j, Sh ••• Str with 1% + h + • • • + lr = k > h ^ k ^ ••• ^ I, ^ 1, 2 ^ r ^ k% and put (12)
fa
= «*_i/i* .
LEMMA 3:
h g k^'Nf1
(13)
(* = 1, 2, • • •)•
PROOF: The assertion is true in the case k = 1; assume A; > 1 and apply induction. The numbers a* = k~2'N\~l satisfy the inequalities
£*£? = ( r 1 + r 1 ) W S ^'JVT1 < 1
(Jtji.UD,
dk+l
and consequently (14)
5*3,-, • • • Sif g j - 2 ' i V r
(1 ^ ji + • • • + j , = j < k; f 2; 1).
By (12), there exists a decomposition h = tk-iK^H •••«»«
(ffi + • • • + 9a = k > gi ^ • • • ^ ga ^ 1).
In the case <jfi > fc/2, we use this formula with gi instead of k and find a decomposition 5„ = etl-iShlht
•••«»,
(h + • • • + hf, = gi > h S • • • ^ hp ^ 1);
if also /ti > k/2, we decompose again Shl = «»!_!«,•,«;, • • • 8iy
(t'i + • • • + i 7 = h > ii £ • • • ^ z'T ^ 1)}
and so on. Writing k0 = k, ki = gx, k2 = hi, • • • , we obtain in this manner the formula r
8* = n («*,-iA,,) p-0
with fc = A;0 > fci > • • • > kr > k/2, where A„ denotes for p = 0, • • • , r a certain product 5it • • • 5,-; and (kp - kp+i
(p = 0, •••, r - 1)
Ji + • • • + 3, = I
*'
(P = »•),
all subscripts j \ , • • • , j / being ^ fc/2. The number / depends upon p; let / = s for p = r. Using (13) for the s single factors of Ar and applying (14) for the estimation of Ap (p = 0, • • • , r — 1), we find the inequality
n AP ^ Nkr~° ( n jt n ( ^ - *„))"*, P-0
p_l
1^=1
J
where 1 ^ jq ^ fc/2 (g = 1, • • • , s) and ji + • • • + j , ~ k,.
By Lemma 2,
n *»,_! < N\+I Ik n (fcp-i - fcp) p-0
and consequently
^
+1
k
p=l
i
V Sk
with t — r + s, xp = fcp_i — kp, yq = j q .
N\-kk"h
< N[+1N\-'8'{i-l)
By Lemma 1,
* ( ^ 1 ) " 1 = 1,
and (13) is proved. PROOF OF THE THEOREM: Since the power series (1) has a positive radius of convergence, there exists a positive number a, such that | an | ^ a" -1 (n = 2, 3, • • •)• The functional equation (3) remains true under the transformation f{z) —> af(z/a), v(f) —> a
Instead of (4), we consider the functional equation
(15)
2>*T*r* = E r + Z ^ r , k-2
1=2 \
r-2
/
where 772, m , • • • are positive parameters. Then the coefficients 71 = 1, 72, 73, • • • are uniquely determined by the formula fk = ihl 2
(16)
yiji2
••• yir
{k = 2, 3, • • •),
where Zi, • • • , I, run over all positive integral solutions of h + • • • + Zr = fc (r = 2, • • • , fc). Writey k = <jk in the case t\k = «i~ii (& = 2,3, • • •), a n d 7k = T* in the case ij* = 1. The inequality (17)
£Ti ^
5*7*
is true for fc = 1. Applying induction, we infer from (12) and (16) that ok ^
tk-ink ] C r ' i T ' 2 " ' "
T
'r
=
f>kTk ;
hence (17) holds for all values of fc. On the other hand, the power series k—\
satisfies the equation
* - f = (1 - *)-y, whence 4^=l + f-(l-6f
+ f2)*;
consequently ^ converges in the circle | f | < 3 — 2 \ / 2 . By (4), (15) and (17), \ck\
^ hrk
(fc = 2 , 3 , •••)•
It follows now from Lemma 3, that the Schroder series p(f) converges in the circle | f | < (3 - 2v / 2)2~"" 1 . I N S T I T U T E FOR ADVANCED STUDY
Wolf Prize in Mathematics, Vol. 2 (pp. 599-646) eds. S. S. Chern and F. Hirzebruch © 2001 World Scientific Publishing Co. CURRICULUM VITAE
Birthdate:
September 21, 1935
Birthplace:
Moscow, USSR
EDUCATION
B.S.
Moscow State University, 1957
Ph.D.
Moscow State University, 1960
Doctor Degree
Moscow State University, 1963
POSITIONS
1960-1971
Scientific Researcher, Laboratory of Probabilistic & Statistical Methods, Moscow State University
1971-Present
Senior Researcher, Landau Institute of Theoretical Physics, Academy of Sciences
1971-1993
Professor, Moscow State University (part of position)
1993-Present
Professor of Mathematics Department, Princeton University
LECTURES
1962
Invited Speaker, International Congress of Mathematics, Stockholm
1970
Invited Speaker, International Congress of Mathematics, Nice
1978
Loeb Lecturer, Harvard University
1981-1986
Plenary Speaker, International Congresses on Mathematical Physics, Berlin (1981), Marseilles (1986)
1989
Distinguished Lecturer, Israel
1990
S. Lefshetz Lectures, Mexico
1990
Plenary Speaker, International Congress of Mathematics, Kyoto
1993
Landau Lectures, Hebrew University of Jerusalem
1983
Foreign Honorary Member, American Academy of Arts & Science
1986
Boltzman Gold Medal
1989
Heineman Prize
1990
Markov Prize
1991
Member of Russian Academy of Sciences
1992
Honorary Member, London Mathematics Society
1992
Dirac Medal from ICPT in Trieste
1993
Doctor Honoris Causa of Warsaw University
1993
Foreign Member of Hungarian Academy of Sciences
1997
Wolf Foundation Prize in Mathematics
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70. (with S. Burkov) Phase diagrams of the one-dimensional lattice models with longrange antiferromagnetic interaction, Uspehi, 38 ((1983), N4), 205-225. 71. (with S. Burkov) Rotation numbers that do not relate to KAM theory, Funct. Anal. Priloz, 18 ((1984), N4), 73-74. 72. (with N. Khimchenko) On the description of classical reflectionless potentials, Rep. Math. Phys., 20 ((1984), N l ) , 53-63. 73. (with E. B. Vul and K. M. Khanin) Feigenbaum universality formalism, Uspehi, 39 ((1984), N3), 3-37. 74. (with E. Dinaburg) An analysis of ANNI-model Math. Phys., 98 ((1985), N l ) , 119-144.
and
thermodynamic
by Peierls contour method, Comm.
75. An answer to a question by J. Milnor, Comm. Math. Helv., 60 ((1985), N3), 173-178. 76. Lectures in Probability Theory, Part I, II, MGU-press, 1985, pp. 128, 106. 77. (with E. Presutti and M. Soloveichik) Hyperbolity and MUer-morphism for a model of classical statistical mechanics, Prog, in Physics, 10 (1985), Birkhuser. 78. (with C. Boldrighini, A. Pellegrinotti, E. Presuitti, and M. Soloveichik) Ergodic properties of semi-infinite one-dimensional system of statistical mechanics, Comm. Math. Phys., 101 ((1985), N3), 363-382.
79. Structure of the spectrum of a difference Schroedinger operator, Func. Anal, and App., 19 ((1985), Nl), 42-48. 80. (with M. Soloveichik) One-dimensional classical massive particle in the ideal gas, Comm. Math. Phys., 104 ((1986), N3), 423-443. 81. (with K. Khanin) A new proof of Herman theorem, Comm. Math. Phys., 112 ((1987), N l ) , 89-101. 82. Anderson localization for one-dimensional difference Schroedinger operator with quasi-periodic potential, Jour, of Stat. Phys., 46 ((1987), N5), 861-909. 83. (with K. M. Efimov) Hydrodynamic modes for a Lorentz gas with a periodic configuration of scatters, Some Problems in Modern Analysis, MGU (1984). 84. (with N. I. Chernov) Ergodic properties of some systems of two-dimensional and three-dimensional balls, Uspehi, 42 ((1987), N3), 153-174.
disks
85. (with E. Dinaburg) Contour models with interaction and their applications, Selecta Math. Soviet, 7 ((1988), 3), 291-315. 86. (with Ya. B. Pesin), On stable manifolds for a class of two-dimensional diffeomorphisms, Topology and Geometry-Rohlin Seminar, Lecture Notes in Mathematics, Springer-Verlag, 1346 (1988). 87. (with A.B. Soshnikov), Refinement of Wigner semi-circle law near the boundary of the spectrum, Functional Analysis and Applications, (1998). 88. (with L. Bunimovich) Spacetime chaos of coupled map lattices. Non-linearity, ((1988), N4), 491-516.
1
89. (with V. A. Chulaevski) Anderson localization for the I-discrete Schroedinger operator with two-frequency potential, CMP, 125 ((1989), N l ) , 91-112. 90. (with K. M. Khanin) Smoothness of conjugacies of diffeomorphisms with rotations, Uspehi, 44 ((1989), N l ) , 57-82.
of the circle
91. (with C. Series) Ising models on The Lobachevsky plane, Comm. Math. Phys., 128 ((1990), N l ) , 63-76. 92. Kolmogorovs works on ergodic theory, Ann. Prob., 17 ((1989), N3), 833-839. 93. (with Ya. Pesin) Space-time chaos in the system of weakly interactive systems, J. Geom. Phys., 5 ((1988), N3), 483-492.
hyperbolic
607 94. (with L. Bunimovich, N. Chernov) Markov partitions for the two-dimensional perbolic billiards, Uspehi, 45 ((1990), N3), 97-134.
hy-
95. Hyperbolic billiards, Proceedings of the International Congress of Math., SpringerVerlag, 1991. 96. Two results concerning asymptotic behavior of solutions of the Burgers with force, Journal of Statistical Physics, 64 (1991 (N 1/2)).
equations
97. (with L. Bunimovich and N. I. Chernov) Statistical properties of two-dimensional hyperbolic billiards, Russian Math. Surveys, 46 ((1992) N4), 43-92. 98. Finite-dimensional
randomness, Russian Math. Surveys, 46 (1991 (43)), 147-159.
99. Poisson distribution in a geometrical problem, Advances in Soviet Mathematics, 3 1991. 100. (with Ya. B. Pesin) Space-time chaos in systems of weakly interacting maps, Advances in Soviet Mathematics, 3 (1991).
hyperbolic
101. (with V. A. Chulaevski) Anderson localizaiton for the Schroedinger operators with quasiperiodic potentials, Reviews in Mathematical Physics, 3 (1991 (N3)), 241-285. 102. (with S. K. Nechaev) Asymptotical properties of random walks with topological constraints, Theory of Probability and Applications, 38 (1993 (N2)), 331-345. 103. Distribution of some functionals of the integral of the Brownian motion, Theo. and Math. Physics, 90 (1992 (N3)), 323-353. 104. Topics in Ergodic Theory, Princeton University Press, 1994. 105. Random Walk with Random Potential, Prob. Theory and Appl., 38 (1992 (N2)), 457-459. 106. (with K. M. Khanin) Mixing of some clases of special flows over rotation, Functional analysis and applications, 26 (1992 (N3)), 1-21. 107. (with D. V.Kosygin and A. A. Minasov) Spectra of Laplace-Beltrami Liouville surfaces, Russ. Math. Surveys, 48 (1993 (N4)), 1-126.
Operators on
108. (with N. I. Chernov, G. Eyink, and J. Lebowitz) Steady-state electrical conduction in the periodic Lorentz gas, Comm. Math. Phys., 154 (1993), 569-601. 109. Probabilistic Approach to Statistics Vol. 28, No. 2., (1994), 108-113.
of Convex Polygons, Funct. Anal, and Appl.,
608 110. (with P. Bleher and D. Kosygin), Distribution of Energy Levels of a Quantum Free Particle on a Liouville Surface and Trace Formulae, Comm. in Math. Physics, 170, No. 2, (1995), 375-403. 111. A remark concerning random walks with random potentials, Fundamenta Mathematica, 147, No. 2, (1995), 173-180. 112. (with K. Khanin, J. Lebowitz, A. Mazel), Self-avoiding random walks in five or more dimensions: an approach using polymer expansions, Russian Math. Surveys, 50, No. 2, (1995), 403-434. 113. (with E. Weinan & Yu. Rykov), Generalized Variational Principles, Global Weak Solutions and Behaviour with Random Initial Data for Systems of Conservation Laws Arising in the Adhesion Particle Dynamics, Comm. Math. Phys., Vol. 177, No 2, (1996), 349-380. 114. A Remark Concerning the Thermodynamic Limit of the Lyapunov Spectrum, International Journal of Bifurcation and Chaos, Vol. 6, No. 6, (1996), 1137-1142. 115. (with W.E, K. Khanin, A. Mazel), Probability Distribution Functions for the Randomly Forced Burgers Equation, Phys. Rev. Letters, Vol. 78, No. 10, (1997), 1904-1907. 116. Parabolic Perturbations (In Press).
of Hamilton-Jacobi
Equations, Fundamenta Mathematica,
117. (with A.B. Soshnikov), Central Limit Theorem for Traces of Large Random Symmetric Matrices with Independent Matrix Elements, Bol. Soc. Braz. Math., Vol. 29, No. 1, (1998). 118. Simple Random Walk on Tori, (submitted for publication in Journal of Statistical Physics). 119. Asymptotic Behavior of Solutions of ID-Burgers Equation with Quasi-periodic Forcing, (submitted for publication in Topological Methods in Non-linear Analysis). 120. Convex Hulls of Random Processes, Publications of AMS, (In Press).
609 M A R K O V P A R T I T I O N S A N D C - D I F F E OM OR P HIS M S Y a . G.
1.
Sinai
Pertinent
Information
on C - D i f f e o m o r p h i s m s
and F o r m u l a t i o n
of t h e
Results
Among all diffeomorphisms,C-diffeomorphisms [sometimes also called U-diffeomorphisms*] are d i s tinguished by the maximum possible instability properties. Specifically, if T is a C-diffeomorphism of class r C of some compact Riemannian manifold M of class C°°, x£M, then for any point x1 near x the distance b e tween T n x and T n x' increases with n at an exponential rate (either as n — + « o r a s n - * - » ) . It is presumed that the r e a d e r is familiar with the conventional definition of the C-diffeomorphism [3, 4, 5). The following fundamental property of C-diffeomorphisms is used in the present article: For every point x the set of points x' for which d(T n x, T n x') — 0 as n — °° (d is the metric on the manifold) forms a k-dimensional submanifold r( c )(x) of class C r _ 1 ; the set of points x' for which d(T x, T n x') —0 as n -— *> forms an /-dimension submanifold r( e )(x) of class C r - 1 ; here: a) the numbers k and I do not depend on x, and n = k + I; at every point x the distance between the tangent spaces to r( c >(x) and r( e )(x) is greater than a fixed positive constant; b) the set of submanifolds r( c )(x) forms a continuous k-dimensional foliation &{k), and the set of submanifolds T(e\x) forms a continuous ^-dimensional foliation S ( " ; each of these foliations is invariant: T©<*)=e(*>, r©(,> = ®(/,; c) there exist positive constants a > 0, X c < l, X e > 1, such that if d c (d e ) is an induced metric on the layer r( c > (r.(e>, then dc(T'lx, rV)
~ y €D n c , and the correspondence TT : x -*y is a one-to-one mapping of D*c onto Dnc. We call this mapping IT a canonical isomorphism. ,
ft
A
More generally, let D' c and D"c be two arbitrary CLL. We assume that Dc ~ (J Dc,„ Dc = (J DcA and that there exist chains of CLL
D'c,i.a = Dc,,, D'c,i,lt ..., Dc.t.s = DZti , such that D'a.f
and Dc.,./+i lie in one
* Publisher's note. Moscow State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 2, No. 1, PP. 64-89, January-March, 1968. Original article submitted October 20, 1967.
61
small spherical neighborhood Oij and are canonically isomorphic therein (s does not depend on i, and D ' c j , E"fc,i a r e pairwise nonintersecting for different i). Then D' c and D"c are called canonically isomorphic. Canonical isomorphism between ELL is defined analogously. If D'c and D'c are canonically isomorphic, then. T n D' c and T n D' c are also canonically isomorphic for every n. A canonical isomorphism is continuous, because the foliations
dDcCZdO . We call the
Definition 1.2. A subset f / c O is called a parallelogram if all its CLL (ELL) a r e admissible and canonically isomorphic with one another. Parallelograms U exist. It suffices for every point x to pick accessible CLL D c (x) and ELL D e (x) of sufficiently small dimensions and to set U = \J De(z) = '(J Dc(z) , where D e (z)[D c (z)]isanELL [CLL] zeO c M
z€De(x)
canonically isomorphic to De(x) [Dc(x)l and passing through z6D c (x)[zeDe(x)]. Every parallelogram U can be so represented as long as x is any point belonging to it and Dc(x), De(x) are local layers in U passing through it. For a parallelogram U its boundary r(U) =
11 4 (z)u
U
D
^(z) = r P rt/) II r c (£/).. We define a p a r -
tition of M into parallelograms Ulf . . . , U r as a system of parallelograms such that M = (J 0it Ut fl U) CI r (£/,-) fl T (£//)
. If a is a partition into parallelograms, then r e ( a ) = [j r e ((/,), r c
In the present article we describe a new method in the Markov partition. Definition 1.3. A partition a into parallelograms is called a Markov partition if for some m > 0
re (r-\x) c re <«), rc (rm«) c rc (a). In a recent paper Adler and Weiss [2] give a very simple example of a Markov partition, when M is a two-dimensional torus and T is its algebraic automorphism. In this case (see Fig. 1), the partition consists entirely of two parallelograms, and m = 1. The sides of these parallelograms are formed of segments lying on the proper lines of our automorphism that pass through O. THEOREM. Every C-diffeomorphism has a Markov partition. Moreover, for any z > 0 it is possible to construct a Markov partition such that the diameter of every parallelogram is not greater than e. This theorem is proved in the second part of the present paper. In the first part we deduce some results pertaining to C-diffeomorphisms, using only the extence of a Markov partition into parallelograms of sufficiently small dimensions, If a is a Markov partition, then the partition o^-.-a • 7 - 1 a- ... •T~m+lar whose elements have the form V = Uit fl T~1Uil f] • • -fl T~m+lUim-v i s a i s o Markovian, with m = 1.
Fig. 1
62
Let oft be a Markov partition with m = 1. One of its advantages is that any intersection Vit f] TVil f] ... fl T"Vtft =f= 0 when and only when the pairwise intersections V,s f) TVis+i (see Section 3), x = 0, 1, . . ., n - 1, are not empty.
611 We introduce the matrix of intersections II = || % || } where ir^ = 1, if Vt f\ TVj=j=C3 , TTJJ — 0 otherwise. We examine the space fin of all infinite sequences w = {... i_ n . . . i0 . . . i n . . . } for which ffigis+i = 1 ft>r all s. We assign a weak topology to Qj[. The mapping fljj **M, where mapping of flrj onto all M. On a certain set Ml
of the second category this is a one-to-one mapping,
The fundamental class of C-diffeomorphisms is distinguished by the following condition: For some k the matrix n k consists of positive elements. C-diffeomorphisms meeting this condition are called transitive. In order for T to be transitive, it is necessary and sufficient that every layer r< c ) and every layer H e ) be everywhere dense (see Section 3). In the present article we investigate various invariant measures for transitive C-diffeomorphisms. We consider first the existence of invariant measures associated with smoothness. Markov partitions make it possible to represent every C-diffeomorphism as a Markov chain, which has, of course, a continuous state space. Specifically, the state of our Markov chain is provided by the 0
semiinfinite inters ;ction
f] T Vin
• Inasmuch as a t is Markovian, it follows that if this intersection is
not empty, it represents a CLL D c in Vj.. It turns out (Section 2) that it is possible on every such D c to specify j7 (• | D c ), uniformly equivalent to a normed Riemann volume on D c ,and a defining "transition probability," from D c to D*C is equal to /T(TVi | D G ), and 0 in all other c a s e s . F o r the transition probabilities thus defined the well-known Chapman-Kolmogorov equation from the theory of Markov processes is valid. In Section 4, we study the Markov chain constructed in Section 2 for transitive C-diffeomorphisms. It turns out that it satisfies a very good regularity condition of the Doeblin type familiar in the theory of Markov chains (see [6]). This enables us to construct an invariant measure for our Markov chain. A s e c ond look at the C-diffeomorphism T yields the following theorem: THEOREM 1. For every transitive C-diffeomorphism T there exists a measure Jxc positive on every open set and having the following properties: 1) £*c is invariant relative to T, and T as an automorphism of a space with measure is a K-automorphism. 2) For every measurable partition f of a space M, the elements of which a r e CLL C£, the conditional measure juc{ • 1 C£) induced b y ^ c o n almost every C^ is equivalent to a normed Riemann volume onC£. The properties 1) and 2) of the measure fic a r e uniquely defined. Replacing the contractile foliation by an expansible foliation, we construct a measure /Te having analogous properties. It follows from Theorem 1 that if T has an invariant measure associated with smoothn e s s , it i s congruent with both fic and ^ e - Clearly, it i s possible to find examples in which the measures /*c and /Te a r e distinct and, therefore, singular with respect to the class of smooth m e a s u r e s . The algebraic automorphisms of compact nilmanifolds and their small perturbations serve as c u r rently known examples of transitive C-diffeomorphisms. The basic problem in the theory of transitive C-diffeomorphisms is to prove that every such C-diffeomorphism is topologically associated with an a l gebraic counterpart. In the fifth and last section we prove a number of theorems bearing on this fundamental problem. THEOREM 5.1. On every layer r ( c ) of a contractile foliation it is possible to specify a
Ad Tic)
H C (7-M) = ^ C ( ^ -
aj) If v4cr t c ) is canonically isomorphic to X c f ' c ) , then nc (A) =/i c (A) [it is assumed here that Pc( A ) is a measure on r ( c ) and that AC(A) is a measure on r ( c ) ] .
63
Replacing the contractile foliation with an expansible foliation, we obtain a measure /Je on the layers of the expansible foliation and a corresponding constant he. It turns out that h c = he = h. The properties a t and a2 demonstrate that the measures /J.C and fJe have certain homogeneity p r o p e r ties as natural measures induced by the Haar measure for algebraic automorphisms. The constant h obtained in Theorem 5.1 is related to various properties of the diffeomorphism. example, the following is true:
For
THEOREM 5,3. The topological entropy of a diffeomorphism T is equal to In h. Theorem 5.1 makes it possible to generate a topological classification of the C-diffeomorphisms of a two-dimensional torus. THEOREM 5.2. Let M be a two-dimensional torus. Every G-diffeomorphism of M is topologically conjugate with an algebraic automorphism. In general, the transformation formulated in Theorem 5.2 is singular, i.e., it takes a Lebesgue m e a sure into a singular measure. We state one necessary and sufficient condition in order for it to be absolutely continuous. Specifically, let h'^ and h1^ be metric entropies of a diffeomorphism T when the fic and /Te constructed in Theorem 1 a r e adopted as the invariant m e a s u r e . THEOREM 5.5. If h = h1^ = h ^ , then a homeomorphism taking T into an algebraic C-diffeomorphism is absolutely continuous. In fact, it can even be shown that it is a diffeomorphism of the class C r , but we will not concern ourselves with this. We bring attention, finally, to Theorem 5.4, which b e a r s on algebraic automorphisms of an n-dimensionaltorus. THEOREM 5.4. Let M be an n-dimensional torus, T its algebraic automorphism, a C-diffeomorphism. Then T is metrically (i.e., in the measure-theoretic sense) conjugate with a Markov' chain having a finite number of states. This Markov chain has the property of maximum entropy among all Markov chains for which the transition probabilities a r e positive when and only when they a r e positive for our chain. Apropos chains of this type, see P a r r y [7]. The author is grateful to G. A. Margulis for the elimination of a number of e r r o r s in the original version of the article. We mention also G. A. Margulis's alternative approcah to the introduction of the measures fic and ne constructed in Theorem 5.1. 2.
C o n s t r u c t i o n of
Measures
an I n c r e a s i n g
Partition
and S y s t e m s
of
Conditional
for C - D i f f e r o m o r p h i s m s
As shown in [8], for every measure-preserving ergodic diffeomorphism T associated with smoothness and having a transversal foliation Z, there exists a measurable partition £ with the following three proper properties: 1) T£ > t . 2) I1T £ = e mod 0, where e is a partition into single points. k
3)
0 7* ^
=
^2'
wnere V
Z
is a
measurable hull of a partition of Z into complete layers of the
transversal foliation Z. Making use of the Markov partition, we show that an analogous partition exists for every C-diffeomorphism, without assumingthe existence of an invariant measure. Thus, let T be a C-diffeomorphism, a an arbitrary partition into parallelograms Uj, U2, . . . . We assume for now that they are closed and that V, f] Uj = dUL f] dUj . LEMMA 2 . 1 . There exists a number E > 0, depending only on T, such that if the diameter of every LL in Ui is not greater than e, then the partition a is a generating partition. This means that for any two points x and y, x * y, there exists a k, such that T k x and T y belong to different parallelograms Uij, Ui2,
uix n uia = e- . 64
613 Proof. We pick a number d > 0 having the following characteristics: If two points z t and z2 lie on one layer of an expansible (or contractile) foliation and the distance between them in the metric of the layer is contained between d and d max (X e , A^ 1 ), then the points zt and z2 belong to different parallelograms Ui, and the distance between in the metric M is larger than some dt > 0. The numbers d and d t depend only on T if the diameters of the LL in Ui a r e sufficiently small. Let there be given two arbitrary points x and y belonging to the same Ui. We first examine the case when x and y do not lie on the same CLL in Ui. We find a point z, such that x and z lie on one ELL in Ui, while y and z lie on one CLL in Ui (if x and y lie on the same ELL, we let z ~ y). We pick a k > 0, such that the distance in the metric of the layer between T k x and T k z is between d and dX e - Then, by our choice of the number d, the distance between T k x and is greater than dj. On the other hand, the distance between T k and T k is not greater than the distance between them in the metric of the contractile layer and, hence, is not greater than the maximum diameter of the CLL in Ui. If this diameter is smaller than d t /4, then d(T k x, Tky) > d(T k x, T k z) - d(T k z, T k y) > dj - j 1 - | di.
Consequently, T k x and T k y lie in different p a r a l -
lelograms Ui. The case when x and y lie on the same ELL is treated analogously, replacing the expansible layer by a contractile layer. This proves the lemma. It follows from Lemma 2.1 that the infinite intersection
f] TnUi
consists of at most one point.
In-
—00
asmuch as the parallelograms Uj intersect, certain points may be represented in this form nonuniquely. It is readily seen that the set of nonuniqueness points has the form E = {J Tn (U dUi) , Now let a be a Markov partition satisfying the conditions of Lemma 2.1, and let m be the number involved in its definition. We set ail = a • T ^ a • . . . • T~ m + 1 a . The partition at is also a partition into parallelograms. We designate the latter Vlf V2, Since • r e ( r m a ) c r e ( a ) , I ^ K ) = r e (a) U re(T~la) [J ... U r e (7^ m+1 a) ( then r e t ^ a , ) c Te («i), r e ( r " , a 1 ) c r c ( 7 ^ 1 a 1 ) c r e ( a 1 ) etc., r e (r - B a 1 ) C r e (a,) for any n > 0. We now bring into the discussion the partition a~ - JJ T~"a H 1~"Uir. . It is clear that JJ TkaT ~ JJ T*cx = e
t
the elements of which have the form
T c t - = Ta • a~ > a - . Lemma 2.1 implies that on the uniqueness set
. I t turns out that a- has a very simple form in the case of a Markov partition.
LEMMA 2.2. If x(j Vi, then an element Ca -(x) of the partition or, containing the point x, represents a CLL D c (x) in Vi. Proof. We denote by £ a measurable partition, which on every Vi represents a partition into closed CLL. Plainly, ai == £ . If C^ is an arbitrary element of the partition £ , then T" n C^ is representable as a sum of integer-valued elements C'£ of the same partition £ , for otherwise there would exist a C'£ forwhich C'l H T~nCl^= 0, C'i\T-"Cl=i=0 . But then part of B(T~nC£ ) would lie strictly inside C*£ , and this is i m possible, inasmuch as d(7~ n C0€r p (r - n o I )cirp(a l ) . Consequently,
r-"£ < t r- 0 l < r-"£ < E The truth of the equality J[ T~ka = \
H
«r = ft r~*a' = ft r ~' a < s-
is established most easily by referring to the proof of Lemma
2.1. It was shown there that if two points x and y belong to different elements of the partition £ , then there exists a k > 0, such that T*x£Vtl, t'ytV^ for Viv f] Vit = 0 . B u t then x and y lie in different elements of the partition T ^ O j . Consequently, a^ > £ , i.e., off = a- = £ , and the lemma is thus proved. We have therefore constructed a partition or having the first two properties assigned to the partition t at the beginning of this section.
65
.
614 We now show that it is possible to define a natural "good" normed measure on every
Ca-.
LEMMA 2.3. On every C a _ it is possible to define a normed measure p( • | c a - | having the following properties: o^) The measure ^ is equivalent to the measure generated on a CLL by the Riemann volume; m o r e over, there exist constants Cy and C2, depending only on T, such that a, (A n C-.)
oc(C„_)
~
o , ( / l n C„_)
'
oc(Ca_)
Here tr c denotes the Riemann volume on the CLL; the density / of the measure with respect to normed CTC satisfies the Lipschitz condition with a fixed constant. a 2 ) The measures ? a r e compatible in the following sense:* We consider T~lC„- = C r -i a _ and d e note by n (• I Cj—i^) the measure generated on T~lCa- — C r -i a - by the natural transference of the m e a sure [I(|C a -) to C^,^ , i.e., p(A\T~'-Ca-)='\i(TA\Ca-) ; then for every A c Ca- c C r -i a _
ii(/i i c ^ j = ^(-41 c;-> ii(c;-1 c r _ la j. The properties a j) and a 2) of the measure Jx a r e uniquely defined. Proof. Taking Ca-(x) , we construct 7™C«- (*) = Cr„0_ (r 1 *) . Clearly, T n specifies a one-to-one mapping of Ca-ix) onto C^.tT" 1 *) . On every Cr/,a_(r*x) we consider the normed m e a s u r e generated by the Riemann volume, along with its image on Co:-(x) under the action of T~ n . We designate this image ;un. Fo-r every n the measure Mn is equivalent to the Riemann volume on Ca -(x) because T n is a continuous mapping. We denote the density of the measure | i n with respect to the normed volume
where Ac(y) is the Jacobian transformation of the Reimann volume on a CLL passing through y into the Riemann volume on a CLL passing through Ty. Of course, Ap(y) < 1. By virtue of the smoothness of T and the properties of a contractile layer, for certain constants C ^ < °° and 0 < p t < 1
I Kil*!") -11 < C"Vc( TV, fyj < C'"A^, where A c is the maximum diameter of the CLL Ca— It follows from this that f„toi)//«(&)
= C W . If we pick the y, for which
f„ (yj = 1 ( K (Ca-))-' jj f* W *»c0_ W = 1, we finally obtain c
a-
l/C'"„()
(2)
We now show that the functions fn converge uniformly to Ca-. We do this by analyzing the image of a normed measure on CTmQ,-(Tmx) under the action of T - ( n " m ) . This is a normed measure on Cxncr(T n x). Arguing as before, we find that the ratio of the values of the density of this measure at two points y l t y2 £ C T m a _ (T m x) is equal to Ac (y,) . . . 4C (T"-"'-ly2) 4C (I/,) . . . 4C (r-" 1 " 1 !/,)
* The relation introduced below is the analog of the Chapman-Kolmogorov equation in the theory of Markov processes.
66
The latter ratio does not differ from unity by more than C' Ac>."'; C ( " is another constant. Consequently, there occurs on Cr,„0_(T™x) a measure whose density (with respect to the normed Hiemann volume) does not differ from unity by more than Ca,AcK' • Under the action of T m the density of this measure and unity, i.e., also a density with respect to the normed volume, are multiplied by the same number. Hence, we obtain |-^--l|
=
/(x)
ana
" denote the measure
[L(A\Ca-) =—-— °
a
~
V
f(x)dac(x)
Af\Ca-
induced by that limit by $(• |C„-) • It follows from (1) and (2) that C t ^ J s C2, and J satisfies the Lipschitz condition with a fixed constant, inasmuch as this is true f o r / n (x). Moreover,
\7(x)-f«(x)\
(3)
Thus atf is proved. We turn now to the property a 2 ) . We have ? (A 1 Cr„la_ (x)) =^{TA\ Ca- (Tx)) - lim ^ (TA \ Ca- (Tx)) = lirn^ \y.'n (TA \ CTa- (Tx)) \i« (CTa- (Tx) | C„- (Ty))).
Here fi*n is the nominal measure induced on C T Q : - by the measure £t n on C a - . But it quickly follows from the definition of fxn that H« (TA | Cr„- (Tx)) = ii..! (A | Ca- (*)) = 1ifl_1 (A \ Cr), and passage to the limit as n — °° gives the required relation. It now remains for us to demonstrate the uniqueness of the measure Jx. Making use of the fact that / satisfies the Lipschitz condition, we have
(. (A i c„- w) = f(T"A i rca- w) = jr ( r A \ c ^ . ( n » \
/ca-(r»„W*c(y)l
\
ftrlI.„(s)J«.(!/)r'
and |ji(/4|Cd,-M) —^(>4|C„-(x))K2con5tdiamCr„a_(7™jc)-»0
«
n-»oo.
If there existed a measure £ having the same properties, then for it also | jl (A | C„- (.v)) - |i„ (41 C«- (*)) | < 2 const diam (Cr„„_ (T"x)), whence we obtain
|T(4|Ca-(x)) = JI(.4|LV(*))
. The lemma is thus proved.
It is not a difficult matter to refine Lemma 2.3, showing that the density /(x) has on every CLL a smoothness class smaller by unity than the layer of our foliation.
67
616 In this section we also wish to discuss the dependence of the density fca-(x) on Ca-. We examine the partition a °_n = a • T _ 1 a • . . . • T - n a = at • r[-ial • . . . • T - n + m a , for n > m . This is a partition into smaller parallelograms than the parallelograms Vi representing the elements of the partition a t . Since a is a Markov partition, every element AlL„c V, of the partition a1n consists of CLL in Vj. Consequently, if CpWand C e ') a r e , respectively, the CLL and ELL defining the V;, then every AG_n is defined by a subset of C e W whose diameter does not exceed AeXe~n, where A e is the maximum diameter of the Ell in all the Ui. Hence, it follows in particular that different CLL C^-gAl,, and C„ -6 A°-„ are canonically isomorphic. LEMMA 2.4. Let A°_n be an element of the partition a°_n, and let C'a-, C a -€ Al„ . We denote by n:C„ -»C a the canonical isomorphism between C ' a - and C'a-. Then there exist constants C3, x > 0, such that I P (^ I C'_)
I _, „ „ , | ji (n(/l) I C'a_)-1
Proof. Consider r ' c'a-
and T ' C"a-
. It follows from Lemma 2.3 and from (3) that for any
q > 0
Consequently, ji 1 M]^5-)(l-ci , 'A c »»)
SMlO ^ M/,lOj1i;c<"AA*> " ^ ~z :— ^ '
v, <* w i O (t+ c" \ K )
fi (n MI i c;_)
(i, (3x(-4>i c;_> ( i - cw.^xj)
By the definition of Hq (see Lemma 2.3),
M, (<4 I Ca) =
We set q = |~.l"j
and compare
;-2 °c C*c„_)
, |i,(n (/4) C„-) =
ac(T°C'a-) and n (T'C.-);
VC'a- and T'CJ-, as well as r ' ( z i n C a - )
and T'(n(A)f]Cl-),
o (T°(4 fl C„-)). and
<j(7* (JI(4) ("| C„-)) . Clearly,
a r e canonically isomorphic. Therefore,
oc(rc;->= $ /wdoc._w, oc(r«(Anc;-))= i^V
;—2— . u0 (r'c,,,)
5
/w^-w,
TI i.i(^jnca-)
where the expression for I(x) is given in [4] [Eq. (5.3)]. It follows from this expression and from [10] that if d e (x, ir(x)) == d, then | I(x) - l| s C' T 'dx for certain constants c ' 7 ' , x > 0. Inasmuch a s the point 7i(x) lies in the same element of the partition a* for x£TqCa- , we can let A^\~n/1 represent d. We can regard the dimensions of the original parallelograms Ui as so small that C" , A c < 1 /2, C l "A e < 1 / 2 . Then, assembling 6 the ensuing inequalities, we have for a certain constant C 3 ,depending on c( ) and C^7', j!(.4|C)
ll(nM)|C;_)
|i,4|C0_)
l+C'«'A c ^ / !
|i, (II (/I) | O
1~CU>AX"
< — " ' ' -S-S < 1 + C3max (Ac, ^ ) [max (>.c, J.-')!™". (l-C">A,A c ")(i-c 1 ".i ( ^'"'-') A lower estimate is deduced analogously. This proves the lemma.
617 3.
Symbolic Dynamics
for C-Diff eomo r p h i s m s
In the preceding section we analyzed a system of closed parallelograms Ui of sufficiently small d i ameter Ui f| U; = <>Ui |~l <XJ( . We now consider the open kernels of these parallelograms forming the actual partition of the set Mj = M E, where E = (J 7" (|j ay,) . We denote these open kernels by the same lett e r s Ui. Lemma 2.1 states that every point xeM, is uniquely representable in the form x = •'fj T"V,n —00
= fl T"U, . which is equivalent to T~°x 6 U(|1, — co < n < <x> . The set Mt is an everywhere dense set of the type G e . If we now go from theMarkavpartitiona tothe Markov partition ai = a • T~'a • . . . •T" m + 1 a into parallelograms Vi, V2, . . . , Vr, then analogously x = fl T"V,n — C\ T"Vllt f ° r x6M 4 . Definition 3.1. An intersection matrix n = || n,; || irjj = 1 if V, fl TVi + 0 , otherwise Try = 0 .
of the partition at is defined as a matrix for which
THEOREM 3.1. Consider the space £2n of infinite sequences u = {... i_ n . . . io • • • »n • • •} .
such
tnat
7TJ j = 1 for all s.and introduce a weak topology in Q\i. The mapping ^ n - * M, operating according to the s s-K formula ip{w) = f| T"V,n
, is a one-to-one mapping of £2fl onto all M. On the set Mj, it is a one-to-one
1
mapping, and
W e now
show that { Zn . . . lns} € 8 is not empty when and only when T"'V,t fl . . . fl T kV,t + 0 .
It is sufficient for the proof of this assertion to investigate cylinders of the form { Z0, lu .. .ln}- F o r n = 1 the assertion follows from the definition of £2 and n . Assume that it has been proved for some n. Inasmuch as a j is Markovian, the intersections V,, fl TV,, consist of integer-valued ELL in V/„. Moreo v e r , if D'e and D% are arbitrary ELL in V r , the intersections TD e fl Vi. and TD"ff] Vi, are both empty or both nonempty at the same time. In fact, if A = {D'c: TD^C\ V'I. + 0> is not congruent with all of Vjj, then A fl T^'V,, = V,, fl T~lVi. is a parallelogram, part of whose boundary T c lies inside V;,, contrary to the definition of the Markov partition. Furthermore,
v,. n TV,, n ... n r" + v,„ + i =v,,nr
tends to zero as n - " . The
theorem is thus proved. Definition 3.2. A C-diffeomorphism T is called transitive if there exists a k,j for which all elements of the matrix I r ^ a r e positive definite. It follows from the foregoing that the transitivity condition is equivalent to the assertion that U, fl 1*'U, + 0 for any i and j .
618 We now expound necessary and sufficient conditions for the transitivity of T. THEOREM 3.2. A C-diffeomorphism T is transitive when and only when every layer of every foliation thereof is everywhere dense. Proof. We call a foliation almost periodic if for any e > 0 there exists and R(e), such that every sphere of radius R ( E ) on any layer of our foliation forms an e-net in M. We show first that T is transitive when and only when both foliations are almost periodic. We denote by U_,5 (o;) the set of points separated from the boundary T(a) by a distance no smaller than 6 . Let 6 be so small that U-t (a) f]Ui=f'0 for every i. We assume now that both foliations a r e almost periodic. We pick an arbitrary point x and find an R, such that the sphere D c (x ; R) of radius R on a contractile layer intersects all sets £/_» (a) f] U,. Expanding D c (x; R) slightly, we find that for the expanded set D' c (x; R) its intersection with every Ui would be a sum of CLL D c (z) from Ui, i = 1, 2 Taking a sphere D e (x; Sj) of sufficiently small radius 6t, we draw CLL canonically isomorphic to D' c (x; R) through all points x6D e (x; 6j). For the ensuing parallelogram W the connected components of the intersection W f| £/<=^»0 and represent parallelograms for which the CLL are CLL in Ui; the number of these parallelograms is equal to the number of CLL Dl(x; R)(~\Ui . Let us consider T^W. We assume k to be large enough that 1) diamcfT^Dc) < (5/2 for every CLL D c 6 W; 2) T*£>e(x; 6,) H (U-»(«)n U/) + 0 f o r e v e r y j . Fulfillment of the latter condition is guaranteed by the a l most-periodicity of the foliations. We now show that 7 * ( / , n " ( + 0 for arbitrary i, j . We pick some p a r allelogram Wj representing the connected component of the intersection W f)Ut . It is enough to show that T*Wj f]Ui=f=0 . Inasmuch a s the ELL De in Wi a r e canonically isomorphic to De(x; 6 t ) and this i s o morphism is established by means of the CLL D c , it follows that T-D e is canonically isomorphic to T k (De(x; 6)), and the distance between corresponding points does not exceed 6/ 2. Hence, it follows that if TkDe(x;ii)f)(U-i(a)r]Ul) + 0, then r*Def"| U, + 0 for every ELL D e C W, , i.e., 7*117, f\ U,j=0 • Consequently, T is transitive. We assume now that T is transitive. We investigate the partition T _ t a • . . . • T*ia: . For sufficiently large t the diameter of each of its elements A_t is not g r e a t e r than e/2. We pick a point xGMj. Then the CLL C„-(jr) = nV"t/,„ . -<-*. „
t
We examine Cj— t-k„a- (*) = f) T ^A„ . Inasmuch as T is transitive, for any nonempty A_t the i n t e r s e c tion C j - i - i ^ . W n A - i + 0
. Consequently,
C,_(_n.„_(*)
T~'~l" (C- (T,+*'JC)) , the diameter of the CLL Cj-t-t^
forms an e/2-net.
Since
(f+ft ,
maximum diameter of the CLL in all Ui. Consequently, the sphere .Dc(x; AcX^ If x^M,, we investigate
T'+k,xiU,
£,—<_*,„- (x) —
A c xr" + i ''
(x) is not greater than
°)
where A0 is the
forms an e/2-net.
. We find an interior point xjeM, in Ui. The CLL Dc(T'+i,x)
and
CLL D c (xi) in Ui a r e canonically isomorphic. Then T~"+*,,DC (Tl+I"x). and 7*~"+J,,Dc (x,) are canonically isomorphic. If t = t(E) is sufficiently large, the distance between corresponding points in the metric of the expansible layer is not greater than e/2. Since, xieMj, it follows that + ,1
proof, C7._„+i>)a47~" * j:l)
forms an E/2-net. Consequently,
T'
v+t,)
Tt+k'xi£Ml
Dc (T
,+I,
'x)
, and , by the e a r l i e r
is not g r e a t e r than Acxr"+*-1-
This proves that the contractile foliation is almost periodic. Expansible layers a r e investigated analogously. We now show that the almost-periodicity of a foliation follows from the everywhere-density of each of its layers. Let e > 0 be given. Then for every x6M there exists an R ( E , x), such that the sphere of radius R ( E , x) with center at the point x will form an e -net. Picking R' (e, x) = 2R (E , x), we find a neighborhood U x of the point x, such that for every y6U x the sphere of radius R'(E, x) with center at the point y will form an e-net. This is admissible by the continuity of the foliation. The set of neighborhoods U x forms a cover of M. Choosing a finite subcover from it, we find a single R for all x. This proves the theorem. Throughout the remainder of this article we consider only transitive C-diffeomorphisms.
70
For xeM t the element C ff .(x) has the form Ca-(x)= f]TnUi n
is nonempty as long as
f] T Ui ^ = 0 feo+l
. Any intersection Ca-{x)f] f! TnUi
, and it consists of a finite number of points. Such an intersection
"
represents an element of a measurable partition into Ca-(x). The basis of this partition comprises all p o s t
sible finite intersections
f] T"Ui/t, />& 0 + l . We let
@£+1
represent the a -algebra generated in M by
these sets. We examine as a measure on C a -(x) the conditional measure o(- \Ca-(x)) generated by the Riemann volume
4
^
a{A\Ca-{*'))
^
5
The proof of these inequalities by and large reiterates the proof given in [4] for the absolute continuity of a canonical isomorphism. We, therefore, present only the highlights. We observe that a{- \Ca-) is uniformly equivalent with respect to Co;- to the Riemann volume (see [4], Lecture 5). For a volume element dcr£n) on the layer T~ n C 0 ,-(x) and its image da£°) on C (x1) which contains the point x, dal0) 1
Ac ( r - * ) • . . . • Ac (x)
We pick points x 1 ^ . . . . x ' s , and x ^ , . . . , XQ representing the intersections
Ca-(x') f] Pi T"Utn
an
d
Ca-(x") H fl T^^in » respectively. Proceeding as in [4], we verify that the ratio of the products A c for different points x'j, x"j is uniformly bounded from above and below. Inasmuch as the layers T~nCa-(x')
and
T~~aCa-(x') are uniformly smooth with respect to n, it is possible to pick "identical" d ( j c
Construction
of a n I n v a r i a n t
Measure
(Proof
of T h e o r e m
1)
In Section 2 we constructed a partition a~ into CLL Ca- and the measures |T(- |C„-) on the elements of this partition. We note that or is not a partition in the fullest sense of the word; the intersection Ca- fl Ca- = dCa- H dCa- can also be nonempty. Of course, there is a real partition on the set Mj =
M\Ea~
Below, when speaking of M as a space with measure, we are concerned with the measure generated by the normed Riemann volume o, unless special mention is made to the contrary. •Let a be a Markov partition, U(, U2, . . . its elements. We now consider open U'j, U'j=Ui. O n t h e s e t M j the parallelograms U'j form a real partition. Inasmuch as LL a r e admissible in Uj (see Section 1), for every xeM, we have ac(MtJ£,_•„-(*)) = fi(Mv|Cr_„a_ (*)) = 1 . We designate a,' = T 'a- ... . Tl'a. for l\ ~l2 (the possibility of lt and l2 being equal to infinity is not excluded). It is clear that partition
a!,[ by
M'
a',;
is a partition of M, and that T'a\\ = a',\X\ . We denote the elements of the
.
Let A be an arbitrary element of the partition ists a lim jl(.4 | CT-*rW)
di\ . We now w i s h t o s h o w t h a t f o r e v e r y x g M , t h e r e e x -
independent of x. Since |T(,4|Cr_0_(jr)) = yi(T~''A\CT-n-:,r(T~\)),
it suffices to in-
vestigate those A which are elements of the partition a°(.
71
620 Let t be sufficiently large and fixed (the limitations on the values of t a r e indicated in the course of the proof). The collection of numbers
jl (AL,+I |C T -, tr . (*)) represents the probability distribution of Mn(x)
on the elements of the partition t*Zt + i. We prove the following lemma. LEMMA 4.1.- Let <5 > 0 be given. Then there exist t„(6) and n^t, 6) = n„, such that for any t > t 0 ; "l. "2 > "oft, o); *,, x 2 6M, VarftI» l (x I )-ir„ 1 (x,))<«. Proof. We have H(Al, +1 |C r _„„_M)
2
The collection of numbers
MazL_ I |c rw , a .w)-jI(Ai (+ . 1 |A=i,_ I nc r .„ ll .w).
j»(Ali_, \CT~n(x)) = ^.{T1 AZ[i-l\Cr-n+ia.(T'x))
(4)
is nothing more than
p„^t(T'x)
Furthermore, the collections of numbers ji (AL/+,1 AZ',,^ fl ,_„„_(*)) = ? (T'Aii+i I T'AZ'„_, fl c r-»+' 0 - C7"'^
ma
Y
be regarded as matrix elements p„,x (Al<+11 AL<+1), A°-(+1 = T'AZi/-i of some stochastic matrix P n x . Using this matrix, we can rewrite (4) in the form
M*) = i;„-i(T'*)'v* = i'»-„(f*)/•„_„•<,-P...,-... =;;„_,((r'-r)p„_rf.rrt, • ... -p„.,; p=mt«get.
<5>
Consequently, our problem reduces to the investigation of a product of stochastic matrices. We establish first that the matrices Pn,x depend little on n and x. To this end, we define a constant matrix P = |[jp(AL,+11 Al, +I )|j , with which we compare the matrices Pn,x, a n d we show that for certain constants Ce < °° and X < 1
| P.,, (Ai, + , I * ! , + , ) _ , I
I p(A°_, +I |4!L, +1 )
,
(6)
I
(these inequalities also imply that the matrix elements P n
x
and p go to zero simultaneously). Wefirstwrite
p„., (Al,+11 AlHl) = p (Ai, + , | 7-'Al,_, 0 C,_„a_ W) ? <4-'+. I C r-'«-' ? (C7~'„- I C r-»a- (*,)
S
MCr-'a-l c r-»a-W>
2J
In each Alt + 1 we pick an arbitrary element Ca-eA!L(+1
and designate it
Ca- {h-t+i)
. We set
p (AL,+11 Ai, +I ) = iT(T'AL,+, | C„- (Al,+,)). This defines the stochastic matrix P. If y£CT-,r, basis of Lemma 2.4
then
i»(r'AL,+JCn_(r'i,)) ? C'Al, +l I C„- <£!_,+1))
ji(Al( + i|C r _/ 0 .(y)) = (T(r'AL,+1|C0-(T'i/)) . On the
_y(,T'A°_,+l\Cr{Tly)) P (4i, +1 1 A°_,+1)
does not differ from unity by more than C6A , where CG = C3 max (Ac, Ap), X = [max (Xc, XJT1)]*'2 - This proves (6).
72
We now establish that the matrices P n 2
x
have one special property: We let P ^ = P n _t f ' x ' p n , x
=
1
P = l| P " (A^-I+J I A-(+01| , representing the square of the matrix P, and we show that there exist constants CT> C',, independent of t, such at for any A°_t+1 and A°_t+1
„»> ( A y,l^, +1 ,
;
j-(^ + ,iaL, + ,>
c
We point out first of all that if t is sufficiently large, then all pi'i(Alf +l | A°_/+1)
(7)
and, hence,
1
p" (Al,+11 A°_,+1) are positive. Thus, p<» = |T(7"'Alf+11 AL,+1 fl C,_„a_ (*)) = £ ( r ' + " Al, + 1 | r 3L, + . n C a - (r*)). Let T n xfUj. Now, if Al, + I +
and t is sufficiently large, then, as shown in Section 3, in the
proof of Theorem 3.1, U,-fl T"A°-,+l ("] 7"' + "A-; + i + 0 Ca-(y) D ^"A^,+l fl r" + "A!_ /+l C„-(i/)nr"Al,+ln
.
It was also shown there that the intersections
are canonically isomorphic for all y6Ui f\ Mt.
r"+"A,_,+1^0
Consequently,
andpWx>0. '
In view of (6), it is sufficient in (7) to prove the latter inequality.
Let
P = ||p (A!_,+1151,+1)|| = || jT (T"M.l+l | Al, +l ) || and
° ( r " A -'+ i I *-'+'> a (*-<+.)•
a (r'A°_,+I) = 2 For all
£L, +l
we have
o(T*'AL, + ,| AL, + 1 )>0 " (T"A'_I+1
. Furthermore,
I AL,+1) =
: r
i-r-
t
o(Sl, +1 ) =„J We now observe that, inasmuch as t > k^ (see the end of Section 3), for any C'Q,- and CnaC , < " ^ In particular, setting
1
0
Ca- = Ca~ (&-t+i) , we have cr (T"A"_l+l) > C4a (7"' A°_,+11 C - (5!.,+.)),
and, by virtue of Lemma 2.3 and the absolute continuity of the canonical isomorphism (Section 1), f (T"A%+11C„- (AL,+1)) < O c (7"'A!Ll+i1 C„- (£-,+,)). Consequently, l»(7-"Al, + ,|C n -(A°., + ,))
The lower inequality is derived analogously. We now compare we have p ( " (Ai, +1 1 5L,+1) =
^
p (Al/+i | Al, +1 ) and p""(Ali+i| A-i+Cl
P (A-i+i I S!.(+1) p (A"_,+11 Al, +1 ).
622 On the other hand, p(Al /+1 1 51, +1 ) = j I ( r , ' 4 l , + 1 | C „ - ( 3 i , + I ) ) =
Clearly, ing to
T' Al, + 1 (") C«- (Al<+i) = 7"'
A!_/+i
2
i I ( 7 '"^+.|r'Al, + 1 nC„-(A- ( + 1 ))|r(7-'A^ + 1 |CV(A-, + 1 ))
. w h e r e CJ-(5L,+1)
is an element of the partition a" belong-
. Therefore,
jr (7-*'AL, +1 I r' A°_, +1 n c«- (Ai, +1 )) =;; cr1 A°_, +1 I C;~
(A% + 1 ».
But, by virtue of Lemma 2.4, !*<&•->+, I £.-<*-,+,»
l'(A,_,+1|Ca.(4l,+1))
Consequently,
?* , (4l, + 1 Ml, + 1 )
P (A?-,+i I *!,+,)
We state the next proposition in the form of a separate lemma. It represents a modification of the ergodic theorem for Markov chains under conditions of the Doeblin type (see [6]). The author is grateful to R. A. Minlos for a substantial simplification in the original version of this lemma. LEMMA 4.2. Consider the class of probability distributions p. on the elements A°_j.+1 of the partition a ^ t + 1 . There exists a constant C9, depending only on T, such that for any two probability distributions M' and nn Var (n'PSi - n'Pi" ) < (1 - C„) Var (|i' - (O, ,Var (u'P2 - y."P') < ( 1 - C,) Var fti' _ n"). Proof. We prove only the latter inequality. The first has an analogous proof. We designate \i.' = \L'P2, |p = |i-p>
and introduce the sets
It is clear that
fl'"uBw
B"> = {Al, +1 : jr'(Al ( + 1 )> |P(Al, +1 )). B<" =
= M. Therefore, either a- ( T ^ ' B " ^ ' / ^
dition prevail for definiteness. Then, letting
Var (i? - j?) = 2 2
+
2
denote summation over positive terms, we write
(f" (Al(+1) - i? < Al,+1)) = 2 £ +
< 2 2
+
.
or a(T"B'") > '/, . Let the former con-
2
?"' < A - ' T ' I 5 - ' T ' ) G1" (S-<+'> - ^' ( 5 -'+0)
(M' (Al,+1> -1»"
By virtue of (7), we have p (fl"! | Al, +1 ) >C,o ( T V 1 ) > 0.5 C, . Hence, we obtain Var (j? — ] ? ) < ( ! — 0.5C,) Varfti' — |»"). This is then the required inequality, as soon as we set C9 = 0.5C 7 . The lemma is proved. We conclude now with the proof of Lemma 4.1. Let ^ 0 be an eigenvector of the matrix P. We show that the neighborhood Oa,. = {n : Var ((i — n0)<8/2) is invariant with respect to all stochastic matrices p ^ if t is sufficiently large. Making use of (6), we write
74
pft(Ai, + ,|Ai, +l) ?»
=
aL<+i ^
MSL,+1|Sl/+1)7(4l,+llil,+I)
"-'+1
for sufficiently large t and, analogously,
p">(A!. (+1 |4l, +1 ) Hence, it follows that VarftiPi*1, — \ip') < 3C,x' for any p.. From Lemma 4.2, we now obtain v = Var ( r f , - (iJ = Var (fl* - (i„) + 3 C.J.' «i(1 - C.) Varfti- |i„) + 3C.X'.. Let t be so large that 0.5C96 > 30^.
Then
» < ( 1 - C , ) | + 3C,X' = A - i . C , 8 + 3 C . X ' < | , i.e., ^iP™,6 0«/„
and our assertion is proved.
We saw [see (5)] that for p < t, n - ct + p, with e an integer, ?,,„ = i V - . , - , • P , + p . r — , ' Pt+V.r-**
•-••
P
"-
It is readily inferred from Lemma 4.2 that there exists an n0, independent of x^M^ such that Consequently, Lemma 4.1 is proved.
H-„ , £06
,
It is a direct consequence of Lemma 4.1 that for all sets A representing elements of some partition a'*
, there exists an x-independent limit \\mjiL (A | Cr._„a_ (*)) = ^ (A) = £(A),
positive for every A. We now show that the formulated limits \i (A) a r e invariant with respect to T. We have from the property a2) of Lemma 2.3 £{A) = fix(A) --=-- ^.Tx (JA) = jl (TA). Thus, the limits fi (A) are invariant with respect to T. It is logical to adopt the limits fi (A) as the values of the invariant measure we seek. We now investigate the problem: In what sense do the M(A) actually generate a measure defined on the cr-algebra of all Borel subsets of M? It follows at once from the definition that Ji is finite-additive. We look once again at the space fin that we constructed in Section 3. The relation £({/, , l,n)) = \i(Ti'Uli fl • • • fi TlttUti ) specifies a compatible system of finite-dimensional distributions on cyclindrical subsets of the space fij|. According to the well-known Kolmogorov theorem, (7 can be continued to a denumerable-additive measure d e fined on the (7-algebra of all Borel subsets of fi. We now recall that the mapping y : fin — M is one-to-one on Mj and continuous. Therefore,
75
624 We begin by noting that the conditional measure induced by /I on an element a~ of the space Ml coincides with
f ( . |C„-) • Thus, we let
C„- f~] /M, of a partition
Ca-(x) = n * - . • For almost every x, with r e -
spect to the measure M, and y€M 1( we have from Lemma 2.4
2 lim lim »-«> ,^»
iI(4i|Ca.)1i(C0-|C1..,a_(»))
c„-ec ^
—-— — Mc<.-lcr->><.-«>
< |i (Ai 1 Ca- (*)) • lim (1 + C,V) < |i (Aj | C„- (*))
with an analogous estimate on the other side. Consequently, jT(Ai | Ca- (*)) is congruent with the measure constructed in Lemma 2.3 almost everywhere with respect to /x, hence, it is equivalent to ffe|CaWe now formulate the required system of sets Oi for £ e \ £ c = J T ° ( y(F e (C/,)\ r c (t/,))) . It i s sufficient to formulate it for one r e (Ci)\r c ((/,) , all other sets being treated analogously. Inasmuch a s all Ui consist of admissible local layers, for every CLL D c we have IT (fe ((/,) | Dc) = 0. We pick an x6Ui and a CLL Dc(x) representing a sphere of radius 6" (in the metric of the layer) sufficiently large that some neighborhood Fe(Ui) f] Dc(Xi) is contained in Dc(x). By the regularity of the measure *J(-[DC) there exists a diminishing sequence of open sets Of c D= , such that a (Of) -* 0 and fl Of z> Te (£/,-) fl ft . But then the open sets Oj=U»(Of'|C,-)
, where the Ca- a r e such that the closure
Ca-C\(re(U,)\Tc(U,))
and the canon-
ical isomorphism image of the sets 0 | c ' , viz. JI (Of | Ca-) . c o v e r r e (ty,)\r c (£/,) . We now have jl (0/) = $ RO,|C„-) <£ = ^ ( n (0f)\ Ca-)djL < const o (Of) - 0. Consequently, n (p. 0/) —* 0 for * / -> oo. We direct our attention now to r c ( U i ) . We saw (Section 1) that l\(Ui) = U BcM
. where D|i)is any
P
ELL in Uj. Moreover, the closed set r c (Ui) intersects with other r c (J/,.) (3Ur f| dU,^0) ber kj, such that for every ELL De in Uj, j = 1
the intersection
connected component (for any j and j ' ) . We examine T~ k l Uj. Clearly,
. We find a num-
T~k'De P Vr consists of more than one 1*'Ui P U,-=/=0. . We set 0\ equal to
the sum of those connected components of the intersections T*6V fl ^/ whose closure contain T*ir c (Ui) and Oj = T" 'O'). We Investigate the structure of the set Oj. We may regard all Uj' as contained in some fixed spherical neighborhood W. We choose a complete ELL De in W. The intersection D e n O t consists of a finite number of open sets (one for each Ujt). Every point x6 De 0 Tc(^A) will have some neighborhood consisting of points belonging to Oj and of points belonging to r c ( U i ) . Therefore, if we add the set r c (Uj) to 0 1 ( we obtain an open set Of containing Tc(Ui). It is clear that Ca-(x) f] Ol = Ca-(x) P 0t for x6Mj. Now that we have the set Of, we consider every connected component Ojji of the intersection Th'0[ p Ut and separate the part whose boundary intersects with T 2 ^ir c (Ui). it r c ( U i ) . As before, we obtain an open set O*, such that etc., analogously.
W e ta
^e
tne
rc([/,-)c0\c0\
union of all T" 2 ^IOJ;I and add to . We construct OJczOJ, 0 j c ; 0 j ,
Our task now is to show that Ji(Ofi) — O as n - « . We establish that for some constant X < 1 the inequality u (C£+1)
holds. Let us consider the connected component P of the intersection 7"*'"0^ f] U, .
Here P comprises canonically isomorphic ELL. The intersection r' ; i { r t + l ) 0^i fl T*'^ * s generated as follows. Pick an ELL De C P; then T*'De is the sum of ELL lying in all possible Uj. We pick those D J c ^ ' D , , , whose boundary intersects with
76
T l/ ' +m T c ([/,)
. Clearly, in this case the boundary of D' e must at least
625 partially intersect with the boundary d (Tfc,De) . By the choice of k1( however, among the ELL conic ^1 stituting T ' P there are necessarily D ' e that lie wholly within T D e . Consequently, r*''" + 1 , 0j + , f"l T"'P is k _ a proper subset of T i p , and it suffices for us to show that the inequality y.(r*'l"+110l+1 P[1*'P)<, >-n(7**'P). . is valid for some \ < 1. We note that T ' p is a parallelogram; its ELL have a diameter limited to a constant K independent of n. If we examine different CLL D c belonging to T ' p , then for certain constants K( and K2. Hence, it follows that K3 < p (Q')/p ( Q ' X Kt Q of the intersection belong to
/f,
for the connected components
1*'P f) U, for suitable constants K3 and K4. Inasmuch as scarcely all the components
7'M" H'o'„+„ , we have the following, denoting by 2
summation over those connected components
which belong to T*An+l)0'n+1: ^(7-*'Pnr*^+ 1 >o; +l )
£'MQ)
KS
t
where N is the total number of connected components. Clearly, N does not depend on n, and our assertion is proved. Thus, the measure"/! is regular. By its construction the measure ~ix is invariant with respect to T. It induces a conditional measure fx on almost every Ca-. Since we have lim jl{A \ CT-na- (*)) = (*(^) for every x6M t , the transformation T with invariant measure n is a K-automorphism. We prove the uniqueness of £ . Let there be another measure A having the same properties. We pick a Markov partition a and the measure induced by it on Ca-. It will satisfy the properties a t ) and a2) of Lemma 2.3, hence, it will coincide with the measure /T constructed in that connection. But then ~jl = £, b e cause the values n are defined according to M • The property 2 needed in Theorem 1 is a direct consequence of the fact that it is fulfilled for the partitions T1^'. This completes the proof of Theorem 1. 5.
Several Theorems
on t h e C o n s t r u c t i o n
of
C-Automorphisms
Let T, as before, be a transitive C-diffeomorphism. THEOREM 5.1. On every layer r^ c ) of a contractile foliation it is possible to specify a cr-finite m e a sure / i c having the following properties: aj) There exists a constant h c > 1, depending only on T, such that for every set Acz r'c' az) If A c r ( t l
is canonically isomorphic to A"CZ T(c) , then M C ( A ' )
=
Mc(A").
1
Similarly, replacing T by T" , it is possible to specify a measure / i e on the layers of an expansible foliation and a corresponding constant he. It turns out that h c = he. Proof. Consider a Markov partition a, and let Qj = a • T-1Q! * . . . * T ~ m + 1 a , where m is the number involved in the definition of t£e Markov partition. Let Vj, V2, . . . , Vr be parallellograms representing the elements of the partition Qfj. It follows from a^ that the quantity \ic (C a -) should take on a constant value for all C a - e Vi. Therefore lic(Ca-)=}ic(ih Let us try to find the numbers ^ c ^ and the number h c . It follows from a2) that (ic^ must satisfy the following equation: Ac|iW-2|ie%,
i= I
r,
(8)
j
or, in vector form
77
626 We see from this that the vector / J C {(i c ^ , . . . ^ c ' r ' } is an eigenvector of a matrix n having nonnegative components, and h c is the corresponding eigenvalue. If n ° has positive-definite elements, then this eigenvector is unique correct to a multiplier, and the number h c is unique as well. We note, incidentally, that h c is an algebraic number. We assign definite values to the numbers (i c W. Now we let Cn-(Z.Ca-
be an element of the partition
Tor. We define McfC-por) so as to meet S4) and a 2 ). To do this, we need tjp set u0(Cra-) = /iTV„
have already been determined. We set n c (C r , + , a _) r=
(Cj»„-).
We show that such a definition is proper. We observe that
2
Pjf-ra-) = frr'2 Cc (7—Cix-) = nc (Cj-) ,
since U 7—Cr«- = T" 1 C-, and for C^- we have the system (8), (8'). We assume that the following relations hold for all n ^ r^:
MC!«-.0-)=
MCr«.-). M<W) = ^'M^'Cr^-)-
2
Then from the definition for n -t- n0 + 1, we have
2 But
fc «w,„-) = AT' 2 Co (^"c^+i.-)
7" (U Cr„a_) = T-1Cr"a- = Cri-i-a-. hence the latter sum on the right is equal to nc (Cr"-iu-), and by the
induction hypothesis Cc (Cr«-ia-) ^= hcy.c (Crna-l
. We obtain
2 c
Mcr..+.a-) = MC?«.-).
r"+'a- cc r»a-
Consequently, M C is a finite-additive function on the system of sets Cr,,a^(ZCa- . As in the preceding section, it has been established that ^ c generates a regular Borel measure on every C^,-. Continuing it additively, we obtain a measure [ic on every layer of the contractile foliation. Inasmuch as ^ c | C Q - complies with the condition a^, / i c will then meet the same condition on every layer. All that remains is to verify a 2 ). If A1 and n (A') lie inside the same Vi, the equation MC(A') = M0(7r(A')) follows from the construction of nc. Now let A'(-V,, n(A') = A" e V:\Vi • Then i
i
We set A'; = 7r_1{Ap. We bring into the discussion any CLL A'." d Vj canonically isomorphic to A"; and thus to A';. We then show that/i c (A!) = /iC(A-") = Mc(A?).
Since every canonical isomorphism decomposes
into a chain of mappings of the type A':— A m , this will ultimately lead to the property a 2 ). F i r s t let A'j be an element of the partition T 1 ^ - for some n > 0. Now T~"A] -- C„- Q V;t T~"Aj" t: I';,
, then A*" is also an element of the partition T11^-, and ^{T^'Aj) = u-c(T A,- )
. If by definition.
From a.,), however, we have pc{A]) =^ nc(Aj"). We investigate the case when T" n A'j and T""nA»" do not lie in the same parallelogram V;. A' = U Cr«.a-. " i > " . T"%
- U T-"'C'T:uu- -• \jC'a-
. Now, if C" = n (Cr«m-)C A'i" , then
Let
n (T-n'C'T»,a -)
- .-T(C^) -= T~"C" . If C'Q,- and T" n C m lie in the same parallelogram V;, then we are entitled to use our
78
earlier argumentation, deducing that C m is an element of the partition T la~ and fc(C") = nc(Cr«,„-) . Conquently, the points xfA'j.for which T~"CT„a_ (x) and T~" (n(Cr«0.(*))) belong to different parallelograms,for all n fall outside the scope of our investigation. Here dx(T~"x,7~"n(x))—>0 C+ (r~"it (*)) + 0
as n — •», Ca+fT"""*)!"!
and enters into r c ( a , ) . It is easy to show (in fact, the corresponding arguments a r e e x -
actly analogous to the proof of regularity of the measure Mc in Section 4) that for the set B of such x we have Mc(B) = 0, /Jc(7r(B)) = 0. It turns out, therefore, that A'; = B uU CT«a- , where the summation is taken over the open elements of the partitionT^,- that do not intersect B. Inasmuch as |\(fl) = (xc(n(B)) = 0 and uc(Cria-) = He(Cr
79
628 Then the proof of 2.1 c a r r i e s over without any modification, and the open covering a\ by the sets V ; is a generator in the sense indicated in the statement of Lemma 2 . 1 . It follows from this and from the results of [1] that the topological entropy can only be calculated with respect to the covering a\ . We now investigate the finite intersections (A')o = Vj, f] TV'it f] • • • H T"Vi
» representing elements of
the covering ( a ^ = a[ f) Ta[ f) • • D Tnau and, in addition to these, the elements A" =Vit f] 7V,-, 0 • • • 0 T"Vln of the partition (a 1 )J=a 1 -7 ,-1 a l • ...•• T~nav. . We show that there exists a constant K, independent of n, such that every element (A') n contains at most K elements of the partition a§. From this we a r r i v e at the statement of theorem. We analyze the subcovering of the covering (a')*J consisting of elements (A')o = Vt, D ?Vi, Cl • • • f] TnV[ such that A3 = Vf, fl TV,-, [")-.- r\T*Vin=f=0. Since distinct AJ1 intersect only on the boundary, this will indeed be a covering of M. The number of elements of this covering is equal to N ^ ) , i.e., the number of nonempty elements of the partition a n . On the other hand, if p is a subcovering of ( « ' ) n with the minimum power N(y3), then KN(£) == N ( a 0 n ) . Finally, we have K ^ N f a J ^ N(/3) < N ( a " ) . Consequently, the limiting behavior o f - l n N(0) is the same as that o f - I n NfaJ 1 ). Let N{a", Vi) be the number of elements AJf of the partition a", such that T ^ A J 1 C Vi- The vector N n with components N ( a n , Vi) = Nn{Vi) satisfies the set of equations N n + 1 = N n Il, where, as we recall, n is the intersection matrix; TTJJ - 1, if Vi (~\TVj ^= 0 , otherwise TTJJ = 0 . But then, as we are well aware, lim — lntf (c£) =lim - I n J NR (V,-) = In ft. Consequently, the topological entropy is equal to In h. We now have to find the constant K. We observe that if the dimensions of the parallelograms Vi are sufficiently small, then every V'j and every T ^ V ' j can be included inside a sphere of radius e . Let O e (i) be a sphere of radius e containing V'j. We draw a complete CLL in 0 E ( i ) , and find out how many elements Ca- €V'j of the partition a~ intersect with it. We designate the maximum number of these elements with respect to the entire complete CLL by Kj. We now observe that if we pick some complete CLL in T - 1 0 £ (j), and find for it the number of elements Ca.- 6 V*i contained therein, again finding the maximum, we then obtain the same constant Kj. It is seen at once that Kt is not greater than the maximum number K2 of p a r a l lellograms Vj lying in one V ' j . The following is easily proved by induction: Consider V,-, f]...f\TnV,n ; inside VjCl Vi, this intersection is contained within the closure of the union of a collection of elements A^ whose number does not exceed Kj, Hence, it follows that Vio f] ... f) T"V'tn is contained within the closure of the union of a collection of elements A° whose total number does not exceed KtK2. Consequently, every element of the covering ( a ' j ) ^ contains at most KjK2 elements Aj1 of the partition (a^J 1 . The theorem is completely proved. For any C-diffeomorphisms in the algebraic case the Markov constructions we formulated acquire a curious metric characteristic. THEOBEM 5.4. Let M be an n-dimensional torus, T its algebraic automorphis, a C-diffeomorphism. Then T is metrically (i.e., in the measure-theoretic sense) conjugate with a Markov chain having a finite number of states. This Markov chain has a special property, to be spelled out in the proof, Proof. We investigate a Markov partition a t for T. It is a generator for T in the measure-theoretic sense. With it a torus M with Lebesgue measure is mapped mod 0 into a space of sequences w. We show that the image of the Lebesgue measure for this mapping specifies a stationary Markov chain in U. It is readily seen that on every Ca~ the conditional measure JX( • \ Ca-) induced by the Lebesgue m e a sure is a simply uniform measure. Within one Vi the canonical isomorphism between distinct CLL Catakes one conditional measure into another owing to the algebraic quality of T.
80
629 But then n (TVjl Ctt-) is identical for all C^- 6 Vi, i.e., it depends only on Vj. We can therefore let \i{TVj\Ca-) ~ x,,
. The numbers xij are then the transition probabilities of our Markov chain with states Vi.
If Li is a k-dimensional area of any Ca- 6 Vi, then x,,- --= LjL^hT1, • , where h, {Li} are the eigenvalue and components Of the eigenvector of the intersection matrix II. As shown by Parry [7], a Markov chain having these transition probabilities is a unique stationary measure in the space 9, with maximum entropy. The theorem is proved. We turn once again to the C-diffeomorphisms of a two-dimensional torus. The measure ^ c constructed in Theorem 5.1 can be singular with respect to the Riemann length on a layer of a contractile foliation. We recall that this layer is a smooth submanifold of class C r . Consequently, a homeomorphism that takes T into an algebraic diffeomorphism can be, and frequently is, singular, i.e., nondifferentiable. We now state a necessary and sufficient condition for it to be differentiate. Let h be the topological entropy of T. We adopt the invariant measures M* and fj." that we constructed in the preceding section for an expansible and a contractile foliatioL, respectively. Then the transformation T, as a measure-preserving transformation, has an ordinary, i.e., metric, entropy for each measure. We designate them h ' m and h'm. THEOREM 5.5. If h = h ' m = h" m , then a homeomorphism taking T into an algebraic C-diffeomorphism is absolutely continuous. Proof. Let T a be an algebraic C-diffeomorphism of a two-dimensional torus, and let U be a homeomorphism taking T a into T. We adopt a Markov partition a t for T a . Its image fa = Voi1 is a Markov p a r t i tion for T. Inasmuch as h' m (T) = h ' m ( £ ; T) = h, where h ' m ^ j ; T) is the stepwise entropy of the partition /3 J, computed according to the measure £ , it follows from a theorem of P a r r y [7] that £ induces in the space fi a Markov chain with conditional probabilities xij, computed according to the intersection matrix n, as in Theorem 5.4. But this measure induces in Q a Lebesgue measure on a torus for the mapping therein of M with the diffeomorphism T a . Hence, it follows that U takes a Lebesgue measure into the measure /J. . We let ju0 designate the image of the Lebesgue measure. Since Mo = £\ it follows from Theorem 5.1, therefore, that ^ 0 induces a measure equivalent to length on the CLL. Replacing T by T" 1 , we infer that ^ 0 coincides with the invariant measure constructed for an expansible foliation and inducing a measure equivalent to length on the ELL. Consequently, ^o induces a length-equivalent measure on the CLL for any p a r tition and induces a length-equivalent measure on the ELL for any partition. We show that ^ 0 is thus a measure equivalent to the Lebesgue measure a on M. Let A a V; and a (A)= 0. We investigate the partitions a~ |Vi and a+\ Vj. Inasmuch as a (A) = 0, we have u (Aj) = 0 for A t = {x :\x(A\ C„T (A)) =f= 0} . The set A t comprises ELL. It follows from the absolute continuity of the foliations that the intersections A^ f] Ca- are canonically isomorphic for distinct CQ:- and A M (A\ |C
[ a {A | Ca+ (x)) da , we deduce that or (A | C0+ (.v))
since nQ induces a length-equivalent measure on almost every C a +, then Ho(-4 H A?j = 0, and, finally, /x0(A.) = 0 .
= 0 on every
Ca+ $ Al • But
M-0 (-^ I Cc+(*)) = 0. Consequently,
Thus, we have obtained /i 0 « cr. There now exists, invariant with respect to T, a set of positive cr-measure on which the measure /i 0 is concentrated. This set should necessarily comprise mod 0 both complete layers of a contractile foliation and complete layers of an expansible foliation. By virtue of a b solute continuity, the set must coincide with M correct to a factor of cr-measure zero. Thus, IJ.Q is equivalent to cr, and the theorem is proved. LITERATURE 1. 2. 3.
CITED
R. Adler, A. G. Konheim, and M. H. McAndrew, "Topological entropy," Trans. Am. Math. S o c , 114, 309-319 (1965). R. Adler and B. Weiss, "Entropy, a complete metric invariant for automorphisms of the torus," P r o c . Nat. Acad. Sci. USA, 57, No. 6, 1573-1576 (1967). D. V. Anosov, "Geodesic flows on closed Riemannian manifolds of .negative curvature," Trudy Matem. Inst. im. V. A. Steklova, 90 (1967).
81
4. 5. 6. 7. 8. 9. 10.
D. V. Anosov and Ya. G. Sinai, "Certain smooth ergodic systems," Uspekhi Mat. Nauk, 22, No. 5, 107-172 (1967). V. J. Arnold and A. Avez, Problemes ergodiques de la meoanique classique, Gauthier-Villars, Paris (1967). J. L. Doob, Stochastic P r o c e s s e s , Wiley, New York (1953). W. Parry, "Intrinsic Markov chains," T r a n s . Am. Math. S o c , 112, 55-66 (1964). Ya. G. Sinai, "Classical dynamical systems with a denumerable-tuple Lebesgue spectrum. II," Izv. Akad. Nauk SSSR, Ser. Matem., 30, 15-68 (1966). Ya. G. Sinai, "A lemma from measure theory," Matem. Zametki, 2, No. 4, 373-378 (1967). D. V. Anosov, "Tangential fields of transversal foliations in C-systems," Matem. Zametki, 2, No. 5, 539-548 (1967).
STRUCTURE OF THE SPECTRUM OF THE SCHRODINGER OPERATOR WITH ALMOST-PERIODIC POTENTIAL IN THE VICINITY OF ITS LEFT EDGE Ya. G. Sinai
1.
UDC 517.43+517.9
Statement of Problem and Formulation of the Results
Paper [1] dealt with the structure of the spectrum of the one-dimensional Schrodinger operator with an almost-periodic potential for large values of E; it was shown that the spectrum contains a Cantor set such that the measure of its complement decreases as E grows. Close results were obtained in [2 ]. Subsequently, the results of [1, 2] were improved in [3], and then were extended to the difference Schrodinger operator in [4]. The present paper deals with the structure of the spectrum of the difference Schrodinger operator in the vicinity of its left edge. Judging by the methods it uses, it is a direct continuation of [I]. Since many arguments and estimates actually repeat those of [1], we omit them here. First, let us describe the operators of interest to us. We confine ourselves to the main example and then indicate its straightforward generalizations. The example we have in mind L. D. Landau Theoretical Physics Institute, Academy of Sciences of the USSR. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 19, No. 1, pp. 42-48, January-March, 1985. Original article submitted February 7, 1984.
34
© 1985 Plenum Publishing Corporation
632 arises in the analysis of what is called the "Frenkel—Kontorova model" in the theory of phase transitions of type "commensurability—incommensurability" (see the remarkable work of Aubry [5]). Specifically, we are concerned with the stationary difference sine-Gordon equation —«„+! -f- 2u„ — wn-i = X sin 2nun, — oo <^n <^ oo.
(1)
In the KAM (Kolmogorov-^Arnold—Moser) theory [6] it is known that for numbers OJ that are approximated by rationals badly enough (the precise conditions are given below) there are numbers Xo(w) = Xo > 0, such that for every X with lXl < Xo one can find a periodic function k (
(2)
L is precisely the difference Schro'dinger operator that will concern us below. An immediate generalization is obtained on replacing sin 2-TTUJJ by V (un) , where V is a periodic function of period 1, analytic in some strip which contains the real axis. Another problem that leads to the study of the spectrum of operator L is the so-called phonon spectrum problem for the configurations of the Frenkel—Kontorova model (see [5]). Differentiation of (1) with respect to
a e C (Sl).
The sequence v = (vn} is called a Bloch sequence if v n = e
a(nw), where
We can state now the main result of this work. THEOREM 1.
Suppose that UJ is such that minlco
J- >-Ar*^
with A, v > 0.
There is an
ZQ > 0, which depends on X and V, such that if for arbitrary EI, 0 < ei < EO. A(EI) designates the set of all e e? (0,ej for which the equation Lv = EV admits two independent solutions that are Bloch sequences, then Z (A (ei)) ^> Ei— expj—e2+av (In — j l , where 1 denotes the Lebesgue measure and di > 0 is a constant. In contrast to [1], where the estimate was universal, here it depends on the nature of the approximation of OJ by rational numbers. As in [1, 2], the proof of the theorem is carried out by passing to a certain reducibility problem for a difference equation with almost periodic coefficients and using the methods of the KAM theory.* 2.
Reducibility for e = 0
For e = 0 we investigate the properties of the sequences v for which Lv = 0, i.e., those whose terms satisfy for all n the equation vn+1 — 2vn + un_i + 2nX (cos 2nun) vn — 0. We already know one sequence satisfying (3): vn = 1 + -j— k((an) . a (=£($*).
(3)
It has the form v n = a(nto),
If v n = b(naj), wherei&(q>) eC(5') , then (3) means that b ((n + 1) to) — 2b (rm) + b ((n — 1) o>) + 2nk (cos 2n (noi + k (im))) b (tm) = 0.
W
Since the sequence {nw} is dense in (0, 1), it follows from (4) that function b must satisfy the equation (41)
b (q> +
633
and the t w o - d i m e n s i o n a l in t h e form
vector
,
/2-2nXcos2.T(ip + t(ip))
b (qp) =
(b (q>), b (
b (
-1 \
The s o l u t i o n
of
(4')
c a n be
written
C'°) (q))"* (
(5)
THEOREM 2. There exists a two-dimensional nondegenerate matrix Q (*?)• such that
-1(9 + o>)Cl°»(
(6)
, X ->• 0.
The assertion of the theorem means that system (5) is reducible to triangular form. Proof•
The indicated reduction is effected in two steps.
Step 1 . Set
we
Then denoting r(
have
K «P+»> c<« („ * . ( „ ) - ( £ , w ( ;£i. u ) r .)=c« („. Step 2.
We s e e k
the needed m a t r i x
in the
form
9(9) = / + (o „,(„))• , > ,
(
, w. ,
,_»
1 ^ W + ' W H + - W 1 -
»1 (T + " ) r (
r ( , — ) ( ! + * { , + -))
r(
S e t tz;, ((p) = r ( < p — 0 ) ) — 1, V ~ J r(ffi) r (*P— tion ° wx (tp) — wx which admits a solution, provided that X -*• 0 is readily derived from the fact This theorem shows that there are there are solutions that grow linearly 3.
w) dtp.
To determine wj we have the usual homology equa-
(
Transformation of the Problem of Finding the Bloch Eigenfunctions
to a Reducibility Problem Consider the matrix
Suppose that we have already found a nondegenerate complex matrix Q (
?-'(9 + o,)C'«W9(m) = ^ P °_J)
36
(7)
634 for some r e a l "quasimomentum" p . This means t h a t g n (cp + co) exp = (2 — e — 2nX cos 2n (
0
) , we have
w-c+-rc-we-(,)-(::)-.(tistsi)We s h a l l n o t w r i t e h e r e t h e exact e x p r e s s i o n s f o r &a (
What i s e s s e n t i a l
to us i s t h e f a c t
I
t h a t 8 U = O (X), 6 „ = O (X), 6„ = 1 + O (X), and 6 „ = 1 + O (X) . the new m a t r i c e s
"-ti?
Set ltJ = $ 6tJ (cp) dcp and i n t r o d u c e o
::£)• ^-I.(««W-^I.
v
Let us examine t h e m a t r i x C ' i n more d e t a i l . I t s e i g e n v a l u e s a r e of t h e form p+ = 1 ± i / e p 2 , where p2 ^ M ^ 2 1 a s e ->• 0 , with c o r r e s p o n d i n g e i g e n v e c t o r s e+ = ( 1 , i a + / e ) , where a+ ^ p2Y _1 a s e -»• 0 . We f i r s t c o n s i d e r a c o n s t a n t l i n e a r change of c o o r d i n a t e s which r e d u c e s C ^ 2 ' to diagonal form. The m a t r i x Q^1^ of t h i s s u b s t i t u t i o n h a s t h e form
Then
A.
A p p l i c a t i o n of t h e Technique of t h e KAM Theory
We a r e now i n t h e p o s i t i o n of a p p l y i n g t h e s t a n d a r d t e c h n i q u e of t h e KAM t h e o r y . The next s u b s t i t u t i o n i s c o n s t r u c t e d e x p l i c i t l y . We seek i t i n t h e form QP) (
( 8
6i? (9) = 0; 1
l+i^-u,(9)-^?(-P + ») + - r J 7 — - « S W - 0 ; 1 —iylft i —»y»pi 1-i/ep,- 4? ,„(,),„, , „, , , ,VI « / m(9) X _ (9) - ,„(»,„ "4? (9 + «>) + \',- .«ff = n 0. 1 + i V ^ f t " " V"" ~ " w r " ' ^ 1 + i / e f t Equations ( 8 . 1 ) and ( 8 . 2 ) a r e r e c o g n i z e d a s o r d i n a r y homology e q u a t i o n s .
•' >
( 8
<"u (9) - «ff (9 + co) + y T (1 - i / i p , ) - 6 g (9) = 1 ?
( 8
"
•2} 3 )
(8.4) They a r e s o l v a b l e
I
provided t h a t \ fiii (cp) dip = 0. But t h e l a s t r e q u i r e m e n t i s met i n view of o u r c h o i c e of m a t r i x r"(2) ** • The attempt to solve the second pair of equations leads to constraints on the admissible values of e, which in our approach single-out the forbidden zones. If wii(y)=2ievtinvw?)(n) in
37
635 and 6if(9) = Se ori "' , 6g ) (B) , then
•oil' (») = - f e(i - i /i> 8 )-' C + ' K f p , , ^ - 1 6 < « i
(n)i
,
«ff (n) = - / i ( l + i VTpJ-i f-^p^--^ "«)" 6l;> (»). The relations written above show clearly how the forbidden zones arise; our method does not work there.
Further, — ~ ' Ltp*
By construction, 6 • • = U. 1J
.1 = 1
and the numbers e
are
I i+iVsPa 1
dense on the unit circle. Therefore, there are values of c for which the n-th denominator is anomalously small, and these must be eliminated from our considerations. Since functions Oij (
Os)
and the following conditions are satisfied: 1) the matrix-function A*-3' is analytic in a strip n p of width p s about the real axis and continuous up to the boundary; also IIA ^s ^ IITT ^ mQ. . r
J
u
pS
b
2) For some c o n s t a n t cs ;> mj1_>0''4 the following i n e q u a l i t i e s hold for | n | ^ Ns = — 6J1 In m Ps+i — PB = 28a : | gtninu + zJzs | > ca ( | n | + 1)~2 # 3)
I f AW (q>) = II Si}' (9) II ,
i Chen $&» ( 9 ) ^ 9 = 0. 0
Then t h e r e e x i s t s a c o n s t a n t m = m (x, a) ^> 0, such t h a t for m i ^m< 1+x > J one can perform a change of v a r i a b l e s £<>> (
> = („" °J, A<"(9)=£1«B,""PA<»(n), A«t(9)= £
^""A"'(„).
The needed estimates are also obtained in the same way as in [1]. After changing the variables, we write the right-hand side as the sum of a diagonal matrix and an error, and then add the mean values of the diagonal elements of the error to the diagonal matrix. From now on the argument goes as in [1]. Thus, if for some e and all s one can perform inductively the indicated changes of variables, our difference system takes in the end the form i<-»(9 + co) = (^ _ J i where zffi and z m are complex conjugate numbers. Notice that IzmI = 1. This follows readily from the fact that the determinant of the original matrix C<e> (
636 Consequently, z « = exp {2nip}t
g^ = exp {—2nip}.
Now let us estimate the removed "forbidden zones."
Suppose E I is given.
smallest ni in absolute magnitude such that I e2j"nit,)— 1 |
Choose the
From the assumption that OJ
I
is incommensurable it follows that |«i| ^ A v ex a_l"2v. Next, consider all n, Inl > I ni I , and for each number e 7 r i n w — a neighborhood 0n s of radius cs(lnl + 1 ) ~ centered at this number. We note that s satisfies the inequality Inl ^ const s a (1 + x ) s . The same arguments as in [1] show that, on the axis of quasimomenta p, to this neighborhood there corresponds an interval of length at most const | exp {Zninut} — 1 | c, (| n j + 1)~2. Therefore, the total length-of the removed interval does not exceed
{ — conste~S+^/ln -i-Y^l t \ E / J
This estimate is equivalent to that given in the statement of the theorem. 5.
Concluding Remarks
1. Our arguments work also in the case of the usual Schrddinger operator with almostperiodic potential, whenever the ground state is a Bloch function with a function a {(f), analytic in some neighborhood of the torus. In the case of an infinitely and finitely smooth function one must very likely apply the more refined technique of Moser in KAM theory (see [9]). The estimates will also depend on the smoothness class. 2. Of extraordinary interest is the investigation of the operator L in the case where the solution of Eq. (1) lies on a cantorus, i.e., on a Cantor set that remains after the destruction of an invariant curve in the KAM theory. The existence of such cantori was proved by Mather [10], Aubry [ 5 ] , and Percival [11]. The indicated problems will be considered in detail in a separate publication. LITERATURE CITED 1. 2. 3. 4. 5. 6.
7.
8. 9. '0. '••
E. I. Dinaburg and Ya. G. Sinai, "On the one-dimensional Schrb'dinger equation with a quasiperiodic potential," Funkts. Anal. Prilozhen., 9, No. 4, 8-21 ( 1975) . E. D. Belokolos, "The quantum particle in a one-dimensional deformed lattice. Estimate of the sizes of the lacunas in the spectrum," Teor. Mat. Fiz., ^5_, No. 3, 344-357 (1975). H. Russmann, "On the one-dimensional Schrddinger equation with a quasiperiodic potential," Ann. New York Acad. Sci., 357, 90-107 (198u). J. Bellisard, R. Lima, and D. Testard, "A metal—insulator transition for almost Mathieu model," Commun. Math. Phys., 88_, No. 2, 207-235 (1983). S. Aubry, "The twist map, the extended Frenkel-Kotorova model and the Devil's staircase," Physica D_7_, 240-258 (1983). J. Moser, "Lectures on Hamiltonian systems," Mem. Am. Math. Soc., No. 8 1 , 1-60 (1968); C. L. Siegel and J. Moser, Lectures on Celestial Mechanics, Sees. 32-36, Springer-Verlag, Berlin-Heidelberg-New York (1971). V. F. Lazutkin and D. Ya. Terman, "Percival's variational principle and commensurateincommensurate phase transitions in one-dimensional chains," Commun. Math. Phys., 9 4 , No. 4, 511-522 (1984). S. M. Kozlov, "Reducibility of quasiperiodic operators and averaging," Tr. Mosk. Mat. ^ , 99-123 (1983). J. Moser, "A rapidly convergent iteration method and nonlinear differential equations. I, II," Ann. Scuola Norm. Sup. Pisa, Ser. Ill, 2£, 265-316 and ^ 0 , 499-535 (1966). J. N. Mather, "Existence of quasiperiodic orbits for twist homeomorphisms of the annulus," Topology, 2\_, bbl-hbl (1982). I. C. Percival, "Variational principle for invariant tori and cantori. Nonlinear dynamics and the beam-beam interaction," AIP Conf. Proc., No. 5 7 , 302-310 (1979).
39
A RANDOM WALK WITH RANDOM POTENTIAL* YA. G. SINAlt (Translated by E. E.
Dyakonova)
Abstract. Recurrence properties are investigated for a model of one-dimensional random walk with random potential arising in polymer physics [1]. Two theorems giving the distinctive localization of probability distributions of the walk are presented. Key words, random walks with random potential, statistical weights, statistical sums, the distribution localization A beautiful model of one-dimensional random walk was offered in [1], [2] for some physics polymer problems and the question about its properties such as recurrence was posed. The problem is to study the trajectories of a simple random walk ui("> = {w(fe), 0 Sj k ^ n}, u/(0) = 0, u(k + 1) - w(fc) = ± 1 supplied with a statistical weight 7rn(a,<">) = e x p | ^ 6 ( f c ) t / ( a , ( f c ) ) | p ( ^ " ) ) = ±
exp j e ^ S ( * ) £ / ( w ( f c ) ) j .
Here b(k) is a sequence of independent random variables taking on values ±1 with probability i ; the function U(x) takes on three values: U{x) = - 1 for x < 0 and £/(0) = 0, t/(x) = 1 for x > 0; e is a nonzero parameter. We introduce the statistical sum Z<0'"> = J ^ ( n ) 7r„(u)(">), where
•Received by the editors October 30, 1992. TLandau Institute of Theoretical Physics of the Russian Academy of Sciences, Kosygin str. 2, Moscow, 117940, Russia.
638 A RANDOM WALK WITH RANDOM POTENTIAL
3S3
p„(w'">) = (,£(0'"))_17rn(u>(n>) and the corresponding probability distribution Pn. Of course, Pn depends on the properties of the sequence {b(k)} and the problem is to investigate the asymptotic behavior (as n —» oo) of the distributions of the random variables w(fc) for typical sequences {b(k)}. The main result of this note are two theorems describing the distinctive localization of these distributions for e = 0. We observe that [1] solved the case of the periodic sequence {b(k)} = {+1; —1; + 1 , —1,...} explicitly by means of the transfer-matrix and also found the localization for e = 0. It is convenient to assume that {6(fc), k § 0} is a part of a two-sided infinite sequence b = {b(k), - o o < k < oo}. We introduce the Bernoulli probability distribution P on the natural
\b(-l) + --- + In what follows we denote by ZQ
1,m
b(-k)\
*'(6) the statistical sum
where ^ ( * " i ' " > 2 ' stands for summation over all the trajectories of simple random walk within the time interval m\ g k g mi such that u(mi) = u(mj) = 0. Of course, both m i . m j should be either even or odd. LEMMA 1. There exist Si(e) > 0 and m"{b) > 0 such that Z^~m'0)(b) g exp{«i(e)m} for all m j> m"(b). We prove the lemma a little bit later. Let m' = m'(n\ b) < n be such that for all m>m! |b(n) + 6 ( n - l ) - r - - - + b ( n - m ) | g m 2 / 3 . If there is no m'(n, 6) < n satisfying the previous condition, we put m'(n;b) = n. Let m" — m"(n;b) < n be such that z^~m,n) g exp{«i(e)m} for all m" <^ m n. We put m " = n if the last condition does not hold. We set now i/(6) = max(m'(6),m"(f>)) and vn(b) = max(m'(n;6),m"(fc;n)). LEMMA 2. There exists an no(b) such that vn(b) = u(T"b) for all n g no(6). In fact, there exists an no(6) such that m'(n;6) = m(T"b) and m"(n;b) = m"(QTn6) for all n > no (6). For m' this statement is elementary and for m " it follows from Theorem 1 below. Assuming that Lemmas 1, 2 have been established, we prove Theorem 1. Let n ^ n0(b) and |s| ^ u(Tnb). By definition Pn(s) = P„(«(n) = s ) - ^ £ ( s ) , r „ ( u / " > ) . Here Yy stands for summation over all the trajectories w
PW=
"
(»)
Ei(^)'
YA. U. SINAI
384
where Z, ' is the sum over the trajectories u / n ) with a given t. Evidently, n and s may be assumed to be even. In this case 2(0,0 g Z ^ . 0 z ( ' . " ) i a n d Z, = ZQ 'Zj , where Z j ' ' " ' is the corresponding sum over the trajectories of simple random walk that leave 0 at time t, hit point s at time n, and do not cross 0 within the time interval (t,n). For this reason pn(s) g £ " r o * Z < t ' n ) ( z £ t , " ) ) - 1 . It is clear that rr{t,n)
Z\ ' ' = exp
L J2 b(k)sign»|«C'")(.),
where g(''") is the probability that the trajectories of simple random walk that leave 0 at time t, hit s at time n and do not cross 0 within the interval (t,n). The rough estimate q(«.") g const (n - t ) - 1 / 2 is sufficient for us. Now we can write n—8
p„(s) g ^ e x p | e | n - t| 2 / 3 + const + - lg(n - t) - «i(e)|n - t| j . t=o We may always assume that n — t ~2. const. Then the right-hand side of the last inequality does not exceed exp{—5(e)|s|}. The theorem is proved. Proof of Lemma 1. We fix a number A = A(e), whose value will be specified below. It is sufficient to consider only m proportional to A, i.e., m = rA. For such m we have (-JA.-(J-I)A) z<-""°>(6)>;Qz;oo •
J'=I l>
where Z^a' ' " ' is the statistical sum over the trajectories such that w ( - j A ) = u(-(j 1)A) = 0, w(m) = 0, for - j A <m<~(j1)A. Then (j-l)A-l
exp It
Y,
,
W
,
+exp
-
-(j-l)A-l
-e
£
6(fc)
J ,(A)
where g(A) ~ const A - 3 / 2 . Therefore, -(j-l)A
logZ0-^--»-^) :
J2 **>
- | l o g A + 0(l).
k=~jA ,It is easy to see that logZ^1 « ~ 1 ' ' constitute a sequence of independent (with respect to j) and identically distributed random variables and their logarithms have expectations of the form 62(e) VA - § log A + O(l) for some 62(e) > 0. For A large enough this quantity exceeds some 63(e) > 0. Now the lemma follows by the strong law of large numbers. Lemma 1 is proved. Remark 1. From the proof of Lemma 1 some explicit estimates of the distribution of u(b) can be obtained. Remark 2. It is natural t o treat u(T"b) as the radius of localization at time n. By the Birkhoff-Khinchin ergodic theorem, for almost all 6, the frequency has a limit with which u(T"b) takes on any given value m. Improving the reasoning and estimates somewhat used in the proving Theorem 1, one can obtain estimates of the distribution of random variables u>(fc) for 0 < k < n. The following theorem holds. THEOREM 2. There exist a random variable P(fe),constants 7 > 0,6{e) > 0,and a number m(6) such that Pn{a>(k) = s } g e x p { - « ( < r ) | s | } for \s\ > u(Tkb)
for almost all 6,n g ni(b) and lg 7 n g k g n - lgT n.
We omit the proof of the theorem. 385 It would be interesting to show that lira„_ 0 0 Pn (u>(k) = s) exists for any fixed k. A c k n o w l e d g m e n t s . I am grateful to S. K. Nechaev, D. B. Mazel, K. M. Khanin, and H. Spohn for numerous discussions of the problem considered.
REFERENCES [1] A. Yu. GROSBERG, S. F. IZRAILEV, AND S. K. NECHAEV, J. Stat. Phys. (to appear). [2] T. GAREL, D. A. HUSE, S. LEIBLER, AND H. ORLAND, Localization transition of random
chains at interfaces, Europhys. Lett., 8 (1989), pp. 9-13.
641
Probabilistic Approach t o t h e Analysis of Statistics for Convex Polygonal Lines Y a . G. Sinai*
U D C 514.172.45+519.2 To A. M. Vershik on his 60th §1.
birthday
Introduction
In this p a p e r we show t h a t t h e problem on t h e statistics of convex polygons solved by I. B a r a n y a n d A. M. Vershik in [1-3] admits a relatively simple analysis by traditional probabilistic m e t h o d s , which provides a slight improvement of t h e results in [2, 3]. Consider a planar convex polygon T issuing from zero such t h a t t h e vertices of T are points of the lattice Z 2 a n d t h e angle between each side of V a n d t h e horizontal axis is in t h e interval [0, TT/2]. T h e convexity implies t h a t the slopes of t h e consecutive sides form a strictly increasing sequence. T h e space of such polygons will be denoted b y I I . T h e m a i n observation t h a t p e r m i t s one t o apply probabilistic methods is t h a t II admits a n alternative, r a t h e r simple description. Namely, consider t h e set X of all pairs i = ( i i , x 2 ) of coprime positive integers. For convenience, we include ( 1 , 0) a n d ( 0 , 1) in X. We set T(X) = X2/X1.
T h e n T ( ( 1 , 0 ) ) = 0 , r ( ( 0 , 1 ) ) = o o , a n d r ( x ' ) ^ r ( x " ) if x' ±
x".
L e m m a 1. The space H is in one-to-one correspondence with the space Co{X) of finite nonnegative integer-valued functions i / ( i ) on X (v(x) is said to be finite if v(x) ^ 0 only for finitely many x). P r o o f . For any v € C0(X), consider t h e points x ( j ) = (x['\x^) a t which I / ( X ( J ) ) > 0 . W e c a n always assume t h a t r ( x ' J ' ) < T ( X ' J + 1 ' ) . Obviously, t h e polygon with consecutive sides i / ( x " ' ) x " ' is convex. T h e converse statement can b e proved in a similar way. T h e proof is complete. Lemma 1 implies t h a t t h e polygons T 6 II m a y be thought of as finite configurations of ideal gas of particles located at points of X. This remark emphasizes t h e statistical-physical n a t u r e of the situation and elucidates t h e essence of t h e m e t h o d s applied in t h e sequel. We now state t h e main results of t h e present paper. Let I I ( m i , m 2 ) denote t h e set of polygons with endpoint (1711,7712). Obviously, T € I l j m ! , m 2 ) if a n d only if 5 3 » e x " ( ( ^ l 1 xi))xi = m i anc' £/ cx u({xi > x2))x2 = m2 • T h e n u m b e r of different polygons T 6 n ( m j , 7712), i.e., t h e cardinality of t h e set l l ( m i , m 2 ) , will b e denoted by A f ( m i , "12) • T h e o r e m 1 . Let m j —* 00 and TU2 —* 00 so that n\2Jm\
- * c , 0 < c < 00.
l n ^ r m , m2) = m ^ m ^ ' l a K ^ / C ^ ) ) 1 / as m i —> c o , where f is Riemann's
zeta
3
Then
+ o(l)]
function.
Theorem 1 was in fact proved in [2, 3], b u t o u r proof differs from t h a t in [2, 3]. Let P T O l i m 2 denote the uniform probability distribution on n ( m i , m 2 ) . We can now investigate various statistical properties of t h e polygons T € n ( m i , m 2 ) with respect to this distribution under t h e passage t o t h e limit described in T h e o r e m 1. Consider t h e rectangle with vertices ( 0 , 0 ) , ( 1 , 0 ) , ( l , c ) , (0, c) a n d t h e curve Cc defined by t h e equation (c(i + I2) 2 = 4c<2 , i.e., t i = (2^/ct^ — t2)/c. Note t h a t for this curve we have (dt2/dti)\t1=o =0 and {dt2/dti)\tl-i — 0 0 . Take an arbitrary 5 > 0 a n d consider t h e strip U{,mi
= {x = ( x i , x 2 ) I | x 2 - m i £ c ( x i / m i ) | < Smi} .
^The work was partially supported by the "Fundamental Research" foundation of the Russian Academy of Sciences. Department of Mathematics of Princeton University. Landau Institute of Theoretical Physics, Russian Academy of Sciences. Translated from Funktsional nyi Analiz i Ego Ptilozheniya, Vol. 28, No. 2, p p . 41-48, April-June, 1994. Original article submitted January 6, 1994. 108
© 1 9 9 4 Plenum Publishing
Corporation
642 T h e o r e m 2 . For any S > 0 the probability
Pmi , m 2 { r C Us,mi}
ierwis to 1 as m i —> oo.
Theorem 2 shows that for large m i the probability distribution P m , m 2 is concentrated on the polygons taken by the scale transformation x\ = Xi/mj, x'2 = xi/mi into an arbitrarily small neighborhood of the curve £2 = £>c{x\). In other words, under this scale transformation the shape of t h e polygons V becomes deterministic. This assertion, as well as Theorem 3, was proved in [2] and [3]. Denote by v(T) the number of vertices of a polygon T . T h e o r e m 3 . For any S > 0 the probability
P
f|"(n
m m2
" llm^ 3
«="* (C^sKP)) 1 "
<'}
tends to 1 as m j —*• o o . T h e following theorem is a version of the central limit theorem for t h e problems under consideration. We assume t h a t a r a n d o m polygon V is represented by a function x2 = T(xi). Under the normalization it goes into the polygon determined by the equation x2 = ~ r ( m i £ i ) . We consider the difference — T(mit) — Cc(t) a n d normalize it, i.e., pass to t h e r a n d o m process
i»,c) = [ i r M - a ) ] » ' ! / 1 . T h e o r e m 4 . For any set of moments of time 0 < t\ < • • • < tr < 1 the joint distribution of the random variables i / m i ( ( j ) , 1 < j < r, induced by the distribution Pmi,m-i converges to a nondegenerate limit Gaussian distribution as m j —* o o . In fact, our argument also implies the Donsker-Varadan invariance principle for the random process 77 m (t), b u t we d o not go into detail here. Our proof of Theorems 1-3 is based on the representation of II as a large canonical ensemble of statistical mechanics and on a two-parameter probability distribution QZl,z3 (0 < zi < 1, 0 < z2 < 1) on I I , which plays t h e role of t h e large canonical Gibbs distribution. By L e m m a 1, this probability distribution may be regarded as being defined on the space of functions v € Co(X). T h u s , we set
«.„«M=
n [(*r,4i)"i*)(i-*r,*j')]= n (*v*z')*'y n &-*?*?)• x=(xi ,i 2 )eX
x=(xi,Z2)€X
i=(xi ,x?)€X
Obviously, each r a n d o m variable u(x) has an exponential distribution with p a r a m e t e r z\xz^ relative to t h e distribution QZl >Z2 , a n d for different x t h e r a n d o m variables v{x) are mutually independent. Furthermore, since t h e coordinates of the endpoint of V are Ylxex v{x)xi •> ^2xax v{x)x2 •> w e have Q I 1 , „ ( n ( m 1 , m 2 ) ) = Y.'Q'I.'^")
= 2 i"" 2 2 m 3 Y[(l-zI1>z?)Sf(mum2)}
(1)
where J^ denotes s u m m a t i o n over all v with YLx&x v(x)xt — m i a n < ^ S ) i e x "(x)x2 — mi • Moreover, v x x u x x = m for Y^ { ) i — " i i and Yl ( ) i 2 t h e conditional distribution induced by t h e distribution Q 2 1 ,z2 is the uniform distribution P m i , m 2 . We can say t h a t P m i , m 2 is the microcanonical distribution with respect t o t h e distribution Q:i y i 2 . These relationships form a basis for the subsequent argument. I wish to t h a n k A. M. Vershik and I. Barany for excellent lectures presenting their theorems on the statistics of convex curves a n d surfaces. This paper has appeared as a result of pondering on t h e probabilitytheoretic aspects of these problems. For m a n y years A. M. Vershik and I have intended t o write a joint paper, b u t this has not happen yet. Hopefully, we are now closer to t h a t goal. 109
643 §2. Proof of Theorems 1, 2, 3 , and 4 It follows from (1) that jV(m,, m 2 ) = 2 f "•>*-"" J ] (1 - 2 f '2j= ) - ' Q , 1 , ! 3 ( E K * ) n = ™i, £ " ( z ) * 2 = m 2) •
(2)
As was noted, the conditional distribution induced by ( J , l i f t on the set of polygons with given JTJi-^x) X! = mi and ^ ^(x) X2 = ™2 is the uniform distribution Pmi t m 2 . Therefore, if the probability of an event with respect to the distribution Q,,,,, is much less than the probability on the left-hand side in (1), then the event has a small probability with respect to ? „ , , „ , as well. This remark provides a basis for the proof of Theorems 1-3. The main idea is to find the values of the parameters 2i and z2 for which the probability on the left-hand side in (1) is maximal. It is natural to choose z\, zi so that the relations
^,^fE"WI')="'''
E'i,'A J2vWxy=m*
(3)
hold for the mathematical expectations
£2 = **,*,( E W ) = E r $ S We now write z\ and z-i in the form. ^ = 1 — 8\jm^ , zi = 1 — 8ijmx and assume that <$! and <52 depend on mj but lie within some fixed limits: 0 < const < <5i, 8% < const. If this is the case, then z?> = (1 - SJml'3)^ /3
*;»> = (1 - S2/m\ )""
=
1 /i SXp{-S1m 1 (l 2 3
= exp{-S2cm / (l
+ o(l))}, + o(l))},
(4) (5)
where o(l) tends to zero uniformly for all <$i, (52 under consideration. Similarly, set x\ = m /
n(l- 2l '^) = exp{E4l-(l-4 J ) m; ' 3 "(l-4 ? )'" :/3 ' 2 )} = e x p | A m J / 3 | /" /
l n ( l - e - s ' , ' - ' s = ' 2 ) d t 1 d t 2 (l + o ( l ) ) |
(6)
with the same remark on o ( l ) . The factor X = (C(2)) _1 occurs since we consider only the pairs x = (xi, x 2 ) where i ] and x2 are coprime. The value of A is equal to the density of such pairs among all the pairs (xi, x 2 ). The integral on the right-hand side in (6) can be calculated explicitly. The substitution r'j = 6iti, t'2 = <S2r2 gives
The last integral is equal to —C(3), which can be verified by the direct series expansion of the logarithm. Thus, v('5i.'52) = -C(3)/'51<52C(2). Furthermore,
E fi
l
_m s- «»(i-<./">i /3 )-' ,, "(i-hi^r^rn-^ - " > ]
/
.
( fX
= i n i / i
Hi
s
1 - (1 - S1/m\' )"»i
773
TTi
"ii ~
TTz
3
hlm\' )<"«»
1 ^ 7 7 7 ^ ^ ( 1 + 0(1))
= _ mi (J_, ( r^ 2) ) (1 + 0(1)) =
^ | ( 1 + o(1)).
644 Similarly, we obtain
^ = - m i (i^' i 2 ) ) ( 1 + o ( 1 ) ) = £™ ( 1 + 0 ( 1 »
If we set E\ = m\ and E2 = cm\, relations (3) pass into
•
then these relations become equations for £1 and 62.
_ ,
C(3) <S?i2C(2)
= 1,
Obviously,
_„
C(3) SiSU(2)
as nil - 1 0 0 , whence ($1 = (C(3)c/C(2)) 1 / 3 , S2 = ( C ( 3 ) / C ( 2 ) c 2 ) 1 / 3 . Thus, the desired quantities z j , z 2 are of the form
respectively. T h e next remark is t h a t in the ensemble with probability distribution Q Z l l * 2 the sums an< ^2xeXu(x)x^ ^ S r e X I / ( : c ) ; r 2 of independent r a n d o m variables obey the local limit theorem of probaajl1 bility theory as m\ —• 00. It follows t h a t if we set d = YlxeX v{x)x\ ^ C2 — T2X£X t/{x)x2 j then QzltzziCi
-rni,(2
=
m2)
const V^i.^Ci •
DZXtZ^2
where DZl t Z2 (£_,), j = 1, 2 , are the variances of the r a n d o m variables £j with respect to the distribution QZi iZ2 . It is easy to check t h a t t h e variances are increasing as const -m^ (also, see the sequel). Returning to (1), we obtain M{mi,
m 2 ) = exp{m 2/3 [<S, + S2c -
V(S,,
52) + o ( l ) ] } = e x p { m y 3 [ 3 ( C ( 3 ) c / C ( 2 ) ) 1 / 3 + o ( l ) ] } .
This completes the proof of Theorem 1. Let us now prove T h e o r e m 3. Using the probability distribution Q* 1( « a w i t h parameters z\ and z2 obtained in the proof of T h e o r e m 1, we introduce r a n d o m variables £x(v) by setting / 1
if i/(x) >
^ 0
if v\x)
=
The random variables ^x[y) are independent for different x , and YIXGX of the polygon T associated with v. T h e mathematical expectation is
ls
^i^)
^
e
nuinDer
of vertices
1/3,
i€X
iex
I Eex X
v
m
i
v
'
o1/3
(1+ »(!))
2/3
+
-*'"(«2fc; "")
s 2/3
U(2)C (3) J 2
1/3
+o(l)
T h e assertion of T h e o r e m 3 now follows from the classical law of large numbers of probability theory. More precisely, to obtain t h e result pertaining to t h e probability distribution P m i , m 2 , we must estim a t e the probability in the law of large numbers by using t h e direct estimation of the fourth moment ^*ii*a [ S i G j f (f*(' / ) — z ? I , 2 r 2 2 )] • ^ e om ifc the details of t h e calculations since they are quite simple. In statistical mechanics t h e corresponding assertion is well known as t h e equivalence of the large canonical and micro canonical ensembles.
645 Let us proceed to the proof of Theorem 2 on the self-averaging of the shape of a random curve. Fix a sequence 0 < rj < r 2 < • • • < r/v • In what follows, we shall set TJ —» 0, T^ —» oo, and maxi<j
ci'V)=
**"(x)-
£
* = i,2.
x€X : rj<x 2 /xi
Obviously, Ci (") (Ci (")) ' s ' n e increment along the axis x\ (x 2 ) of the part of the polygon where the slopes of the segments lie between Tj and TJ+I . The random variables Q are mutually independent for different j with respect to the probability distribution QZI )Z2 • Their mathematical expectations have the form Fr^(,A-rr,
M l ~ *i/mi'3)"-"(l -
V "'lC'Tj+l C(2)
-[/0
1 _ e -( ft+ ^ J )«,"
ft
fr/rnl")-*mr2/i
']( i+ °( m "^'- T -i)))-
(?)
The last integral can be calculated explicitly and is equal to const /(<5i + S2TJ)3 . If max(rj + 1 — r,) —» 0, then relation (7) passes into the differential equation dti
const
d(dt2/dh) that is,
{s1 +
d2t2/dt\ = const(S! +
s2dt2/dt1y S2dt2/dt1f.
Its solution corresponding to the curve passing through the points (0, 0) and (1, c) is (cti + t2)2 = 4ct2 . Thus, if r is a random polygon in the ensemble Co(X) ~ II, then the mathematical expectation of the polygon 7 = ^-r(mi
* , . , * « - m, ( Z a
1 _ ( 1 _, l / m j/» r i-u ( 1 _ 4 l / r a ;/vs^
Let us find the variances of the random variables p2(s). Note that the random variables Q1 are statistically independent, and each random variable Q3' is a sum of independent random variables u(x)x2 . Hence,
D,ltZ2p2(s)=
J2 x : xi/xi <s
xlD>l,'MX),
646 and owing to the relation £> 2 l , Z 2 i/(x) = (1 + < ' z j 2 - z\Zlz\x*)l{l
- z f ' z j 2 ) 2 , we have
x : x 2 / n <5
4/3
/*j[l + (1 ~ f./m; /, )"i / '"(l - Ja/m^)"i /, "]m 1 - a/3
v
^ JT.A
[l-d-f./ml/V^U-^/ml/'K-'.] 2
«j(l - fi/m;")-»-i"''(l - fa/m^,)-»"I/,'»mrV '
[i - (i - ^ / m i - v 5 ' 3 ' ^ - «52/m;/3)""!/3'=]2 ^
/ 33
/•/• /•
/•
f|(l + e-*.'.-fe'» -
~dwJb<,s,
e-2<.«.-2^'»)
-«n;-^iip-
(1
- * i * 2 ( i + »(i))
/3
= m* cr(S)(l+o(l)). To the random variable P2(s) we can apply the classical central limit theorem of probability theory, which states that the normalized r a n d o m variable (pm{ {s) — E,ltnp\ni(a))/m1' all2(s) obeys the Gaussian distribution law as m i —> 00 . To prove a similar assertion for the probability distribution Pmi ,m2 we must use the multivariate central limit theorem for the joint distribution of the r a n d o m variables £x = ^ x\ u(x) an( i C2 = X ] X 2 I / ( I ) > Pi{s) i Pi(s)T h e appropriate technique is widely known, and we do not go into detail. The joint distributions of the r a n d o m variables £1, £ 2 and of several r a n d o m variables pi(sk), P2($k), k = 1 , . . . , K, are investigated in a similar way. Note t h a t t h e normalized differences Pi(sk+i)
- p . ( - s t ) - EIllZk(pi(st+i) \fD^,z:l(pi(Sk+i)
-
- pi(sk))
._
Pi(sk))
are mutually independent for different k . This completes the proof of Theorem 4. §3. Concluding R e m a r k s Obviously, the m e t h o d suggested can be used for investigating the statistics of closed convex polygons lying inside rectangles with vertices ( i m j , ±7712). T h e derivation of the "action functional" in [2] for the statistics of convex polygons in a neighborhood can be reduced to finding the probabilities of large deviations for the probability distribution Q21 lZ2 • We omit the details. References 1. I. Barany and A. M. Vershik, "On the number of convex lattice polytopes," Geometric Funct. Anal., 2, No. 4, 381-393 (1992). 2. A . M . Vershik, "The limit shape of convex lattice polygons and related topics," Funkts. Anal. Prilozhen., 2 8 , No. 1, 16-25 (1994). 3. I. Barany, T h e Limit Shape Theorem for Convex Lattice Polygons, Preprint, M a t h . Institute of Hungarian Academy of Sciences (1994). Translated by N. K. Kulman
Wolf Prize in Mathematics, Vol. 2 (pp. 647-702) eds. S. S. Chern and F. Hirzebruch © 2001 World Scientific Publishing Co.
CURRICULUM VITAE
Born, January 13, 1931, Belgium EDUCATION A.B. 1951, M.A. 1953, Ph.D. 1955, University of Chicago FACULTY POSITIONS: 1956-58
Massachusetts Institute of Technology, Instructor
1958-63
University of Chicago, Assistant Professor, Associate Professor
1963-
Princeton University, Professor
1968-71; 1985-87
Princeton University, Chairman, Mathematics Dept.
FELLOWSHIPS, ETC.
1955-56
NSF Postdoctoral Fellowship
1962-63; 1971-72
NSF Senior Postdoctoral Fellowship
1961-63
Sloan Foundation Fellowship
1976-77; 1984-85
Guggenheim Fellowship
1989-90
von Humboldt Award
1974
National Academy of Sciences
1982
American Academy of Arts and Sciences
1988
Peking University, Honorary Ph.D.
1992
University of Chicago, Honorary D.Sc.
1993
Schock Prize, Swedish Academy of Sciences
1999
Wolf Prize
648
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150. (with D. H. Phong) Operator versions of the van der Corput lemma and Fourier integral operators, Math. Research Letter, 1, (1994), 27-33. 151. (with D. H. Phong) On a stopping process for oscillatory integrals, The Journal of Geometric Analysis, 4, (1994), 105-120. 152. (with J. D. McNeal) Mapping properties of the Bergman projection on convex domains of finite type, Duke Math. J. 73, (1994), 177-199. 153. (with D. Muller) On spectral multipliers for the Heisenberg and related groups, J. Math. P u r e s e t Appl. 73, (1994), 413-440. 154. (with D. H. Phong) Models of degenerate Fourier integral operators and Radon transforms, Annals of Math. 140 (1994), 703-722. 155. (with A. Nevo) A generalization Math. 173 (1994), 135-154.
of Birkhoff's
pointwise
ergodic theorem, Acta
156. (with D. Muller and F. Ricci) Marcinkiewicz multipliers and multi-parameter ture on Heisenberg (-type) groups, I Invent. Math. 119 (1995), 199-233.
struc-
659 157. Oscillatory integrals related to Radon-like transforms, Journal of Fourier Analysis and Applications, Kahane Special Issue, (1995), 535-551. 158. Spectral multipliers and multiple-parameter structures on the Heisenberg group, Proc. Cont. "Journees Equations aux Derivees Partielles", St. Jean de Monts, 29-Mar-2 Juin 1995, XVI, 1-15, (1996). 159. (with D. Miiller and F. Ricci), Marcinkiwicz multipliers and multi-parameter ture on Heisenberg-type group, II, Math. Zeit. (1996), 267-291. 160. (with J. McNeal) The Szego projection on convex domains," (1997), 519-533. 161. (with A. Nevo) Analogs of Wiener's (1997), 565-595.
ergodic theorem,"
struc-
Math. Zeit., 224,
Annals of Math., 145,
162. (with D.H. Phong) The Newton polyhedron and oscillatory integral operator," Acta Math., 179, (1997), 105-152. 163. (with D.H. Phong) Damped Oscillatory Integral Operators with Analytic Advances in Mathematics, 134, (1998), 146-177.
Phases,
164. Singular Integrals: The Roles of Calderon and Zygmund, Notices of the AMS, 45, (1998), 1130-1140. 165. (with C.E. Kenig) Multilinear L t r s , 6, (1999), 1-15.
Estimates
& Fractional Integration,
Math.
Res.
166. (with D-C Chang, G. Dafni) Hardy Spaces, BMO, & Boundary Value Problems for the Laplacian on a Smooth Domain in RN, Trans. Amer. Math. Soc, 351, (1999), 1605-1661. 167. (with D.H. Phong, J.A. Sturm) On the Growth & Stability of Real-Analytic tions, Amer. Jour. Math., 121, (1999), 519-554.
Func-
168. (with S. Wainger) Discrete analogues in harmonic analysisl, Amer. Jour. Math., 121, (1999), 1291-1336. 169. (with M. Christ, A. Nagel, & S. Wainger) Singular and maximal Radon Annals of Math., 150, (1999), 489-577.
transforms,
Actes, Congres intern, math., 1970. Tome 1, p. 173 a 189.
SOME PROBLEMS IN HARMONIC ANALYSIS SUGGESTED BY SYMMETRIC SPACES AND SEMI-SIMPLE GROUPS by E. M. STEIN CONTENTS
1. Introduction: Euclidean background
173
Part I. — Analysis on the boundary 2. Examples of boundaries 3. Singular integrals on nilpotent groups 4. Some applications
175 175 176 178
Part II. — Analysis on the boundary, higher rank case 5. The Siegel upper half-space 6. Poisson integrals 7. Matrix space M„(R)
180 181 182 184
Part III. — Analysis on the group 8. Euclidean Fourier transform 9. Other problems on the group manifold References
185 185 187 188
Our purpose is to survey some recent contributions and also to suggest several avenues of further development in the area of analysis indicated by the title of this talk.
1. Introduction: euclidean background. We begin by saying a few words about the classical case corresponding to IR1. In order to facilitate the presentation that follows we single out three main concerns of that theory as points of reference. These are A. The Fourier transform B. The Hilbert transform, n
Ax - y)
-dy
y
C. Harmonic and holomorphic functions in the upper half-plane, U\ = { (x, y),y>0,xe
U1 } .
E. M. STEIN
174
By B we mean of course the whole apparatus that goes with the Hilbert transform, including maximal functions, operators of fractional integration (Riesz potentials), etc., and by C such things as Fatou's theorem, Poisson integrals, Hardy spaces, etc. Now the upper half-plane is the arena of action of the group SL(2, U) of fractional linear transformations; it is the symmetric space of that group. In this setup the harmonic analysis is taking place, in effect, on the space IR1 which is the boundary of the symmetric space ('). There are two points of view we may take about extending these theories, and in particular A, B and C, in the context of symmetric spaces and semi-simple groups. The first point of view, and the one I have already suggested, is to start with a semisimple group and its corresponding symmetric space (of non-compact type), and consider a " boundary ". One then performs the harmonic analysis on the boundary, relating it of course to the objects on the group or symmetric space, such as harmonic or holomorphic functions on the symmetric space, or the theory of unitary representations of the group, etc. The first point of view will be taken up in Parts I and II below. The second point of view is that of considering the (semi-simple) group itself as the primary object of the analysis what we have in mind will be described later, but the best known example that one may cite is that of the " Plancherel formula " for the group (2). We shall be dealing with other problems, however. A few more words about the Euclidean background may be in order. Much of what is indicated by our points of reference A, B, and C can be extended to the context of Euclidean IR". We shall here comment only on the singular integral operators generalizing B (3). Our concern is then with operators of the form
/ -
Tf-
K(y)f(x
- y)dy,
«"
where K is a suitable singular kernel. Under appropriate conditions of existence these operators can also be realized as multiplier operators, namely (Tf)~(x) = m(x)f(x), where * denotes the Fourier transform, and m is in effect the Fourier transform of the kernel K. In the well-known and important case studied by Mihlin and Calderon and Zygmund K(x) is, besides some regularity, homogeneous of degree — n, and has mean value zero on the unit sphere. The multiplier m is then homogeneous of degree 0. The Mihlin-Calderon-Zygmund theory and its variants take care of one important class of singularities of the kernel, but there are many other types of singularities and the study of their corresponding operators represents serious difficulties which are still unsurmounted. I cite an example which is both fundamental for the Euclidean theory and has some bearing on our later discussion. PROBLEM 1 (4). — Consider the case of T when the multiplier m is the characteristic (') For the theory in the closely related and analogous setting where the unit disc replaces the upper half-plane, see ZYGMUND [36], (2) See GELFAND and
NEUMARK [7], and
HARISH-CHANDRA [10],
[II].
(3) See e. g. STEIN [29], and the references given there. (4) For some recent progress in the direction of the solution of this problem, see FEFFERMAN [4], (Added in proof). A counterxample for p =£ 2 has been found by FEFFERMAN.
SOME PROBLEMS IN HARMONIC ANALYSIS
175
function of the unit ball in W. It is known that T is not bounded on LP(R"\ when 1 < p < 2n/(n + 1), or 2n/(n — 1) < p < oo. Is it bounded when 2n/(n + 1) < p < 2n/(n - 1) ?
PART I. — ANALYSIS ON THE BOUNDARY
2. Examples of boundaries. We shall come more quickly to the main points if instead of giving a systematic discussion of the class of spaces X which arise as " boundaries " of non-compact semisimple groups or symmetric spaces, we list some typical examples (5) (6). One type of boundary (that could properly be called the maximal distinguished boundary) arises from an Iwasawa decomposition of G as KAN. Then the boundary in question of the symmetric space has two essentially equivalent realizations; either in its non-compact form, when it is isomorphic to the nilpotent group N, or in its compact form as K/M; M is the centralizer of A in K. One example of this is (2.1)
G = SU(n-l),
G/K is the complex n-ball, K/M is its boundary 2n — 1 sphere. Here X is isomorphic with N, and is defined below; it is the genuine boundary of the realization of G/K as a Siegal domain of type II, equivalent to the complex ball via a Cayley transform (7). X is { (z, a>), z e C " " 1 , coe U1 } , with the multiplication law (z x , co,) °(z 2 , a>2) = (zj + z 2 , m1 +ft)2— 2 Im Zi-z2). Another example of a maximal distinguished boundary is (2.2)
G = SL(n, U),
and X is isomorphic with N = n x n strictly upper triangular matrices of G. Notice that when n = 3 in (2.2) we get a boundary which is isomorphic with the one that arises in (2.1) for n = 2. The problems that will arise however will be quite different since in the context of (2.1) we are dealing with a rank one situation, and in (2.2) we are in the higher rank case. Other examples, which do not arise from the Iwasawa decomposition, are: (2.3)
G = Sp{n, U),
G/K is the Siegel upper half-space = { x + iy, x, y real symmetric n x n matrices, (5) See however the general theory of SATAKE [24], FURSTENBERG [5] and C. C. MOORE [21].
(6) We shall consider primarily the realizations of thfc boundaries in their non-compact form, as nilpotent groups. (7) For the realization of bounded Cartan domains as Siegel domains of type II, see PJATECKII-SAPIRO [22].
E. M. STEIN
176 and y is pos. def.}. structure.
Here X = set of real sym. n x n matrices, with the additive
(2.4)
G = SL(2n, U),
but if portioned into n x n blocks, then the appropriate boundary is isomorphic with I x ] " > as x ranges over Mn(U) = n x n real matrices. Thus X can be taken to be MJU), with its additive structure Notice that X, in both (2.3) and (2.4), is a Euclidean space (of dimensions n2 and respectively); but the problems of interest in these examples will not be the Euclidean ones alluded to in section 1.
3. Singular integrals on nilpotent groups. In generalizing the Euclidean theory to the nilpotent groups which arise as boundaries two fundamental notions need to be introduced: that of dilations (8), and that of a norm function (9). The first concept generalizes the standard dilations in IR given by scalar multiplication, i. e. x -> <5x, 5 > 0, x e IR", and is prompted by the observation that broadly speaking, much of the usual harmonic analysis on W is not only translation invariant, but also dilation invariant. The precise definition of dilations is as follows. We assume that with our group X (which is nilpotent and simply connected) we are given a one-parameter group of automorphisms of X, namely { ad }0<5
set K. A more useful, and somewhat stronger assumption, and the one we shall adopt here, is that when we consider its effect on the Lie algebra of X, namely a j , then aj = 5A, where A is diagonable v/ith all positive eigenvalues. Given such a one-parameter group of dilations we introduce a norm function x -> | x | on X as follows. We have | x | = | x _ 1 |, also: (3.1) (3.2)
|x |
|x|>0 is C00 on the set where
|x | > 0
the measure (3.3)
dx
M
is invariant under dilations; here dx is Haar measure on X. For the purposes of Part I we add the important assumption: (3.4) if and only if x is the group identity. (8) See STEIN [27], (9) See KNAPP and STEIN [16].
| x | = 0,
SOME PROBLEMS IN HARMONIC ANALYSIS
177
This is equivalent with the statement that the sets {| x | < C } are bounded. We shall see that whether we impose (3.4) or not makes a crucial difference in the theory. We cite two quick examples. First in R" <xd(x) = 5.x, and | x | = ||x||", where || . || is the usual Euclidean norm. Secondly for the boundary X corresponding to the unit ball cited in (2.1), we may take OL6(X) = (dz, d2a>) if x = (z, to), and |x| = (|z|4 +
ffl2)"'2.
Armed with the above notions, we come now to some of the results that can be proved. First, there is an elegant analogue of the Hardy-Littlewood maximal theorem. Let K be any bounded subset with non-empty interior on which the dilations «s are contractive in the sense that aa(K) <= K, if 5 < 1; e. g. K = { x, | x \ < 1 }. Write Ks = ond(K), and let (3 • 5)
(M/)(x) = sup — — | f(xy) \ dy »o m(Ks) JKd where dy = dm denotes Haar measure. Then M satisfies all the usual properties of the maximal function. As a consequence whenever / is integrable (3 • 6)
m _L_ lim ——
f
| f(xy) - f(x) \dy = 0,
.o m(K3) }Ki
for a. e. xe X. We shall come to the applications of the maximal function and (3.6) momentarily. We discuss next a basic class of singular integrals, written in the form
(3-7)
f
f(x-y)^r.dy
Jx
\y\
where the function fl is homogeneous under <xd of degree 0, that is Q(aa(x)) = ii(x), and £2 is suitably smooth away from the group identity. While the integrals have an interest for all complex values of s, and can indeed be studied as meromorphic functions of s, the range when Re (s) = 0 is the most critical, and we shall thus impose that restriction for the rest of this section. Assuming then that Re (s) = 0, and / is bounded and sufficiently smooth, then the integral (3.7) can be defined in several ways. First if the mean-value of £2 vanishes, i. e. 0(x)dx = 0, 0| <M=S02
then as a principal-value integral (3.7')
ft
lim £-»0
*
u\>t
^
A
\y\
or more generally, if the mean-value of Q vanishes or if 5 ^ 1, then the integral exists as (3.7")
lim Re(s')>0
ft
^ ^
A tVS\
\y\
(10) If s — 1 and the mean-value of Q is nonzero, then the integral cannot be denned without a non-trivial normalizing factor; such a factor has the effect of making it a constant multiple of Ax).
E. M. STEIN
178
The above limits exist for every x and also in the L2(X) norm. If we denote the limiting operator by / -+ T{f), then the first result is its extensibility to a bounded operator on L2(X\
(3.8)
Iir(/)||2<^||/||2.
Unfortunately this fundamental result cannot be proved by following the standard arguments of the Euclidean case of IR", because what would amount to a calculation in terms of the Fourier transform (in the sense of the unitary representations of the group X) seems to lead to unmanageable computations. The one attack which has succeeded in proving (3.8) was suggested by a method originally applicable only in W. It turns out that even in the general case T can be written, in effect, as an infinite sum of uniformly bounded operators (3.9)
T=
f
Tj,
\\Tj\\
j=-0O
where the 7} are almost orthogonal in the sense (3.10) \\TfTk\\
First, the specific
form of the kernel -—-p^ allows a variety of modifications in form. Secondly, I •* I and more interesting, is the fact that the same theory can be carried out in a setting which replaces the existence of dilations by appropriate substitute conditions on the open sets Ks = {x: \x\ < 8}. This generalization is used if one wants to find the analogues of the above maximal function and singular integrals on the compact version of X, which is of course related to X via a Cayley transform.
4. Some applications. We shall now discuss several applications of the theory sketched above. 1. One can construct the intertwining operators for the principal series of representations by means of the operator (3.7). Let G = KAN as before, then the representations induced by irreducible representations of the subgroup MAN are the principal series. Thus there is natural action of G on the boundary X (where X is isomorphic with N), which action generalizes the usual action of SL(2, U) in U1 given (") See KNAPP and STEIN [16]. Earlier ideas of this kind are due to M. COTLAR. (12) This is due to RIVIERE [23], KORANYI and VAGI [18], COIFMAN and DE GUZMAN [3].
SOME PROBLEMS IN HARMONIC ANALYSIS
179
by fractional linear transformations (13), and in terms of which the principal series can be defined. Now the action of the elements of M on X are particularly simple, and these transformations have Jacobian determinant equal to one. This allows us to define the Jacobian determinant corresponding to each element of the Weyl group of G. The square roots of the reciprocals of these Jacobian determinants each provide us with an example of a norm function. It is to be emphasized that each satisfies the properties (3.1), (3.2) and (3.3) for appropriate " dilations " coming from the subgroup A, but in general not the crucial compactness property (3.4). However, in the case of rank one (when dim A = 1), the non-trivial element of the Weyl group gives us a norm function (satisfying also (3.4)), and the dilations are provided by the conjugations of X given by A. All the intertwining operators are then of the form (3.7), after suitable normalization. This construction provides the basic information as to irreducibility and analytic continuation (that is existence and unitarity of the complementary series). The general case, when G has higher rank, can also be treated to some extent, since the intertwining operators can then be written as products of rank-one intertwining operators (14). 2. A special case of the intertwining operators, which arise for a particular representation of the group SU(n, 1) (discussed with its boundary in (2.1)) is the Cauchy integral for the complex ball. In the unbounded realization of the ball, if one takes the Cauchy-Szego kernel which represents H2, then as boundary integrals one is lead = constant x ( | z | 2 + iw)~", and
the singular integrals (3.7) with |x|
| X I = ( I Z | 4 + CD2)"'2 , where (z, w)eC~1 x R\ and aa(z, w) = (<5z, 52co) (15). 3. In this application the space X = W, but the dilations are not the usual ones. These are now given by at{x) = (d^x^, S"2x2,. . ., <5""x„), with x = (xlt. .., x„), where at > 0. We can put | x | = inf { X > 0, £ xf/A2"' < 1 } l a ' . i=l
Then the theory described above reduces essentially to the Euclidean theory of singular integrals with separate homogeneity due to Jones, Fabes and Riviere, Lizorkin and Kree (16). Notice that this has many points in common with example 2 just cited, in that the degree of singularity of the kernels depends on the different directions of approach to the group identity. The present application differs from the preceding, however, in that the convolution is commutative. (13) This comes about by identifying (modulo sets of measure zero) G/MAN with 9N, where 8 is the Cartan involution, and then identifying X with 9N. (14) For details concerning the above application to intertwining operators, see KNAPP and STEIN [16]. Some earlier works in this subject may be found in KUNZE and STEIN [20], and SCHIFFMAN [25]. See also the recent paper of HELGASON in Advances in Mathematics, vol. 5, 1970, 1-154. (15) (16)
See GINDIKIN [9] and KORANYI and VAGI [18]. See e. g., KREE [19].
180
E. M. STEIN
Examples 2 and 3 suggest the following problem which, as should be understood, we state only rather vaguely. PROBLEM 2. — Construct appropriate algebras of singular integrals (or more generally pseudo-differential operators), together with their symbolic calculus, which algebras are to incorporate such examples as 2 and 3 as their building blocks. It is strongly indicated that such algebras should have applications to various non-elliptic problems, in particular in complex analysis, such as behavior near a pseudo-convex boundary and properties of solutions of d problems. 4. As a final application, in this case of the maximal function (3.5) we mention some results dealing with harmonic functions on the symmetric space G/K and centering about Fatou's theorem and Poisson integrals. In the case of bounded functions, the generalization of the boundary behavior guaranteed by the classical Fatou theorem turns out to be a direct consequence of two facts: a) Furstenberg's representation of bounded harmonic functions as Poisson integrals, and b) the maximal function, and in particular (3.6) (17). However, in the case of Poisson integrals in general (e. g. of U functions), much remains to be done. The problems involving Poisson integrals will be discussed more fully when we treat the higher rank case below.
PART II. — ANALYSIS ON THE BOUNDARY; HIGHER RANK CASE We shall discuss now the situation when the assumption (3.4) concerning the norm function is not satisfied, that is when the sets { x: | x \ < c } are no longer bounded. Very often in this case the group of automorphisms of X which preserve the measure dx -—- is larger than a one-parameter group, and so in considering the appropriate dilaI •* I tions it is not entirely natural to limit oneself to a fixed one-parameter group of dilations as we did in Part I. It is for this reason that we refer to the situation when (3.4) is not satisfied as the higher rank case. The rank-one case treated above provides us—at least on the formal level—with an idea of the kind of problems that may be of interest in the general case. However, those results have only a limited applicability in the present context; one instance of this is the decomposition of intertwining operators for the principal series as products of rank-one intertwining operators, already mentioned. In general, however, new and different methods surely need to be developed here. We shall organize our presentation by discussing several different but related problems which reflect the fragmentary state of our knowledge at this stage. (17) HELGASON and KORANYI [12]. This has been superseded by a later results of KORANYI and KNAPP and WILLIAMSON. See [171.
SOME PROBLEMS IN HARMONIC ANALYSIS
181
5. The Siegel upper half-space. We are dealing with the example cited in (2.3). X is the space of n x n real symmetric matrices under addition, which is the Bergman-Shilov boundary of the Siegel domain = { x + iy; x, y real n x n symmetric, y pos. def. }. The action of Sp(n, R) imposes the following structure on X. The dilations are provided by the mappings: ax'-a, where a e GL(n, R), and for norm function we take | x | = | det (x) ^ £l(x) Let us first look at the analogues of the integrals (3.7) with the kernels \x\- where £2 is homogeneous in the sense that £2(ax£a) = Q(x), a e GL(n, U). These integrals have a long history, going back to Siegel, Bochner, and others. We indicate an interesting example arising from the Cauchy kernel. Consider the H2 space of holomorphic functions f(x + iy) on the Siegel upper half-space, those which satisfy | f(x + iy) \2dx < oo.
sup
x
y>0
Such functions have boundary values, namely lim f(x + iy) = f(x) y-*0
2
L (X) 18norm. then (18\ ( ) (5.1)
exists in the
Their integral representation in terms of their boundary values is f(x + iy) = c
(det (t + iy))~^
f{x - t)dt
Jx
where c is an appropriate constant. The boundary value functions form a closed subspace of L2(X), and the orthogonal projection on this subspace is formally given by an operator of the form (3.7), where now
2
. Rigorously the operator is given as the limit as y -* 0 |x| in (5.1), and more particularly as (5.2)
= c (det (x))
lime
E-.0 £>0
(det (t + isl))
- n - l 22
f(x - t)dt.
This operator then is clearly a natural generalization of the Hilbert transform to the present context. A host of questions arise for it, but only a few have an answer at present. We indicate one such unsolved problem: PROBLEM 3. — The operator (5.2) is a projection on L2(X). other LP(X) space!
Is it bounded on any
The close relation of this problem with problem 1 (in section 1) can be aeen as follows. The operator (5.2) is a multiplier operator corresponding to the characteristic function of the cone of positive definite real n x n matrices. When n = 2 this cone is equivalent with a circular cone in IR3, and the intersection of that cone with an appropriate plane is a disc in R2. Thus by a theorem of de Leeuw, a positive resolution of problem 3 for any p, when n = 2, implies the same for problem 1 when n = 2. ( 18 ) See BOCHNER [1].
182
E. M. STEIN
Part of the difficulty in dealing with integrals such as (5.2) lies in the fact that the singularities of the kernel, that is where | x \ = 0, are a whole variety and not merely a point. However, one is not always stimied by this obstacle. An example of this is the Poisson integral, closely related to (5.2); it is given by (5.3)
Py(t)f(x - t)dt
where
n+1
Py(x) and
c (det (2y)) 2 | det (x + iy) \
p
feL (X)
It can be shown that as y -> 0 " regularly ", then the integral (5.3) converges to / almost everywhere, even for feL}(X) (19). This result is rather delicate because as y -> 0, the singularities of the kernel Ps(x) again appear on the variety ] x | = 0. It shows us that the hope of carrying out a theory for integrals of the type (5.2) may not be entirely forlorn. Our discussion for the Siegel upper half-space may be generalized as follows. We consider any bounded symmetric domain of Cartan and realize it as a tube domain when this is possible, or in general as a Siegel domain of type II (20). The Cauchy kernel has also been determined (21), and we can of course pose the analogue of problem 3 (For the complex ball the answer is in the affirmative for 1 < p < oo, by the discussion of section 4). Finally there is an analogue of the Poisson kernel, and the result sketched above is known to hold in that generality (19).
6. Poisson integrals. We have already alluded to Poisson integrals at several occasions, and we shall now discuss them in their generality. Briefly the setting is as follows. For any symmetric space G/K, the class of harmonic functions are those annihilated by all G-invariant differential operators which annihilate constants. Equivalently, these functions can be characterized by the mean-value property. Now every harmonic function which is appropriately bounded at oo can be represented as a Poisson integral, which is in effect a convolution on the group X isomorphic to N. By the mean-value property the Poisson kernel P can be described as follows. We have already pointed out the existence of a natural correspondence between X and the compact homogeneous space K/M, if one leaves out an appropriate set of measure zero (22). If we transplant Haar measure of K/M to X we get a measure of the form P(x)dx, where dx is Haar measure on X. Now the subgroup A acts on X by automorphisms x ->• axa'1, aeA. Let a3 be a one-parameter subgroup of these automorphisms which are dilations in sense defined ( ,9 )
STEIN and N. J. WEISS [33].
(21)
See GINDIKIN [9],
(20) See footnote (7).
(") See footnote (13).
SOME PROBLEMS IN HARMONIC ANALYSIS
183
in section 3. It is then easy to see that for any fe LP(X), 1 < p < JX>, the Poisson integral (6.1)
x
Hy)f(x-aJLy)dy
converges to / in the LP(X) norm, as 5 -* 0. The main real-variable problem can then be stated as follows. PROBLEM 4. — Does the integral (6.1) converge almost everywhere, as S -> 0, for any feL"(X), 1 < p ? One gets an idea of the resistive nature of the problem by observing the increase in difficulty met in passing from the classical case of the upper half-plane, to the case of the product of half-planes contained in the theorem of Marcinkiewicz and Zygmund (23). The farthest advance of the problem at present is the solution of a closely related variant for the symmetric spaces which are bounded domains, already alluded to in section 5. That variant differs from the present one in that it refers to a different boundary of the symmetric space in question, one that can be viewed as a quotient space of the maximal distinguished boundary occurring in problem 4 (24). There is a reason why problem 4 in its general setting seems more complicated than the analogue already obtained for the case of bounded domains. To oversimplify matters a little, it is as follows: the locus of singularities in the latter problem (e. g. { det (x) = 0 } ) is generated by straight lines issuing from the origin. Along these lines the theory for Ul is applicable and then the result follows by a rather delicate calculation which is akin to " integrating " over appropriate lines. In the general case, however, straight lines would have to be replaced by other curves; these curves are the orbits of points under one-parameter groups of dilations. The above raises a simply-stated (and possibly fundamental) problem which we shall discuss only in the context of OS" . Let y(t) be the curve y(r) = sign (r) (At 11 \" , A2 \t \"2,. .. ,A„ \t |°") where A^,..., A„ are real, and at > 0. Consider the analogue of the Hilbert transform
(6.2)
{Tf)(x)= [" f(x + y(t))^ J — oo
t
(Notice that if a t = a2 . . . = a„, then this reduces essentially to the classical Hilbert transform along the direction defined by (Alt A2,..., A„)). Consider also the associated maximal operator (6.3)
(M/)(x) = sup j \\f(x
+ y(t))\dt
PROBLEM 5. — Is there an LP(R") theory for T and M? An analogous result for nilpotent groups (in particular for M) could be applied to the solution of problem 4. (") See ZYGMUND [36], Chapter 17.
(24) This incidentally raises the question of giving an intrinsic characterization of the functions which arise as Poisson integrals for the other boundaries.
E. M. STEIN
184
There is one hopeful indication that may be mentioned concerning problem 5. A calculation carried out by Wainger and myself (see [31]) shows that the operators (6.2) when suitably defined is bounded on L2(R"), (and the bound does .not depend on Ait A2,- •., An).
1. The matrix space M„(U). We shall now consider the example (2.4), with X = M„(U) the n x n real matrices, and G = SL(2n, R). Here we take as dilations the mappings x -> axb~l, with a, be GL(n, R), and as norm function | det (x) |" . This example has obviously some resemblance to that of the Siegel upper halfspace in section 5, but it differs from it in that the space Mn(R) has not only the obvious additive structure (its group structure), but upon removal of a set of measure zero what remains also has a multiplicative structure {Glin, R)). The situation has an analogy with that of a field (e. g. R1) where one of the concerns is with the interplay of an additive and a multiplicative harmonic analysis. The additive harmonic analysis here is that given by (7.1)
P(f)= f
e^"^f(y)dy,
while the multiplicative analysis (the analogue of the Mellin transform) is given by the unitary (infinite-dimensional) representations of GL(n, R). This interplay is at the bottom of the results detailed below (See also section 8). The most direct analogue of the integral (3.7) arises if £2 = 1. We consider therefore (7-2)
/,(/)=[
f{x-y)-^r,
The L 2 theory of this integral is contained in the following statement (2 5 ). Suppose / is C00 and has bounded support. (7.2) initially defined as an absolutely convergent integral when Re (s) > 1
has a meromorphic continuation into the whole comn plex plane, and when Re (s) = 0 the operator / -> Is(f) is unitary modulo a multiplicative constant. More precisely, with 7*(s) = f\ «(ns - j + 1),
<x(5) = T t ^ - T j j V
we have that when Re (s) = 0, Is is a multiplier operator with multiplier y„,(s) | x \~s. The above also has the following consequences: (a) The facts just stated can be reinterpreted by saying that the Fourier transform of the distribution | x | - 1 + s is y^(s) \ x \~s, where both distributions are defined by analytic continuation. This functional equation is closely related to the functional equations of generalizations of the zeta function, and is therefore of interest in several number-theoretic questions (see also the generalizations in (8.3) below). (25) See the references cited in footnote (").
SOME PROBLEMS IN HARMONIC ANALYSIS
185
(b) The operators (7.2) also serve as intertwining operators, but not for the principal series. They arise typically in the " degenerate series ", in this case for the group SL(2n, U). (c) If we write A(s) = y*'(s).^, and B(s) as the multiplication operator by | x | " s , then as we have seen A(s)B(s) is unitary when Re (s) = 0. In addition /4(s)B(s) has an analytic continuation as bounded operators (on L2(M„(R)), in the strip 0 < Re (s) < l/2n. This fact is important in constructing certain uniformly bounded and unitary (complementary series) representations of the group SL(2n, R). There are many variants and generalizations of the above that can be suggested; we shall discuss briefly one typical of those we have in mind. The underlying space X will be IR" and we will pick a fixed non-degenerate quadratic form Q on it, which for simplicity we normalize as Q(x) = x\ + x\ . .. + x2 — xl+ j . . . — x2. We introduce the norm function | x | = | Q(x) |" /2 . The analogue of the integral (7.2) is the integral (7.3)
f(.x-y)\Q(y)\-"{1-s)dy
/ s (/) =
It has well-known analytic continuations, going back to M. Riesz and Gelfand and Graev (26). We let B(s) denote the operator of multiplication by | x \~s = | Q(x) \~"s'2. PROBLEM 6. — Are the IsB(s) bounded operators on L2(X) in some strip of the form 0 < Re (s) < c? ( 27 ). An interesting approach to this problem might be to study the decomposition of the action of 0(n, Q) on L2(U"), since after all, the operators IsB{s) commute with this action (28).
PART III. — ANALYSIS ON THE
GROUP
8. Euclidean Fourier transform. The interplay of the additive and multiplicative harmonic analysis on M„(U), mentioned in the previous section, will now be outlined. We take the additive Fourier transform given by (7.1). A simple change of variables leads to a slight modification of itself, which we shall call &* where now (8.1)
&*(f) = e*f
with the convolution taken on the group GL(n, U), and e(X)
(26) See GELFAND et al. [8].
=
e
2*l"<*-l)|x|-»/2_
(21) When Q is definite, the answer is yes, with c = 1/2. The cases n = 1 and 2 are in KUNZE and STEIN [20]; their method essentially applies to all n, but in the definite case only. When n — 4, k = 2, we are back to M2(K), so c = 1/4. (28) Part of the decomposition of the action of 0(«, Q) on L2(R") is in the book of VILENKIN [34].
186
E. M. STEIN
The properties of &* are then twofold: &* is unitary on L2(GL(n, R)), and secondly &* commutes with both left and right group multiplication, i. e. with the action f(x) - • / ( a - 1 * ) , f(x) - f(xb), a, b e GL(n, R). (The original & had this commutation property only when both a and b were orthogonal). &* is therefore a central operator on L2(GL(n, R)). From this it follows by a general form of Schur s lemma that whenever x -* p(x) is an irreducible unitary representation of GL(n, U) we may expect that (8.2)
p ( ^ * ( / ) ) = y(p)p(f)
whenever / and #"*(/) are in L^GLin, R)) n L2(GL(n, R)). Here y(p) is a constant factor which depends only on the representation p. This identity is formally equivalent with the statement
(8• 3)
KjTF^) = Ts(P)p('X"1}' X '"'
where the factor ys{p) can be immediately read off from the factor y(p). When p is the trivial representation, then ys(p) reduces to the factor y^(s) of the previous section. The other cases where the factor ys(p), (and thus y(p)) has been computed explicitly are those for the representations p which arise in the decomposition of L2(GL(n, R)), (i. e. those which occur in the " Plancherel formula " for the group). In this case, because of the unitary character of &*, all the factors y(p) have absolute value one. It is particularly simple to describe these factors in the analogous case corresponding to Af „(C). In that case if the representation is induced from the character of the triangular subgroup which has value
for a triangular matrix with eigenvalues ($!,.. .,<5„), then
•M = fi { W ^ ± ^ \ T ^
}
The formulae in the case M„(R) have a similar appearance but are more complicated because there are now [n/2] + 1 different series of representations which occur in the L2 reduction of GL(n, U) (29). The mapping / -• p(f) may be viewed as the natural generalization of the Mellin transform (to which it reduces when n = 1). The explicit determination of the factors y(p) which occur in (8.2) gives the desired multiplicative analysis of the additive Fourier transform in M„(R). This " Mellin transform " analysis of & is the main tool in the proof of several results of the previous section, in particular those stated in paragraph (c). (29) The results sketched above, and those in section 7, were first obtained in the complex case (corresponding to M„(Q); see STEIN [26]. In the real case they were obtained by GEIBART [6], but in the meanwhile several of these problems had been dealt with from a different point of view by GODEMENT (unpublished), and JACQUET and LANGLANDS [14].
have also obtained extensions to the ^-adic analogue, when n = 2.
These authors
SOME PROBLEMS IN HARMONIC ANALYSIS
187
A related question arises by analogy with the ordinary Fourier transform on R". The fact that the Fourier transform commutes with rotations leads to a well-known decomposition of L2(R"), compatible with the Fourier transform. The various invariant subspaces are defined in terms of spherical harmonics, and the restriction of the Fourier transform to each can be described in terms of appropriate Bessel functions (30). The theory of higher Bessel functions, in the setting of matrix spaces, has been started by Bochner (31), but much still remains to be done. This discussion is the background for the following problem. PROBLEM 7. — Describe the action of the Fourier transform & on L2(M„(R)) when restricted to the subspaces invariant under the action f(x) ->• f(a~lxb), a,beO(n), in terms of appropriate generalizations of spherical harmonics and Bessel functions.
9. Other problems on the group manifold. The last general question we shall deal with is the following. Is it possible to develop a systematic generalization of some of the objects dealt with in Parts I and II, such as Hilbert transforms, boundedness of various convolution operators, multipliers, etc. but on the semi-simple group itself, and not on one of its boundaries. For compact groups, the answer is surely yes (32). However, for non-compact groups, the situation seems to be far from clear. Part of the difficulty of the problem there, and also its interest I believe, is that unlike the classical case the group Fourier transform of an LP function, 1 < p < 2, is actually analytic in some of its parameters. It is thus more like the classical Laplace transform than the classical Fourier transform. The analyticity of the Fourier transform is intimately connected with the possibility of analytic continuation of the representations of the non-compact semi-simple groups, but even this subject is far from understood (33). To get a better inkling of the nature of these questions, we pose the simplest convolution problems. Suppose we know the LP classes of two functions / and g, what is the class of / * g ? There is a very general answer, valid for any locally compact unimodular group, and it is given by Young's inequality and its variants. Young's inequality is ll/*gllr
where
i = - + - - 1.
The variants of Young's inequality (which include the theorem of fractional integration for R" of Hardy, Littiewood and Sobolev) arise when we replace these norms by " weak-type " norms. For U" these inequalities are in the nature of best possible; for semi-simple groups this is far from the case. In fact the evidence already at hand, and described below, suggests the following L 2 convolution problem for semi-simple groups. (30) See e. g. STEIN and WEISS [32], Chapter IV. (31) See BOCHNER [2] and HERZ [13]. (32) See STEIN [28], where part of this has been carried out; see also COIFMAN and DE GUZMAN [3] and N. J. WEISS [35]. (33) See KUNZE and STEIN [20], and the survey article, STEIN [30].
E. M. STEIN
188 PROBLEM 8. — Suppose
G is semi-simple
and has finite center.
ii/*sii2<^,imiPiisii2,
if
Prove
that
I
This problem involves only the relative sizes of | / | and \g\, and thus, one would think, should be resolvable without any detailed study of the g r o u p Fourier transform of / or of analytic continuation of representations. Paradoxically however, that a p p r o a c h is the only one that has h a d any substantial success so far. T h e answer to problem 7 is k n o w n to be affirmative in the following cases ( 3 3 ). (i) G = SL(2, U) (ii) G is any complex classical group, i. e. SL(n, C), SO(n, C), or Sp(n, C) (iii) G is any semi-simple group, but the function / is assumed to be bi-invariant, i. e. f(k1xk2) = f(x), when / q , k2e K, and K is a maximal compact subgroup of G.
REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
[17]
S. BOCHNER. — Group invariance and Cauchy's formula in several variables, Ann. of Math., 45 (1944), pp. 686-701. —. — Bessel functions and modular relations of higher type and hyperbolic differential equations, Com. Sem. Math. Lund (1952), pp. 12-20. R. COIFMAN and M. DE GUZMAN. — Singular integrals and multipliers on homogeneous spaces, to appear. C. FEFFERMAN. — Inequalities for strongly singular convolution operators, Acta Math., 124 (1970), pp. 9-36. H. FURSTENBERG. — A Poisson formula for semi-simple Lie groups, Ann. of Math., 11 (1963), pp. 335-386. S. GELBART. — Fourier analysis on GL{n, R), Proc. Nat. Acad. Sci., 65 (1970), pp. 14-18. I. M. GELFAND and M. NEUMARK. — Unitdre Darstellungen der Klassischen Gruppen, Berlin, 1957. I. M. GELFAND et al. — Generalized functions, vol. 1, New York, 1964. S. G. GINDIKIN. — Analysis in homogeneous domains, Uspekhi Mat. Nauk, 19 (1964), pp. 3-92. HARISH-CHANDRA. — The Plancherel formula for complex semi-simple groups, Trans. Amer. Math. Soc, 76 (1954), pp. 485-528. —. — Harmonic analysis on semi-simple Lie groups, Bull. Amer. Math. Soc, 76 (1970), pp. 529-551. S. HELGASON and A. KORANYI. — A Fatou-type theorem for harmonic functions on symmetric spaces, Bull. Amer. Math. Soc, 74 (1968), pp. 258-263. C. HERZ. — Bessel functions of matrix argument, Annals of Math., 61 (1955), pp. 474-523. H. jACQUETand R. LANGLANDS. — Automorphic forms on GL(2), Lecture notes, No. 114, Springer-Verlag. A. W. KNAPP. — Fatou's theorem for symmetric spaces I, Ann. of Math., 88 (1968), pp. 106-127. — and E. M. STEIN. — Singular integrals and the principal series, Proc. Nat. Acad. Sci., 63 (1969), pp. 281-284 ; II, ibid., 66 (1970), pp. 13-17 ; also « Existence of Complementary Series » in Problems in Analysis: Symposium in Honor of Salomon Bochner, Proceedings to appear in 1971. — and R. E. WILLIAMSON. — Poisson integrals and semi-simple groups, to appear in J. d'Analyse Math.
676 SOME PROBLEMS IN HARMONIC ANALYSIS [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]
189
A. KORANYI and S. VAGI. — Integrales singulieres sur certains espaces homogenes, C. R. Acad. Sci. Paris, 268 (1969), pp. 765-768. P. KREE. — Sur les multiplicateurs dans FLp, Ann. Inst. Fourier, 16 (1966), pp. 31-89. R. A. KUNZE and E. M. STEIN. — Uniformly bounded representations, Amer. J. of Math., 82 (1960), p. 162 ; II, ibid., 83 (1961), pp. 723-786 ; III, ibid., 89 (1967), pp. 385-442. C. C. MOORE. — Compactifications of symmetric spaces, I, Amer. J. of Math., 86 (1964), pp. 201-218 ; II, ibid., pp. 358-378. I. I. PJATECKII-SHAPIRO. — Geometry of classical domains and automorphic functions, Fizmatgiz (1961) (Russian). N. RIVIERE. — On singular integrals, Bull. Amer. Math. Soc, 75 (1969), pp. 843-847. I. SATAKE. — On representations and compactifications of Symmetric Riemannian spaces, Ann. of Math., 71 (1960), pp. 77-110. G. SCHIFFMAN. — Integrales d'entrelacement, C. R. Acad. Sci. Paris, 266 (1968), pp. 4749, pp. 859-861 ; also These Paris (1969). E. M. STEIN. — Analysis in matrix spaces and some new representations of SL(N, C), Annals of Math., 86 (1967), pp. 461-490. —. — The analogues of Fatou's theorem and estimates for maximalfunctions (C. I. M. E., 3" Ciclo, Urbino, 1967), Cremonese, Roma (1968), pp. 291-307. —. — Topics in harmonic analysis related to the Littlewood-Paley theory, Annals of Math., study No. 63, Princeton, 1970. —. — Singular integrals and differentiability properties of functions, Princeton, 1970. —. — Analytic continuation of group representations, Advances in Math., 4 (1970), pp. 172-207. — and S. WAINGER. — The estimation of an integral arising in multiplier transformations, Studio Math., 35 (1969), pp. 101-104. — and G. WEISS. — Introduction to Fourier Analysis on Euclidean Spaces, 1971 — and N. J. WEISS. — On the convergence of Poisson integrals, Trans. Amer. Math. Soc, 140 (1969), pp. 35-54. N. A. VILENKIN. — Special functions and the theory of group representations, Amer. Math. Soc. Transi, Providence, 1968. N. J. WEISS. — Article in preparation. A. ZYGMUND. — Trigonometric series, Cambridge, 1959.
D e p a r t m e n t of Mathematics Princeton University Fine Hall Princeton, N . J. 08540 (U. S. A.)
677 Proceedings of the International Congress of Mathematicians Berkeley, California, USA, 1986
Problems in Harmonic Analysis Related to Curvature and Oscillatory Integrals E. M. STEIN Introduction. A series of developments in harmonic analysis in the last ten years have led us to recognize the increasing importance of certain primitive geometrical ideas such as some notions of "curvature," and certain "metrics" related to the analysis of vector fields and nilpotent groups. Often the exploitation of these properties is intimately connected with oscillatory integrals. These developments have a close relation, both in terms of motivations and applications, with such areas of mathematics as analysis on semisimple Lie groups and symmetric spaces, several complex variables, partial differential equations, and the theory of Fourier integral operators. Because of obvious limitations of space we shall say little about these relations and applications, 1 but instead we shall concentrate on surveying the developments linking curvature, oscillatory integrals, nilpotent groups and real-variable theory. The objects of study. We shall identify three interrelated analytical constructs, which for the purposes of this survey may be considered the central objects of study. These are: "averages of functions," "singular integrals," and "oscillatory integrals." Each of these has its own long and rich history, which we will not try to give; what does interest us here are the closer connections and new points of view which have developed about them. A few introductory words about each: Averages of functions. A basic example occurring at the earliest stages of analysis is that of the fundamental theorem of calculus. Other classical examples arise in the solution of the heat equation d2u/dx2 — du/dt; also the Poisson integral solving the Dirichlet problem, etc. The basic real-variable properties of such averages were subsumed in the 1930s by the Hardy-Littlewood-Wiener maximal function denned in R " by 1 f ( M / ) (x) = sup —— / n r
Vnr
\J\yl
f(x-y)dy
(0.1)
For us the most important change in perspective that occurred is that now one realizes the increasing interest of taking averages over lower-dimensional 'See, however, some of the literature cited below. © 1967 International Congress of Mathematicians 1986
196
678 PROBLEMS RELATED TO CURVATURE AND OSCILLATORY INTEGRALS
197
varieties. E.g., in (0.1) one replaces the balls of radius r by a suitable family of lower-dimensional varieties. The study of such averages is of interest not only for its own sake, but because it is relevant to other problems, such as the behavior of the classical average (0.1) as n —» oo, behavior of solutions of wave equations, etc. Singular integrals. The classical singular integral operator (arising from the Cauchy integral and from the theory of elliptic differential equations) may be written in the standard form T(f)(x)= f
f{y)K{x-y)dy
(0.2)
where the kernel K is homogeneous of degree —n, smooth away from the origin, and has vanishing mean-value. However the requirements of several complex variables and large classes of "subelliptic" equations have forced us to shift our interest to operators with kernels K(x, y) (instead of K(x-y)) with the following features: (a) When the singularity of K(x, y) is along the diagonal, it is no longer "isotropic," but the behavior of K near the diagonal must be controlled by certain "distance" functions and their volume-forms; (b) One must also envisage the situation when the singularity is not just along the diagonal, but along a larger subvariety. Oscillatory integrals. Oscillatory integrals are of many different types and not easily classifiable. Suffice it to say here that the example par excellence is the Fourier transform
/_> J e-2™
(0.3)
JR"
Other variants are obtained by replacing f(x)dx by a fixed density (then the behavior of /(£) as |f| —• oo is the point of interest), or by replacing the phase 2nx • £ by a suitable generalization (then the boundedness properties of the resulting operator is what matters). Several principles. We shall now formulate three broadly-stated (but vague) principles which may clarify the underlying approach in the development below. As with all assertions of this nature, these general observations can be taken in part as interpretation of results already obtained, and also in part as a heuristic guide to future research. (i) For the study of operators whose kernels have singularities of nonclassical type, such related geometric notions as curvature, associated quasimetrics, and their volume-forms, play a basic role. (ii) Properties of curvature are exploitable by the use of oscillatory integrals. (iii) What is often crucial about the curvature in question is its nonvanishing of infinite order. The kind of phenomena that arises, then, can usually be modeled in R " in terms of polynomial behavior, or ultimately in terms of analysis on nilpotent Lie groups. Organization of this survey. We have organized this survey in parallel with the logic of the above general principles. Thus in Part I we consider the situation in R n where the relation between curvature and oscillatory integrals is the clearest.
198
E. M. STEIN
Next, in Part II we deal with the situation for nilpotent groups. This can be taken as a model for the "general" situation treated in Part III (which we refer to as the situation of "variable coefficients"). Some related results are briefly described in Part IV. Part I. R n . 1. Maximal spherical averages. The most graphic example of the role played by curvature in real-variable theory and harmonic analysis occurs when we consider averages of functions—where these averages are not taken over solid balls as in the classical theory ((0.1) above) but, instead, where the averages are taken over the boundaries of those sets; or alternatively, one considers what happens to the standard averages in R n , but when n —• oo. For an appropriate function / given on R n , we define the mean-value of / over the sphere centered at x of radius t by
Mt(f)(x)= [
f{x-ty)da{y).
(1.1)
•>l!/| = l
(Here da denotes the normalized uniform measure on the unit sphere.) The controlling device in the study of limt_,o Mt (/) (x) is the maximal function M(/)(x) = s u p | M t / ( x ) | , (1.2) t>o which is well denned at least when / is continuous. In this context the fundamental result is THEOREM 1. inequality holds:
Suppose n > 2, and p > n/{n — 1). ||M(/)||LP<>MI/||L*.
Then the a priori (1-3)
(See Stein [99], Stein and Wainger [105] for n > 3, Bourgain [4] for n = 2.) REMARKS, (a) One should point out first that inequalities like (1.3) can fail utterly if we replace the sphere in definition (1.1) by other surfaces which are smooth, but for which no curvature is assumed. To see this, observe first that no nontrivial result can hold for the spherical maximal function when n = 1 since in this case Mt{f)(x) = ±(f(x + t) +f(x-t)), and / need merely be unbounded near one point to defeat any Lp inequality. Similar observations show that (1.3) fails for p < n/(n — 1), for the spherical maximal function, and that (1.3) may fail for any p < oo if we merely flatten the sphere near one point. (b) The ideas in Theorem 1 have connection with several other questions in analysis which we mention briefly. First, there is a striking analogy with the behavior of the classical Radon transform defined by
/ - R(f)(a,t) = /
f(x)dx,
J(x,a)=t
with \a\ — 1, t > 0, and where one obtains control of s u p t > 0 \R(f)(a, t)\ when p > n/(n — 1). However, paradoxically, this analogy holds for n > 3 only; the
PROBLEMS RELATED TO CURVATURE AND OSCILLATORY INTEGRALS
199
Radon transform result fails when n = 2 in view of the Besicovitch-Kakeya set. See Marstrand [59], Falconer [27], Stirchartz [108], and Oberlin and Stein [75]. Secondly, there is a close connection with convergence to initial values for solutions of the wave equation, and more generally other hyperbolic equations. For this see Stein [99], Greenleaf [37], Ruiz [86], and Sogge [91]. The proofs of results like Theorem 1 require two sets of ideas, among others: oscillatory integrals and their relation with curvature, and the use of square functions. We now turn to a brief discussion of each. 2. Oscillatory integrals and curvature. Since the family of mean-value operators Mt given by (1.1) are convolutions, it is natural to analyze them by the Fourier transform. Thus (M t (/))^(^) = /(£)<*(*£)> w i t h *(£)= f
2 ixlL
e-
"
do{x).
J|i|=i
It so happens that a is expressible in terms of Bessel functions (i.e.,
m
= 0(|£|-(n-1)/2),
as \(\ - oo,
essentially if and only if S has nonvanishing Gauss curvature at each point of the support of ip. For future reference we now state two further results relating (some kind) of nonvanishing curvature and decay of the Fourier transform. For the first result, Sk will denote a smooth A;-dimensional manifold embedded in R n , dfi = ipda, with ip € CQ°, and da the induced measure on Sk. We shall say that Sk is of finite type at a point XQ G Sk, if Sk has only a finite order of contact with any affine hyperplane through xo- Then one can observe the following (see Stein [103]) LEMMA 1. With Sk and dfi as above, there exists an e > 0, e = e(Sk), that / i ( 0 = 0 ( | £ | - £ ) , as |£| - oo.
so
This result holds in particular when Sk is real-analytic and does not lie in any affine hyperplane, as was previously shown by Bjorck [2].
681 200
E. M. STEIN
Suppose, however, that we wish to reinstate the full decay ( 0 | £ | - ' " - 1 ^ ' 2 ) , at least for hypersurfaces in R n . Then this is possible if we insert a mitigating factor involving the Gaussian curvature. For our second result, S denotes an arbitrary smooth hypersurface in R n , K{x) its Gaussian curvature at x E S, and dfi = ip da as before. LEMMA 2.
For N sufficiently large,
e-**i*-e\K (jjjjv d(J^ /is. In fact (2.3) holds when N > 2n - 2.
=
o(|£|-(»-i)/2).
(2.3)
This lemma is in Sogge and Stein [92]. Another type of estimate for (2.2) which gives 0 ( | £ | - ( " - D / 2 ) , ^ |£| _* 0 O ; i n « m o s t " directions can be found in Randol [80] and Svenson [112]. See also the estimate in §16, which involves certain other geometric notions of interest. In connection with Lemma 2 we state the following PROBLEM, (a) What is the least N for which (2.3) holds? (b) More interestingly, what is the "largest" function (in place of \K(x)\N) for which the integral (2.3) is Oflf l ^ " " 1 ) / 2 ) ? 3. Square functions. The decay properties of the Fourier transform of a surface-carried measure can be exploited by the use of certain square functions. The simplest version of these which arises when considering spherical means is the one given by S(/)(I)_,
,
-K"
|."W)W dt
2
\V2
tdt)
.
(3.1)
By Plancherel's formula, and a decay property like (2.1), it can be shown that | | S ( / ) H L 2 < c\\f\\L2, when n > 4. Then since tnMt
rtd(snMs)ds ds
Jo
/•' Jo
n_.
ftdMa
, Jo
ds
it follows easily that M(/) < c{S(f) + Af(/)}, which shows how the case p = 2, n > 4 of Theorem 1 can be proved. More refined versions of this argument are needed to deal with the situation p = 2, n > 3. An example of what can be achieved by such refinements is as follows. Suppose m(£) is a C1 function on R " which satisfies, for some e > 0, |m(OI + | V m ( O I < c ( l + | e | ) - 1 / 2 - £ . Define Mtf by Mt(fr(0 LEMMA.
=
(3.2)
rn(t£)f(Z).
| | s u p t > 0 | M t ( / ) | || La < A||/|| L 9 .
This version is in Sogge and Stein [92]. Other versions are in Bourgain [3], Carbery [10], and Rubio de Francia [85]. The result above is sharp in the sense that it fails when e = 0.
PROBLEMS RELATED TO CURVATURE AND OSCILLATORY INTEGRALS
201
The Lp theory p ^ 2 is actually implicit in this approach to the L2 theory via square functions. In fact an examination of the above shows that there is some slack in the argument, and a "larger" or rougher version of the operator M is still bounded on L2. Using the L2 credit just established we can combine it with the fact that a "better" version of X can actually be treated (for all Lp) by standard means. Note, however, that the case n = 2 is more intricate here. As Bourgain shows, the standard Lp estimates have to be replaced by some tricky geometric arguments. Thus the general scheme for proving maximal Lp inequalities (to which we will return again) can be summarized: first use square functions (for L 2 ); then interpolate (for Lp). The paradigm for this argument was actually developed in the more abstract context of a general maximal theorem for symmetric diffusion one-parameter semi-groups. (See Stein [95], [96].) The application of square functions in one form or another is crucial for several other results we will discuss. See also §9 below. 4. Further results for surface averages. Having outlined the ideas in the proof of Theorem 1, we come now to some variants and consequences. The first deals with a generalization of Mt where the sphere is replaced by a more general smooth hypersurface S. If we define dfj,(x) = I/J(X) da as in (2.2) we can write
Mt{f){x) = j f{x-ty)dfi(y), and M(f) = s u p t > 0
t > 0,
\Mt(f)\.
THEOREM 2. Suppose that the Gauss curvature of S does not vanish of infinite order at any point of S. Then there exists a po = po{S), po < oo, so that ||.M(/)||LP < ^ P I I / | | L P
whenever p0 < p < OO.
(4.1)
Observe that the assumptions are verified if S is a compact real-analytic hypersurface. The proof of this theorem follows from the decay estimate (2.3) and an interpolation argument with L°° (where the nonvanishing of infinite order is exploited). Note that allowing the curvature of S to vanish of infinite order at one point may invalidate (4.1) for all p < oo. For details of the proof of Theorem 2, see Sogge and Stein [92], for n > 3, and Bourgain [5] for n = 2. Earlier relevant ideas are in Greenleaf [37], and Cowling and Mauceri [21]; see also Cowling and Mauceri [22]. In this connection it is natural to raise the following question: PROBLEM, (a) It may be conjectured that the maximal inequality (4.1) holds for a nontrivial range of exponents if and only if at each point, S makes a contact of finite order with its tangent hyperplane. (b) Related to this is the problem of finding the smallest value of po = Po{S) for which (4.1) holds. 5. Results for R n , n —• oo. The results and techniques described above in connection with the nonorthodox maximal functions apply also to the usual
202
E. M. STEIN
maximal functions and other standard operators in harmonic analysis in R n , in studying the question of what happens to the bounds of the operators when we let n - x x ) . We begin with the usual (centered) maximal function defined by M( n )(/)(z) = sup-
/
f(x ~ y) dy
(5.1)
J\y\
with vn the volume of the unit ball. THEOREM 3 . (a) ||M( n )(/)|| p < A p ||/|| p , 1 < p < oo, where the constant Ap may be taken to be independent of n. (b) For the weak-type inequality we can assert that m{x\M{n)f(x)
>a}< cn/a\\f\\^,
with cn = 0{n).
The idea of the proof of (a) is that we can transfer the spherical maximal inequality (1.3), for a given p and a fixed dimension, to higher dimensions, without increasing the bound. (This transference does not work for the standard maximal function!) Part (b) requires a separate argument which utilizes the general maximal ergodic theorem. Further details are in Stein and Stromberg [104]; see also Stein [100-102]. Another result of this type deals with the basic singular integrals, the Riesz transforms, Rj, j = 1 , . . . , n, defined by Rj{f)~{£) = (*£j/lf!)/(£)• In analogy with the above one can deal with the problem for the bounds for \R{f)\ = (X)"=i l-Rj(/)| 2 ) 1//2 m terms of appropriate square functions—in this case, the n-dimensional extensions of the ^-functions of Littlewood and Paley. The functions in question are 0i(/)(*)=(/
A OU M
m
2
\ I/*
tdt)
,
g{f){x) = j°° \Vu{x,t)\2tdt)1'2
with u(x, t) the Poisson integral of / . PROPOSITION. When 1 < p < oo, the Lp norms off, R(f), g(f), and ^ i ( / ) are all equivalent, with bounds that can be taken independent of n. The equivalence of norms (with bounds independent of the dimension) of / , g(f), and gi(f) are implicit in the treatment given for these square functions in Stein [96] and [97]; see also P. A. Meyer [62]. The inequalities involving the Riesz transforms follow because of the pointwise estimate 9i(Rf)(x)
< g(f)(x).
(5.3)
(A different approach to the bounds of R{f), which exploits the "method of rotations" has been developed by Duoandikoetxea and Rubio de Francia [25].) After this digression we return to Theorem 3 and ask whether centered balls can be replaced by dilates of a fixed convex symmetric body B. That is, for B fixed we consider MB defined by 1 If ( M B / ) ( x ) = sup——r- / f{x-y)dy r>o m{rB) \JrB
684 PROBLEMS RELATED TO CURVATURE AND OSCILLATORY INTEGRALS
203
where rB = {rx,x € B}, and m(rB) is its volume. We raise the question of whether or not we can make W estimates for MB which are independent of B and the dimension n. THEOREM 4. (a) | | M B ( / ) | | P < 4 P | | / | | P , if\
There exists an L €E GL(n, R), L = L{B), so that if m(£) = then
\m(t) - 1| < c\£\,
\
(5.4)
|(e,Vm(0)|
204
E. M. S T E I N
etc.—are taken with respect to a fixed submanifold. There is some analogy to matters described above, but also important differences, as we shall see. The theory for the fixed submanifold began with the case when the submanifold was a curve. It nourished with the early work of Nagel, Riviere, and Wainger [66, and 67]; it was initially developed under the twin incentives of coming to grips with the "method of rotations" for nonisotropic singular integrals and better understanding the boundary behavior of Poisson integral on symmetric spaces. (See the account in Stein and Wainger [105]; some earlier motivation is in Stein [98].) We shall state our results in two forms. First in their "global versions," and then in their "local" versions. The global theorems are, implicitly, consequences of the local versions; their real role is to serve as models for the general theorems. We have formulated the global theorems separately to emphasize a principle formulated earlier, namely, that the properties related to curvature not vanishing of infinite order should be modeled by polynomial behavior. Fix a polynomial mapping P : Rfc —• R n , and let K be a standard CalderonZygmund kernel in Rfc, i.e., K is homogeneous of degree —k, smooth away from the origin, and has vanishing mean-value. (Here the image of P in R " plays the role of our submanifold.) We then define the singular integral T acting on appropriate functions on R n by r ( / ) ( i ) = p.v. f
f{x-P{u))K{u)du.
(6.1)
Similarly we define the maximal function M by M{f){x)
=
sup r -k 0
f(x - P(u)) du
f
(6.2)
J\u\
In order to formulate the local versions we replace the polynomial mapping P by a smooth embedding p of the unit ball B\ in Rfc, into R n ; i.e., Rfc D B\: p —• R". We also suppose that p(0) = 0. The key assumption to make is that p{Bi) is of finite type (in the sense of §2) at the origin. We then define, in analogy with the above, 7\(/)(x) = p.v. f
f(x - p(u))K(u)du,
(6.1')
J\u\
Mi(f){x)
= sup r -k 0
/
f(x - p{u)) du
(6.2')
|. u |l < r
THEOREM 5. (a) T is bounded on Lp, 1 < p < oo; the bounds for the operator do not depend on the coefficients of the polynomial P, but only on the total degree of P. (b) T\ is bounded on Lp, 1 < p < oo, if we assume p is of finite type at the origin. There are similar results for M and Mi in the range 1 < p < oo. Extensions and variants of this result in the context of nilpotent Lie groups are in §8 below.
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205
REMARKS. Some degree of curvature (in the guise of a "finite type" condition or an attenuated convexity) seems necessary for the conclusions of part (b) of Theorem 5. In fact, there is an example of a smooth curve in the plane— which is infinitely flat at the origin—for which the inequalities fail for all p < oo. However, in marked distinction with the situation in §§1 and 4, there are (besides the straight-line) curves which are infinitely flat at the origin for which there are positive results for all p. Further details are in §15 below. We sketch briefly some of the ideas of the proof of Theorem 5. The fact that the local maximal operator Mi is bounded on L2 can be seen by using a square function argument, not unlike that in §3. To define the requisite square function we proceed as follows. Let p(u) ~ YlaaUa be the Taylor development of p at the origin; here aa are vectors in R*. Using these vectors we can find disjoint (possibly empty) subspaces V\, V2,. •. of Rfc so that Vi + Vb H h Vm is spanned by {o Q }| a |<m- Since p is of finite type at the origin, J2j=i V, = Rfc for some N. Next define dilations 6t by 6t{v) = Pv, v € Vj, t > 0. Then 6f1p(tu) —> po(u) as t - > 0 , where po is a polynomial, which again is of finite type at the origin. Next choose 4> e Cg°(R fc ), V € C^°(R n ), <\> > 0, V > 0, with fRk
f
f(x-p(2-iu))
N*(f)=
JR*
[
f{x-82-]{y))^{y)dy.
JR«
The square function in question is then given by » 1/2 \}>0
The fact that ||5(/)||x,2 < > 1 | | / | | L 2 follows easily by appealing to (a slight variant of) the decay estimate given by Lemma 1 in §2. The Lp theory for Mi can then by carried out in the same spirit as the case of the curve: What is important here is that there is an "e slack" in Lemma 1 which can be exploited. It should be pointed out that the L2 result for the operator T in (6.1) is equivalent, via Plancherel's theorem, to the following estimate for oscillatory integrals: let PQ denote an arbitrary real-valued polynomial on Rfc; then p.v. /
eiPo^K(u)du < A
(6.3)
JR.*
where the constant may be taken to depend only on the degree of Po, and not otherwise on the coefficients. For this, see Stein [103]. Part II. Nilpotent Lie groups. 7. Nilpotent Lie groups: background. Let us briefly describe some of the standard aspects of analysis on nilpotent groups that are relevant here. 2 Let TV be a simply-connected nilpotent Lie group, which we will always identify with its Lie algebra n via the exponential map. We assume a fixed direct sum 2
In addition to the papers cited below the reader may consult the survey of Goodman [36].
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E. M. STEIN
decomposition of n = J2T=i Vr •) a r i d a sequence of strictly positive numbers ai,...,am, with which we define dilations {St}, t > 0, on n by the formula St(v) = ta'v, if v € n.. In general such dilations are not automorphisms of the algebra n; however, when they are we will refer to them as automorphic dilations. The group N together with these dilations will be called a homogeneous group. An example is the Heisenberg group H1, identified with R 3 = {(x,y,z)}; its multiplication law is {x, y, z) • {x', y', z') = (x + x',y + y', z + z' + x'y - xy').
(7.1)
We mention here two different types of dilations on H1: (a)
6t: (x,y, z) - • (tx, ty, t2z),
(b)
6t: {x,y,z) ->
(tx,ty,tz).
The first dilations are automorphic; they are intimately connected with the geometry of Hl induced by its realization as the boundary of a domain in C 2 . The second dilations (which are not automorphisms) may be said to be the vestiges remaining from elliptic theory, insofar as it applies in view of the dNeumann problem for this domain. We begin by concentrating on homogeneous nilpotent groups. The more general situation will be taken up in §8. Incidentally, the groups that arise in the examples cited after (7.3) are all homogeneous. The structure of a homogeneous group naturally leads to some basic geometric constructs: a (left-invariant) quasidistance p(x, y) which is homogeneous under dilations, and defined byp(x,y) = \y_1 • x\, where | • | is a suitable homogeneous norm function on N; also the volume function V(t), giving the volume of the ball of radius t (with distance measured by p). Because of homogeneity, V{t) = ct®, t > 0, where Q is the homogeneous dimension of N. We can then define the "standard singular integrals" on N in close analogy with the classical case of R " (as described in (0.2)). Thus we consider convolution operators T: f —• / * K, where the principal-value kernel K satisfies (a)
K is smooth away from the origin,
(b)
K(6tx)=t-QK{x),
(c)
K has vanishing mean-value.
t>0,
(7.3)
Examples where such operators occur are: (1) Intertwining operators (see Knapp and Stein [54]); (2) The Cauchy-Szego operator for the generalized upper half space (see Koranyi and Vagi [55]); (3) Operators arising for the critical estimates for Db, the Kohn-Laplacian (see Folland and Stein [34]); and, more generally, in the analysis of the Hormander [48] operators, as studied in Rothschild and Stein [84]. However it is not our intention to dwell here on these "standard" operators, which are by now well understood. Rather we turn to operators on nilpotent groups whose kernels are carried on lower-dimensional'submanifolds in analogy with results in Part I. We are in fact forced to consider these generalizations
PROBLEMS RELATED TO CURVATURE AND OSCILLATORY INTEGRALS
207
when we make a detailed analysis of the d-Neumann. The reason for this is that the explicit formulae that arise (see Henkin [45], Lieb [57], and also the survey of Krantz [56] for the operators solving d; for the 3-Neumann problem see Phong [76], Harvey and Polking [43], Lieb and Range [58], Stanton [94], and the survey of Beals, C. Fefferman, and Grossman [1]) lead to kernels which are products of the two kinds of homogeneities (7.2). How do we come to grips with these two conflicting homogeneities? It turns out that the kernel K{x,y,z) = L{x,y)A(z)
(7.4)
on H1, where L(x,y) is a standard Calderon-Zygmund kernel on R 2 , and A{z) is the Dirac delta-function, is simultaneously homogeneous with respect to both types of dilations, each of the critical degree! So it is not surprising that the analysis of the operator T, given by Tf = f * K, should be important for the <9-Neumann problem. The L2 theory of T is (because of the particular form (7.4) of K and the multiplication law (7.1)) most easily treated by taking the Fourier transform in the ^-variable. This reduces the L2 theory of T to that of certain oscillatory operators which, put in somewhat more general form, are as follows: Fix m, let B(-, •) be a real-valued bilinear form on R m x R m , K a standard CalderonZygmund kernel on R m , and consider the operator T given by (T/)(z) = p.v. /
ew^K(x-y)f(y)dy.
(7.5)
For the Heisenberg group H n we take m = 2n, B(x, y) = \{x, y), where (•, •) is the symplectic form on R 2 n x R 2 n which defines the multiplication laws on H n , with —oo < A < oo. In this case the operators (7.5) reduce to "twisted convolution" operators on the Heisenberg group—operators which have an interest in their own right (see, e.g., Segal [89], A. Grossman, Loupias, and Stein [40], Howe [52], Mauceri, Piccardello, and Ricci [60]). To summarize this background section: In analogy with Part I we have isolated certain convolution operators carried on lower-dimensional submanifolds and the oscillatory integrals they lead to; the curvature condition in this context is subsumed in the fact that ( , ) is nondegenerate. A more geometric description of this curvature is as follows: for each point P = (x, y, z) in H1 we associate the two-dimensional (affine) plane Mp, where Mp is the left group translate of M0 by P, with Mo = {(x,y,0)}. The value of T(f) at P is given by an integration over Mp; the curvature here is the "rotation" of the plane Mp as P varies. The theory of the operators T and (7.5) can be found in Geller and Stein [35] and Phong and Stein[77-79]. Further related results are in §§8 and 11 below. 8. Some general results for nilpotent Lie groups. We now come to a series of results which will extend and unify much^f the theory in R", presented in §6, with the ideas for nilpotent Lie groups discussed in §7. Here N will be an arbitrary simply-connected nilpotent group, and {6t}, t > 0, a fixed family of dilations, not necessarily automorphic. We fix a real-analytic submanifold V of N/{0} which, for simplicity, we assume connected, and also
208
E. M. STEIN
that V is homogeneous in the sense that St{V) = V, all t > 0. We shall assume we are given a distribution K on N which is invariant under dilations in the sense that (K,f(x)) = (K, f(St(x))), all t > 0, and which agrees, away from the origin, with a measure on V with smooth density (with respect to induced Lebesgue measure da on V), i.e., of the form ip(x)da{x), with rp smooth and having compact support when restricted to {|x| = 1} n V. We define T by Tf = f*K.
(8.1)
In the same setting we can consider an analogous maximal function. Let dp, — xr da, where T is the characteristic function of a compact subset T of V. We define dpt by (d/j,t,f{x)) = (dp,f(Stx)), t > 0. Set (Mf)(x)=
THEOREM 6. LP(N) to itself,
sup \(f*dpt)(x)\. (8.2) t>o Both T and M, defined by (8.1) and (8.2), are bounded on Kp.
A special case of this result deserves to be formulated separately. Suppose K is a principal-value kernel, supported on all of N, which satisfies (7.3). In this case we set Tf = f * K, and similarly if T is an open subset of N of compact closure we write = sapl [ f(x-(My))-1)dy t>o \Jr T and M so defined are both bounded on LP(N),
(Mf)(x) COROLLARY 1.
(8.3) 1 < p.
This shows that the assumption that the group be equipped with automorphic dilations, previously considered essential in proving such results, is in reality unnecessary. Notice however that the balls which are implicit in the definition of (8.3) do not satisfy the Vitali covering property (i.e., if TV is not a homogeneous group, then \y~x -x\ = p(x, y) does not satisfy the quasitriangle inequality). Thus the standard methods do not apply here. Another corollary of the proof is a generalization of Theorem 5, part (a). For this purpose let P be a polynomial mapping from Rfc to N, and K a standard Calderon-Zygmund kernel on Rfc. COROLLARY 2.
The operators f{x-(P(u))-l)K{u)du
/-P-v. I and f —• sup r 0
k
f
'\u\
/{x-ipwr^du
are bounded on LP(N), 1 < p < oo. Their bounds may be taken not to depend on the coefficients of P, but only on its degree. A final corollary provides a unification with the oscillatory integrals in (6.3) and (7.5) and Corollary 1.
690 PROBLEMS RELATED TO CURVATURE AND OSCILLATORY INTEGRALS
209
COROLLARY 3 . Let P(x, y) be a fixed, real polynomial on N x N. Then the operator f -+ p. v. / jr^Kiy-1
• x)f(y) dy
(8.4)
JN
is bounded on LP(N) to itself, 1 < p < oo; again the bounds depend (insofar as P is concerned) only on its total degree. Besides the literature already cited, treatment of some cases of operators (8.1), (8.2), and (8.4) may be found in Strichartz [109], Miiller [63, 64], Christ [14], and Greenleaf [39]. The general theorem as stated, and its corollaries, are contained in the recent work of Ricci and the author [81-83]. We shall now briefly describe the three main ideas involved in the proof of the theorem. First, one passes from the nilpotent group N with its nonautomorphic dilations 6t, to a larger nilpotent group G which has corresponding automorphic dilations, so that N is a quotient of G. In fact we take G to be the "free" nilpotent group of sufficiently high step whose Lie algebra g has as its generators a basis of n. The boundedness properties of the operators T and M may then be reduced to the boundedness properties of (approximately constructed) corresponding operators T" and M' on G. This procedure is in reality a streamlined form of the "lifting" method which will be discussed again in §10. Secondly, the "curvature" properties of our manifold V are expressible by the fact that it is real-analytic and may be assumed to generate the group N. The following lemma takes advantage of this situation. LEMMA. Suppose V is a k-dimensional real-analytic submanifold in a connected Lie group G; suppose that V generates G. Let dfj, be a measure supported in V with a smooth, compactly supported density. Then du = d/i * d/j, * • • • * dfi ^2
(9.1)
and this kernel is often better than K because the integration in (9.1) can have both a smoothing effect and can take into account cancellation properties of K. Classical examples arise when dealing with unitary operators such as the Fourier transform, the Hilbert transform, etc. A trickier example is the theorem
691 210
E. M. STEIN
of Kolmogorov-Seliverstov-Plessner (see Zygmund [115]) which shows, incidentally, that the computations which prove the boundedness of TT* may be quite different from those that show it for T*T. Other examples arise in the theory of semigroups (see Stein [96]) and for Fourier integral operators (see Hormander [49]); closer at hand are the oscillatory integrals (7.5), (8.4), and (11.4) and (12.3) below, and the treatment of Hilbert transforms and maximal functions for "variable curves" (discussed further in §11), as carried out in Nagel, Stein, and Wainger [69]. (ii) Almost orthogonality. The idea of TT* does not always suffice to do the job. It can be refined as follows: One has boundedness of T in L2 if we can find a decomposition of T as Yl Tj, with the norms of the Tj uniformly bounded, and an approximate mutual orthogonality of the different pieces Tj\ the latter is expressed by the requirement that the norms of both T{T^ and T*Tj tend to zero suitably when |i — j | —>• oo. This is the lemma of almost orthogonality of Cotlar and the author. It was applied to prove the L2 boundedness of the standard singular integrals on nilpotent groups discussed above (see Knapp and Stein [54]), and has since found many applications and modifications. One such variant of relevance here (due to Christ [14], and in another form to Greenleaf [38]) is the observation that since the norms of the Tj are bounded, the control of T{T* can be reduced (by applying the principle of TT* repeatedly) to the control of (T*Ti)kT*, for any fixed (large) integer k. This is of particular use in conjunction with arguments such as the lemma in §8 above. (iii) More square functions. We have already seen several variants of square functions (in §§3 and 4) and discussed their usefulness in proving Lp inequalities. Here we shall describe a version closely related to the almost-orthogonal decompositions above. Suppose we can write T = 53Tj, with Tj factorable as Tj = B*Aj. We then define two square functions Si and 52 : S i ( / ) = (E> l-^j/l 2 ) 1 ^ 2 ) 82(f) = (E> \Bif\2)1/2It then follows that the boundedness of T on U> is a direct consequence of the boundedness of / -+ S i ( / ) and g —• 82(g), on Lp and Lv respectively. This type of decomposition and use of square functions plays an important role in the theory of singular integrals of Coifman, Macintosh, and Meyer [16], and David and Journe [24]; the reader should also consult David [23]. Part III. Variable coefficients. 10. Geometry and analysis of vector fields. The passage to variable coefficients via nilpotent groups is nowhere better illustrated than in the study of analysis and geometry of vector fields. The idea of using nilpotent groups was initiated in the study of db and the Kohn Laplacian, and then extended to deal with the operators analyzed by Hormander [48]. Let us describe it briefly. Let Xi,..., Xm be m smooth real vector fields in (a neighborhood of the origin of) R n . We assume that these vector fields and their commutators of order not greater than k span near the origin. It is then possible to "lift" these vector fields by placing them in a larger space, R " x R"' (with additional variables
PROBLEMS RELATED TO CURVATURE AND OSCILLATORY INTEGRALS
211
t € R " ) so that the lifted vector fields are of the form n'
n
Xi=Xi + 'J2aij{x,t) — ,
i = l,...,m,
(10.1)
j
3=1
and so that the Xi have the twin properties: (i) The Xi and their commutators of length not greater than A; span near the origin in R™ x R n '; (ii) The Xi are "free," that is, these vector fields and their commutators satisfy the same relations that the elements of the free nilpotent Lie algebra 7m,k satisfy. (The Lie algebra is obtained as the quotient of the free Lie algebra of m generators Yi, Yi,..., Ym, divided by the ideal generated by all commutators of length > k.) Besides this lifting there is a basic approximation property. First write the vector fields Y%,..., Ym in canonical coordinates (yi,..., yn+n') Yi =
^bij(y)d/dyj. i
Then for each fixed (xo, to) € R n x R " near the origin we can find a coordinate system (j/i,... ,yn+n') centered at (x 0 ,to) so that Xj=Yj
+ Rj,
j = l,...,m,
(10.2)
with Rj vanishing of appropriately high order as (x, t) —> (xo, to) (i.e., as (j/i,..., Vn+n') -> 0). The above two-fold procedure allows one to reduce the study of, e.g., the operator J2^j t o that °f X}^?2> which in turn is modeled on the left-invariant (and homogeneous) operator ^2 Y? o n the nilpotent group G whose Lie algebra is 7mtk- This procedure of appealing to analysis on nilpotent Lie groups goes back to Folland and Stein [34] and Rothschild and Stein [84]; it has since undergone substantial development. (In this connection see, e.g., the survey of Helffer and Nourigatt [44].) In addition one may seek a better understanding of the geometry of the vector fields, also, for example, the size of the fundamental solution of the operator ^2X?. To do this two basic notions are needed: an appropriate "metric" (or quasidistance) and the volume formula for the resulting balls. Given the vector fields Xi,... ,Xm, we can then define a naturally induced metric: the distance p(x, y) is the least time it takes to travel from x to y along a curve pointing in the directions of X\,... ,Xm, always moving at "unit speed." More precisely, p{x,y) = infimum of T so that 7(0) = x, ~i(T) = y, where l{t), 0 < t < T, is a rectifiable curve so that i(t) = Y^JLi aj(t)Xj{l(t)), with
M O I < 1In order to work with this distance it is quite useful to describe it in several equivalent forms. We shall limit ourselves here to one such form, linking it with the exponential mapping, and we shall also describe a formula for Vx{8) = volume of the ball {y\p(x,y) < 6}. For each ordered /-tuple of integers I = (I'I , . . . , ij), |/| = /, where 1 < i3 < m, and / < k, we define the vector field Xi by Xj = [Xi1 [Xi2 • • • [Xi,_ , Xit] • • • ]].
212
E. M. STEIN
Next, for each n-tuple of such J's, {h,h,..., In} = J, write |«7| = | i i | 4- 1^21 + ••• + |/„|, and define Xj by Xj(x) = det(Xj1,Xi2,... ,XIn)(x). The Xj may be thought of as generalizations of the Levi invariants, to which they reduce in special cases. THEOREM 7. exp(^2jaiXi)(x),
(a) For 6 sufficiently small, p{x,y) < 6 if and only if y = with aj constants satisfying \ai\ < c<5lJ'.
(b)Vx(6)~Zj\*j(x)\6lJlFrom part (b) we see, in particular, that the volume of the balls have the important doubling property. Next we state the estimates for the parametrix K(x,y) of the operator
THEOREM
8.
\xll---xtM^y)\
(10.3)
p(x,y).3
REMARK. Notice that for homogeneous nilpotent groups the forms of all these estimates are implied by scale-invariance. The two theorems above, and further results along these lines, are in Nagel, Stein, and Wainger [70, 71]; see also Nagel [65]. There is an alternate theory leading to Theorem 8, among other results. It is due to C. Fefferman and Phong [31], C. Fefferman [30], Sanchez-Calle [87], and C. Fefferman and Sanchez-Calle [32]. This approach has the advantage of being able to treat the general subelliptic operators which are of the form J2aij{x)d2/dxidxj, with {aij(x)} semidefinite and smooth. On the other hand, the first approach described applies to the general Hormander operator J2 ^] + Xo and, for that matter, to any subelliptic operator which is a "homogeneous" polynomial in the vector fields, and it gives analogues of (10.3) for their parametricies. 10. Singular Radon transforms. This topic may be viewed as the culmination in the progression from singular integrals in the translation invariant case of R " treated in §6, to the case for nilpotent groups, as in §§7 and 8, and thence to the general "variable coefficients" situation. Here the complete picture, both with respect to the optimal formulation of the general assertions and to the details of the expected proofs, is not yet fully worked out. So we will describe only an important special case which is of particular interest because of its relevance to the <9-Neumann problem. Let M be a (compact) manifold of dimension m + 1, m > 1. Then, imitating the situation described at the end of §7, we assume that for each P 6 M we are given a submanifold Mp of dimension m, with P G MP\ also for each P we are to be given a Calderon-Zygmund kernel density K(P, •) on Mp, with singularity 3 Except of course in the case where n = 2 and when X\ • • • Xm is asserted only for r > 1.
already span; then (10.3)
PROBLEMS RELATED TO CURVATURE AND OSCILLATORY INTEGRALS
213
at P, so that the mappings P —• MP and P —• K(P, •) are smooth. We then define the singular Radon transform T (mapping C°°(M) to itself) by (Tf)(P)=
f
K(P,-)fP(-)
(11.1)
where fp denotes the restriction of / to Mp. The curvature condition that intervenes here turns out to be related to that occurring (in Guillemin and Sternberg [41]) in the invertibility problem for the more standard versions of the Radon transforms (e.g., when K(P, •) in (11.1) is everywhere smooth). Let $(P,Q) — 0 be a defining function for the relation Q e Mp which is nondegenerate (i.e., d p $ ^ 0, d,Q$ ^ 0). Then the "rotational curvature" form (P) is the bilinear form defined for {vi,v2) € Tp(Mp) x TP(Mp) by L$(«i,t>2) = (dpQ$\p=QVi,V2), where the Hessian dPQ$ is the differential of the mapping $ —* dp$(P,Q) (the differential is a linear map from TQ(M) to Tp{M)*). The curvature condition is: The rotational curvature form L is nondegenerate for each P G M.
(11.2)
THEOREM 9. The operator T defined by (11.1) is bounded on LP{M) itself, 1 < p < oo, if the curvature condition (11.2) is satisfied.
to
(See Phong and Stein [77, 78]; an earlier approach for the case of variables curves in the plane is outlined in Nagel, Stein, and Wainger [69].) REMARKS. (1) There are also Lp inequalities for a related maximal function which is associated to T in the same way as Mi is associated to Ti in (6.1') and (6.2'). (2) A connection between operators of the type (11.1) and Fourier integral operators with singular symbols associated to a pair of Lagrangians has been pointed out by Uhlmann [114]. For the latter class of operators see Guillemin and Uhlmann [42], also Melrose and Uhlmann [61]. (3) To clarify the curvature condition, we point out two cases: First, in the translation-invariant case in R n + 1 the condition means that M 0 has nonvanishing Gauss curvature at the origin. Second, when D is a smooth domain in C™+1, M = boundary of D (m = 2n), then for a naturally attached family {Mp} the condition is equivalent to the nondegeneracy of the Levi form. The idea of the proof of the theorem is first to localize to a coordinate system, (t,x) G R x R m , so that if P = (t,x), Q = (s,y), then the relation Q e Mp is given by s = t + S(t, x, y), with S(t, x, x) = 0, and so (11.2) becomes
where $(x,y) = S(t,x,y). After a Fourier transform in the t-variable, one can ultimately reduce the problem of the L2 boundedness of (11.1) to oscillatory integrals of the form h:f->[
e>x*^K(x,y)f(y)dy.
(11.4)
214
E. M. STEIN
Here K is a compactly supported kernel of a standard pseudodifferential operator of order p,, with — m < p, < 0. LEMMA. / / $ satisfies (11.3), and —m < p, < 0, then for the operator norm of I\ we can make the estimate I | J A I I L ^ < ^ ( I + |A|)" / 2 .
(11.5)
Note the relation between the oscillatory integral (11.4), and those in (7.5), (8.4), and (12.3) below. The special case arising when K is the kernel of an operator in Op(Sj~£°) is in Hormander [50], but the order of decrease in (11.5) cannot be improved beyond (1 + | A | ) _ m / 2 . For the relation of Theorem 9 and the d-Neumann problem, see Phong and Stein [79]. 12. Averages over variable hypersurfaces. We return to our first topic, maximal surface averages discussed in §§1 and 4, and try to obtain diffeomorphicinvariant versions of these results. Our setting is as follows: For each x e R™ and e > 0, we are given a smooth hypersurface SStX C R". The family of surfaces {S £ , x } will be described as images of a fixed hypersurface S: We assume that S£tX = p£>x(S), where pe>x is an imbedding map for each (e, x), and such that pSiX(u) is smooth for (e,x,u) e [0, oo) x R n x 5 . Fix a cut-off function ip € C Q ° ( R " ) and define the mean-value operator M£{f){x)=
f_f{x + ep£tX{u))i>{x)d<j{u), (12.1) Js with da a measure with smooth density and compact support on S. The relevant curvature conditions are (a) The hypersurface S0,x, for each x € R n , has Gauss curvature not vanishing of infinite order at any point. (b) Same as (a), except the Gauss curvature is assumed to be nowhere vanishing. Notice that these conditions are diffeomorphic invariant, in the sense that if $ : R n —* R n is a diffeomorphism, then $(x + epe,x(u)) = y + ep£,y(u),
if $(z) = y
where p satisfies the same conditions ((12.2)(a) or (b)) as p. T H E O R E M 10.
Let
M(f)=
sup | M e o ( / ) | . 0<£<£0
If SQ is sufficiently small, the mapping f —• M(f) is bounded in Lp to itself in the following situations: (i) when (12.2(a)) is satisfied, when n > 6, and p > p0, for a finite po = Po({se,x}); (ii) when (12.2(b)) is satisfied, when n > 3, p > n/(n — 1). For part (ii) see Greenleaf [37], but the argument given there does not extend to cover case (i). As to the proof of that (see Sogge and Stein [93]), while
PROBLEMS RELATED TO CURVATURE AND OSCILLATORY INTEGRALS
215
it is not, strictly speaking, modeled on nilpotent groups, it has nevertheless essential features in common with the argument of Theorem 9 for singular Radon transforms. In fact, one proceeds, after an appropriate coordinate choice and taking n = m + 1, to study the key square function via the Fourier transform in one variable. (It is at this stage of the proof that the restriction n > 6 enters.) One is again led to oscillatory integral operators elX*^\H(x,y)\N
h-.f^f
(12.3)
m
JB.
where $ is a real function, H(x,y) is its Hessian = det(d2$/dxtdxj), and m m ip € Cg°(R x R ) . In effect, the properties of $ and its Hessian express the curvature conditions imposed. In analogy with Lemma 2 in §2 and the lemma in §1 one can prove the following estimate for the L2 norm of the operators I\: LEMMA.
For N sufficiently large \\h\\L^L*
+ \\\)-m/2.
(12.3)
In fact, (12.3) holds if N > 5m/2. Theorem 10 suggests the following PROBLEM. Can we extend both (i) and (ii) to n > 2? Part IV. Some further results. We describe very briefly some related areas of research in harmonic analysis where oscillatory integrals (and curvature) play an important role. 13. Restriction theorems and Bochner-Riesz summability. The appelation "restriction theorem" refers to the a priori inequalities for restriction of the Fourier transform in R n to submanifolds S of the form
( ^ 1/(01'<M0)
9
<^||/||LP,
(13.1)
which go back to the observation (unpublished) of the author that such estimates do hold whenever the Fourier transform of da has a decay at infinity like 0(|£|~ £ ). Thus by Lemma 1 in §2 there are such restriction theorems for all 5 of finite type. Bochner-Riesz summability is closely connected with the problem of finding Lp estimates of the Fourier multiplier operators on R™ given by m(fl = ( l - K | 2 ) i ,
(13.2)
and
S&fr=m(t/R)f(t). The results in C. Fefferman [28], Carleson and Sjolin [11], and Tomas [113] related restriction theorems for the sphere with the behavior of the multiplier (13.2), and obtained sharp Lp estimates for both problems when n — 2, and also partial results when n > 3. The first proofs of some of these results were
216
E. M. STEIN
reformulated in Hormander [50] in terms of boundedness properties of certain oscillatory integrals as mappings from L p ( R n _ 1 ) to L 9 (R"), with q = ( ^ f ) p ' . One considers the operators (Txf)(x)=
eiX*l*rtrl>{x,y)f(y)dy,
f
xeRn,
(13.3)
where ip € C Q ° ( R " X R n - 1 ) ; for $ one makes an assumption similar to everywhere nonvanishing Gaussian curvature. Then one has the inequalities ||7A(/)||L<(R")
< AA-^H/HiPfRn-i).
(13.4)
These hold for 1 < p < 4, when n = 2, as shown by Carleson and Sjolin, and for n > 3, when 1 < p < 2 (see Stein [103]). The main problem left open is the range 2 < p < 2n/(n — 1). An alternate approach for n = 2 (which emphasizes the geometry of rectangles in Fourier transform space) is in C. Fefferman [29]; it was pursued by Cordoba [17] who found related I? estimates for rectangular maximal functions in R 2 (see also §14 below). Further, Carbery [8] used this approach to prove sharp estimates for the key square function ,•00
\l/2
Of /
ISk-S^dR/Rj
in R 2 , for 2 < p < 4, obtaining a maximal theorem for Bochner-Riesz summability. Earlier use of this square function is in, e.g., Stein and Weiss [106, Chapter 7]. For related results in this area see also Igari [53], Cordoba and Lopez-Melero [20], Cordoba [18], Carbery [9], Christ [13], Seeger [88], and Sogge [90]. 14. Directed maximal functions; in particular, lacunary directions. For any unit vector 0 in R n define the directed maximal function M$ by Me{f){x)
=
sup— / h>0 &"•
f{x-t$)dt
.
J-h
In several problems one is interested in the boundedness properties of MY where Mr{/) = sup$er Mg(f), with T a given set of directions. (When T = S n _ 1 , n > 2, there are no Lp inequalities for p < oo because of the Besicovitch-Kakeya set.) There are two situations when Mr has been studied. First, when T is uniformly distributed, i.e., 3e > 0, so that if 0,0' G T, 0 ^ 0', then | 0 - 0 ' | > e. Then it can be conjectured that ||Afr(/)|| p < AClogl/^^H/Hp, if p > n, for some N = N(p, n). This has been shown by Cordoba [17] for n = 2 in connection with the multiplier (13.2); see also Stromberg [111]. The second situation occurs when T is lacunary, e.g., in 2 dimensions, with T = {0*;}, 0fc makes an angle 2~k with the x-axis. Then l|M r (/)||p < 4,11/Hp,
Kp
(14.1)
PROBLEMS RELATED TO CURVATURE AND OSCILLATORY INTEGRALS
217
This can be proved by using square functions and decay properties of the Fourier transform; see Nagel, Stein, and Wainger [68]. The argument combines a Littlewood-Paley inequality
(Ei^/i2)
1/2
<Mf\
where Pfc(/)~(£) = Xs t (£)/(£)>
and
?'
1 < p < oo,
Sk denotes the sector
sfc = ttl|(eA)|-KII<2-fcK|}, together with a "bootstrap": if (14.1) holds for p0 then
(EI^M 2 )
1/2
< A
(Ei/*i2)
1/2
holds if l / p < | [ l + l/po]. Earlier results for p > 2, by covering arguments, are in Stromberg [110], and Cordoba and R. Fefferman [19]. Further results relevant to MT in the uniformly distributed case are in R. Fefferman [33] and Christ, Duoandikoetxea, and Rubio de Francia [15]. Bootstrap arguments similar to the one above are exploited in Duoandikoetxea and Rubio de Francia [26]. 15. Hilbert transform along flat convex curves. We return to §6 and here we point out that even when the submanifold is infinitely flat at the origin, results such as Theorem 5 may still hold. We shall consider curves 7(f) = (t, 72(f)) in R 2 with t —• 72(f) convex, continuous, for 0 < t < 00, 72(0) = 7 2 (0) = 0, and 72(f) odd (i.e., 7 2 (-f) = -72(f)). One considers the Hilbert transform and maximal function along 7, oo
/ M(f){x)
= sup
l
/Or-7(f) W ,
(15.1)
/(x- 7 (f))df
(15.2)
-00
-j
h>0
Then a necessary and sufficient condition for (15.1) to be bounded on £ P (R 2 ) is that the curve satisfies the "doubling-time" condition: with h(t) = f72(f) 72(f) there is a constant D > 1 so that h(Dt) > 2h(t), t > 0. The result for p = 2 (as well as related results for "even" curves, some higher-dimensional variants, and (15.2) for p = 2) are in Nagel, Vance, Wainger, and Weinberg [72-74]. The general Lp result and the Lp inequalities for M are in Carlsson et al. [12]. The Lp argument makes use of a bootstrap which has some similarity to that in §14 above. 16. Fourier transform of surface-carried measures—a vignette. We close this survey by returning to the problem of §2 and presenting a recent elegant result of Bruna, Nagel, and Wainger [7], which provides a striking illustration of the interrelation of many of the notions considered in this essay: the decay of the Fourier transform, curvature, and quasimetrics, and their associated volume forms.
218
E. M. STEIN
Let S denote the boundary surface of a smooth convex compact set in R™, and the question is the decay at infinity of (2.2), the Fourier transform of a smooth surface-carried measure. We assume also that S is of finite type at each point, in the stronger sense, that for each XQ € S each tangent line passing through XQ makes only a finite order contact with S. Now define a family of balls {B{XQ, 6)}, for each x0 e S, 6 > 0, by B(x0,8) = { i £ S\(x-x0,i/Xo)
< 8},
where vXo is the inward-pointing normal to 5 at xQ. (B(xo,8) consists of those points in S which lie between the hyper-planes determined by TXo (S) and TX0{S)+8uX0.) Now it can be shown first that this family of balls satisfy the basic properties of the Vitali covering lemma: i.e., the quantity p{x,y) = inf{8\y € B(x,8)} is a quasimetric on 5 , and the volume form Vx{8) — volume of B(x,8), has the doubling property. Next, for f ^ 0 let x+(£) and x~(£) denote the points on S so that
"*+ = e/iei,
"*- = -€/iei-
Then the desired estimate is \m\
+ Vx-(6)) with 5 = l/|e|.
(16.1)
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112. I. Svenson, Ark. Mat. 9 (1971), 11-22. 113. P. Tomas, Proc. Sympos. Pure Math., Vol. 35, Part 1, Amer. Math. S o c , Providence, R. I., 1979, pp. 111-114. 114. G. Uhlmann, personal communication, 1986. 115. A. Zygmund, Trigonometric series, Cambridge Univ. Press, London and New York, 1959. P R I N C E T O N U N I V E R S I T Y , P R I N C E T O N , N E W J E R S E Y 08544,
USA
Wolf Prize in Mathematics, Vol. 2 (pp. 703-754) eds. S. S. Chern and F. Hirzebruch © 2001 World Scientific Publishing Co.
Note concerning Jacques Tits Presentation This Note consists of two parts. Part I is based on an opuscule Titres et travaux scientifiques written in 1972 and divided into three sections modified as follows: Sec. 1.1 — a brief curriculum vitae which has been updated in year 2000; Sec. 1.2 — a description of the contents of the papers written until 1972; it is unchanged except for minor corrections; Sec. 1.3 — a list of publications which has been completed in March 2000. Part II is the reproduction ne varietur of mimeographed notes entitled Groupes et geometries de Coxeter dated 1961 (Ref. 43 of Sec. 1.3), which never appeared in print. This is the first paper ever written on arbitrary Coxeter groups (the terminology was coined there); it played a rather important, though somewhat hidden role in the early history of those groups. Indeed the most commonly used reference concerning them is Chap. 4 of Bourbaki, Groupes et Algebres de Lie, and the mimeographed notes in question were precisely written to serve Bourbaki's work on that volume. But while Bourbaki's book, and in particular its "Note historique", fully acknowledges Tits' contributions to the subject, it does not explicitly mention the preprint reproduced here, which is the original source for those contributions.
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1.1. Curriculum Vitae Vie et carriere: Ne a Uccle (Belgique) le 12 aout 1930. Doctorat en Sciences Mathematiques, mai 1950. Boursier du FNRS (Belgique) de 1948 a 1956; Assistant (1956-1957) puis Charge de cours (1957-1962) a I'Universite de Bruxelles; Professeur a I'Universite de Bruxelles (1962-1964) puis a I'Universite de Bonn (1964-1974); Professeur associe (1973-1974) puis Professeur au College de France, titulaire de la chaire de Theorie des Groupes depuis 1975. Sejours de longue duree a l'ETH de Zurich (1950, 1951, 1953), a 1'IAS de Princeton (1951-1952, 1963, 1969, 1971-1972), aux University de Rome (1955, 1956), de Chicago (1963), de Californie a Berkeley (1963), de Tokyo et de Kyoto (1971), a I'lHES de Bures-sur-Yvette (sejours frequents de 1960 a 1973), a Tianjin, Shanghai et Beijing (1987), a I'Universite Yale (1966-1967, 1976, 1980, 1984, 1990, 1995), a I'Universite d'Ottawa (1998). Membre du Comite de redaction d'une dizaine de periodiques et collections d'ouvrages scientifiques; redacteur en chef des Publications Mathematiques de I'lHES de 1980 a 1999. Distinctions: Prix scientifique interfacultataire L. Empain (1955); Prix Wettrems de l'Academie de Belgique (1958); Prix decennal de mathematique du gouvernement beige (1965); Grand prix des Sciences mathematiques et physiques de l'Academie des Sciences (1976); Cantor-Medaille de la Societe Mathematique Allemande (1996). Conferencier invite aux Congres internationaux des mathematiciens de Stockholm (1962), Nice (1970), Vancouver (1974). Weyl Lectures (IAS, Princeton, 1972); Ritt Special Lectures (Columbia University, 1976); Mordell Lecture (Cambridge, 1978); Hardy Lectures (Grande-Bretagne, 1983); I. Schur Memorial Lectures (Tel Aviv, 1984); Wolfgang Pauli Vorlesungen (Zurich, 1985); conference inaugurale du Colloque Mathematique de Berlin reunifle (avril 1990); Lezioni Leonardesche (Milan, 1990); Pitcher Lectures (Lehigh University, 1990); Weisfeiler Memorial Lecture (Penn. State University, 1990); Adrian Albert Lectures (University of Chicago, 1995); etc.
705 Docteur honoris causa des Universites d'Utrecht (1970), de Gand (1979), de Bonn (1988) et de Louvain (1992). Membre de la Deutsche Akademie der Naturforscher Leopoldina (1977); Membre correspondant (1977) puis membre (1979) de I'Academie des Sciences; Membre etranger de I'Academie Royale Neerlandaise des Sciences (1988); Membre fondateur de l'Academia Europaea (1988); Associe etranger de I'Academie Royale de Belgique (1991); Foreign honorary member de 1'American Academy of Arts and Sciences (1992); Foreign associate de la National Academy of Sciences des Etats-Unis (1992); Honorary member de la London Mathematical Society (1993); Membre de I'Ordre "pour le merite" fiir Wissenschaften und Kiinste (RFA, 1995).
1.2. R e s u m e d e s t r a v a u x a n t e r i e u r s a 1 9 7 2 (ref. [1] a [92])
Introduction La theorie des groupes p e u t etre sommairement definie comme une theorie de la symetrie, de l'indiscernabilite et de l'homogeneite; le lien entre ces notions est clair: u n objet possede une certaine symetrie si des angles de vue differents en donnent des images indiscernables, un milieu est homogene si ses points sont indiscernables. L'idee apparait deja dans la mathematique grecque, ou les figures a h a u t degre de symetrie jouent u n role essentiel. Elle est aussi sous-jacente dans le principe d'inertie de Galilee: l'indiscernabilite des divers etats de mouvement rectiligne et uniforme n'est autre que l'invariance de la dynamique pour u n certain groupe. Cependant, la notion meme de groupe n ' a ete degagee qu'au X I X e siecle, dans les t r a v a u x d'Evariste Galois sur les equations algebriques. Galois observe que la complexite d'une telle equation est liee a l'indiscernabilite de ses solutions; celle-ci est mesuree par u n «groupe de substitutions)) mais c'est le groupe «abstrait» correspondant qui resume les proprietes essentielles de l'equation. C'est encore a l'occasion de recherches sur les equations, differentielles cette fois, qu'ont ete introduits les groupes «continus» (Lie) et les groupes algebriques (Picard-Vessiot, Maurer). Mentionnons aussi le point de vue de F . Klein qui, cherchant un classement systematique des geometries etudiees de son temps, en arrivait a identifier, de facon u n peu abusive, les concepts de geometrie et de groupe continu de transformations. Suivant u n processus bien connu en mathematique, et sans doute dans les sciences en general, il arrive souvent qu'une notion ou propriete qui joue d'abord u n role auxiliaire devienne u n objet d'etude en soi et finisse par depasser en importance le probleme particulier dont elle est issue. Nee de la theorie des equations, la theorie des groupes s'est vite erigee en branche autonome. Elle ne s'est pas developpee en vase clos mais a penetre au contraire plusieurs domaines des mathematiques, de la physique et de la chimie theorique, soit qu'elle y ait trouve des applications (groupe fondamental, representations unitaires, spectroscopic moleculaire, classification des particules elementaires, . . .), soit que des problemes poses par elle y aient acquis une grand importance (groupes de transformations en topologie differentielle, groupes arithmetiques et representations unitaires des groupes p-adiques et arithmetiques en theorie des nombres, . . .). La partie pour ainsi dire la plus pure de la theorie des groupes est la theorie des groupes finis. Ses problemes et ses methodes lui sont specifiques et, jusqu'a ces dernieres annees, elle a ete relativement isolee du reste des mathematiques. Cette situation est d'ailleurs en train de changer: les aspects arithmetiques des questions que pose par exemple la classification des groupes finis simples deviennent de plus en plus apparents. Dans l'etude des groupes infinis, les questions les plus interessantes concernent generalement des groupes munis de structures additionnelles ou bien telle ou telle classe de groupes importants pour d'autres parties des mathematiques; ainsi, la theorie se ramifie en
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plusiers branches, d'ailleurs liees: groupes classiques, groupes topologiques, groupes de Lie, groupes algebriques, groupes arithmetiques, etc. Une autre division se presente lorsque Ton considere la plus ou moins grande richesse des groupes en sous-groupes distingues. Aux deux poles de la theorie, on trouve d'une part les groupes commutatifs et plus generalement les groupes resolubles, et de l'autre ceux qu'on appelle simples, de facon un peu paradoxale car 1'impossibilite de «decomposer» ces groupes, parfois tres enchevetres, ajouterait plutot a leur complexity. La theorie des groupes resolubles fait souvent figure de science auxiliaire, dont les resultats sont interessants dans la mesure ou ils sont utilises pour la theorie des groupes simples. Quant a celle-ci, un mathematicien connu, feru de structures deformables, 1'a un jour qualifiee par boutade de «supercristallographie». Depouillee de sa nuance pejorative, la comparaison est bonne: les groupes simples ont une rigidite et une individualite qui rappellent celles des systemes cristallins et, pour revenir a la mathematique grecque, des polyedres reguliers. II est d'ailleurs significatif que des groupes apparentes a ceux des reseaux cristallins ou des polyedres reguliers, les groupes de Coxeter, jouent un role essentiel dans la theorie des groupes algebriques simples. Cette individualite des groupes simples explique 1'importance qu'ont pour eux les problemes de classification et donne son aspect «concret» a leur etude: paraphrasant G. H. Hardy, on peut dire que pour pratiquer la theorie des groupes simples, il est necessaire d'avoir chacon d'eux pour ami personnel. Mes travaux se rapportent, pour la plupart, a la theorie des groupes et de leurs espaces homogenes. Les groupes simples, plus specialement les groupes de Lie et les groupes algebriques simples, y occupent une place importante. Le point de vue adopte est souvent geometrique ou combinatoire, plut6t que purement algebrique.
I. Transitivite et espaces homogenes Une bonne partie de mes premieres recherches et quelques travaux ulterieurs concernent la caracterisation axiomatique et l'etude de diverses classes d'espaces a l'aide de concepts lies k l'homogeneite et l'isotropie. Les resultats qui font l'objet des §§ 1 a 3 ci-dessous ont un trait commun: on pose au depart des conditions combinatoires, geometriques ou topologiques simples, d'apparence generale, et on ne trouve en fin de compte que quelques espaces, fortement structures et lies a des notions ou proprietes algebriques connues (corps, formes quadratiques, principe de trialite, etc.). 1. Groupes multiplement transitifs Un groupe de permutations G d'un ensemble E est dit n fois transitif si, etant donnes arbitrairement deux w-uples ordonnes de points distincts de E, il existe un element de G appliquant le premier de ces n-uples sur l'autre. Lorsque cet element est unique, on dit que G est strictement
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n fois transitif. Le probleme, non resolu, de determiner tous les groupes finis n fois transitifs pour » > 2 a occupe les theoriciens des groupes depuis longtemps, surtout apres la decouverte par E . Mathieu de deux groupes cinq fois transitifs aux proprietes etonnantes. Si k est un corps commutatif, on sait que le groupe des transformations homographiques x h-> (ax + b) (ex + d)-1 de la droite projective k <"> {00} est strictement trois fois transitif. Dans le but, propose par P . Libois, d'axiomatiser la geometrie projective a une dimension, j ' a i donne [1,2,10] diverses conditions pour q u ' u n groupe strictement trois fois transitif soit isomorphe, comme groupe de permutations, a u groupe homographique d'un corps commutatif; il suffit par exemple que le stabilisateur d'un couple soit commutatif, ou encore que t o u t element du groupe qui permute deux points soit d'ordre 2. Les t r a v a u x cites contiennent aussi une etude des groupes strictement trois fois transitifs en general et montrent l'equivalence entre cette etude et celle de certaines structures algebriques generalisant les corps commutatifs. Ces resultats, appliques au cas d'un ensemble E fini, redonnent [3] la classification des groupes strictement trois fois transitifs finis due a H . Zassenhaus. L'article [9] concerne le cas ou E est une variete connexe: les groupes homographiques reels et complexes s'averent etre les seuls groupes strictement trois fois transitifs «continus», ce qui repond a une question posee par B. de Kerekjarto et resolue par lui en dimensions 1 et 2. Les groupes strictement deux fois transitifs «continus» sont determines par des methodes analogues dans [11] et [26]; ce sont les groupes de transformations affines x M» ax -f- b des presque-corps localement compacts connexes, qui sont determines par la meme occasion: corps des nombres reels, corps des nombres complexes, corps des quaternions et variantes «tor dues » de ce dernier. P a r la suite, utilisant des moyens plus puissants (structure des groupes localement compacts et classification des groupes de Lie simples), j ' a i generalise ces resultats en dormant la liste de tous les espaces homogenes GjH, ou G est localement compact non totalement discontinu et H u n sous-groupe ferme de 0, tels que G opere effectivem e n t et deux fois transitivement sur G/H (cf. [27], IV. F , et [22]); le resultat est particulierement simple pour les groupes trois fois transitifs: les seuls groupes a y a n t cette propriete sont les groupes conformes et leurs composantes neutres (groupes conformes directs). Pour n > 4, le probleme de la determination des groupes strictement n fois transitifs est resolu au chapitre I V de [10]: si G n'est pas un groupe symetrique ou alterne, on a n = 4 et \E\ = 11 ou bien n = 5 et \E\ = 12, et les groupes correspondants sont les deux «petits» groupes de Mathieu. Ce resultat avait ete obtenu par C. J o r d a n sous l'hypothese supplementaire que E est fini. Les groupes projectifs a n dimensions sur u n corps commutatif ne sont pas strictement n -\- 2 fois transitifs mais il s'en faut de peu. Le memoire [10] donne aussi diverses caracterisations de ces groupes basees sur une notion de groupes «a peu pres (strictement) n fois transitifs »; la recherche d'autres exemples de tels groupes m'a conduit a redecouvrir le
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groupe de Mathieu M22 ([10], p . 99). Plus tard, des considerations du meme ordre m'ont permis [57] de dormer la caracterisation commune simple suivante des trois «grands» groupes de Mathieu. Soient n u n entier et E u n ensemble fini dont on distingue certaines parties, appelees blocs, telles que n points quelconques de E appartiennent a u n et u n seul bloc. Disons qu'une partie de E est libre si elle ne contient pas n + 1 points a p p a r t e n a n t a u n meme bloc, et supposons que le groupe des automorphismes du systeme soit transitif sur les parties libres ordonnees kn -\- l e t a w + 2 elements. Alors, a des cas triviaux pres, E est un plan projectif arguesien ou l'un des trois systemes de Steiner associes aux «grands» groupes de Mathieu. On sait que les groupes de Mathieu peuvent s'obtenir par extensions transitives successives a partir de deux groupes classiques: un sousgroupe d'indice deux M10 de PrL2(Fa), operant sur la droite projective, et PSL3(F4), operant sur le plan projectif. (On appelle extension transitive d'un groupe G de permutations d'un ensemble E, u n groupe G' de permutations de E^*> {*} tel que le stabilisateur de * dans G' soit G.) H . Zassenhaus a montre que si k est u n corps fini et si u n sous-groupe G de PrLn(k) contenant PSL„(k) possede une extension transitive, alors n = 2et\k\ < 4 ou G = Mw, ou bien n = Z,\k\ = 4 e t [ G : P S Z 3 ( F 4 ) ] = 1 ou 2. Par la suite, de nombreux t r a v a u x ont ete consacres a la nonexistence d'extensions transitives pour divers groupes classiques finis. Dans cet ordre d'idees, j ' a i demontre u n lemme geometrique dont se deduit facilement la non-existence d'extensions transitives pour de nombreux groupes de permutations non necessairement finis. E n particulier, la plupart des resultats connus dans ce domaine sont retrouves plus simplement et sous des hypotheses plus generates; ainsi, le theoreme de Zassenhaus reste vrai sans que 1'on doive supposer k fini ou meme commutatif (le cas n = 2, qui requiert d'ailleurs des methodes assez differentes, est traite dans [90]). Une partie E d'un espace projectif P sur u n corps commutatif est appelee u n ovoide si toute droite la rencontre en deux points a u plus et si, pour p eE, les droites D telles que D r\ E soit reduite a p forment un hyperplan, dit tangent a, E en p. L'interet de cette notion provient n o t a m m e n t de I'existence, en caracteristique 2, d'ovoi'des «exotiques» qui ont pour groupes d'automorphismes les groupes de Suzuki (cf. [41,46] et le § 22 ci-dessous). Des caracterisations geometriques des «ovoides de Suzuki » font 1'objet de [63]; dans le cas fini, ce sont, avec les quadriques ovales, les seuls ovoides doublement homogenes, c'est-a-dire possedant u n groupe d'automorphismes deux fois transitifs. L'etude des ovoides doublement homogenes infinis fait apparaitre une «monstruosite» liee aux corps non parfaits de caracteristique 2 : I'existence sur de tels corps d'ovoides de toutes dimensions dont les hyperplans tangents sont tous paralleles pour un choix convenable de l'hyperplan a l'infini dans P [45].
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2. Probleme de Helmholtz-Lie Espaces homogenes et isotropes Le principe de relativite en relativite generate On appelle probleme de Helmholtz-Lie le probleme de la caracterisation commune des geometries euclidiennes et «non-euclidiennes» par des proprietes de «libre mobilite» de leurs groupes de mouvements. Precisant des idees de Helmholtz, Lie avait etudie ce probleme a I'aide de sa theorie des groupes continus de transformations. Les resultats de Lie etaient locaux et analytiques. Les reprenant d'un point de vue global et topologique, A. Kolmogorov proposait en 1930 l'axiomatique suivante. Soient E un espace topologique localement compact, connexe, metrisable et G u n groupe transitif d'homeomorphismes de E tel que E possede une structure uniform* invariante par 0. Si plt . . ., pm sont des points de E, G(plt . . ., pm) designe l'intersection de leurs stabilisateurs dans G. Pour n e N*, enoncons l'axiome (Mn)
Quels que soient les points p1, . . . „ pn e E, de deux orbites distinctes de G(p1,. . . pn) dans F = G(plt . . ., p»_i) . pn, l'une separe l'autre de pn sur F.
A. Kolmogorov affirme que si (Mn) est satisfait pour t o u t n, l'espace E possede une metrique euclidienne ou non-euclidienne telle que G soit le groupe Isom E de toutes les isometries. Les paires (E, G) telles que G soit complet, en u n sens evident, et satisfasse au seul axiome (Jfj) sont determinees dans [27], IV. E. L'examen de la liste montre que (Mx) et (M2) suffisent a entrainer la conclusion de Kolmogorov, a ceci pres que G peut n'etre qu'un sous-groupe d'Isom E (en fait, a une exception pres qui ne satisfait deja pas a (M3), le complete de G est toujours soit Isom E soit sa composante neutre). Connaissant les paires (E, G) qui satisfont a (Jfj), on en deduit aussitot les espaces metriques localement compacts connexes tels que Isom E soit transitif sur les ensembles de couples de points a distance donnee: ce sont les espace euclidiens, les spheres et les espaces elliptiques et hyperboliques reels, complexes, quaternioniens et «octaviens», munis d'une distance fonction de la distance usuelle. Des resultats partiels dans cette direction avaient ete obtenus par H . C. Wang. Lorsqu'on suppose que E est une variete differentiable et que G est u n groupe de Lie operant differentiablement sur E (cas auquel on se ramene en fin de compte en ufcilisant la structure des groupes localement compacts), le probleme dont il vient d'etre question est u n cas particulier du suivant, egalement resolu dans [27], IV. D : determiner les paires (E, G) telles que G soit transitif sur l'ensemble des elements de contact (point et direction) de E, c'est-a-dire sur le fibre en espaces projectifs «quotient)) du fibre tangent. Les methodes utilisees pour resoudre ces questions, methodes basees sur la classification des groupes de Lie simples, sont susceptibles de nombreuses autres applications. II m ' a p a r u interessant d'etudier en detail le probleme suivant. Soit E u n univers de la relativite generate, c'est-a-dire une variete connexe de dimension 4 munie d'une forme
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differentielle quadratique ds de signature -| . L'ensemble D des elements de contact de E se decompose en trois parties D+, D0, Z)_ formees respectivement des elements de temps (ds2 > 0), de lumiere (ds2 = 0) et d'espace (ds2 < 0). Disons que E est t-isotrope (resp. 1-isotrope; e-isotrope) si le groupe de ses isometries est transitif sur D+ (resp. D0; ZL). Les univers 1-isotropes et e-isotropes sont enumeres dans [35] et le probleme analogue pour les univers conformes, c'est-a-dire munis non plus d'une metrique mais seulement d'un champ de cones de lumiere y est aussi resolu. Le cas de la t-isotropie est le plus interessant, car celle-ci est en quelque sorte u n principe de relativite restreinte «ponctuel»; il s'agit done de determiner tous les univers de la relativite generale qui satisfont ponctuellement au principe de relativite restreinte. Mais ce cas est aussi plus difficile parce que, contrairement a la l-isotropie et a la e-isotropie, la t-isotropie n'implique pas la semi-simplicite du stabilisateur d'un point; cela m ' a conduit a etendre a certains groupes non semi-simples la technique des systemes de racines de W. Killing et E . Cartan. J ' a i ainsi p u montrer que t o u t univers t-isotrope est a courbure constante. La liste des solutions globales, trop longue pour etre reproduite ici, fait apparaitre des formes d'univers interessantes au point de vue cosmogonique. D'autre part, la methode s'applique encore lorsqu'on m u n i t l'univers de structures additionnelles, ce qui met en evidence par exemple le fait que, dans u n univers a courbure constante negative, il existe des faisceaux lumineux (systemes de rayons lumineux fibrant l'espace) tels qu'en distinguant l'un d'eux, on ne detruit pas la t-isotropie. 3. Varietes complexes compactes homogenes Toute variete complexe compacte homogene E possede une et une seule fibration a fibre F parallelisable et dont la base est une variete projective rationnelle homogene. C'est le resultat principal de [47] qui donne aussi une construction effective des varietes E pour lesquelles F est u n tore. Deux applications immediates sont la classification des varietes complexes compactes homogenes simplement connexes, que H . C. W a n g avait obtenu par d'autres methodes, et celle des varietes complexes compactes homogenes non parallelisables de dimension inferieure a 3 : si elle n'est pas produit de deux varietes homogenes de dimensions plus petites, une telle variete est u n espace projectif, une quadrique de dimension 3, la variete des drapeaux d'un plan projectif, une variete de Hopf, une variete de Calabi-Eckmann ou une variete fibree en tores a deux dimensions au-dessus d'une droite projective. 4. Etude d'espaces homogenes Des principes generaux pour l'analyse geometrique d'un groupe de transformations (6, E) «plus que transitif» sont esquisses au chapitre VI de [10] et au chapitre I I I de [A]. Retenons-en deux notions: u n axe de (0, E) est l'ensemble des points fixes d'un sous-groupe de G; les orbites de (G, E) sont les orbites des fixateurs (centralisateurs) des axes.
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Meme si G possede une infinite de sous-groupes non conjugues, il arrive souvent qu'il n'ait qu'un petit nombre de classes de conjugaison d'axes et que la structure formee par les axes et les orbites resume bien les proprietes geometriques essentielles de E. A titre d'exemples, [A] donne la description du systeme des axes et orbites des espaces intervenant dans les theoremes de classification de [27], I V (voir les §§ 1 et 2 ci-dessus). Les plans projectifs sur les algebres d'octaves et certains espaces derives de ceux-la font l'objet de [15, 16, 28] et de plusieurs sections de [82]; d'autres descriptions d'espaces homogenes se trouvent dans divers articles, n o t a m m e n t dans ceux deja cites aux §§ 1 et 2 ci-dessus. Outre l'etude des groupes d'automorphismes, des involutions, eventuellement des polarites, etc., une methode souvent mise en ceuvre est la consideration de structures induites sur certains sous-espaces et l'expression intrinseque des relation entre deux sous-espaces. U n exemple fera mieux comprendre ce que cela signifie: les droites du plan projectif P sur une algebre d'octaves a division sont des espaces conformes orientes a 8 dimensions; par dualite, il en est de meme du pinceau des droites contenant un point p, et la bijection d'intersection de ce pinceau sur une droite ne contenant pas p est une application conforme renversant l'orientation; si D et D' sont deux droites distinctes, considerees comme espaces conformes, leur inclusion a P induit une relation de trialite entre leurs espaces tangents au point D r\ D', qui sont des espaces euclidiens a 8 dimensions (cf. [82], 9. 11). Les isomorphismes exceptionnels entre groupes de Lie simples de petits rangs, y compris u n isomorpbisme entre deux formes reelles de Dt qui ne semble pas avoir ete remarque par E. Cartan (cf. [27], p . 249), ont pour consequences de nombreux isomorphismes entre espaces homogenes, enumeres dans [27], tableau V I I . D'autres aspects du principe de trialite font l'objet des articles [32] et [33]: le premier montre comment les identites de Moufang dans les algebres d'octaves sont liees a des proprietes geometriques des quadriques a 6 dimensions; le second est consacre a l'etude des «trialites», transformations d'ordre 3 qui sont au principe de trialite ce que les polarites (d'ordre 2) sont a la dualite. 5. Le plan de Cremona Reprenant des idees de P . Libois et P . Defrise, j ' a i etudie la geometrie birationnelle du plan sur u n corps algebriquement clos, dans le b u t de lui donner des fondements geometriques «purs», invariants par le groupe de Cremona. L'axiomatique que j ' a i donnee p a r t d'un groupe ordonne F, dont les elements positifs sont appeles figures, et d'une partie P de l'ensemble F+ des figures, dont les elements sont les points. Ce sont la les seuls elements primitifs de l'axiomatique: le groupe des automorphismes de F conservant F+ et P est le groupe de Cremona etendu par le groupe des automorphismes du corps de base. La relation d'ordre dans F est la contenance, et Vintersection de deux figures, c'est-a-dire leur borne inferieure, est supposee exister toujours. U n point p peut etre contenu dans u n autre q: c'est la notion de «point infiniment voisin». II
- 12 peut aussi y etre contenu plusieurs fois; cela correspond a la theorie des «points proches» d'Enriques, qui fournit une partie du systeme d'axiomes. Deux points p, q sont dits transversaux si leur intersection est un point infiniment voisin du premier ordre (c'est-a-dire maximal parmi les points proprement contenus) de chacun d'eux; alors p + q est appele une demi-droite. Ces demi-droites, qui correspondent aux generatrices des modeles quadriques du plan (courbes exceptionnelles c telles que c • c = 0) ont des proprietes remarquables dont la mise en evidence est sans doute Fun des resultats interessants de cette etude. Par exemple, le groupe des transformations de Cremona qui conservent une demidroite d, un groupe «de dimension infinie» lie au groupe des transformations de Jonquieres, est transitif sur les points simples de d, c'esta-dire les points p tels que d contienne p mais non 2 p. En particulier, d — p est alors un point transversal a p; cette propriete est prise comme axiome. II y a aussi I'axiome d'Euclide: si p € P et si d est une demi-droite telle que p r\ d — 0, alors il existe une et une seule demi-droite d' contenant p et telle que d r\ d' = 0 . Enfin, le theoreme de Noether sur la decomposition des transformations de Cremona en produit de transformations quadratiques prend ici la forme d'un axiome de minimalite du modele. Ces recherches m'ont conduit a etudier aussi d'autres espaces lies au plan cremonien: les modeles homogenes du plan (surfaces rationelles normales), l'espace des points infiniment voisins d'ordre donne d'un point, l'espace des points simples d'une demi-droite, etc.
II. Groupes topologiques; groupes et algebres de Lie La theorie generale des groupes topologiques et la theorie de Lie, souvent utilisees dans les recherches resumees en I, n'ont ete pour moi qu'un sujet d'etude marginal, sauf lorsqu'elles s'integraient dans la theorie des groupes algebriques (cf. IV). Les contributions que j'y ai apportees a l'occasion de cours ou en reponse a des questions qui m'ont ete posees sont passees en revue ci-apres. 6. Eevetements En theorie des groupes topologiques, on utilise habituellement les notions de connexite, de simple connexite, etc. heritees des espaces topologiques sous-jacents. On obtient cependant un expose plus general et souvent plus simple en utilisant des notions adaptees a la categorie ou on travaille. Ainsi, un groupe topologique se comporte comme un groupe connexe, disons qu'il est GT-connexe, des qu'il est engendre par tout voisinage de l'element neutre ; de meme, il est GT-simplement connexe si tout revetement (au sens des groupes topologiques) est trivial, etc. C'est le point de vue adopte dans [B], ou est aussi defini, pour tout groupe topologique G, un homomorphisme universel G -> G qui coincide avec le GT-revetement universel lorsque celui-ci existe.
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7. Application exponentielle Integration de representations lineaires Soit G un groupe differentiel banachique dont le centre est simplemerit connexe et supposons l'algebre de Lie g de G munie d'une norme | | telle que \[x, y\\ < \x\ • \y\; alors l'application exponentielle est injective sur la boule de rayon n de g (resultat obtenu en collaboration avee M. Lazard: cf. [64]). Soit Q une representation lineaire d'une algebre de Lie g de dimension finie dans une 6-algebre (algebre «a bornes»); alors, I'ensemble des elements h de g tels que o(h) soit integrable est une sous-algebre de g (resultat obtenu en collaboration avec L. Waelbroeck: cf. [77]). 8. Automorphismes a emplacements homes des groupes de Lie Soit G un groupe de Lie connexe. L'article [55] decrit tous les automorphismes ix de G tels que I'ensemble {x(g)g~1\g e G} soit relativement compact; cela determine en particulier le sous-groupe B (G) des elements de Crdont la classe de conjugaison est relativement compacte. On constate par exemple que si G est simple non compact, oc est necessairement I'identite (et B(G) = {e}). II s'ensuit qu'une isometrie 9? d'un espace symetrique irreductible non compact et non euclidien telle que la distance d'un point x et de son transforme q>(x) soit bornee est I'identite. Ces resultats ont pour corollaires plusieurs theoremes connus sur les groupes de Lie et les espaces symetriques. 9. Constantes de structure des algebres de Lie semi-simples La structure des algebres de Lie semi-simples complexes a ete elucidee par W. Killing et E. Cartan. Leurs travaux ne fournit cependant pas de presentation explicite de ces algebres: si g est une algebre de Lie complexe semi-simple, f) une sous-algebre de Cartan, 0 le systeme de ratines correspondant, et ea un vecteur propre de f) correspondant a la racine a, on a, pour a, b, a + b e&, [ea, e&] = Cabea+b avec co6 e C*, mais la theorie de Killing et Cartan ne dit rien sur la valeur des constantes cab- Un pas decisif vers la solution de ce probleme a ete fait par C. Chevalley qui a montre que, pour un choix convenable des ea, la valeur absolue de cab est le plus grand entier n tel que a -\- b — nb soit une racine. Reste la question du signe des cab- Celle-ci est ramenee dans [67] a l'etude d'un certain groupe fini N, extension du groupe de Weyl de 0 par un 2-groupe abelien elementaire. A toute racine a est associee canoniquement une paire Ma d'elements de N; le choix d'une base de Chevalley revient au choix d'un element dans chaque Ma et les signes des Cab sont alors donnes par des formules dans N. Ce resultat permet de donner une demonstration elementaire de l'existence de l'algebre de Lie ayant un systeme de ratines donne: les relations de commutation etant connues explicitement, il suffit en effet de verifier I'identite de Jacobi.
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10. Tables Le syllabus [74] a ete ecrit a l'oeoasion d'un seminaire reunissant des mathematiciens et des physiciens, et est concu en vue des applications. Son but est de permettre la determination aisee de tous les groupes de Lie simples ou toutes les representations lineaires de tels groupes satisfaisant a telle ou telle condition concernant la dimension, les formes invariantes, l'ordre du centre, etc. III. Immeubles et BN-paires Les immeubles (la terminologie est de N. Bourbaki), sorte de repliques oombinatoires des groupes algebriques semi-simples isotropes, sont nes de la recherche d'un procede systematique pour 1'interpretation geometrique des groupes semi-simples et, plus particulierement, des groupes exceptionnels. lis se sont ensuite reveles un instrument utile pour l'etude des groupes algebriques (cf. §§ 12 a 14). D'autre part, la mise en evidence, par R. V. Moody et K. Teo, de I?iV-paires (done, indirectement, d'immeubles: of. § 13) dont le groupe de Weyl n'est pas fini ou de type affine laisse entrevoir des perspectives de developpement qui vont au-dela de ces applications. 11. Geometric des sous-groupes paraboliques Les geometries etudiees dans [27], III, premiers exemples d'immeubles, sont des generalisations naturelles de la geometrie projective complexe a n dimensions. Celle-ci peut etre construite a partir du groupe G = PSLn+1 (C) de la facon suivante: soienti? le groupe des matrices triangulaires et Plt . . ., Pn les sous-groupes maximaux de G contenant B. Alors, les varietes lineaires de l'espace projectif a n dimensions sont les points des espaces GjPi et deux varietes sont incidentes, e'est-a-dire que l'une d'elles contient l'autre, si les classes laterales qui les representent ont une intersection non vide. Pour deerire les generalisations en question, fixons d'abord une terminologie de base suggeree par Fexemple precedent. On considere des geometries r, constituees par une famille (Vi)i e / d'ensembles et, pour toute paire d'indices i, j e /, une relation entre Vi et Vj, la relation d'incidence. Le rang de r est par definition le cardinal de I. Si J est une partie de / , une famille (Pj)j e j de points pj e Vj incidents deux a deux est appelee un drapeau de type J. Deux drapeaux, de types J et J', sont dits incidents s'ils forment ensemble un drapeau de type J w J', et on appelle ombre d'un drapeau D sur un ensemble de drapeaux, 1'ensemble des elements de celui-ci qui sont incidents a D; ainsi, les varietes lineaires de l'espace projectif sont les ombres des drapeaux sur 1'ensemble des points de l'espace. Une geometrie F peut aussi etre vue comme un complexe simplicial dont les sommets sont les points des Vi et dont les simplexes sont les drapeaux. Etant donnes un groupe G et une famille (Pi)i £ / de sous-groupes, on en deduit une geometrie F (G; Pi) formee des ensembles Vi = GjPi et des relations d'incidence gP%<-\ g'Pj =£ 0.
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Prenons maintenant pour G u n groupe de Lie complexe semi-simple et pour Pi les sous-groupes maximaux de G contenant u n sous-groupe resoluble connexe maximal (sous-groupe de Borel) B de 6. Comme les sous-groupes de Borel sont tous conjugues, nous assooions ainsi a t o u t groupe G du type considere une geometrie r = /"((?). L'etude de ces geometries fait Pobjet de [27], I I I , et [23, 28, 29]. Elles ont beaucoup de points communs avec les geometries projectives: pour ne donner q u ' u n exemple, l'ensemble des ombres de drapeaux sur la variete des drapeaux d'un type donne est ferme pour l'intersection. Mais l'observation essentielle de [27], qui rend les F(G) utiles pour l'interpretation geometrique des groupes semi-simples, est le fait que les I\G) correspondant aux divers groupes G sont liees entre elles et que la nature de ces liens se lit commodement sur les graphes de Dynkin des groupes. Plus precisement, les Pi, done les V{, sont canoniquement indexes par les sommets du graphe de Dynkin A de G; cela e t a n t : (-B)
sip e Vt, la sous-geometrie de r formee par les ombres de p sur les Vj (j =f= i) est la geometrie F(H) d'un groupe H dont le graphe de Dynkin s'obtient en retirant de A le sommet i et les traits qui y aboutissent.
Dans le cas des groupes classiques, cela met en relation des proprietes geometriques connues (par exemple, des proprietes des varietes lineaires d'hyperquadriques) avec les graphes de Dynkin de ces groupes. Mais la propriete (R) est surtout, comme on le verra plus loin, u n outil de r6currence efficace pour l'etude des -T(Cr); des applications aux groupes de t y p e E sont donnees dans [28, 29]. U n resultat de [27], apparente au «theoreme de Borel-Weil», met aussi en relation la geometrie JT(CT) avec les representations projectives de G: pour toute representation projective irreductible, il existe une et une seule partie J de / et u n seul plongement G-covariant de la variete des drapeaux de type J de F(G) dans l'espace de la representation. Dans cet ordre d'idees, notons que l'importance des r(G) s'est trouvee confirmee par un resultat de H . C. W a n g caracterisant leurs varietes de drapeaux comme les seules varietes Kahleriennes compactes simplement connexes homogenes, et surtout par les t r a v a u x de A. Borel et C. Chevalley qui font jouer a ces varietes u n role primordial dans la generalisation de la theorie de Killing et Cartan a un corps algebriquement clos quelconque. Dans [30, 39, 40], la theorie des r(G) est generalised d'abord aux groupes de Chevalley puis aux groupes algebriques semi-simples isotropes sur u n corps k quelconque: G est a present le groupe des points rationnels sur k d'un tel groupe algebrique et les Pi sont les groupes de points rationnels des &-sous-groupes paraboliques maximaux contenant u n &-sous-groupe parabolique minimal donne. Ajoutons que la geometrie r(G) peut aussi etre definie pour u n groupe classique G sur un corps gauche quelconque; si par exemple G est le groupe unitaire d'une forme hermitienne non-degeneree, dont il faut seulement supposer que son
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indice de W i t t n est fini, on prend pour Pi les sous-groupes m a x i m a u x contenant le stabilisateur d'un drapeau forme d'espaces totalement isotropes de dimensions 1, 2, . . ., n (cf. per ex. [82]). 12. Geometries polyedriques Immeubles Les articles [30] et [40], deja cites, contiennent l'esquisse d'une theorie axiomatique des r(G) basee sur les considerations suivantes. Soit m e N w{oo}, m > 2. Une geometrie de rang 2 (Vlt V2) est appelee u n m-gone generalise ([33], appendice) si, etant donnes deux elements x, y de V1 ^ V2, il existe au moins une suite xjc = y de longueur k < 2 m telle que deux elements cons6cutifs quelconques de la suite soient incidents, et a u plus une suite de longueur k < 2 m — 1 avec cette propriete. Une matrice de Coxeter est une matrice carree M = {mij)i,jei telle que my e N u {oo}, mu = 1 et my > 2 si i ^ j ; on la represente par u n graphs A = A (M) dont les sommets sont les elements de / , les sommets i et j e t a n t relies par un trait de multiplicite my — 2. II s'avere que toutes les geometries r(G) de rang 2 sont des m-gones generalises, et que si G est u n groupe de Chevalley de type Ax X Au A2, B2 ou G2, on a m = 2, 3, 4 ou 6 respectivement. Cela suggere d'associer a t o u t graphe A = A (M) une classe ^(A) de geometries, les geometries polyedriques de type A, dont les ensembles constitutifs V% sont indexes par / et qui sont definies comme suit: si / = {1, 2} et m12 = m, la classe &(A) est constitute par les m-gones generalises; les autres classes @(A) s'en deduisent par u n axiome d'induction sur le rang, caique sur la propriete (R) du n° 11, et une condition de «simple connexite», plus technique et qui n'a qu'un role secondaire. L'interet de cette definition reside dans le fait que, malgre la generalite apparente des axiomes, les geometries d'une meme classe ^(A) sont tres semblables entre elles d'un point de vue combinatoire. Ainsi, pour A = An, @{A) est la classe des geometries projectives a n dimensions. Plus generalement, disons qu'un element d'une geometrie est determine par deux drapeaux a, b s'il p e u t etre caracterise a u moyen d'assertions ou n'interviennent que a, b, la relation d'incidence et les types des elements variables quantifies (ex.: deux points determinent la droite qui les joint); alors, dans les geometries polyedriques, l'ensemble des elements determines par deux drapeaux est toujours fini, et les configurations possibles de ces ensembles sont les memes pour toutes les geometries appartenant a une meme classe 'S(A). L'exemple A = ,F 4 , important pour les groupes exceptionnels, est traite dans [40] et, avec plus de details au § 73 de H . Freudenthal, Linear Lie groups. De meme que le polygone ordinaire a n cotes est le plus simple des w-gones generalises, de meme chaque classe ^(A) contient une geometrie particulierement simple, a savoir le «polyedre regulier» A (A) = T(W; Wi), ou W = W(A) est le groupe de Coxeter correspondant a la matrice M et Wi le sous-groupe engendre par les generateurs rj de W avec j #: i (cf. [50] et le § 16 ci-desosus). Notons en passant que cette
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observation, et le fait que la «droite projective » A (At) est u n ensemble a deux points, font voir le groupe de Weyl d'un graphe de Dynkin A comme le groupe de Che valley de type A sur le « corps » a u n element [30]. Non seulement A(A) est u n modele pour la classe @{A), mais chaque membre F de cette classe contient A (A) a de nombreux exemplaires, et c'est la raison pour laquelle r herite des principales proprietes combinatoires de A {A). Cette constatation heuristique trouve son expression precise dans l'axiomatique des immeubles. Pour formuler celle-ci, nous passerons a present du langage des geometries a celui, mieux adapte, des complexes de chambres (cf. [59, 82]). Un complexe de chambres de rang n est u n complexe simplicial de dimension n — 1, dont tous les simplexes maximaux, appeles chambres, sont de dimension n — 1, et tel que deux chambres quelconques peuvent etre jointes par une galerie, c'est-a-dire une suite finie de chambres telles que deux elements consecutifs de la suite aient une cloison (face de dimension n — 2) commune. Les morphismes de tels complexes sont supposes appliquer les chambres sur des chambres. Soit A u n complexe de chambres tel que toute cloison appartienne a deux chambres exactement. U n complexe i"1 est appele u n preimmeuble (weak building) de type A s'il possede une famille j / de sous-complexes isomorphes a A, appeles appartements, tels que deux simplexes quelconques a, b, de T soient contenus dans u n appartement et que si A1 et Az sont deux appartements cont e n a n t a et b, il existe un isomorphisme de Ax sur A2 qui fixe chaque sommet de a et de b. Si de plus toute cloison appartient a trois chambres au moins, r est u n immeuble. Dans ce cas, A est necessairement le complexe A (A) d'un graphe [59]. Les preimmeubles de type A (A) ne sont autres que les geometries polyedriques de type A. U n theoreme facile mais fondamental est le theoreme de retraction: si A 0 est u n appartement et C < A0 une chambre du preimmeuble F, il existe une retraction Q: r -> A0 telle que p - 1 (C) = C (cf. [50, 59, 82]). Des corollaires immediats de ce theoreme montrent que les proprietes combinatoires de A du t y p e decrit plus h a u t (configurations des elements determines par deux drapeaux) se retrouvent dans JT. Cela est d'ailleurs vrai aussi pour d'autres types de proprietes. Supposons par exemple que A = A (A) soit u n complexe fini. Alors, A, ou plutot le complexe geometrique qu'on en deduit en remplacant les simplexes «abstraits» par des simplexes affines, possede u n metrique naturelle qui en fait une sphere euclidienne. Cette metrique et la notion de points diametralement opposes se transposent aussitot a r, et il resulte du theoreme de retraction que deux points p, q qui ne sont pas diametralement opposes sont joints par une geodesique unique contenue dans t o u t appartement contenant p et q. On en deduit que le complementaire de l'ensemble des points diametralement opposes a p est contractile, puis que 7 1 a le type d'homotopie d'un bouquet de spheres. L. Solomon a remarque que ce resultat fournit une construction simple de la representation de Steinberg d'un groupe fini G muni d'une IW-paire (cf. § 13); il suffit de considerer l'operation naturelle de G sur H n _ i (-TfCr]),
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ou n — 1 est la dimension de r(G). D'autres applications des immeubles aux groupes algebriques sont donnees par D. Mumford a u chapitre 2 de sa «Geometric invariant theory*. U n theoreme bien connu de geometrie projective met en correspondance bijective les espaces projectifs de dimension donnee n > 3 et les corps. Sa generalisation aux immeubles est 1'un des objectifs de [82]. On considere seulement des graphes A qui sont graphes de Dynkin de systemes de racines irreductibles de rang > 3. Alors, les seuls immeubles des types A (A) correspondants sont essentiellement les immeubles r(G) associes aux groupes classiques et aux groupes algebriques simples de rang relatif > 3. Ainsi la classification des immeubles redonne en parti culier celle des groupes algebriques en question. Deux sous-produits du resultat, qui n'apparaissent pas dans l'enonce precedent, meritent d'etre mentionnes. D'une part, 1'acception du terme «groupe classique» doit etre etendue de facon a inclure les groupes orthogonaux correspond a n t a une notion nouvelle de forme quadratique sur les corps gauches a involution (voir aussi le § 20); d'autre part, la classification des immeubles de type Ft met en evidence des groupes simples nouveaux, lies a des phenomenes d'inseparabilite et a l'existence d'isogenies exceptionnelles. Quant aux immeubles de rang 2, qui ont pour cas particuliers les plans projectifs, ce sont des objets trop generaux pour qu'il y ait u n sens a vouloir les classer; t o u t au plus peut-on envisager la determination de tous les immeubles finis, mais elle semble hors de portee a l'heure actuelle. On n'a done ici que des exemples, et aussi u n theoreme negatif, du a W. Feit et G. H i g m a n : si A est u n polygone a 2m cotes, il n'existe pas d'immeuble fini de type A pour m # 2, 3, 4, 6, 8. Les notes [82] contiennent aussi la determination de tous les automorphismes de rimmeuble r(G) lorsque 6 est u n groupe classique ou le groupe des points rationnels d'un groupe algebrique simple de rang relatif > 2. E n gros, u n tel automorphisme est toujours semi-lineaire ou semi-algebrique, e'est-a-dire compose d'un automorphisme d u corps de base et d'une transformation lineaire ou d'un morphisme. C'est la generalisation aux immeubles du «theoreme fondamental de la geometrie projective», et aussi des resultats bien connus de W. L. Chow, J . Dieudonne et L. K. H u a sur les varietes d'espaces totalement isotropes. Si Ton prend pour G u n groupe exceptionnel, cela montre aussi que la theorie des immeubles a atteint son b u t primitif: l'interpretation geometrique de ces groupes. 13. BN-paires Nous avons vu qu'on peut associer u n immeuble r(G) a certains groupes G. L'axiomatisation de ces liens conduit a la notion de SiV-paire (cf. [48, 56, 82]). Deux sous-groupes B, N d'un groupe G forment une BN-paire (ou, selon N . Bourbaki, u n «systerae de Tits») s'il existe u n immeuble T et une operation de G sur F transitive sur les couples (A, C) formes d'un appartement A et d'une chambre C de A, et telle que B soit le stabilisateur d'une chambre C 0 et N le stabilisateur d'un appartement
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contenant C0. Cela se traduit, en termes de groupes, par les conditions suivantes: B et N engendrent G, H = B ^ N est distingue dans N et le groupe W = NjH, appele groupe de Weyl, possede un systeme generateur B tel que, pour t o u t r e B, et t o u t w e W, on ait rBw < BwB <s BrwB et rBr ^ B. L'interet de cette definition est de resumer en peu d'axiomes plusieurs proprietes importantes des groupes algebriques simples. Ainsi, les assertions suivantes en sont des consequences faciles (cf. [48, 59]): on a BWB = G, et l'egalite BwB = Bw'B avec w,w' &W entraine w = w' (decomposition de B r u h a t ) ; les sous-groupes de G contenant B sont en correspondance canonique bijective avec les parties de R, ils sont deux a deux non conjugues et chacun d'eux est son propre normalisateur; W est u n groupe de Coxeter (resultat obtenu independamment par H . Matsumoto). D'autre part, u n groupe avec BN--p&iie a tendance a avoir peu de sous-groupes distingues; c'est en gros ce qu'expriment les resultats du § 2 de [56], etablis en vue d'applications aux groupes algebriques et aux groupes finis (cf. §§ 19 et 22). Grace a u resultat mentionne plus h a u t (§12) sur la classification des immeubles de rang > 3, on connait tous les groupes finis avec BlV-paires de type irreductible et de rang > 3 [82]. Des resultats partiels sur les .BiV-paires de rangs 1 et 2 ont ete obtenus par P . Fong, C. Hering, W. Kantor et G. Seitz. Dans les dernieres annees, la notion de .Bi^-paire a joue u n role important en theorie des groupes finis simples, n o t a m m e n t dans les t r a v a u x de C. Curtis et son ecole sur les representation lineaires, et dans les recherches sur la caracterisation de groupes simples par des proprietes de centralisateurs d'involutions, de sous-groupes de Sylow, etc.: on peut parfois ramener des problemes de ce type aux theoremes de classification dont il vient d'etre question en m o n t r a n t que les hypotheses faites sur un groupe imposent a celui-ci de posseder une -BiV-paire.
14. Immeubles et BN-paires
de type affine
Le champ d'application des .BjV-paires, introduites d'abord a 1'occasion de l'etude des sous-groupes paraboliques de groupes algebriques, s'est etendu lorsque N . Iwahori et H . Matsumoto ont montre que les groupes de Chevalley simplement connexes sur u n corps local possedent une seconde i W - p a i r e , dont le groupe de Weyl est cette fois un groupe affine engendre par des reflexions. P a r descente galoisienne a partir de leur resultat, F . Bruhat et moi [70, 71, 72] avons montre Fexistence d'une telle jEW-paire dans t o u t groupe semi-simple simplement connexe sur u n corps local a corps residuel parfait (voir aussi le § 18). Dans la premiere partie de [88], nous etudions les UiV-paires de type affine (c'est-a-dire dont le groupe de Weyl est u n groupe affine engendre par des reflexions) et leurs immeubles. Comme dans le cas d'un groupe de Weyl fini (cf. § 13), u n tel immeuble ^ possede une distance qui en fait u n espace metrique complet. U n resultat important, utilise n o t a m m e n t
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pour la descente galoisienne susmentionnee, est le theoreme de point fixe: tout groupe borne d'isometries de J possede un point fixe. Ce theoreme, et l'usage qu'on peut en faire pour la determination des sous-groupes compacts maximaux des groupes p-adiques, suggerent de voir l'immeuble d'un groupe simple p-adique comme l'analogue naturel de 1'espace symetrique d'un groupe simple reel. Ce point de vue a trouve confirmation dans les travaux de A. Borel, H. Garland et J.-P. Serre sur la cohomologie des groupes arithmetiques, et dans des recherches en cours concernant l'analyse harmonique sur les immeubles et, en particulier, sur les arbres (travaux de P. Cartier). Dans cet ordre d'idees, citons encore une application due a A. Borel: celui-ci definit la representation speciale d'un groupe simple p-adique en transposant a l'immeuble affine du groupe la construction de L. Solomon de la representation de Steinberg (cf. § 13). Dans [88], nous associons aussi un « immeuble » ^ a toute donnee radicielle valuee (cf. § 18). Si la valuation n'est pas discrete, J n'est plus un complexe simplicial. On peut s'en faire une idee en considerant le cas d'un groupe algebrique simple sur la cloture algebrique K de Qj,. Le corps K etant limite inductive de corps locaux «de plus en plus ramifies », l'immeuble du groupe est limite inductive d'immeubles ordinaires de plus en plus ramifies, c'est-a-dire dont les chambres sont de plus en plus petites. Un tel immeuble possede encore des chambres et des facettes, qui ne sont plus des ensembles mais des filtres. Comme pour les immeubles ordinaires, deux facettes appartiennent toujours a un appartement, mais il s'agit cette fois d'un theoreme difficile et non plus d'un axiome. 15. Arbres Les arbres dont tout sommet est d'ordre 3 au moins sont les immeubles de type affine de rang 2, d'ou leur place ici. Soient A un arbre quelconque, G le groupe de ses automorphismes et G+ le sous-groupe engendre par les stabilisateurs des sommets d'ordre > 3. Alors G+ est un groupe simple et G/G+ est produit libre d'un certain nombre Woo de groupes isomorphes a Z et d'un certain nombre n2 de groupes d'ordre 2. C'est le theoreme principal de [81], qui contient aussi un procede de construction d'un arbre A a partir d'un ensemble I et d'une matrice irreductible M = (m,ij)i,j 6 i ou les m# sont des cardinaux soumis a la seule condition que my = 0 implique m^ = 0. L'interet de cette construction reside dans le fait que certaines proprietes de G, et notamment les nombres n^ et n2, se lisent facilement sur la matrice M. On montre ainsi que Woo et w2 peuvent prendre n'importe quelle valeur. Lorsque les sommets de A sont d'ordre fini, G et G+ sont des groupes localement compacts denombrables a l'infini, et il resulte de la construction en question qu'on trouve, par application du theoreme principal, une infinite non denombrable de groupes localement compacts simples non isomorphes.
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16. Groupes de Coxeter Soient I, M et A comme au § 13 et W = W{M) le groupe defini par des generateurs r% (i e /) et les relations (r^j)™" = 1 pour my ^ cx>. H. S. M. Coxeter a determine toutes les matrices M telles que W(M) soit fini et il a etudie ces groupes finis et leurs representations comme groupes engendres par des reflexions; par la suite, ses resultats ont ete simplifies et precises par E. Witt. Le developpement de la theorie des immeubles m'a amene a etudier les groupes W(M), pour M quelconque, que j'ai appeles groupes de Coxeter. II apparait [43] qu'une bonne partie des resultats de Coxeter et de Witt restent valables pour tout groupe de Coxeter W. En particulier, W possede une representation lineaire Q comme groupe engendre par des reflexions affines; l'injectivite de cette representation est le resultat principal de [43]. L'image Q{W) a un domaine fondamental C qui est un cone simplicial, mais a la difference du cas fini, la reunion des « chambres », Q(W). C, n'est pas necessairement l'espace tout entier mais seulement un cone convexe. Le complexe obtenu en coupant le systeme des chambres par une sphere n'est autre que le complexe A (A) du § 12. Les methodes de demonstration topologiques utilisees par Coxeter et Witt ne s'etendent pas au cas general; elles doivent etre remplacees par des methodes combinatoires basees sur la consideration des galeries (cf. § 12) et d'une propriete de pliage qui caracterise d'ailleurs les groupes de Coxeter: soit A un complexe de chambres; pour que A soit le complexe A (A) associe a un groupe de Coxeter, il faut et il suffit que pour toute chambre G et toute cloison D de C, il existe une et une seule autre chambre C ayant aussi D comme cloison, et un endomorphisme idempotent n de A fixant les sommets de D, appliquant C sur C et tel que toute chambre soit l'image par TI de zero ou deux chambres (cf. [82]). Le normalisateur N d'un tore maximal T d'un groupe algebrique semi-simple est une extension du groupe de Weyl W par T. L'operation de W sur T est bien connue mais, pour en deduire N, il faut encore determiner la classe de l'extension (element de H2(W, T)). Dans [66], l'etude de cette extension est ramenee a celle d'une autre extension X de W, definie «universellement» pour tout groupe de Coxeter. Pour M comme ci-dessus, soient T(M) le groupe defini par des generateurs u (i e I) et les relations r
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L'article [78] donne une solution du probleme des mots pour les groupes de Coxeter et esquisse, en guise d'application, une nouvelle demonstration de la classification des groupes de Coxeter finis. IV. Groupes algebriques Comme on l'a vu, les imnieubles ont leur origine et leurs principales applications dans la theorie des groupes algebriques. Mes recherches concernant ces groupes ont souvent ete suggerees ou meme necessities par les t r a v a u x geometriques resumes plus h a u t (III). 17. Structure Representations lineaires Les theoremes fondamentaux sur la structure des groupes algebriques lineaires connexes sur u n corps algebriquement clos sont dus a A. Borel et C. Chevalley. E t u d i a n t le cas d'un corps de base k quelconque, A. Borel et moi avons obtenu des analogues relatifs des principaux theoremes absolus. Nous montrons par exemple que si 67 est u n groupe algebrique lineaire sur k, les tores deployes sur k maximaux de 67 sont conjugues par des elements de 67(k), que cela est aussi vrai pour les sousgroupes resolubles &-deployes maximaux et, si G est reductif, pour les ifc-sous-groupes paraboliques minimaux. Supposons que G soit connexe et ne possede pas de &-sous-groupe unipotent fc-deploye distingue non trivial (par exemple, que G soit reductif), et soit T u n tore deploye sur k maximal de 67. Alors, I'ensemble 0 des poids non nuls de T dans G est u n systeme de racines (non necessairement reduit) dont le groupe de Weyl W est le quotient du normalisateur N de T par son centralisateur Z; de plus, N = N(k) • Z, done W = N(k)/Z(k). A chaque racine a est associe un sous-groupe unipotent TJ(a) normalise par T et si B designe le groupe engendre par Z et par les U(a) correspondant aux racines a positives (sous-groupe &-parabolique minimal lorsque G est reductif), alors B(k) et N(k) forment une i?i\^-paire dans G(k), avec les consequences que cela comporte (cf. § 13). Ces resultats, et d'autres, du meme ordre, qu'il serait trop long de passer en revue ici, sont etablis dans [61,84, 89] pour u n groupe 67 reductif ou resoluble. Les cas general ne se ramene pas a ceux-la par u n simple «devissage» parce que le radical unipotent de 67 n'est pas necessairement defini sur k. La theorie des representations lineaires des groupes semi-simples complexes, due a E. Cartan et H . Weyl, a ete etendue partiellement a u n corps algebriquement clos quelconque par C. Chevalley; celui-ci montre qu'une representation lineaire rationnelle irreductible d'un groupe semi-simple 67 est caracterisee par son poids dominant et que I'ensemble A des poids dominants possibles est «le meme» que sur les complexes. Le § 12 de [61] et l'article [85] sont consacres a la «relativisationo de ces resultats. Considerons par exemple le cas d'un groupe 67 de type interieur sur u n corps k. II s'avere que pour t o u t A eA, la representation de poids dominant A p e u t etre realisee sur k comme
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representation dans u n groupe lineaire GLm(D) ou D = (}(?,) est une algebre a division bien determinee. De plus, l'application {3 de A dans le groupe de Brauer de k est u n homomorphisme de monoJdes s'annulant sur les poids qui sont combinaisons lineaires entieres de racines, et est lie a u n invariant cohomologique bien connu de G. Lorsque k = R, on retrouve, sous une forme rendue plus comprehensible par la generalite du point de vue, les resultats de E. Cartan sur les representations lineaires reelles des groupes de Lie simples reels. Les notes de cours [C] contiennent quelques resultats nouveaux sur la structure des groupes commutatifs et resolubles. Citons par exemple: une demonstration elementaire du fait que, sur u n corps algebriquement clos, u n groupe lineaire de dimension u n est isomorphe au groupe additif ou au groupe multiplicatif; l'existence dans t o u t &-tore d'un point rationnel engendrant u n sous-groupe dense, a condition que k ne soit pas extension algebrique d'un corps fini; diverses caracterisations des groupes unipotents «antideployes» (wound). La theorie des elements unipotents reguliers des groupes semisimples a ete faite par R. Steinberg. E . Brieskorn, R. Steinberg et moi avons etudie en collaboration les elements unipotents sous-re.gulie.rs d'un groupe simple G, c'est-a-dire les elements dont le centralisateur a la dimension rg G + 2 (pour les elements reguliers, cette dimension est rg G). Ces elements sont tous conjugues et la variete des points fixes d'un tel element dans GjB (B u n sous-groupe de Borel) est formee de courbes rationnelles dont le «schema d'intersections» se deduit immediatement du graphe de Dynkin de G et est ce graphe lui-meme lorsque toutes les racines de G ont meme longueur (cf. la conference de E. Brieskorn au Congres International de Nice, 1970). 18. Groupes reductifs sur les corps locaux Les groupes reductifs sur les corps values complets font Fob jet de recherches en cours depuis plusieurs annees, en collaboration avec F . Bruhat. Notre point de depart a ete le resultat d'lwahori et Matsumoto deja cite (§14). E n gros, notre b u t est de faire la theorie de ces groupes vus comme des etres (de dimension infinie) definis sur le corps residuel. Sous cet angle, les resultats presentent une analogie frappante avec la theorie des groupes reductifs evoquee au § 17. Bornons-nous ici au cas d'un groupe simplement connexe sur un corps de base local a corps residuel k parfait. Aux sous-groupes de Borel, sous-groupes resolubles connexes maximaux, correspondent ici les sous-groupes d'lwahori qui sont proresolubles connexes (pour une topologie a definir) m a x i m a u x ; ils n'existent eventuellement qu'apres extension de k, c'est-a-dire apres extension non ramifiee du corps local, mais ils existent toujours sur k lui-meme lorsque ce dernier est fini. P a r analogie avec les sous-groupes paraboliques, definis comme contenant u n sous-groupe de Borel, nous appelons parahoriques les sous-groupes contenant u n sousgroupe d'lwahori; ce sont aussi des groupes proalgebriques sur k, et les sous-groupes A;-parahoriques minimaux sont tous conjugues. De plus,
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un tel sous-groupe forme avec le normalisateur d'un tore deploye maximal une .BiV-paire dont le groupe de Weyl n'est plus fini, mais est un groupe affine engendre par des reflexions: c'est la .BiV-paire dont il a ete question au § 14. Aux tores, au systeme de ratines (absolu ou relatif), au graphe de Dynkin, . . . correspondent ici les minitores, un systeme de ratines affines (infini), un graphe de Dynkin affine, . . . Les sous-groupes parahoriques sont le pivot de cette theorie. En premier lieu, les sous-groupes parahoriques maximaux s'averent etre les sous-groupes bornes maximaux; ainsi est resolu, en particulier, le probleme de la determination des sous-groupes compacts maximaux des groupes semi-simples sur un corps localement compact non discret. En outre, les sous-groupes parahoriques qui sont, d'une part, proalgebriques sur k et, de I'autre, lies au groupe G par la BN-structure, sont le joint qui permet de relier la theorie des groupes algebriques sur k a la structure de G. C'est ainsi que, dans le cas d'un corps k fini, nous pouvons deduire le theoreme de M. Kneser sur la nullite de H 1 du theoreme analogue (theoreme de Lang) pour les groupes algebriques sur un corps fini, et que la classification des groupes semi-simples sur les corps locaux localement compacts se ramene, comme dans le cas d'un corps fini, a la simple determination des automorpbismes des graphes de Dynkin. Ces resultats sont annonces dans les notes [70] a [73] (cf. aussi [75]). Les fondements «abstraits» de la theorie sont etablis dans [88]. Nous y etudions les PJV-paires de type affine (cf. § 14) et les donnees radicielles valuees. Si 0 est un systeme de ratines, une donnee radicielle de type 0 dans un groupe est un systeme (Ua) de sous-groupes indexes par 0, soumis a des axiomes caiques sur les principales proprietes de U(a)(k) d'un groupe algebrique reductif (cf. § 11). Une valuation d'une telle donnee est un systeme de filtrations 990:E/a—>R *~> {oo} satisfaisant a certaines conditions. Toute valuation discrete donne lieu a une -Bi^-paire de type affine. Une partie des resultats mentionnes plus haut peut deja s'etablir sur cette base axiomatique. Par exemple, un groupe G avec .BjV-paire (B, N) de type affine irreductible possede une et une seule bornologie compatible avec la loi de groupe, telle que B soit borne et que G ne le soit pas; pour cette bornologie, tout sous-groupe borne maximal contient un conjugue de B. Dans le cas des valuations denses, on a un resultat analogue qui ne s'exprime d'ailleurs bien qu'en termes d'immeubles. Une partie importante de [88] est consacree a l'etude d'une classe de sous-groupes, les groupes P/, qui generalisent les sous-groupes parahoriques; dans le cas d'un groupe reductif defini sur un corps value d'anneau d'entiers 0 et qui se deploie sur une extension non ramifiee de ce corps, les P / sont les groupes de points entiers pour certaines ^-structures qui peuvent etre caracterisees par des proprietes de lissete et de « bonne reduction ».
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19. Le groupe des points rationnels d'un groupe algebrique simple groupe « abstrait»
comme
La theorie des groupes classiques, telle qu'elle est exposee par exemple dans le livre de J . Dieudonne, s'interesse principalement a deux types de questions: structure (suites de composition et, en particulier, simplicite); automorphismes et isomorphismes. Dans les articles [56] et [92] (ce dernier ecrit en collaboration avec A. Borel), les problemes de simplicite et la recherche des automorphismes et des isomorphismes sont portes sur le plan des groupes algebriques; les theoremes etablis ont pour cas parti culiers la plupart des resultats connus dans ces domaines, pour a u t a n t que les groupes consideres soient algebriques et isotropes, les groupes anisotropes necessitant des methodes differentes. Soit G u n groupe algebrique absolument presque simple defini sur u n corps k. On note G+ le sous-groupe de G(k) engendre p a r les points rationnels des £-sous-groupes unipotents deployes de G. Le theoreme principal de [56], consequence de resultats generaux sur les BN--paiies (cf. § 13), affirme que, sauf dans u n petit nombre de cas dont la liste est connue, t o u t sous-groupe non central de G(k) normalise par G+ contient G+. E n particulier, le quotient de G+ par son centre est simple. Dans [92], nous considerons non seulement les automorphismes et isomorphismes mais, plus generalement, les homomorphismes a image dense: soient G' u n groupe absolument simple adjoint defini sur u n corps k', H un sous-groupe de G(k) contenant G+ et oc: H -> G'(k') u n homomorphisme tel que oc(G+) soit dense; alors, il existe u n unique homomorphisme de corps
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la conjecture de J . Milnor et J . Wolf sur la croissance d'un groupe de t y p e fini dans le cas particulier des groupes lineaires. 20.
Classification Formes U n theoreme bien connu de E. W i t t permet de caracteriser une forme quadratique par son indice et sa partie anisotrope. On a u n resultat analogue pour les groupes algebriques semi-simples: un groupe G defini sur u n corps k est caracterise a isogenie centrale pres par u n indice et un noyau anisotrope: l'indice est consitue par le graphe de Dynkin A de 0, une operation du groupe de Galois de la cloture separable de k sur A et un ensemble J de sommets de A; le noyau anisotrope est u n groupe semisimple anisotrope dont le graphe de Dynkin s'obtient en retirant de A les sommets a p p a r t e n a n t a J et les traits qui y aboutissent. Ce resultat fait Fob jet de [36] et [68], ou sont aussi enoncees des conditions necessaires et suffisantes pour qu'un indice et u n noyau anisotrope donnes caracterisent effectivement u n groupe G, ainsi que la liste des indices possibles sur un corps « general», sur u n corps localement compact non discret et sur u n corps de nombres. Remarquons que dans le cas reel, u n groupe simple est entierement determine par son indice et que Ton retrouve ainsi la classification des groupes de Lie reels simples. 21. Applications de la classification des groupes algebriques a l'algebre Une remarque fondamentale de A. Weil ramene la classification des groupes algebriques des types classiques A, B, C, D a celle des algebres simples a involution, sauf toutefois pour les groupes des types B et D sur u n corps de caracteristique 2. Pour traiter ces cas, il faut faire appel a la notion de forme quadratique introduite dans [76]: soient E une algebre simple munie d'une involution a de premiere espece et A l'ensemble des « formes alternees » x — a (x) (x &E); on appelle alors forme quadratique u n element de EjA. Lorsque E est une algebre de matrices et a la transposition, on retrouve les formes quadratiques usuelles. Dans [76] sont aussi definis l'indice de W i t t , le discriminant ou l'invariant d'Arf et l'algebre de Clifford paire, obtenue rationnellement comme u n quotient de l'algebre tensorielle de E, d'une forme quadratique; une formule explicite, nouvelle meme dans le cas classique, est donnee pour l'invariant d'Arf. Notons que les formes quadratiques utilisees dans [82] (cf. § 12) sont la generalisation de celles-ci a des corps gauches a involution et des espaces vectoriels de dimension quelconque; d'autre part, l'extension par C. T. C. Wall de cette notion au cas d'un anneau de base joue actuellement u n role important en topologie differentielle. Dans une serie de travaux, C. Chevalley, R. D. Schafer, H . Freudenthal et N . Jacobson ont montre les liens existant entre l'algebre exceptionnelle simple de Jordan, de dimension 27, et les algebres de Lie exceptionnelles _F4, Es, E1, Es, mais ils n'obtiennent que des constructions ad hoc, differentes pour ces quatre algebres. La recherche de formules explicites pour certaines formes de E6 m'a conduit a la decouverte de
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deux precedes de construction d'algebres de Lie a partir d'algebres de Jordan. La premier, decrit dans [49], p a r t d'une algebre de J o r d a n J quelconque et d'une algebre de Lie simple Y de dimension 3 ; si J) est l'algebre des derivations interieures de J, 1'espace L = J <2> Y + D est muni d'une structure d'algebre de Lie naturelle, donnee par des formules simples. La reciproque de ce resultat fournit une caracterisation des algebres de J o r d a n qui explique le role joue par celles-ci dans l'etude des domaines bornes symetriques (travaux de M. Koecher). La seconde construction (cf. [54, 65] et R. D. Schafer, An introduction to nonassociative algebras, chap. IV) associe a tout couple forme d'une algebre alternative A de degre < 2 et d'une algebre de J o r d a n J de degre 3 une algebre de Lie L, sorte de produit tensoriel de A et J. Les cas les plus interessants sont ceux ou A est le corps de base k, une extension quadratique (ou k® k), une algebre de quaternions ou une algebre d'oetaves, et ou J est l'algebre de J o r d a n des matrices hermitiennes d'ordre 3 a coefficients dans une algebre de l'un de ces quatre types. Sur un corps algebriquement clos, on trouve ainsi u n carre de 4 x 4 algebres de Lie dont la derniere ligne et la derniere colonne sont (Fit E6, E7, Es). Ce «carre magique» (H. Freudenthal) etait deja connu pour ses proprietes numeriques remarquables, observees experimentalement par H . Freudenthal et moi (cf. [27], I I I ) et qui trouvent ainsi leur explication. Sur un corps non algebriquement clos, la construction donne diverses formes des algebres de Lie exceptionnelles, et en particulier toutes leurs formes sur R, sur les corps p-adiques et sur les corps de nombres (a supposer que le principe de Hasse soit vrai pour Es). P a r t a n t de l'observation que le stabilisateur d'un point generique de l'algebre de J o r d a n simple exceptionnelle J dans le groupe A u t J (de type F4) est u n groupe de t y p e A3, j ' e n ai deduit des constructions explicites de toutes les formes de J sur u n corps quelconque k de caracteristique differente de 2, a partir d'algebres simples de dimension 9 sur k et d'algebres simples de dimension 9 a involution de seconde espece sur une extension quadratique de k (cf. N. Jacobson, Structure and representations of J o r d a n algebras, I X , 12). On en deduit une demonstration simple d'un resultat de A. A. Albert: l'existence de formes a division de J. 22. Groupes finis simples Dans son livre «Linear groups», publie en 1901, L . E . D i c k s o n dressait la liste des groupes finis simples connus. Aucun changement n ' y a ete apporte jusqu'en 1955, date a laquelle un nouveau depart a ete donne par C. Chevalley, apres quoi les decouvertes de nouveaux groupes simples se sont succedees a un r y t h m e rapide. Sont apparues d'abord plusieurs series infinies: groupes de Chevalley, formes tordues de ceux-ci (trouvees par D. Hertzig, R. Steinberg et moi), groupes de Suzuki et de Ree; ont suivi des groupes isoles, les groupes sporadiques, actuellement au nombre de 20 (groupes de Mathieu compris). Mes t r a v a u x dans ce domaine sont les suivants.
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Ayant associe des geometries aux groupes exceptionnels complexes (cf. § 11), j ' a i songe a prendre les principales proprietes eombinatoires de celles-ci pour axiomes et de construire ainsi des geometries analogues, done des groupes exceptionnels, sur u n corps quelconque. Ce programme etait realise pour E6 (cf. [28]) au moment ou les resultats generaux de Chevalley ont rendu inutile la poursuite de cette voie. Les deux series non classiques de «groupes de Chevalley tordus» (groupes 2E6 et 3 D 4 ) sont construits et etudies dans [31] et [33]; simultanement, R. Steinberg obtenait ces memes groupes par une methode uniforme, proche de celle de Chevalley. Des apres la decouverte par Suzuki des groupes qui portent son nom, j ' e n ai donne une interpretation geometrique comme groupes d'automorphismes de certains ovoides [41, 46]. L'expose [41], consacre aux groupes de Suzuki et de Ree, generalise les resultats de Ree (le corps de base n ' y est pas suppose parfait) et contient des donnees nouvelles sur ces groupes, entre autres une description elementaire des groupes 2G2 comme groupes doublement transitifs. E n definissant les groupes 2Ft, R. Ree avait montre qu'ils sont simples lorsque le corps de base a plus de deux elements, mais il restait a determiner la structure de 2F± (F 2 ); il est etabli dans [56] que ce groupe possede u n sous-groupe simple d'indice 2. D'autre part, les resultats generaux de [56] ont comme corollaires immediats les theoremes de simplicite des groupes des series infinies. Dans [79], le groupe de J a n k o d'ordre 604.800 est obtenu comme groupe d'automorphismes d'un graphe a 100 sommets et 1800 aretes, construit a partir de l'hexagone generalise associe a u groupe 6?2(F2) (cf. § 12). Jusqu'alors, l'existence de ce groupe de J a n k o n'etait prouvee qu'a l'aide d'un ordinateur. Enfin, [53] decrivait l'etat de la theorie des groupes finis simples peu a v a n t que Z. J a n k o ait trouve le premier groupe sporadique (groupes de Mathieu mis a part) et l'expose [83] fait le point apres la decouverte du 19 e de la serie; j'essaye d'y mettre u n semblant d'ordre dans u n maquis dont le mystere ne semble pas pres d'etre eclairci.
730 1.3. PUBLICATIONS 1. Articles et ouvrages [1] Generalisation des groupes projectifs, I, Acad. Roy. Belg., Bull. CI. Sci. 35 (1949) 197-208. —, II, ibid., 224-233. —, III, Construction des groupes triplement transitifs finis, ibid., 568-589. —, IV, Proprietes des groupes triplement transitifs finis, ibid., 756-773. Groupes triplement transitifs et generalisations, Colloque d'Algebre et de theorie des nombres du C.N.R.S., Paris, septembre 1949, 207-208. Generalisation d'un theoreme de Kerekjarto, Hie Congres Nat. des Sciences, Bruxelles, juillet 1950, 64-65. Collineations et transitivite, ibid., 66-67. Les groupes projectifs: evolution et generalisations, Bull. Soc. Math. Belg. 3 (1949-1950) 1-10. Sur les groupes triplement transitifs continus; generalisation d'une theoreme de Kerekjarto, Compositio Math. 9 (1951) 85-96. Generalisations des groupes projectifs basees sur leur proprietes de transitivite, Mem. Acad. Roy. Belg. 27(2) (1952), 115 p. Sur les groupes triplement transitifs continus, Comment. Math. Helv. 26 (1952) 203-224. Caracterisation topologique de certains espaces metriques, Nachr. Osterr. Math. Ges. 2 2 / 2 3 (Bericht III. Osterr. Mathematikerkongr., Salzburg, sept. 1952), p. 51. Etude de certains espaces metriques, Bull. Soc. Math. Belg. 5 (1952) 40-52. La notion d'homogeneite en geometrie, Sem. de synthese scient., Universite Libre de Bruxelles, 1952, 1 p. Le plan projectif des octaves et les groupes de Lie exceptionnels, Acad. Roy. Belg., Bull. CI. Sci. 39 (1953) 309-329. Le plan projectif des octaves et les groupes exceptionnels E§ et £7, Acad. Roy. Belg., Bull. CI. Sci. 40 (1954) 29-40. Sur l'article intitule: "Etude de certains espaces metriques", Bull. Soc. Math. Belg. 6 (1953) 126-127. Espaces homogenes et groupes de Lie exceptionnels, Proc. Internat. Congr. Math., Amsterdam, sept. 1954, Vol. 1, 495-496. Etude geometrique d'une classe d'espaces homogenes, C. R. Acad. Sci. Paris 239 (1954) 466-468. Espaces homogenes et isotropes et espaces doublement homogenes, ibid., 526-527. Sur les R-espaces, ibid., 850-852. Transitivite des groupes de mouvements, Schriften Forschungsinst. Math. 2 (1957) {Bericht Riemann-Tagung, Berlin, 1954), 98-111. [23] Groupes semi-simples complexes et geometrie projective, Seminaire Bourbaki, exp. No. 112 (fevrier 1955), 11 p.
731 [24] Sous-algebres des algebres de Lie complexes semi-simples, Seminaire Bourbaki, exp. No. 119 (mai 1955), 18 p. [25] Espaces homogenes et isotropes de la relativite, Helv. Phys. Acta, Suppl. IV, 1956 (Actes Cinquantenaire Theor. Relat., Berne, juillet 1955), 46-47. [26] Sur les groupes doublement transitifs continus: Corrections et complements, Comment. Math. Helv. 30 (1956) 234-240. [27] Sur certaines classes d'espaces homogenes de groupes de Lie, Mem. Acad. Roy. Belg. 29(3) (1955), 268 p. [28] Sur la geometrie des R-espaces, J. Math. Pure Appl. 36 (1957) 17-38. [29] Les groupes de Lie exceptionnels et leur interpretation geometrique, Bull. Soc. Math. Belg. 8 (1956) 48-81. [30] Sur les analogues algebriques des groupes semi-simples complexes, Coll. d'Algebre Superieure du C B. R. M., Bruxelles, decembre 1956, 261-289. [31] Les "formes reelles" des groupes de type EQ, Seminaire Bourbaki, exp. No. 162 (fevrier 1958), 15 p. [32] Sur la trialite et les algebres d'octaves, Acad. Roy. Belg., Bull. CI. Sci. 44 (1958) 332-350. [33] Sur la trialite et certains groupes qui s'en deduisent, Publ. Math. I.H.E.S. 2 (1959) 14-60. [34] Isotropic des espaces de Klein, Coll. de Geometrie differentielle globale du C. B. R. M., Bruxelles, decembre 1958, 153-161. [35] Les espaces isotropes de la relativite, Coll. sur la Theorie de la Relativite du C B. R. M., Bruxelles, mai 1959, 107-119. [36] Sur la classification des groupes algebriques semi-simples, C. R. Acad. Sci. Paris 249 (1959) 1438-1440. [37] Une remarque sur la structure des algebres de Lie semi-simples complexes, Proc. Ned. Akad. Wet. A63 (Indagationes Math. 22) (1960) 48-53. [38] Sur une classe de groupes de Lie resolubles, Bull. Soc. Math. Belg. 11(2) (1959)100-115. [39] Sur les groupes algebriques affins: theoremes fondamentaux de structure; classification des groupes semi-simples et geometries associees, Centro Internazionale Matematico Estivo (C.I.M.E.), Saltino di Vallombrosa, septembre 1959 (Rome, 1960), 111+74 p. [40] Groupes algebriques semi-simples et geometries associees, Proc. Coll. Algebraical and Topological Foundations of Geometry, Utrecht, aout 1959 (Pergamon Press, Oxford, 1962), 175-192. [41] Les groupes simples de Suzuki et de Ree, Seminaire Bourbaki, exp. No. 210 (decembre 1960), 18 p. [42] Sur les groupes d'affinites sans point fixe, Acad. Roy. Belg., Bull. CI. Sci. 46 (1960) 954-956. [43] Groupes et geometries de Coxeter, Notes polycopiees, I.H.E.S., Bures-surYvette, juin 1961, 26 p. [44] Sur une classe de groupes de Lie resolubles, corrections et additions, Bull. Soc. Math. Belg. 14(2) (1962) 196-209.
732
[45] Ovoi'des a translations, Rend. Mat. 21 (1962) 37-59. [46] Ovoi'des et groupes de Suzuki, Arch. Math. 13 (1962) 187-198. [47] Espaces homogenes complexes compacts, Comment. Math. Helv. 37 (1963) 111-120. [48] Theoreme de Bruhat et sous-groupes paraboliques, C R. Acad. Sci. Paris 254 (1962) 2910-2912. [49] Une classe d'algebres de Lie en relation avec les algebres de Jordan, Proc. Ned. Akad. Wet. A65 (Indagationes Math. 24) (1962) 530-535. [50] Geometries polyedriques et groupes simples, Deuxieme reunion du Groupement de Mathematiciens d'expression latine, Florence, septembre 1961, 66-88. [51] A theorem on generic norms of strictly power-associative algebras, Proc. Amer. Math. Soc. 15 (1964) 35-36. [52] Groupes semi-simples isotropes, Coll. sur la Theorie des Groupes Algebriques du C.B.R.M., Bruxelles, juin 1962, 137-147. [53] Groupes simples et geometries associees, Proc. Internat. Congr. Math., Stockholm, 1962, 197-221. [54] Algebres alternatives, algebres de Jordan et algebres de Lie exceptionnelles, Notes polycopiees, IAS, Princeton, mars 1965, 5 p. [55] Automorphismes a deplacement borne des groupes de Lie, Topology 3, Suppl. 1 (1964) 97-107. [56] Algebraic and abstract simple groups, Ann. Math. 80 (1964) 313-329. [57] Sur les systemes de Steiner associes aux trois "grands" groupes de Mathieu, Rend. Math. 23 (1964) {Coll. sur les Geometries Finies, Rome, octobre 1963) 166-184. [58] Geometries polyedriques finies, ibid., 156-165. [59] Structures et groupes de Weyl, Seminaire Bourbaki, exp. No. 288 (fevrier 1965), 15 p. [60] Sur une conjecture de L. Solomon, C. R. Acad. Sci. Paris 260 (1965) 6247-6248. [61] (avec A. Borel) Groupes reductifs, Publ. Math. I.H.E.S. 27 (1965) 55-151. [62] Simple groups over local fields, Summer Institute on Algebraic Groups, Boulder, July 1965, Notes polycopiees, I-G, 6 p. [63] Une propriete caracteristique des ovoi'des associes aux groupes de Suzuki, Arch. Math. 17 (1966) 136-153. [64] (avec M. Lazard) Domaines d'injectivite de l'application exponentielle, Topology 4 (1966) 322-325. [65] Algebres alternatives, algebres de Jordan et algebres de Lie exceptionnelles, I. Construction, Proc. Ned. Akad. Wet. A69 (Indagationes Math. 28) (1966) 223-237. [66] Normalisateurs de tores. I. Groupes de Coxeter etendus, J. Algebra 4 (1966) 96-116. [67] Sur les constantes de structure et le theoreme d'existence des algebres de Lie semi-simples, Publ. Math. I.H.E.S. 81 (1966) 21-58.
733 Classification of algebraic semi-simple groups, Proc. Symp. Pure Math. 9 (1966) (Proc. Summer Inst, on Algebraic Groups and Discontinuous Groups, Boulder, 1965), 33-62. (avec F. Bruhat) Un theoreme de point fixe, Notes polycopiees, I.H.E.S., Bures-sur-Yvette, 1966, 34 p. (avec F. Bruhat) BN-paires de type afHne et donnees radicielles, C. R. Acad. Sci. Paris 263 (1966) 598-601. (avec F, Bruhat) Groupes simples residuellement deployes sur un corps local, ibid., 766-768. (avec F. Bruhat) Groupes algebriques simples sur un corps local, ibid., 822-825. (avec F. Bruhat) Groupes algebriques simples sur un corps local: cohomologie galoisienne, decompositions d'lwasawa et de Cartan, ibid., 867-869. Tabellen zu den einfachen Lie-Gruppen und ihren Darstellungen, Springer Lecture Notes in Mathematics 40 (1967), ii + 53 p. (avec F. Bruhat) Groupes algebriques simples sur un corps local, Proc. Conf. on Local Fields, Driebergen, 1966 (Springer-Verlag, 1967), 23-36. Formes quadratiques, groupes orthogonaux et algebres de Clifford, Invent. Math. 5 (1968) 19-41. (avec L. Waelbroeck) The integration of a Lie algebra representation, Pacific J. Math. 26 (1968) 595-600. Le probleme des mots dans les groupes de Coxeter, 1st. Naz. Alta Mat. Symp. Math. 1 (1968) 175-185. Le groupe de Janko d'ordre 604800, Theory of Finite Groups (A Symp.), Benjamin, New York, 1969, 91-95. (avec A. Borel) On "abstract" homomorphisms of simple algebraic groups, Proc. Internat. Coll. on Algebraic Geometry, Bombay, 1968 (1969), 75-82. Sur le groupe des automorphismes d'un arbre, Essays on Topology, Memoires dedies a G. de Rham, Springer-Verlag, 1970, 188-211. Buildings of spherical types and finite BN-pairs, Springer Lecture Notes in Mathematics 386 (1974), X + 298 p. (deuxieme edition augmented en 1986, X + 302 p.) Groupes finis simples sporadiques, Seminaire Bourbaki, exp. No. 375, fevrier 1970, Springer Lecture Notes in Mathematics 180 (1971) 187-211. (avec A. Borel) Elements unipotents et sous-groupes paraboliques de groupes reductifs, Invent. Math. 12 (1971) 95-104. Representations lineaires irreductibles d'un groupe reductif sur un corps quelconque, J. Reine Angew. Math. 247 (1971) 196-220. Free subgroups in linear groups, J. Algebra 20 (1972) 250-270. Homomorphismes et automorphismes "abstraits" de groupes algebriques et arithmetiques, Actes Congr. Internat. Math., Nice, 1970 (1971), tome 2, 349-355. [88] (avec F. Bruhat) Groupes reductifs sur un corps local. I. Donnees radicielles valuees, Publ. Math. I.H.E.S. 41 (1972) 5-251.
734 [89] (avec A. Borel) Complements a l'article: "Groupes reductifs", ibid., 253-276. [90] Non-existence de certaines extensions transitives. I: Groupes projectifs a une dimension, Bull. Soc. Math. Belg. 23 (1971) 482-493. [91] Une propriete des systemes de racines (appendice a un article de J. Dixmier), Invent. Math. 17 (1972) 174-176. [92] (avec A. Borel) Homomorphismes "abstraits" de groupes algebriques simples, Ann. Math. 97 (1973) 499-571. [93] Homomorphismes "abstraits" de groupes de Lie, 1st. Naz. Alta Mat, Symp. Math. 13 (1974) 479-499. [94] (avec S. Koppelberg) Une propriete des produits directs infinis de groupes finis isomorphes, C. R. Acad. Sci. Paris A279 (1974) 583-585. [95] On buildings and their applications, Proc. Internal. Congr. Math., Vancouver, 1974 (1975), Vol. 1, 209-220. [96] Lecon inaugurale de la chaire de Theorie des groupes, College de France, 1974 (1975) 18 p. [97] Travaux de Margulis sur les sous-groupes discrets de groupes de Lie, Seminaire Bourbaki, exp. No. 482, fevrier 1976, Springer Lecture Notes in Mathematics 567 (1977), 174-190. [98] Two properties of Coxeter complexes (appendice a un article de L. Solomon), J. Algebra 41 (1976) 265-268. [99] Non-existence de certains polygones generalises. I, Invent. Math. 36 (1976) 275-284. [100] Systemes generateurs de groupes de congruence, C. R. Acad. Sci. Paris A283 (1976) 693-695. [101] Classification of buildings of spherical types and Moufang polygons: a survey, Atti Coll. Internat. Teorie Combinatorie, Accad. Naz. Lincei, Roma, 1975 (1976), t.l, 229-246. [102] A theorem of Lie-Kolchin for trees, Contributions to Algebra, a collection of papers dedicated to Ellis Kolchin (Academic Press, 1977), 377-388. [103] Quadrangles de Moufang, I, Prepublication, Paris, 1976, 16 p. [104] (avec W. Feit) Projective representations of minimum degree of group extensions, Canad. J. Math. 30 (1978) 1092-1102. [105] Endliche Spiegelungsgruppen, die als Weylgruppen auftreten, Invent. Math. 43 (1977) 283-295. [106] Sur certains groupes dont l'ordre est divisible par 23, Bull. Soc. Math. Belg. 27 (1975) 325-332. [107] Groupes de Whitehead de groupes algebriques simples sur un corps, Seminaire Bourbaki, exp. No. 505, juin 1977, Springer Lecture Notes in Mathematics 677 (1978), 218-236. [108] (avec A. Borel) Theoremes de structure et de conjugaison pour les groupes algebriques lineaires, C. R. Acad. Sci. Paris A287 (1978) 55-57. [109] Reductive groups over local fields, Proc. Symp. Pure Math. 33 (Proc. Summer Inst, on Group Representations and Automorphic Forms, Corvalis 1977) (1979), Vol. 1, 29-69.
735 110] The work of Gregori Aleksandrovitch Margulis, Proc. Internat. Congr. Math., Helsinki, 1978 (1980), Vol. 1, 57-63. I l l ] Non-existence de certaines polygones generalises. II, Invent. Math. 51(1979) 267-269. 112] (avec C. W. Curtis et G. I. Lehrer) Spherical buildings and the character of the Steinberg representation, Invent. Math. 58 (1980) 201-210. 113] Quaternions over Qf^/5], Leech's lattice and the sporadic group of Hall-Janko, J. Algebra 62 (1980) 56-75. 114] Le principe d'inertie en relativite generate, Bull. Soc. Math. Belg. 31 (1979) 171-197. 115] Four presentations of Leech's lattice, Finite Simple Groups II, Proc. of a London Math. Soc. Research Symp., Durham, 1978, ed. M. J. Collins (Academic Press, 1980), 303-307. 116] Buildings and Buekenhout geometries, ibid., 309-320. 117] Geometrie de I'espace, du temps et de la causalite: la voie axiomatique (extraits d'une conference faite a Halle en 1980; voir ref. [145] ci-dessous), Melanges Paul Libois, Bruxelles, 1981, 291-296. 118] Expose sur les mathematiques, fait a l'occasion du 450e anniversaire du College de France, C. R. de la Reunion extraordinaire de l'Assemblee des professeurs, College de France, octobre 1981, 9-11. 119] Appendice a Particle de M. Gromov: Groups of polynomial growth and expanding maps, Publ. Math. I.H.E.S. 53 (1981) 74-78. 120] Definition par generateurs et relations de groupes avec BN-paires, C. R. Acad. Sci. Paris 293 (1981) 317-322. 121] Algebres enveloppantes et groupes de Chevalley generalises, Actes des Journees 'Groupes et langages', Amiens, mai 1981, 5 p. 122] Groupes a croissance polynomiale (d'apres A. Gromov et al), Seminaire Bourbaki, exp. No. 572, fevrier 1981, Springer Lecture Notes in Mathematics 901 (1981), 176-188. 123] A local approach to buildings, The Geometric Vein, the Coxeter Festschrift (Springer-Verlag, 1981), 519-547. 124] Evariste Galois, son oeuvre, sa vie, ses rapports avec l'Academie, expose fait a l'Academie des Sciences le 7 juin 1983, a l'occasion du 150e anniversaire de la mort d'Evariste Galois (Institut de France et Gauthier-Villars, Paris, 1982), 10 p. 125] Moufang octagons and the Ree groups of type 2F±, Amer. J. Math. 105 (1983) 539-594. 126] On the distance between opposite vertices in buildings of spherical types (appendix to "Some remarks on Tits geometries" by A. E. Brouwer and A. M. Cohen), Proc. Ned. Acad. Wet. A86 (Indagationes Math. 45) (1983) 400-402. [127] (avec F. Bruhat) Groupes reductifs sur un corps local, II. Schemas en groupes. Existence d'une donnee radicielle valuee, Publ. Math. I.H.E.S. 60 (1984) 5-184.
736 [128] (avec F. Bruhat) Schemas en groupes et immeubles des groupes classiques sur un corps local, Bull. Soc. Math. Fr. 112 (1984) 259-301. [129] On R. Griess' "Friendly giant", Invent. Math. 78 (1984) 491-499. [130] Le Monstre (d'apres R. Griess, B. Fischer et al.), Seminaire Bourbaki, exp. No. 620, novembre 1983, Asterisque, 121-122 (1985) 104-122. [131] Symetries, La Vie des Sciences, serie generale des C. R. Acad. Sci. Paris 2 (1985) 13-25. [132] Groups and group functors attached to Kac-Moody data, Arbeitstagung Bonn 1983, Springer Lecture Notes in Mathematics 1111 (1985), 193-223. [133] (avec A. Cohen) On generalized hexagons and a near octagon whose lines have three points, European J. Combinatorics 6 (1985) 13-27. [134] Avatars des grands theoremes de classification d'Elie Cartan (bref resume d'une conference faite au colloque: "Elie Cartan et les mathematiques d'aujourd'hui", Lyon, Juin 1984), Asterisque, numero hors-serie, 1985, 439-440. [135] Immeubles de type affine, Buildings and the geometry of diagrams, Como 1984, Springer Lecture Notes in Mathematics 1181 (1986), 159-190. [136] Ensembles ordonnes, immeubles et sommes amalgamees, Bull. Soc. Math. Belg. 38 (1986) 367-387. [137] (avec J. Dieudonne) La vie et l'oeuvre de Claude Chevalley, La Vie des sciences, serie generale des C. R. Acad. Sci. Paris 3 (1987) 559-565. [138] (avec W. Kantor et R. Liebler) On discrete chamber-transitive automorphism groups of affine buildings, Bull. Amer. Math. Soc. 16 (1987) 129-133. [139] Uniqueness and presentation of Kac-Moody groups over fields, J. Algebra 105 (1987) 542-573. [140] Buildings and group amalgamations, Proc. of Groups St. Andrews 1985, eds. E. F. Robertson & C. M. Campbell, London Math. Soc. Lecture Notes 121 (1986) 110-127. [141] (avec F. Bruhat) Schemas en groupes et immeubles des groupes classiques sur un corps local. Deuxieme partie: groupes unitaires, Bull. Soc. Math. Fr. 115 (1987) 141-195. [142] (avec F. Bruhat) Groupes algebriques sur un corps local, III. Complements et applications a la cohomologie galoisienne, J. Fac. Sci. Univ. Tokyo, Sect. 1A, 34(3) (1987) 671-698. [143] Le module du "Moonshine" (d'apres I. Frenkel, J. Lepowsky et A. Meurman), Seminaire Bourbaki, exp. No. 684, juin 1987, Asterisque 1 5 2 - 1 5 3 (1987) 285-303. [144] (avec P. Lentoudis) Sur le groupe des automorphismes de certains produits en couronne, C. R. Acad. Sci. Paris, Serie I 205 (1987) 847-852. [145] (avec M. Ronan) Building Buildings, Math. Ann. 278 (1987) 101-111. [146] Geometrie von Raum, Zeit und Kausalitat: Ein axiomatischer Zugang, Raum und Zeit (Vortrage anlafilich der Jahresversammlung vom 9-12 april 1980), Nova Acta Leopoldina 54 (Nr. 244) (1987) 101-111. [147] Unipotent elements and parabolic subgroups of reductive groups, II, Algebraic
737 Groups Utrecht 1986, Springer Lecture Notes in Mathematics 1271 (1987) 265-284. Groupes de type E sur les corps globaux, notes polycopiees, College de France, 1988, 4 p. Sur le groupe des automorphismes de certains groupes de Coxeter, J. Algebra 113 (1988) 346-357. Groupes associes aux algebres de Kac-Moody, Seminaire Bourbaki, exp. No. 700, novembre 1988, Asterisque 177-178 (1989) 7-31. Strongly inner anisotropic forms of simple algebraic groups, J. Algebra 131 (1990) 648-677. Spheres of radius 2 in triangle buildings, Finite Geometries, Buildings and Related Topics (Oxford Science Publications, 1990), 17-28. Symmetrie, Miscellanea Mathematica (recueil d'articles dedies a H. Gotze) (Springer-Verlag, 1991), 293-304. (avec G. Lusztig) The inverse of a Cartan matrix, Anal. Univ. Timi§oara, Ser. St. Mat. X X X ( l ) (1992) 17-23. (avec M. Ronan) Twin trees. I, Invent. Math. 116 (1994) 463-479. Sur les produits tensoriels de deux algebres de quaternions, Bull. Soc. Math. Belg., Ser. B 45 (1993) 329-331. Moufang polygons, I. Root data, Bull. Soc. Math. Soc. Simon Stevin 3 (1994) 455-468. Twin buildings and groups of Kac-Moody type, Groups, Combinatorics and Geometry, Durham, 1990, London Math. Soc. Lecture Notes Series 165 (1992), 249-286. Sur les degres des extensions de corps deployant les groupes algebriques simples, C. R. Acad. Sci. Paris 315 (1992) 1131-1138. Ein Fixpunktsatz fiir Gebaude, und Anwendungen (Zusammenfassung eines Vortrags), Jahrestagung der D. M. V., Jena, 1996. (avec M. Ronan) Twin trees, II. Local structure and a universal construction, Israel J. Math. 109 (1999) 349-377. (avec H. Bass) Discreteness criteria for tree automorphism groups (appendix to: Tree Lattices, by H. Bass and A. Lubotzky, to appear in Progress in Mathematics (Birkhauser)). (avec R. Weiss) The classification of Moufang polygons, a paraitre. 2. Resumes de cours au College de France 1. Immeubles: classification et automorphismes, Annuaire du College de France {ACF), 74e annee (1973-1974), 631-637. 2. Groupes reductifs sur un corps local, ACF, 75e annee (1974-1975), 49-55. 3. Rigidite et arithmeticite des sous-groupes discrets de groupes de Lie, ACF, 76e annee (1975-1976), 51-55. 4. Groupes finis simples sporadiques, ACF, 77e annee (1976-1977), 57-66. 5. Polygones de Moufang et groupes de rang 2, ACF, 78e annee (1977-1978), 73-80.
738 6. Problemes de theorie des groupes en relativite Einsteinienne et chronogeometrie, ACF, 79e annee (1978-1979), 65-69. 7. Schemas en groupes sur un anneau de valuation: immeubles affine, ACF, 80e annee (1979-1980), 75-79. 8. Algebres et groupes de Kac-Moody, ACF, 81e annee (1980-1981), 75-86. 9. Algebres de Kac-Moody et groupes associes (suite), ACF, 82e annee (19811982), 91-105. 10. Le groupe sporadique de Griess-Fischer, ACF, 83e annee (1982-1983), 89-102. 11. Immeubles de type affine, classification et application aux groupes finis, ACF, 84e annee (1983-1984), 85-96. 12. Immeubles affines, groupes arithmetiques et geometries finies, ACF, 85e annee (1984-19'85), 93-110. 13. Le groupe de Griess-Fischer: construction et "Moonshine", ACF, 86e annee (1985-1986), 101-112. 14. Suite du precedent, ACF, 87e annee (1986-1987), 90-97. 15. Formes et sous-groupes des groupes algebriques simples sur les corps et les corps locaux, ACF, 88e annee (1987-1988), 85-100. 16. Immeubles jumeles, ACF, 89e annee (1988-1989), 81-95. 17. Suite du precedent, ACF, 90e annee (1989-1990), 87-103. 18. Cohonlologie galoisienne des groupes semi-simples sur les corps de nombres, ACF, 91e annee (1990-1991), 125-137. 19. Groupes algebriques sur les corps non parfaits, ACF, 92e annee (1991-1992), 115-132. 20. Groupes algebriques lineaires sur les corps separablement clos, ACF, 93e annee (1992-1993), 113-130. 21. Groupes algebriques simples de rang 2 et algebres de Clifford de petites dimensions (classification des polygones de Moufang), ACF, 94e annee (1993-1994), 101-114. 22. Polygones de Moufang (suite), description des quadrangles de Moufang connus, ACF, 95e annee (1994-1995), 79-95. 23. Arbres jumeles, ACF, 96e annee (1995-1996), 79-101. 24. Homomorphismes "abstraits" de groupes algebriques, ACF, 97e annee (19961997), 89-102. 25. Immeubles jumeles: theoremes d'existence, ACF, 98e annee (1997-1998), 97-112. 26. I. Elements unipotents et sous-groupes paraboliques de groupes algebriques simples, ACF, 99e annee (1998-1999), 95-107. II. Le plan de Cremona, ibid., 108-114. 27. Groupes de rang 1 et ensembles de Moufang, ACF, lOOe annee (1999-2000), a paraitre. 3. Manuscrits inedits; notes de cours 1. Espaces Homogenes et Isotropes et Espaces Doublement Homogenes, Memoire presente au concours scient. interfacultaire L. Empain 1955, VII + 108 p.
739 2. Liesche Gruppen und Algebren (en collaboration avec M. Kramer et H. Scheerer), notes polycopiees, Bonn, 1965, VI + 220 p.; rendition, Hochschultext (SpringerVerlag, 1983), XIV + 220 p. 3. Lectures on Algebraic Groups, Notes redigees par P. Andre et D. Winter (Yale Univ., 1967), VI + 4 + 68 p. 4. Affine Buildings, Arithmetic Groups and Finite Geometries, Notes' redigees par R. Scaramuzzi et D. White, d'apres un cours fait a l'Universite Yale a l'automne 1984, 117 p.
740
II. G R O U P E S et GEOMETRIES de C O X E T E R par Jacques TITS Les §§ 1 a 3, constituant l'essentiel des presentes notes, sont extraites, sans modification, d'un pro jet de redaction de la premiere partie ("Groupes et Geometries de Coxeter") d'un livre consacre aux groupes verifiant un theoreme de Bruhat et aux geometries qu'on peut leur associer. Cela explique une terminologie qui peut paraitre etrange a premiere vue, et certaines incoherences, telles que l'introduction de notions ou l'enonce de propositions auxiliaires non utilisees par la suite. Le § 0 reunit les definitions des principaux termes utilises; on aura interet a ne pas le lire d'abord, mais plutot a s'y referer chaque fois qu'un terme nouveau s'introduit dans le texte. A titre de bibliographie, on se bornera a mentionner, en tant que principale source d'inspiration, WITT, E., Spiegelungsgruppen und Aufzahlung halbeinfacher Liescher Ringe, Hamb. Abhandl. 14 (1941), 289-322, et a renvoyer, pour toute autre indication, a COXETER, H.S.M. et MOSER, W.O.J., Generators and relations for discrete groups, Ergebnisse der Math., N.F., 14, Springer, 1957. 0. P R E L I M I N AIRES. D E F I N I T I O N S . NOTATIONS Une geometrie est un ensemble ordonne dont les elements sont appeles drapeaux et dont la relation d'ordre, notee o < b, se lit: "a supporte b" ou "a est un sousdrapeau de 6". Les geometries forment une categorie (les morphismes etant definis de facon evidente); on peut done parler de produits directs (qui existent toujours) et de sous-geometries. Un ensemble {aj} de drapeaux est appele une famille incidente s'il possede un majorant; deux drapeaux sont incidents s'ils constituent une famille incidente. La borne superieure (resp. inferieure) d'un ensemble de drapeaux {at}, si elle existe, est appelee somme (resp. p.g.s.c. = plus grand support commun) des c^. Soient a et b deux drapeaux tels que a < 6; si l'ensemble des drapeaux c tels que a + c = b possede un element minimum, il est appele difference de 6 et a, et note b — a. Si une geometrie possede un element minimum, celui-ci est note 0. La hauteur d'un drapeau a est le plus grand entier p tel qu'il existe une suite totalement ordonnee de drapeaux ai, a^... , ap = a, avec a\ ^ 0 (si 0 existe) et a* < aj+i; si une telle suite existe pour tout p, la hauteur est dite infinie. Lorsqu'il existe un 0, on appelle en abrege drapeaux minimaux les drapeaux non nuls minimaux, e'est-a-dire les drapeaux de hauteur 1. Une chaine de longueur n est une suite de n drapeaux tels que deux elements consecutifs de la suite soient comparables. Si Y est une geometrie et a G T, la residuelle de T par rapport a a est la sous-geometrie de T constitute par les drapeaux supportes par a. Etant donnees deux parties A et B de r , I'ombre de A sur B est l'ensemble des elements de B incidents a tous les elements de A
741 Soit S une geometrie. Une geometrie sur S, ou 5-geometrie, est une geometrie r + un morphisme r —> S. On definit de fagon evidente les S-morphismes entre S-geometries. L'image dans S d'un drapeau (d'un ensemble de drapeaux, d'une chaines de drapeaux, . . . ) d'une S-geometrie est appelee I'espece (sur S) de ce drapeau (de cet ensemble, de cette chaine, . . . ) . Une chaine est tendue (sur 5) s'il n'existe aucune autre chaine de meme espece ayant les memes extremites. Une geometrie (resp. une S-geometrie) a operateurs est une geometrie (resp. une S-geometrie) + un groupe operant par automorphismes (resp. S-automorphismes) sur celle-ci; on definit de fagon evidente les morphismes de geometries a operateurs. Les expressions "sur S", ou "a operateurs", seront souvent omises, lorsque le contexte est sumsamment clair. Soient E un ensemble et I une "relation coincidence", c'est-a-dire une correspondance reflexive et symetrique, donnee dans E. Appelons drapeau toute partie de E formee d'elements deux a deux incidents. L'ensemble de ces drapeaux, ordonne par l'inclusion, constitue une geometrie P(E,I), a partir de laquelle on peut reconstituer l'ensemble E (ensemble des drapeaux minimaux de P(E,I)) et la relation d'incidence. Une geometrie qui peut etre definie de cette fagon est dite une geometrie d'incidence. Si la relation / est triviale (i.e. verifiee pour toute paire d'elements de E), P(E,I), qu'on notera alors P(E), est la geometrie des parties de E ordonnees par l'inclusion. Si E est l'ensemble des faces d'un polyedre, et J la relation d'incidence usuelle entre ces faces (l'inclusion), P(E,I) est appelee la geometrie du polyedre en question. Soient G un groupe et G\ un sous-groupe. II sera parfois utile de faire la distinction entre un point a £ G/Gi et la classe laterale de G\ dans G qu'il represente; celle-ci sera alors notee cl(a). Soient G un groupe et {Gi} un ensemble de sous-groupes de G, i parcourant un ensemble d'indices I. On notera G({Gi}) la geometrie dont les drapeaux sont les points de la somme directe des ensembles G/Gi, la relation d'ordre etant definie comme suit: a < b si cl(a) 2 cl(6). Supposons les Gi tous distincts, et introduisons dans I la relation d'ordre suivante: i < j si Gi 2 Gj; la geometrie ainsi definie sera encore designee par / . Cela etant, G({Gi}) est, de fagon naturelle, une /-geometrie a operateurs (le groupe d'operateurs etant G). Soit V un espace vectoriel reel. Un cone simplicial ouvert (c.s.o.) est l'ensemble des combinaisons lineaires a coefficients strictement positifs d'un ensemble de vecteurs lineairement independants; les demi-droites determinees par ces vecteurs sont les aretes du c.s.o. La geometrie des c.s.o. de V est la geometrie a operateurs dont les drapeaux sont les c.s.o. de V, la relation d'ordre etant definie comme suit: a < b si a est une face de b, c'est-a-dire si les aretes de a sont aussi des aretes de b, et le groupe d'operateurs etant le groupe lineaire general (groupe des automorphismes de V). Les applications sont ecrites a droite (sauf lorsqu'il s'agit de simples indexations - I'auteur a d'ailleurs I'intention d'eviter cette incoherence dans la redaction definitive). De meme, les groupes operent a droite.
742 1. G R A P H E S . G R O U P E S E T GEOMETRIES D E COXETER Soit n un entier naturel. Un graphe d'ordre n sera un objet A constitue par un ensemble S(A) de n elements appeles sommets de A, et une loi qui associe a toute paire (i, j) de sommets un entier 5^ egal a L ou > 2 selon que i = ou ^ j . On designera aussi par A la matrice symetrique des <$y. Enfin, on representera encore A par un dessin obtenu comme suit: les sommets de A sont represented par des points, et deux points i, j (i ^ j) sont joints par un trait (S^ — 2)-uple (si Sij = 2, les points ne sont pas joints), ou par un trait simple sur lequel est indiquee la valeur de 6ij (les deux conventions pouvant etre utilisees conjointement). Si A est de la forme
(
Ai A2i A31
A12 A2 A32
A13 A23 A3
---X ••• 1 ••• I '
ou les Aj sont des matrices carrees et ou les Ay sont composees uniquement de nombres 2, on dira que le graphe A est la somme directe des graphes Ai, A 2 , . . . , et on ecrira A = Ai-j-A2+ • • • Un graphe sera dit connexe s'il n'est pas somme directe de deux diagrammes d'ordres strictement positifs. Tout graphe est une somme directe de graphes connexes, appeles ses composantes connexes. On appellera sousgraphe de A, tout graphe Ai dont I'ensemble de sommets S(Ai) est partie de £ ( A ) , le nombre 6^ associe a tout couple i, j G S(Ai) etant le meme pour A et A i . Si i = S(A) — S ( A i ) est I'ensemble des sommets de A qui ne sont pas sommets de A i , on dira aussi, en abrege, que Ai est obtenu a partir de A en lui retirant I'ensemble de sommets i. La geometrie P(E(A)) des parties de S(A) sera encore notee P ( A ) . On appellera groupe de Coxeter de type A le groupe G = G(A) engendre par n generateurs r, (i £ S(A)), en correspondance biunivoque avec les sommets de A, et defini par les relations
(1.1)
far,-)'"
- 1
ou (i,j) parcourt I'ensemble des couples d'elements de S(A) tels que Jy < 00. On notera que pour i = j la relation (1.1) devient r\ = 1. i = {ii,... , ip} et i' = {i[,... , i'n_p} etant deux parties complementaires de E(A), on notera G\ = G^..^ = Gl = G%1""ln-r le sous-groupe de G engendre par les TV (i' G i'). Enfin, on appellera geometrie de Coxeter de type A la geometrie r = T(A) = G({Gi}), ou i parcourt I'ensemble des parties de S(A). 1.2. Exemple. Soient n = 2, S(A) = {1,2} et <Si2 = 5 (i.e. le graphe A se compose d'un trait (5 — 2)-uple). Alors G(A) est le groupe diedral d'ordre 26 et T(A) est la geometrie d'un polygone a 6 cotes. 2. R E P R E S E N T A T I O N S Soient V un espace vectoriel reel a n dimensions, {v;} (i G S(A)) une base de V, V le dual de V, {v^} la base duale de {VJ}, f3&: V x V —>• R la forme bilineaire
743 definie par (I>-,UJ-)PA = - c o s
n/Sij,
et Lj'i la "symetrie" definie par xu'i = x-2-(x,v'i)PA-v'i
(xeV).
2.1. Theoreme. (Witt) II existe une representation u' de G(A) dans V telle que riOj' = cu'i pour tout i G £ ( A ) . Cette condition determine to'. Demonstration. II suffit de verifier que
(2.2)
{u>W« = 1.
Lorsque i = j c'est evident. Supposons done que i ^ j . Le plan V{ engendre par v[ et v'j, et la variete a n — 2 dimensions V^', orthogonale a celui-ci, sont invariants par w£ et u/-, qui induisent sur V2' la transformation identique, et sur V{, considere comme un plan euclidien (avec la metrique definie par la restriction de /3A), les symetries par rapport a deux droites formant entre elles un angle n/Sij. II s'ensuit que les restrictions de (u^o/)'5^ a V{ et a V^ sont l'identite. Or l'espace V est la somme directe de V{ et V^', puisque la restriction de /?A a V[ n'est pas degeneree. D 2.3. Theoreme. (Witt) La forme (3& est invariante par le groupe Gu>'. Si le graphe A est connexe, les seules formes bilineaires symetriques invariantes par Gu>' sont les multiples de /3ADemonstration. La premiere assertion est evidente. Soit f3 une forme bilineaire symetrique invariante par Gu>'. En exprimant que la restriction de /3 au plan determine par v[ et v'j est invariante par w[, on trouve la relation (2.4)
{v'i,v'j)P =
-{v'i,v'i)f3.co&~,
d'ou on deduit en particulier que (2.5)
si%>2,
(v'i,vti(3=(v'j,vW.
Si A est connexe, il est possible, pour tout couple (i,j), de trouver une suite de sommets de A, i = i\, i^,... ,ip = j , telle que Simim+1 > 2 pour tout m (1 < m < p). De (2.5) il resulte alors que {v[,v'j)P a une valeur constante, independante de i, et de (2.4), que (3 est un multiple de /3A• Nous appellerons representation naturelle de G la representation u> de G dans V contragrediente de u>' (i.e. definie par {x,x') = ((x)(guj), (x')(gw')} pour tous x G V, x' G V, et g G G). Le transforme par gu> (g G G) d'un point x de V, ou
744 plus generalement d'un objet x quelconque defini dans V, sera le plus souvent note xg, ce qui revient a considerer que G opere (a droite) sur V. Pour toute partie i = {i\,... ,ip} de E(A), soit C-x = Cj2...jp le cone simplicial ouvert (c.s.o.) ensemble des combinaisons lineaires a coefficients strictement positifs de Vix,... ,Vi (si i = 0, C\ = {0}). Si g € G, le c.s.o. C\g depend seulement du drapeau a = G{ • g, et non du choix particulier de l'element g g cl.(a); nous le noterons au>*. II est facile de voir que (UJ,U)*) constitue un morphisme de geometries a operateurs de T(A) dans la geometrie des c.s.o. de V; nous l'appellerons representation naturelle de T(A). Lorsqu'aucune confusion n'est a craindre, il nous arrivera frequemment de supprimer l'asterisque de u>*. Le cone C[ est invariant par riO> si et seulement si i £ i. On en deduit que 2.6. Theoreme. Les sous-groupes G\ et Gj de G correspondant a des parties distinctes, i et j , de S(A), sont distincts. Cela nous permet de considerer T(A) comme une P(A)-geometrie. En particulier, les drapeaux elements de G/G[ seront dits d'espece i. 2.7. Corollaire. Si i C j , tout drapeau d'espece j possede un et un seul sousdrapeau d'espece i. Si i C j , tout drapeau d'espece i supporte au moins deux drapeaux d'espece j . 2.8. Corollaire. Si i = { i i , . . . , ip}, les drapeaux d'espece i sont de hauteur p. Outre la representation w, nous aurons a utiliser la representation projective naturelle w de G definie comme suit. Soient V l'espace affin deduit de V par "abstraction du 0" (espace homogene obtenu en faisant operer V sur lui-meme par translation) et w£ l'affinite involutive definie par xQ = xul + 2v'i,
x e V'.
Un raisonnement analogue a celui qui nous a permis d'etablir le theoreme 2.1 montre que les ui^ satisfont aux relations (wj^)««=l, c'est-a-dire qu'il existe une representation affine bien determinee u>' de G telle que riQ' = UJ'I pour tout i. La representation Q est alors celle induite par UJ' sur le dual V de V', qui est un espace projectif pointe. Si la forme bilineaire (3& n'est pas degeneree, les hyperplans
Hi =
{x\x€V',(x-v'i,vSfa=0},
ensembles des points fixes des £/, se coupent en un point 0, invariant par Gu>', et la representational' (resp. w) n'est pas essentiellement differente de u/ (resp. u); de fagon precise, l'espace V' (resp. V") peut etre muni d'une structure d'espace vectoriel
745 invariante par GQ' (resp. GQ) telle que la representation lineaire Q' (resp. u>) soit equivalente a ui' (resp. w). Dans tous les cas, V peut etre canoniquement identifie avec l'espace vectoriel tangent &V en son centre (point distingue) et u> est alors la representation de G dans cet espace, induite par w. 3. C H A M B R E S . H Y P E R P L A N S RADICIELS. LE C O N E O(A). FIDELITE D E LA R E P R E S E N T A T I O N u> Conservons les notations du § 2 et posons C = C S ( A ) ! C e s t done le c.s.o. ensemble des combinaisons lineaires a coefficients strictement positifs des vi (i € E(A)). Nous appellerons chambres (de la geometrie Tui*) les images par CJ* des drapeaux de hauteur n de T, e'est-a-dire les transform.es de C par les elements de G, murs d'une chambre les hyperplans de V definis par les faces a n — 1 dimensions de celle-ci, hyperplans radiciels les murs de toutes les chambres, et demiespaces radiciels les demi-espaces fermes limites par des hyperplans radiciels. Notons immediatement que 3.1. L'adherence d'un c.s.o. appartenant aTcj* est I'intersection des demi-espaces radiciels qui le contiennent Nous designerons par Wj l'hyperplan radiciel engendre par les Vj (j ^ i). 3.2. T h e o r e m e et Definitions. Etant donne un hyperplan radiciel W, il existe une et une seule transformation uiw € Gui differente de I'identite et laissant invariants tous les points de W; on I'appellera la symetrie par rapport a W. Le transforme par u>w d'un objet quelconque defini dans V sera dit symetrique de cet objet par rapport a W. Demonstration. On peut, sans nuire a la generality, supposer que W = Wj, auquel cas cji = ri(j jouit des proprietes requises. Reciproquement, soit u>w G GLJ une transformation quelconque possedant ces proprietes. La transformation ui'w g Gui' qu'elle induit dans V conserve chaque droite (affine) parallele a la droite engendree par v[, ainsi que la forme (3A- Tenant compte du fait que {V'^V'^PA ^ 0, on en deduit immediatement que u>'w = LJ[, d'ou Ww = i^iRemarque. Le theoreme 3.2 pourrait aussi se deduire comme corollaire immediat du suivant. 3.3. T h e o r e m e . La representation w* de T est fidele. Plus precisement, les cones images par w* de deux drapeaux distincts de T ont une intersection vide. 3.4. Corollaire. Les representations w et w de G sont fideles. 3.5. Corollaire. G est simplement transitif sur les chambres. 3.6. Corollaire. Les 2n faces d'une chambre donnee quelconque sont deux a deux non equivalentes pour le groupe G.
746 3.7. CoroUaire. L'intersection des adherences amu> d'une famille non vide quelconque de cones amu> (am £ T) est I 'adherence d'une face commune au> de ces cones. Le drapeau a est le plus grand support commun (p.g.s.c) des drapeaux am. 3.8. CoroUaire. Toute famille incidente {a m } de drapeaux de V possede une somme a = Y^am- Si am est d'espece \m, a est d'espece l j i m 3.9. CoroUaire. supportent.
Tout drapeau est la somme des drapeaux minimaux
qui le
3.10. CoroUaire. Soient a, b GT deux drapeaux d'especes i, j respectivement. b < a, la difference a — b existe et est un drapeau d'espece i — j .
Si
3.11. CoroUaire. Si i j , . . . , i p designent des parties quelconques de £(A),
3.12. CoroUaire. Si un element de G laisse invariant un cone au> (a 6 T), il laisse invariants tous les points de ce cone. Les diverses etapes de la demonstration du theorems 3.3 seront mises sous forme de lemmes. 3.13. L e m m e . Soient i, j £ S(A), g £ Gl'i (groupe engendre par n et rj), et D I'intersection des demi-espaces radiciels limites par Wi et Wj et contenant C. L'intersection Wi C\ Cg est I'adherence d'une face d m = n — 2 ou n — 1 dimensions de la chambre Cg. Si Wi = W^g (k € S(A)) (cas ou m = n — 1, i.e. ou Wi est un mur de Cg), n = g~lrkg. Enfin, si Wig ^ W, et Wj, on a Wigf~)D = Wi nWj C\D. Demonstration. Soient V\ la variete lineaire engendree par les v^ (h e S(A), h ^ i,j), c'est-a-dire la variete des points de V invariants par tous les elements de G l ' J , G° le groupe de Coxeter engendre par deux generateurs r° et r"j et defini par la relation {r°r^)5ii = 1,
747 Demonstration. Notons (Im) la proposition enoncee, dans le cas ou g = r^ • • • Tim est le produit de m facteurs rj, et enongons d'autre part la proposition (Ilrn) Soient i, j , h,... ,im £ E(A), g = rh •••rim et g' £ GtJ. Mors, Wig' fl Cg est Vadherence d'une face de Cg, et si Wig' = W^g, g'~xTig' = g~1rkg. (Uo) est une partie du lemma 3.13. Cela etant, la demonstration se fera par induction selon le schema suivant: (7/ m _i) => (Im), et (IIm-i) + ( 7 m ' ) ( m ' — m ) =*" (IIm)- Pour montrer que (77 m _i) implique (Im), il suffit de poser, dans ( 7 / m _ i ) , j = im et g' = rim, et de tout transformer ensuite par nm. II nous reste done a etablir la seconde implication. Nous distinguerons deux cas selon que Cg est ou non contenu dans 1'intersection D des demi-espaces radiciels limites par Wi et Wj et contenant C. Soit Cg C D. II resulte alors du lemme 3.13 que Cg n Wig' = Cg D Wi, ou Cg n Wj, ou encore Cg D Wt n W^. En outre si W ^ ' = Wfcgi, l'hyperplan Wig' est un mur de Cg, done W^g' = Wj ou W} et on a, suivant le cas, toujours en vertu de 3.13, g'~~1rig' = rj OU rj. Mais alors, la propriete a demontrer se ramene a (Im), si on tient compte du fait que 1'intersection des adherences de deux faces de Cg est encore l'adherence d'une face de Cg. Supposons a present que Cg (/ D, et soit p le plus grand entier tel que Cr ip---rim
= rh • • • rip_1rip+1
•••rimrh,
et on se ramene alors a (7/ m _i) en transformant toutes les donnees par rh (qui appartient a Glj). • 3.15. L e m m e . Soit g = r^ • • • rj m S G. 5^ Zes chambres C et Cg sont situees de part et d'autre de l'hyperplan Wi (i.e. ne sont pas contenues dans le mime demiespace radiciel limite par Wi), le produit gri pent s'ecrire sous la forme d'un produit de m — 1 generateurs rj. Demonstration. Soit p le plus grand entier tel que Crip • • • r; m soit contenu dans le demi-espace limite par Wi et ne contenant pas C, et g\ = r, p+1 • • • rim. En procedant exactement comme dans la seconde partie de la demonstration precedente on montre que gilripgi = ru d'ou gn = r;, • • • ripgxri = rh • • • r i p _ 1 r i p + 1 • • • rim . 3.16. Lemme. Si Cg D C j= 0, g = 1.
•
748 Demonstration. Soit CgnC =/= 0. En vertu du lemme 3.14, Cg n'est rencontree par aucun Wi, done Cg C C. De meme, Cg-1 C C, done C C Cg. Par consequent, Cg = C. Supposons que g soit different de 1 et exprimons-le sous forme d'un mot en les rj, de longueur minimum: (3.16.1)
g
=
rh---rim.
Les chambres C et Crim = Cr^ • • •rj m _ 1 sont situees de part et d'autre de Wj m ; par consequent, en vertu du lemme 3.15, g peut s'ecrire sous la forme d'un produit d e m - 2 generateurs rj, ce qui contredit l'hypothese suivant laquelle (3.16.1) est une expression minimum de g. • 3.17. Lemme. Si g £ G, il existe une partie i, eventuellement vide, de E(A) telle que C D Cg = C\ (autrement dit, C D Cg est {'adherence d'une face de C) et on a
Demonstration. La demonstration se fera par induction sur la longueur de g exprimee comme mot en les rj. Tenant compte du lemme 3.16, on peut supposer C PI Cg = 0. Mais alors, il existe au moins un mur Wi de C separant C et Cg. Posons g' = gr{. II est clair que C D Cg est contenu dans Wi, done
(3.17.1)
cnUj = CnCgnWi = cnCgfnWi.
Mais en vertu du lemme 3.15 et de l'hypothese d'induction, il existe une partie j de S(A) telle que
CnC7 = Cj
et
ff'eGj.
En posant i = j PI b{t}, il vient alors, tenant compte de (3.17.1), Cn'Cg~ = Ci
et
g = g'ri£Gi.
•
Demonstration du theoreme 3.3. Ce theoreme est une consequence immediate du lemme 3.17. • La proposition suivante est une consequence immediate du lemme 3.14. 3.18. Proposition. Si un hyperplan radiciel W et un cone aw (a 6 T) ont une intersection non vide, au C W. 3.19. Corollaire. Soient a € T un drapeau et W un hyperplan radiciel. Si aw n'est pas contenu dans W, tous les cones bw images des drapeaux b supportes par a (b > a) sont contenus dans un meme demi-espace ouvert limite par W. Nous designerons par Q. = 0(A) le cone reunion de tous les c.s.o. aw (a 6 F). L'ensemble de toutes les demi-droites de V issues de 0 a une structure de sphere topologique; l'image canonique d'un cone quelconque dans cette sphere sera appelee Vimage spherique de ce cone. II resulte du theoreme 3.3 que.
749 3.20. Theoreme. Les images spheriques des cones au (a € T) constituent une decomposition simpliciale de I'image spherique du cone $7 (A). La nature de cette decomposition peut etre precisee grace a la proposition 3.18 dont on deduit immediatement le 3.21. Theoreme. Les chambres sont decoupees dans Cl par les hyperplans radiciels; de facon precise, ce sont les composantes connexes de I'intersection de CI avec le complementaire de la reunion des hyperplans radiciels. Plus generalement, si W designe la variete lineaire a p dimensions (p < n) engendree par le cone au> image d'un drapeau a S T de hauteur p, les hyperplans radiciels ne contenant pas W decoupent le cone Q, n W en c.s.o. de dimension p, images de drapeaux de Y. 3.22. Theoreme. fi est un cone convexe. En outre, tout segment de droite ferme, contenu dans fi, ne rencontre qu'un nombre fini de cones appartenant a TUJ. 3.23. Corollaire. Les seuls hyperplans radiciels contenant un cone au (a € T) sont les murs des chambres ayant aui pour face. Demonstration du theoreme 3.22. Soient g = r^r^- • -rim € G, x € C et y£~Cg. Nous nous proposons de montrer que le segment ferme [xy] est entierement contenu dans 0 et ne rencontre qu'un nombre fini de cones appartenant a Tui. La demonstration se fera par induction sur m. L'intersection [xy] n C est un segment ferme \xx^\, eventuellement reduit au seul point x. Le point x\ appartient a la frontiere de C, et [xiy] D C = 0. II s'ensuit que si y ^ xi, ce qu'on ne perd rien a supposer, il existe un mur Wj de C contenant x\ et tel que C et }xiy] (intervalle ouvert en X\ et ferme en y) soient separes par Wt. Posons alors gri = g' et 2/r-j = y' e Cg'. Par l'hypothese d'induction et en vertu du lemme 3.15, le segment [xiy'} est contenu dans Cl et ne rencontre qu'un nombre fini de cones appartenant a To;, mais il en est alors de meme pour [xiy] = [x\y']ri, done aussi pour [xy] = [xxi] U [xiy]. D 4. QUELQUES A U T R E S RESULTATS Sommes et produits directs (a) Si A = Ai + A 2 , alors G(A) = G(Ai) x G(A 2 ) et T(A) = T(Ai) x T(A 2 ). (b) Si i et j sont deux parties de S(A) telles que tout drapeau d'espece i soit incident a tout drapeau d'espece j - autrement dit, si G = Gi • Gt (ensemble des produits <7i<72 a v ec g\ G Gi et g
750 1°) G est un groupe fini; 2°) La forme /?A est definie positive; 3°) Cl = V. Lorsqu'elles sont verifiees, l'image spherique de Cl = V a une structure naturelle de sphere euclidienne, et nous dirons alors que A est de type spherique. Les graphes de ce type ont ete determines par H.S.M. Coxeter. (b) Un graphe A sera dit de type euclidien (resp. hyperbolique) si l'interieur du cone Cl est un demi-espace ouvert (resp. l'interieur d'un cone quadratique de signature + + • • • H—), et de type proprement euclidien (resp. hyperbolique) si en outre Cl — {0} est ouvert; suivant le cas, l'interieur de l'image spherique de Cl, ou cette image ellememe, a une structure naturelle d'espace euclidien (resp. de Lobatchevski). Pour que le graphe A soit de type proprement euclidien, il faut et il suffit qu'il soit connexe et que la forme /?A soit semi-defmie (les graphes de ce type ont ete determines par E. Witt). Pour qu'un graphe soit de type proprement euclidien ou proprement hyperbolique, il faut et il suffit qu'il ne soit pas de type spherique mais que tous ses sous-graphes propres le soient (les graphes de ce type ont ete determines par F. Lanner). Les graphes de type euclidien sont les sommes directes d'un graphe de type proprement euclidien et d'un graphe de type spherique. Pour qu'un graphe soit de type hyperbolique, il faut et il suffit qu'il soit connexe, qu'il ne soit pas de type spherique ou proprement euclidien, mais que tous ses sous-graphes propres soient de Fun de ces deux types. (c) Les seuls graphes A tels que l'image projective de la frontiere de l'adherence du cone CI soit une variete algebrique sont les sommes directes de graphes des types considered plus haut. (d) Si A est de type hyperbolique, la forme /3A est non degeneree et de signature +H h— (la frontiere de Cl etant alors 1'une des deux "moities" du cone d'equation (a;, X)/3A = 0 ) . La reciproque n'est pas vraie. En fait, le cas ou /3& est non degeneree et de signature + + • • • -)— apparait en quelque sorte comme le "cas general"; on peut voir, par exemple, qu'il en est toujours ainsi des que tous les <5y sont superieurs a un entier A suffisamment grand (dependant peut-etre de l'ordre n de A - l'auteur n'est pas en mesure de preciser ce point). Geometrie residuelle (a) Soient i une partie de S(A), a un drapeau d'espece i et A; le graphe deduit de A en lui retirant l'ensemble de sommets i. Alors, la geometrie residuelle de T par rapport a o, qui est de fagon evidente une P(Ai)-geometrie, est isomorphe, en tant que telle, a la geometrie de Coxeter T(Ai). (b) Le groupe G{, engendre par les r* (i £ S(A) — i), est defini par celles des relations (1.1) qui concernent ceux-ci (en particulier, il est isomorphe a G(A;)). Automorphism.es,
Isomorphismes
Les seuls P(A)-automorphismes de la geometrie T(A) sont les elements de G(A). Deux geometries de Coxeter T(A) et T(A') sont isomorphes si et seulement si les graphes A et A' le sont.
751 Les geometries de Coxeter sont des geometries d'incidence (a) En particulier, toute famille de drapeaux deux a deux incidents est incidente. (b) En terme de sous-groupes, la propriete precedente se traduit comme suit: si i, j , k sont trois parties de £(A), on a G; • (Gj D Gk) = (Gj n Gj) • (G{ n Gk), ou encore, G[ n (Gj • Gk) = (Gi • Gj) n (G, • Gk)- Ces relations restent verifiees si on remplace un des trois sous-groupes G\, Gj, Gk par un sous-groupe conjugue. Ombres (a) Soient i, j , k C E(A). On dira que j separe i et k si aucune composante connexe du graphe deduit de A en lui retirant j ne renferme a la fois un sommet appartenant a i et un sommet appartenant a k, que k est reduit mod. i si aucune de ses parties propres ne le separe de i, qu'un drapeau est reduit mod. i si son espece Test, on appellera reduction de k mod. i la plus petite partie k' de k separant k et i, et enfin reduction mod. i d'un drapeau d'espece k le sous-drapeau d'espece k' de celui-ci. (b) Soit Ei l'ensemble des drapeaux d'espece i. Deux drapeaux ont meme ombre sur E\ si et seulement s'ils ont meme reduction mod. i; par consequent, les ombres de drapeaux sur E{ sont en correspondance biunivoque avec les drapeaux reduits mod. i. Soient a et b deux drapeaux reduits mod. i, d'especes j et k respectivement; pour que l'ombre de a sur E\ soit contenue dans celle de b, il faut et il suffit que a et b soient incidents et que j separe i et k. (c) L'intersection d'une famille quelconque d'ombres de drapeaux sur Ei est encore l'ombre d'un drapeau sur E\. Pavages et paves (a) On supposera ici que A est connexe et que ses sommets peuvent etre numerates de 1 a n de telle fagon que Sij = 2 pour tout couple (i,j) tel que \i — j \ > 2 (cela signifie que A ne presente ni cycle ni ramification). Soient Ei l'ensemble des drapeaux d'espece 1 (extremite de A) et u> la representation naturelle de V. Pour tout drapeau a d'espece i (1 < i < n), on notera an le cone |J 6 b, ou la reunion est etendue a tous les drapeaux b supportes par a et dont I'espece est constituee exclusivement de sommets < i (i.e. separant 1 et i). Le cone an sera appele le pave d'espece i correspondant a a (pour le choix donne de la numerotation des sommets de A). Le "pavage" ainsi defini jouit des proprietes suivantes: La reunion de tous les paves est le cone fl. Les paves d'espece i sont des cones convexes de dimension i. L'intersection an n bn de deux paves d'especes i et j est un pave en d'espece strictement inferieure a i et j . Le drapeau c est aussi le drapeau reduit dont l'ombre sur E\ est l'intersection des ombres de a et b (autrement dit, n definit un isomorphisme du treillis des ombres de drapeaux sur E\ sur le treillis des paves). La figure affine constituee par un pave d'espece i depend seulement de la partie du graphe A comprise entre les sommets 1 et i.
752 (b) Les resultats precedents redonnent en particulier les resultats classiques sur les polytopes reguliers et les pavages reguliers des espaces euclidiens et de Lobatchevski (cf. Coxeter et Moser, loc. cit.); on peut les obtenir comme cas particuliers de resultats encore plus generaux, mais moins agreables a enoncer, portant sur un graphe A quelconque et une partie privilegiee i de £(A) (jouant le role du sommet privilegie 1 dans le cas particulier envisage plus haut). Morphismes et couples de drapeaux generiques.
Specialisations
(a) Soient A une P(A)-geometrie et
753 1'intersection de fl et d'une famille de demi-espaces radiciels. Lorsque ces conditions sont remplies, nous dirons que V est une sous-geometrie convexe de T. (b) Tous les drapeaux maximaux d'une sous-geometrie convexe de V ont meme hauteur; celle-ci sera appelee la hauteur de la sous-geometrie en question. Les sous-geometries convexes de hauteur n de T sont les images de T par ses -P(A)endomorphism.es idempotents. Sous-geometrie determinee par deux drapeaux (a) On appellera sous-geometrie determinee par deux drapeaux a et b, la plus petite sous-geometrie convexe qui les contient, c'est-a-dire la geometrie constitute par tous les drapeaux invariants par tout P(A)-endomorphisme de F conservant a et b. Un drapeau appartenant a cette sous-geometrie sera dit intermediaire entre a et b. (b) Pour que deux couples de drapeaux (a, b) et (a1, b') soient equivalents, il faut et il sufiit qu'il existe un P(A)-isomorphisme de la sous-geometrie determinee par a et b sur celle determinee par a' et b', qui applique a sur a' et b sur b'. (c) Pour qu'une chaine de drapeaux a\,... ,an soit tendue, il faut et il suffit que, pour tout triple d'entiers i, j , k tels que 1 < i < j < k < n, aj soit intermediaire entre a, et afc. Pour qu'un drapeaux c soit intermediaire entre deux drapeaux a et b, il faut et il suffit qu'il existe une chaine tendue d'extremites a et b qui contienne c, sauf si a et 6 sont des drapeaux minimaux opposes, auquel cas on doit avoir c = a ou b. (d) Soient A de type spherique (i.e. G(A) fini), (a, b) un couple generique de drapeaux, et a' l'oppose de a. La geometrie determinee par a et 6 a meme hauteur que le drapeau a' + b (qui est defini puisque a' et b sont incidents). Si a et b sont minimaux et non opposes, elle se reduit a une chaine dont les elements sont alternativement de hauteurs 1 et 2, et qui est alors la seule chaine tendue d'extremites a et b. Projection (a) Etant donnes deux drapeaux a et b, il existe un drapeau ba, appele projection de b sur a, defini par l'une quelconque des trois proprietes suivantes: (i) ba est maximum dans l'ensemble des drapeaux supportes par a et intermediaires entre a et b; (ii) Si x G auj et y e bw, tout point du segment [xy] sufnsamment voisin de x est contenu dans 60w; (iii) ba est supporte par a et si Va designe la variete lineaire engendree par aw, la projection du cone bw dans V/Va est contenue dans celle de bau).
(b) T a designant la geometrie residuelle d'un drapeau a donne, l'application r -*• Ya qui envoie tout drapeau b sur sa projection est un morphisme (mais non un P(A)morphisme!). (c) Le stabilisateur de ba dans G est l'intersection des stabilisateurs de a et b. On en deduit immediatement que l'intersection d'une famille quelconque de stabilisateurs
754 de drapeaux (c'est-a-dire de sous-groupes conjugues a des sous-groupes de la forme Gi, avec i G S(A)) est encore le stabilisateur d'un drapeau. N . B . Les proprietes de la projection sont un outil essentiel pour la demonstration de la plupart des resultats enonces dans ce § 4 (par suite, I'ordre d'exposition adopte ici ne conviendrait pas a un expose avec demonstrations).
Wolf Prize in Mathematics, Vol. 2 (pp. 755-779) eds. S. S. Chern and F. Hirzebruch © 2001 World Scientific Publishing Co.
Curriculum Vitae
Birth: Paris, 6 May 1906. Student at Ecole Normale Superieure, 1922-1925. Fellowships: Commercy Foundation, Roma 1925-1926; Rockefeller Foundation, Gottingen and Berlin, 1926-1927; Commercy Foundation, Paris, 1927-1928. Ph.D. in Mathematics, Paris (Sorbonne), 1928. Professor at the Aligarh Muslim University, Aligarh (United Provinces, British India), 1930-1932. Lecturer at the College de France (Peccot Foundation), 1932. Lecturer at the Faculty of Sciences, Marseilles, 1932-1933. Lecturer, then assistant professor, then professor at the Faculty of Sciences of Strasbourg, 1933-1939. Grant from the Rockefeller Foundation, U.S.A., 1941-1943. Fellowship: Guggenheim Foundation, 1944. Professor at the Faculdade de Filosofia, Univ. Sao Paulo, 1945-1947. Professor at the University of Chicago, 1947-1958. Professor at the Institute for Advanced Study, Princeton, 1958-1976; emeritus, 1976-1998. Member of the National Academy of Sciences (U.S.A.) and of the Academie des Sciences (Paris). Honorary member of the Royal Society (London). International awards: Wolf Prize (Israel 1979), Steele Prize (A.M.S. 1980), Kyoto Prize (1994). Death: Princeton, 6 August 1998.
756 Weil's Work
The "Repertoire Biographique des Membres et des Correspondants de I'Academie des Sciences" gives the following brief description: "Andre Weil's work ranges through several branches of Mathematics: Algebraic Geometry, Number Theory, Topology and Differential Geometry, Group Theory, and History of Number Theory: 1. Algebraic Geometry: Fermat descent on curves and abelian varieties, Mordell-Weil theorem (1927 and 1951); vector bundles and moduli varieties (1938); function field analogue of the Riemann hypothesis (1940-1948); Foundations of Algebraic Geometry (1946); algebraic theory of abelian varieties (1948-1955); Weil conjectures (1949). 2. Number Theory: Introduction of cohomology (and "Weil groups") in class field theory (1951); complex multiplication and Hecke characters (1955); explicit formulae in analytic number theory (1952 and 1972); adelic methods and generalized Siegel formula (1960-1965); relations between modular forms, Dirichlet series and elliptic curves (1967-1971). 3. Topology and Differential Geometry: uniform spaces (1937); GaussBonnet theorem for polyhedra (1943); introduction of the "Weil algebra" for principal fiber spaces (1949). 4. Group Theory: integration over topological groups (1940); rigidity of discrete subgroups (1962-1964); Weil representation (1964). 5. History of Number Theory: Diophantus, Fermat, Euler, Eisenstein, Kronecker, etc. (1974-1982)." More detailed comments, both on Weil's life and on Weil's work, can be found in the April 1999 issue of the A.M.S. Notices (authors: A. Borel, P. Cartier, K. Chandrasekharan, S. S. Chern, S. Iyanaga, A. W. Knapp, G. Shimura and V. S. Varadarajan). A report by A. Borel (A.M.S. Notices, see above), and an obituary by J.-P. Serre (Biogr. Mem. F. R. S.), are reproduced below.
757 Publications
Numbers between brackets refer to papers reproduced in Weil's Oeuvres Scientifiques, Springer-Verlag, 1979. Boldface numbers are used for books and lecture notes. C.R. stands for "Comptes rendus de l'Academie des Sciences de Paris". [1926] [1927a] [1927b] [1927c] [1928] [1929] [1932a] [1932b] [1932c] [1934a]
[1934b] [1934c] [1935a] [1935b] [1935c] [1935d] [1935e] [1936a] [1936b] [1936c] [1936d]
Sur les surfaces a courbure negative, C.R. 182, pp. 1069-1071. Sur les espaces fonctionnels, C.R. 184, pp. 67-69. Sul calcolo funzionale lineare, Rend. Line. (VI) 5, pp. 773-777. L'arithmetique sur une courbe algebrique, C.R. 185, pp. 1426-1428. L'arithmetique sur les courbes algebrique, Acta Math. 52, pp. 281-315. Sur un theoreme de Mordell, Bull. Sci. Math. (II) 54, pp. 182-191. On systems of curves on a ring-shaped surface, J. Ind. Math. Soc. 19, pp. 109-114. Sur les series de polynomes de deux variables complexes, C.R. 194, pp. 1304-1305. (jointly with C. Chevalley) Un theoreme d'arithrnetique sur les courbes algebriques, C.R. 195, pp. 570-572. (jointly with C. Chevalley) Uber das Verhalten der Integrate erster Gattung bei Automorphismen des Funktionenkorpers, Hamb. Abh. 10, pp. 358-361. Une propriete caracteristique des groupes de substitutions lineaires finis, C.R. 198, pp. 1739-1742. Une propriete caracteristique des groupes finis de substitutions, C.R. 199, pp. 180-182. Uber Matrizenringe auf Riemannschen Flachen und den RiemannRochschen Satz, Hamb. Abh. 11, pp. 110-115. Arithmetique et Geometrie sur les varietes algebriques, Act. Sci. et Ind. No. 206, Hermann, Paris, pp. 3-16. Sur les fonctions presque periodiques de von Neumann, C.R. 200, pp. 38-40. L'integrale de Cauchy et les fonctions de plusieurs variables, Math. Ann. I l l , pp. 178-182. Demonstration topologique d'un theoreme fondamental de Cartan, C.R. 200, pp. 518-520. Les families de courbes sur le tore, Mat. Sbornik (N.S.) 1, pp. 779-781. Arifmetika algebraiceskykh mnogoobrazii (Arithmetic on algebraic varieties), Usp. Mat. Nauk 3, pp. 101-112. Matematika v Indii (Mathematics in India), Usp. Mat. Nauk 3, pp. 286-288. La mesure invariante dans les espaces de groupes et les espaces homogenes, L'Ens. Math. 35, p. 241.
758 [1936e] [1936f] [1936g] [1936h] [1936i] [1937] 1937 [1938a] [1938b] [1938c] [1939a] [1939b] [1940a] [1940b] [1940c] 1940d [1941] [1942] [1943a] [1943b] [1945] 1946a [1946b] [1947a]
[1947b]
La theorie des enveloppes en Mathematiques Speciales, Enseign. Scient. 9 e annee, pp. 163-169. Les recouvrements des espaces topologiques; espaces complets, espaces bicompacts, C.R. 202, pp. 1002-1005. Sur les groupes topologiques et les groupes mesures, C.R. 202, pp. 1147-1149. Sur les fonctions elliptiques p-adiques, C.R. 203, pp. 22-24. Remarques sur des resultats recents de C. Chevalley, C.R. 203, pp. 1208-1210. Sur les espaces a structure uniforme et sur la topologie generale, Act. Sci. et Ind. No. 551, Hermann, Paris, pp. 3-40. Notice sur les Titres et Travaux Scientifiques, Hermann, Paris, unpublished. Generalisation des fonctions abeliennes, J. de Math. P. et App. 17, pp. 47-87. Zur algebraischen Theorie der algebraischen Funktionen, Crelles J. 179, pp. 129-133. "Science Franchise," unpublished. Sur I'analogie entre les corps de nombres algebriques et les corps de fonctions algebriques, Revue Scient. 77, pp. 104-106. Les groupes a pn elements, Revue Scient. 77, pp. 321-322. Une lettre et un extrait de lettre a Simone Weil, unpublished. Sur les fonctions algebriques a corps de constantes fini, C.R. 210, pp. 592-594. Calcul des probabilites, methode axiomatique, integration, Revue Scient. 78, pp. 201-208. L'integration dans les Groupes Topologiques et ses Applications, Hermann, Paris (2nd edition, 1953). On the Riemann hypothesis in function-fields, Proc. Nat. Acad. Sci. 27, pp. 345-347. Lettre a Artin, unpublished. (jointly with C. Allendoerfer) The Gauss-Bonnet theorem for Riemannian polyhedra, Trans. A.M.S. 53, pp. 101-129. Differentiation in algebraic number-fields, Bull. A.M.S. 49, p. 41. A correction to my book on topological groups, Bull. A.M.S. 5 1 , pp. 272-273. Foundations of algebraic geometry, Amer. Math. Soc. Coll., Vol. XXIX, New York (2nd edition, 1962). Sur quelques resultats de Siegel, Summa Brasil. Math. 1, pp. 21-39. L'avenir des mathematiques, Les Grands Courants de la Pensee Mathematique, ed. F. Le Lionnais, Cahiers du Sud, Paris, pp. 307-320 (2nd edition, A. Blanchard, Paris, 1962). Sur la theorie des formes differentielles attachees a une variete analytique complexe, Coram. Math. Helv. 20, pp. 110-116.
1948a,b
[1948c] [1949a]
[1949b] [1949c] [1949d] [1949e] [1950a] [1950b] [1951a] [1951b] [1951c] [1952a] [1952b] 1952c [1952d] [1952e] [1952f] 1952 1953 [1953] [1954a] [1954b] [1954c] [1954d] [1954e]
(a) Sur les courbes algebriques et les varietes qui s'en deduisent, Hermann, Paris; (b) Varietes abeliennes et courbes algebriques, ibid; [2nd edition of (a) and (b) under the collective title " Courbes algebriques et varietes abeliennes", ibid., 1971]. On some exponential sums, Proc. Nat. Acad. Sci. 34, pp. 204-207. Sur l'etude algebrique de certains types de lois de mariage (Systeme Murngin), Appendice a la I e partie de: C. Levi-Strauss, Les structures elementaires de la parente, P. U. F. Paris 1949, pp. 278-285. Numbers of solutions of equations in finite fields, Bull. A.M.S. 55, pp. 497-508. Fibre-spaces in algebraic geometry, in Algebraic Geometry Conference, Univ. of Chicago (mimeographed), pp. 55-59. Theoremes fondamentaux de la theorie des fonctions theta, Sem. Bourbaki No. 16, May 1949, 10 pp. Geometrie differentielle des espaces fibres, unpublished. Varietes abeliennes, in Colloque d'Algebre et Theorie des Nombres, C.N.R.S., Paris, pp. 125-127. Number-theory and algebraic geometry, Proc. Intern. Math. Congr., Cambridge, MA, Vol. II, pp. 90-100. Arithmetic on algebraic varieties, Ann. Math. 53, pp. 412-444. Sur la theorie du corps de classes, J. Math. Soc. Japan 3, pp. 1-35. Review of "Introduction to the theory of algebraic functions of one variable, by C. Chevalley", Bull. A.M.S. 57, pp. 384-398. Sur les theoremes de de Rham, Comm. Math. Helv. 26, pp. 119-145. Sur les "formules explicites" de la theorie des nombres premiers, Comm. Lund (vol. dedie a Marcel Riesz), p. 252. Fibre-spaces in algebraic geometry (Notes by A. Wallace). Univ. of Chicago, (mimeographed) 48 pp. Jacobi sums as "Grossencharaktere", Trans. A.M.S. 73, pp. 487-495. On Picard varieties, Amer. J. Math. 74, pp. 865-894. Criteria for linear equivalence, Proc. Nat. Acad. Sci. 38, pp. 258-260. Variete de Picard et varietes jacobiennes, Sem. Bourbaki, December 1952, 8 pp. Sur la theorie du corps de classes, Sem. Bourbaki, May 1953, 3 pp. Theorie des points proches sur les varietes differentiables, in Coll. de Geometrie Differentielle (Strasbourg 1953), C.N.R.S., pp. 111-117. Remarques sur un memoire d'Hermite, Arch. Math. 5, pp. 197-202. Mathematical Teaching in Universities, Amer. Math. Monthly 6 1 , pp. 34-36. The mathematical curriculum (a guide for students), unpublished. Sur les criteres d'equivalence en geometrie algebrique, Math. Ann. 128, pp. 95-127. Footnote to a recent paper, Amer. J. Math. 76, pp. 347-350.
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1957 1958a [1958b] [1958c] 1958d 1958 [1959a] [1959b] [1960a] [1960b] [1960c]
(jointly with S. Lang) Number of points of varieties in finite fields, Amer. J. Math. 76, pp. 819-827. On the projective embedding of abelian varieties, in Algebraic Geometry and Topology, A Symposium in honor of S. Lefschetz, Princeton Univ. Press, pp. 177-181. Abstract versus classical algebraic geometry, Proc. Intern. Math. Congr., Amsterdam, Vol. Ill, pp. 550-558. Poincare et l'arithmetique, in Livre du Centenaire de Henri Poincare, Gauthier-Villars, Paris, 1955, pp. 206-212. On algebraic groups of transformations, Amer. J. Math. 77, pp. 355-391. On algebraic groups and homogeneous spaces, Amer. J. Math. 77, pp. 493-512. On a certain type of characters of the idele-class group of an algebraic number-field, in Proc. Intern. Symp. on Algebraic Number Theory, Tokyo-Nikko, pp. 1-7. On the theory of complex multiplication, ibid., pp. 9-22. Science Francaise?, La Nouvelle N.R.F., Paris, 3 e annee, No. 25, pp. 97-109. The field of definition of a variety, Amer. J. Math. 78, pp. 509-524. Multiplication complexe des varietes abeliennes, Sem. Bourbaki, May 1956, 7 pp. Zum Beweis des Torellischen Satzes, Gott. Nachr. 1957, No. 2, pp. 33-53. (jointly with C. Chevalley) Hermann Weyl (1885-1955), L'Ens Math. 3, pp. 157-187. (1) Reduction des formes quadratiques, 9 pp.; (2) Groupes des formes quadratiques indefinies et des formes bilineaires alternees, 14 pp., Seminaire H. Cartan, 10e annee, November 1957. Sur le theoreme de Torelli, Sem. Bourbaki, May 1957, 5 pp. Introduction a I'etude des Varietes Kahleriennes, Hermann, Paris. On the moduli of Riemann surfaces (to Emil Artin), unpublished. Final Report on contract AF 18(603)-57, unpublished. Discontinuous Subgroups of Classical Groups (Notes by A. Wallace), Univ. of Chicago (mimeographed). Modules des surfaces de Riemann, Sem. Bourbaki, May 1958, 7 pp. Adeles et groupes algebriques, Sem. Bourbaki, May 1959, No. 186, 9 pp. Y. Taniyama (lettre d'Andre Weil), Sugaku-no Ayumi, Vol. 6, No. 4, pp. 21-22. De la metaphysique aux mathematiques, Sciences, pp. 52-56. Algebras with involutions and the classical groups, J. Ind. Math. Soc. 24, pp. 589-623. On discrete subgroups of Lie groups, Ann. Math. 72, pp. 369-384.
761 1961a [1961b] [1962a] [1962b] [1962c] 1962 [1964a] [1964b] [1965] [1966] [1967a] [1967b] 1967c [1968a] [1968b] 1968 [1970]
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Adeles and Algebraic Groups, I.A.S., Princeton; Birkhauser, Boston, 1982. Organisation et disorganisation en mathematique, Bull. Soc. FrancoJap. des Sc. 3, pp. 25-35. Sur la theorie des formes quadratiques, in Coll. sur la Theorie des Groupes Algebriques, C.B.R.M., Bruxelles, pp. 9-22. On discrete subgroups of Lie groups (II), Ann. Math. 75, pp. 578-602. Algebraic geometry, in Encyclopedia Americana, New York, pp. 455-457. Un theoreme fondamental de Chern en geometrie riemannienne, Sem. Bourbaki, May 1962, 13 pp. Remarks on the cohomology of groups, Ann. Math. 80, pp. 149-157. Sur certains groupes d'operateurs unitaires, Acta Math. I l l , pp. 143-211. Sur la formule de Siegel dans la theorie des groupes classiques, Acta Math. 113, pp. 1-87. Fonction zeta et distributions, Sem. Bourbaki No. 312, June 1966. Uber die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 168, pp. 149-156. Review: "The Collected Papers of Emil Artin", Scripta Math. 28, pp. 237-238. Basic Number Theory (Grundl. Math. Wiss. Bd. 144), Springer (3rd edition, 1974). Zeta-functions and Mellin transforms, in Proc. of the Bombay Coll. on Algebraic Geometry, T.I.F.R., Bombay, pp. 409-426. Sur une formule classique, J. Math. Soc. Japan 20, pp. 400-402. Series de Dirichlet et fonctions automorphes, Sem. Bourbaki, June 1968, 6 pp. On the analogue of the modular group in characteristic p, in "Functional Analysis, etc.", Proc. Conf. in Honor of M. Stone, Springer, pp. 211-223. Dirichlet Series and Automorphic Forms, Lecture-Notes 189, Springer. Notice biographique, in CEuvres de J. Delsarte, C.N.R.S., Paris 1971, t.I, pp. 17-28. L'ceuvre mathematique de Delsarte, ibid., pp. 29-47. Sur les formules explicites de la theorie des nombres, Izv. Mat. Nauk {Ser. Mat.) 36, pp. 3-18. Review of "The mathematical career of Pierre de Fermat, by M. S. Mahoney", Bull. A.M.S. 79, pp. 1138-1149. Two lectures on number theory, past and present, L'Ens Math. 20, pp. 87-110. Sur les sommes de trois et quatre carres, L'Ens Math. 20, pp. 215-222.
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La cyclotomie jadis et naguere, L'Ens Math. 20, pp. 247-263. Sommes de Jacobi et caracteres de Hecke, Gott. Nachr. 1974, No. 1, 14 pp. Exercices dyadiques, Invent. Math. 27, pp. 1-22. Review of "Leibniz in Paris 1672-1676, his growth to mathematical maturity, by Joseph E. Hofmann", Bull. A.M.S. 81, pp. 676-688. Introduction to E.E. Kummer, Collected Papers, Vol. I, pp. 1-11. Elliptic Functions according to Eisenstein and Kronecker, Ergebnisse der Mathematik. Bd. 88, Springer. Sur les periodes des integrates abeliennes, Coram. Pure Appl. Math. 29, pp. 813-819. Review of "Mathematische Werke, by Gotthold Eisenstein", Bull. A.M.S. 82, pp. 658-663. Remarks on Hecke's lemma and its use, in Algebraic Number Theory, Intern. Symposium Kyoto 1976, S. Iyanaga (ed.), Japanese Soc. for the Promotion of Science 1977, pp. 267-274. Fermat et Pequation de Pell, IIPIEMATA (W. Hartner Festschrift), Fr. Steiner Verlag, Wiesbaden 1977, pp. 441-448. abelian varieties and the Hodge ring, unpublished. Who betrayed Euclid?, Arch. Hist. Exact Sci. 19, pp. 91-93. History of mathematics: Why and how, Proc. Intern. Math. Congr., Helsinki, Vol. 1, pp. 227-236. Oeuvres Scientifiques - Collected Papers, 3 vols., Springer. (with the collaboration of M. Rosenlicht) Number Theory for Beginners, Springer. Riemann, Betti and the Birth of Topology, Arch. Hist. Exact Sci. 20, pp. 91-96. Sur l'histoire des equations indeterminees de genre 1 depuis Diophante, Seminar on Number Theory 1980-1981, No. 26, Univ. Bordeaux I, Talence. Euler and the Jacobians of Elliptic Curves, Shafarevich Volume, Birkhauser, pp. 353-360. Number Theory - An Approach through History from Hammurapi to Legendre, Birkhauser, Basel. Souvenirs d'apprentissage, Birkhauser, Basel (translated in English, German, Italian and Japanese).
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Andre Weil and Algebraic Topology ArmandBorel ndre Weil is associated more with number theory or algebraic geometry than with algebraic topology. But the latter was very much on his mind during a substantial part of his career. This led him first to contributions to algebraic topology proper, in a differential geometric setting, and then also to the use in abstract algebraic geometry and several complex variables of ideas borrowed from it. According to [W3], I, p. 562, his first contacts with algebraic topology took place in Berlin, 1927, in long conversations with, and lectures from, Heinz Hopf. The first publication of H. Hopf on the Lefschetz fixed point formula appeared the following year, so it is rather likely that Weil heard about it at the time. At any rate, his first paper Involving algebraic topology is indeed an application of that formula to the proof of a fundamental theorem on compact connected Lie groups (which Weil attributes to E. Cartan, but is in fact due to H. Weyl): Let G be a compact connected Lie group. Then the maximal tori (i.e. maximal connected abelian subgroups) of G are conjugate by inner automorphisms and contain all elements of G ([1935c] in [W3], I, 109-111). The proof is a repeated application of the Lefschetz fixed point formula to translations by group elements on the homogeneous space GIT, where T is a maximal torus. Note that the isotropy groups on G/Tare the conjugates of T, so that an element belongs to a conjugate of 7" if and only if it fixes some point in GIT. Well first points out that T is of finite index in its normalizer N(T). If t <= T generates a dense subgroup of T, then its fixed points are the same as those of T, and a local computation shows their indexes to be simultaneously equal to l o r to - 1 . The Lefschetz number of r is then ^O.But since f is connected to the identity, this number is equal to the Euler-Poincare characteristic x(G / T) of G / T, which is therefore 4 0. As a consequence, any element g e G has a non-zero Lefschetz number, hence a fixed point,
and belongs to a conjugate of T. If T is another torus and r' generates a dense subgroup of 7", then any torus containing t' will also contain V, whence the conjugacy statement. This was the first new proof of that theorem, completely different from the original one, which relied on a study of singular elements (cf. H. Weyl, Collected Papers II, 629-633). It was rediscovered independently, about five years later, by H. Hopf and H. Samelson (Comm. Math. Helv. 13 (1940-41), 240-251). For about ten years, from 1942 on, topology was present in several works of Weil, often pursued simultaneously, which I first list briefly: a) In algebraic geometry: foundations, introduction of fibre bundles, formulation of the Weil conjectures. b) New proof of the de Rham theorems. Together with Leray's work, this was the launching pad for H. Cartan's work in sheaf theory. c) Characteristic classes for differentiable bundles: Allendoerfer-Weil generalization of the GaussBonnet theorem, theory of connections, the ChernWeil homomorphlsm, the Weil algebra. d) Joint work with Cartan, Koszul, and Chevalley on cohomology of homogeneous spaces. e) A letter to H. Cartan (August 1,1950) on complex manifolds, advocating the use of analytic fibre bundles in the formulation of problems such as those of Cousin. There is a last item I would like to add, dating from 1961-62: f) Local rigidity of discrete cocompact subgroups of semisimple Lie groups. On the face of it, it does not belong to algebraic topology, but can be fitted under my general title when stated as a theorem on group cohomology. This formulation was originally an afterthought, but turned out to be important to suggest further Armand Borel is professor emeritus of mathematics at the Insti- developments. tute for Advanced Study. His e-mail address [email protected]. This article is based on a lecture at the Institute for Advanced Algebraic Geometry Study on January 8, 1999, as part of a conference on the Work The algebraic geometry, as developed mainly by the of Andre Weil and Its Influence, January 8-9, 1999. Italian School, did not offer a secure framework for The article is also appearing in the Gazette des Mathematiciens. the proof of the Riemann hypothesis for curves and
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other researches of Weil in algebraic geometry. He had to develop new foundations, with as one of its main goals a theory of intersections of subvarieties. It had also to be over any field. This implied a massive recourse to algebra, but Weil still wanted to keep a geometric language and picture. Until then, only projective, affme, or quasi-projective varieties had been considered, ie. subvarieties of some standard spaces. He wanted a notion of "abstract variety" which would be the analogue of a manifold (albeit with singularities). His first version [Wl] is a bit awkward, as acknowledged in the foreword to the second edition, because no topology is introduced. From {[1949c], [W3], 1,4X1-413) on, however, he uses the language of i lariski topology (introduced in 1944 by O. ZarisicI), and I shall do so right away. Fix a "universal field" K, i.e. an -algebraically closed field of infinite transcendence degree over its prime field. Let V be an algebraic subset of Kn, i.e. an afflne variety. In the Zariski topology, the closed subsets of V" are the algebraic subsets. Hie open, sets are, of course, their complements and are quite big. If V is irreducible, any two nonempty ones intersect in a dense open one, so that the topology Is decidedly not Hausdorff (unless V is a point), which may explain some reluctance to use it initially. To define an (irreducible) abstract variety V, start from a finite collection (V/, fjt)$ (I, j e I), where Vt is an irreducible affine algebraic set, fjt a birational correspondence from Vt to Vj satisfying certain conditions, so that, in particular: fu Is the identity, fij = fj^1, there exist open subsets Djt c Vt such that fjt is a b i r e g u l a r m a p p i n g of Djt o n t o Dy, and fjt = fjk ° fia- Two points Pt e Djt and Pj e Du are equivalent if fjtiPt) = Pj. The "abstract variety" V is by definition the quotient of the disjoint union V of the Vt by that equivalence relation. Note that V Is obtained by gluing together disjoint affine sets. For lack of suitable concepts, it was not possible to start from a topological space and require that it be endowed locally with a given structure, as Is done for manifolds (as was done later by Scire using the notion of ringed space [S2]). As a result, the Vt and fjt are part of the structure, which is rather unwieldy and requires a somewhat discouraging amount of algebra to be worked with. Nevertheless, Weil develops the theory of such varieties and of the intersection of cycles. For the latter, the analogy with the complex case and the intersection product in the homology of manifolds (on which he had lectured earlier at the Hadamard Seminar (IW3J, I, 563)) Is always present. In particular, a key property is the analogue of Hopf s inverse homomorphism (see [W1J, Introduction, xi-xtt). Weil also introduces an analogue of compact manifolds, the complete varieties, which include the projective ones. [W1J supplied the framework for a detailed proof of the Memann hypothesis for curves and for further work on APML
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Weil (left) with Armand Borel in Chicago about 1955. abelian varieties, and it supplied essentially the only framework for algebraic geometry over any field until Grothendieck's theory of schemes (from about 1960 on). Algebraic topology also underlies the formulation of the conjectures in ([1949b], cf. [W3], I, 399-410), soon to be called the Wei conjectures, which suggest looking for a cohomology theory for complete smooth varieties in which a Lefschetz fixed point formula would be valid. This vision, which turned out to be prophetic, was unique at the time. In ([1949c], cf. [W3], 1,411-12), Weil introduces in algebraic geometry fibre bundles with an algebraic group, say G, as structural group. Given a variety B and a finite open cover {Vt}(i e I) of B, assume one is given regular maps stj : Vt n Vj ~* G (i, j € I; su is the constant map to the Identity), with the usual transitivity conditions. Let F be a variety on which G operates. Then a fibre bundle E on B, with typical fibre F, Is obtained by gluing the products Vf x Fby means of the % , as usual. Weil also considers the case of principal bundles (F = G$ acted upon itself by right translations), hi particular, if G = C* is the multiplicative group of nonzero complex numbers, the Isomorphism classes of such bundles correspond to linear equivalence classes of divisors. It also allowed Weil to interpret in a more conceptual way earlier work on algebraic curves (see [W3], I, 531, 541, 570 for comments). A detailed exposition is given in [W2], where the classification of such bundles is studied hi some simple cases. In view of the big size of the neighborhoods on which such a bundle is trivial, it was not a priori clear this would lead to an interesting theory. That it did is one reason why Weil began to gain confidence in the Zariski topology. Of course, Ms definition of fibre bundle was greatly generalized later. Already In [1949c], Weil points out it would be NOTICES OF THE AMS
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765 desirable to have a notion broad enough so that B could be the set of prime spots of a number field. Later (Seminaire Chevalley 1958,1), J.-P. Serre introduced an important generalization of local triviality: a bundle is locally isotrivial if every point has an open neighborhood admitting an unramified covering on which the lifted bundle is trivial. This notion, which encompasses the fibration of an algebraic group by a closed subgroup, led A. Grothendieck to the definition of etale topology. The de Rham Theorems In January 1947 Weil wrote a letter to H. Cartan (fW3), II, 45-47) outlining a new proof of the de Rham theorems, published later in ([1952a], [W3], H, 17-43), the first one since de Rham's thesis. It is limited to compact manifolds, but this restriction is lifted, with very little complication, in the final version. Given a smooth compact connected manifold M, Weil first shows the existence of a finite open cover i f , h e / of M such that any nonempty intersection of some of the U/s is contractible. Let N be the nerve of this cover, and, for each simplex
d-.A™ -A™*1,
S:AP'"^AP*l'i,
where d stems from exterior differentiation and S from the coboundary operator on N, followed by restriction of differential forms. Let f^ (resp. Hp'<>) be the subspace of AVM of elements annihilated by d5 {resp. d or 5). Then Weil establishes the isomorphisms (2) pO,m/H0,m
= HmR(M),
fm,0/f]m,0
(3) fP«IHt>-« = fP+l.i-i/HP + 1 '"- 1
=
Hm{N)
(0 < q < m),
where H$S(M) refers to real de Rham cohomology and Hm(N) to the real cohomology of N. This proves, by induction, that H™R(M) is isomorphic to H m (N), hence to Hm(M), since the Ua are contractible. This last fact is taken for granted in the letter, but Weil also shows in [1952a] by similar arguments that Hm(M) is equal to the m-th singular cohomology of M over the reals. H. Cartan had studied the work of Leray in topology, in particular his wartime paper (7. Math. Pare Appl. 29 (1945), 95-248), and he noticed a similarity with Weil's proof. That was tremendously suggestive to him and quickly gave rise to a flurry of letters to Weil, in which Cartan initiated his theory of faisceaux et carapaces (sheaves and gratings), NOTICES OF THE AMS
of which he gave later three versions (see |B] for references), first following Leray rather closely, arriving eventually at a much greater generality. What Cartan had noticed is an analogy between the proofs of the isomorphisms in (3) and an argument which occurs repeatedly in Leray's paper, to which Leray himself traced later the origin of the spectral sequence (see [B]). However, Weil's argument was completely independent from it: As stated in a slightly later letter to Cartan, Weil did not know that paper and in fact suspected, on the strength of a report by S. Eilenberg in Math. Reviews, that it did not bring much new, if anything. On the other hand, it is quite plausible that the definition of the APM was in part inspired by a short conversation Weil had with Leray in summer 1945, in which the latter spoke of a cohomology "with variable coefficients". In fact, an analogue of A in the theory Leray was developing at the time would be a couverture, N, with coefficients in the differential graded sheaf associated to differential forms. Characteristic Classes In 1941-42 Weil was for some time at Haverford College, Pennsylvania, where he met C. Allendoerfer. This led to their joint work on the generalized Gauss-Bonnet theorem (AW], Given a smooth compact oriented Riemannian manifold M of even dimension m, it expresses the Euler-Poincare characteristic x(M) of M as the integral over M of a differential m-form built from the components of the curvature tensor. Such a formula had already been proved by Allendoerfer and by Fenchel for submanifolds of euclidean space. At the time, the Allendoerfer-Weil theorem was in principle more general, since it was not known whether a Riemannian manifold was globally isomerrically diffeomorphic to a submanifold of euclidean space, though it had been established locally. Because of that, the nature of their proof forced them to prove a more general statement, though I do not know whether the added generality has led to further applications. Recall the Gauss-Bonnet formula in the most classical case: P is a relatively compact open subspace on a surface in R3, bounded by a simple closed curve, union of finitely many smooth arcs. Then the integral of the Gaussian curvature K on P, plus the sum of the integrals of the curvature on the boundary arcs and of the outside angles at the meeting points of those, is equal to 2rr. The Allendoerfer-Weil formula gives a generalization of such a formula for a Riemannian polyhedron. It is proved first for polyhedra in euclidean space. The general case then follows by using a polyhedral subdivision, small enough so that its building blocks can be isomerrically embedded in euclidean space, and by proving a suitable addition formula. In 1944 S. S. Chern produced a proof of the Allendoerfer-Weil formula for closed manifolds {Ann. VOLUME 46, NUMBER 4
766 Math 45 (1944), 747-52) which was much simpler and a harbinger of further developments on characteristic classes. On M choose a vector field X with only one zero, of order \x(M)\ at some point x<>, which is always possible. Let E be the unit tangent bundle to M, p: E -* M the canonical projection, and 0 the Gauss-Bonnet form. The key point is that p*Q = dU is the exterior derivative of some explicitly given form n , the restriction of which to a fibre F represents the fundamental class IF] of the fibre. "The vector field X defines a submanif old V in E, a copy of M - {XQ }, with boundary the unit sphere FQ * p _1 (xo), with multiplicity \x(M)l Hie Gauss-Bonnet formula then folows from the Stokes theorem, applied to V u FQ. Hie relationship between fl, n, and [FQ] is a first example of a notion developed later under the name of transgression in a fibre bundle: a cohomology class p of a fibre F is transgressive if there is a cochain (in the cohomology theory used, here a differential form) on the total space E whose restriction to F is closed, represents J?, and whose coboundary belongs to the image of a cohomology class r) of the base B, under the map induced by the projection p:E — B. The classes p and rj will be said to be related by transgression. This notion, and the terminology, were introduced first by J.-L Koszul in a Lie algebra cohomology setting in his thesis (Bull Soc. Math France 78 (1950), 65-127). In Arm. Math 47 (1946), 85-121, Chem gives several definitions of the characteristic classes cHM) E H2i(M;C)t since then called the Chem classes (1 <, i £ m). In particular, if M is endowed with a hermitian metric, they can be expressed by closed differential forms which are locally defined in terms of the curvature tensor. Again, each one is related by transgression in a suitable bundle to the fundamental class of the fibre. It is at this point that Well comes in. He was familiar with the work of Chern, with the theory of fibre bundles, in particular! with the classification theorem in terms of universal bundles, having written jointly with S. Ellenberg, with some help from N. Steenrod, a report on fibre bundles for BourbaM (which, incidentally, provided much background material for the second Cartan seminar [C2]). He was also aware of Ehresmann's publications on fibre bundles and on the formulation of E. Cartan's theory of connections in that framework, as weU as of Koszul's work towards his thesis quoted above. All this came together in a series of letters to Cartan, Chevalley, and Koszul, of • which the first four were published (almost completely) for the first time, thirty years later ([1949e], in [W3], 1,422-36). Some were shown around at the time, however. In particular, the first one is the basis of Chapter in in [C4J, and this is how its contents became widely known. Aran. 1999
Let G be a compact connected Lie group, § a principal G -bundle, E (resp. B) the total space (resp. base) of §. A connection on § is defined by means of a 1-fonn on E with values in the Lie algebra g of G, satisfying certain conditions. Let IG be the algebra of polynomials on g invariant under the adjoint representation and P e IG a homogeneous element of degree q. Replacing the variables in P by the components of the curvature tensor of the connection, WeU associates to P a differential 2fl-form on M, which is proved to be closed, hence to define an element cp e H2^(M; E). A fundamental theorem asserts that cp is independent of the connection. The proof is short but stunning. Jh the fall of 1949, in Paris, 1 read this letter and said once to Cartan that this proof seemed to come out of the blue and I could not trace it back to anything. "That's genius. You don't explain genius," was his answer. The image of IG under this homomorphism, which became k n o w n as t h e Chem-Weil homomorphism, is then the characteristic algebra of §• At the end of the first letter, Weil states a conjecture Wei! at the Tata Institute of Fundamental relating the primi- Research in iombay, Janyary 1967. tive generators of ff*(G;R) (recall that it is an exterior algebra with a distinguished set of generators, called primitive) to the characteristic algebra by transgression, soon proved by Chevalley. This already provided a generalization of Chern's treatment of characteristic classes of hermitian bundles, modulo some normalization and plausible identifications. In the third letter, which, like the fourth, was addressed to Koszul, Weil makes the analogy closer. Recall that in the classical case the characteristic classes are the images of cohomology classes of a classifying space (a complex Grassmannlan for hermitian bundles), under the homomorphism induced by a classifying map (see [C4J, for example). Weil proposes an algebraic analogue of that situation. He introduces an algebra which, following Cartan [C3], I shall denote W(g) and call the Weil algebra of g. By definition, W(8) = S(g*) e Ag* is the tensor product of the symmetric algebra S(§*) by the exterior algebra Ag* of the dual g* of g. It is graded, anticommutatlve, an element x € g* being given the degree 1 (resp. 2) if it is viewed as belonging to Ag* (resp. S(g*)). The Weil algebra is
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767 further endowed with a specific differential. The latter leaves 5(g*)» R stable, the cohomology of which is isomorphic to IG- The algebra Ag*, endowed with the Lie algebra cohomology differential, is a quotient of Wis). The transgression in W(g) provides a bijection of the space of primitive generators of H* (g) (which is isomorphic to H*(G; R)) onto a space spanned by independent homogeneous generators of /c (the latter is, by a theorem of Chevalley, a polynomial algebra). A connection on 5 provides a homomorphism of W(g) onto a subalgebra of differential forms on E which, after having passed to cohomology, yields the ChernWeil homomorphism. Thus W($) plays the role of an algebra of differential forms on a universal Gbundle, an analogy reinforced by the fact, proved by Cartan [C3J, that W(g) is acyclic. So far, I have focused on characteristic classes. But these letters, combined with Koszul's thesis, led to further correspondence on the cohomology of homogeneous spaces and to more results announced by H. Cartan (Collogue de Topologie, C.B.R.M., Bruxelles, 19S0, 57-71) and J.-L. Koszul (ibid., 73-81). A full exposition is given in [GHV]. Complex Manifolds and Holomorphic Fibre Bundles On August 1, 1950, Weil wrote to H. Cartan a letter about global analysis in several complex variables (unpublished). He first claims that it is high time to stop viewing the object of these investigations as a sort of "domain" spread over n-space or complex projective space. One should look at complex manifolds, noting, of course, that not much can be proved without further assumptions such as compact, Kanler, global existence of holomorphic functions with nonzero Jacobians, etc. Then he points out that analytic fibre bundles underlie some classical problems. For instance, the Cousin data for the multiplicative Cousin problem (find a function with a given divisor of zeros and poles) lead to a principal C*-bundle. For a solution to exist, the bundle should first be topologically trivial. This condition is not always sufficient, but it is on a domain of holomorphy. Pursuing that idea, he conjectures that a complex vector bundle on a polycylinder with structural group a complex Lie group which is topologically trivial should be analytically trivial. Unfortunately, I could only find the first page of this letter in Weil's papers; the original seems to be lost, or at any rate could not be located. The beginning of the last sentence: "Once one has taken the habit to look for fibre bundles in these questions, one soon sees them everywhere (or 'almost everywhere') and there is an enormous gain...", makes one strongly wish to see the rest. These remarks were taken into account by H. Cartan (Proceeding I.C.M., Vol. 1,1950,152-164), who also pointed out that the first Cousin probNOTICES OF THE AMS
lem (find a meromorphic function with given polar parts) leads to a principal complex bundle too, but with fibre the additive group of C. Local Rigidity It is well known that compact Riemann surfaces of higher genus have moduli (noncompact ones too, but I confine myself to the compact case). Such a surface is a quotient Y\X of the upper half-plane X = SL2(R)/SO(2) by a discrete cocompact subgroup r of SL2
,g.gu-3~l,
(a e G)}
contains a neighborhood of x0 in R(E, C). The theorem is further extended to the case where G has some factors locally isomorphic to SL2OR), provided that the projection of Ton any such factor is not discrete. At the time, it was rumored (and VOLUME 46, NUMBER 4
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in fact stated to [SI]) that Calabi had proved local rigidity when X is the hyperbolic it-space (n > 3), but this was not contained in his only publication on that matter [CI], and WeU kept telling me that an essential idea was still missing. But he found it in notes by Kodaira of some 1958-59 seminar lectures by Calabi, and then proved the above results within a few days. The paper [CV] considers first of aE the case where X is an irreducible bounded symmetric domain and shows that its complex structure is locally rigid, provided X is not isomorphic to the unit ball in C n (n s» 2). Both [CI] and [CV] follow the model of the Kodaira-Spencer theory of deformations of complex structures. Local rigidity follows then from the vanishing of a first cohomology group, with coefficients in germs of Killing vector fields in [CI], of holomorphic tangent vector fields in [CV]. In. [1964a] Weil provides similarly a cohomological translation of [1962b] by showing that the proof there implies the vanishing of the first group cohomology space H 1 ^;^) of JTwith coefficients in the Lie algebra § of G, acted upon by the adjoint representation. The proof in [1962b] was already cohomological in sprMt and is described so by Weil in his comments. It is first reduced to the case of a oneparameter group of deformations, defined by a vector field g. Without changing its class modulo inner automorphisms, he replaces 5 by a "harmonic , ' one, i.e. by the minimum of a suitable variation problem. It is then shown to be G -invariant and a direct He algebra computation shows that it is zero if G has no factor which is either compact or locally isomorphic to SL2CR). Weil was in fact not a newcomer to group cohomology. In 1951, he had asked a student, Arnold Shapiro, to prove a certain lemma on the cohomology of finite groups. The latter complied and the lemma came up later in countless variations, all known as "Shapiro's lemma". WeU. never came back to these questions, but several further developments originated in these papers. If instead of 3 we take C acted upon trivially, then Hl(T; C) is trivial if and only if the commutator subgroup of Tis of finite index in. I*. The vanishing of J^flTlC) was proved in many cases, using an approach similar to Weil's, by Matsushima, who extended it further to determine some higher cohomology groups (Osaka J. Math. 14 (1962), 1-20). Later I generalized MatsusMma's theorem to noncocompact arithmetic groups, which yielded the determination of the rational X-groups of rings of algebraic integers (Arm. Set £cole Norm. Sup. Paris (4) 7 (1974), 235-272) and led to the study of Mgher regulators in algebraic ^-theory (Arm. Sci £cole Norm. Sup. Pisa (4) 4 (1977), 613-656). In another direction, N. Mok, Y.-T. Siu, and S.-K. Yeung used a nonlinear version of Matsushima's approach to establish archimedean superrigidity of cocomAPML 1 9 9 9
pact discrete subgroups (Invent. Math. 113 (1993), 57-83). This concludes my survey of algebraic topology in the work of A. Weil. Viewed as part of Ms overall output, it is q u a n t i t a t i v e l y m i n o r . Still, it reaches out to an impressive amount of mathematics, has been very Influential, and testifies to the breadth of his outlook, as well as to his concentration on essential quesWeil 1987. tions. References [AW] A. AIJLENDOERFER and A. WEIL, The Gauss-Bonnet the-
orem for Riemaniiian polyhedra, Trans. Amer. Math Sac. (VI) S3 (1943), 101-129; [W3], I, 299-327, [Bl] A. BOREL, Jean Leray and algebraic topology, Leray's Selected Papers I, Springer and Soc. Math. France, 1997, pp. 1-21. [CI] E. CALABI, On compact, Memannian manifolds with constant curvature, I, Differential Geometry, Proc. Sympos. Pure Math. vol. ID, Amer. Math. Soc, Providence, RI, 1961, pp. 155-180. [CV] E. CALABI and E. VESENTTNI, On compact locally symmetric Kihlerian manifolds, Ann. Math. 71 (I960), 472-507. IC.2] H. CARTAN et a!., Homotopie et espaces fibres, Sem. Eeole Norm. Sup., 1949-50. [C3] H. CARTAN, Notions d'algebre differentielle; application aux groupes de Lie et aux varietes ou opere un groupe de Lie, Colloque de Topologie, C.B.E.M., Bruxelles, 1950, p p . 15-27; Collected Papers III, p p . 1255-67. [C4] S. S. CHERN, Topics in differential geometry, Notes, Institute for Advanced Study, 1951. [GHV] W. GREUB, S. HALPEMN, and R. VANSTONE, Connec-
tions, Curvature and Cohomology, Vol. 3, Pure Appl. Math. vol. 47, Academic Press, 1976. [SI] A. SELBERG, On discontinuous groups in higher-dimensional symmetric spaces, Contributions to Function Theo'ry, Internal. Colloquium TTFR, Bombay, 1960, pp. 147-164. [S21 J.-P. SERRE, Faisceaux algebriques coherents, Ann. Math. 61 (1955), 197-278; Collected Papers I, p p . 310-91. IWl] A. WEE, Foundations of Algebraic Geometry, Colloquium Publ., vol. XXLX, Amer. Math. Soc, New York, NY, 1946; second edition, Providence, RI, 1962. [W2] , Fibre spaces in algebraic geometry (Notes by A. Wallace), Notes, University of Chicago, 1952. [W3] ....,., Collected Papers, 3 volumes, Springer-Verlag, New York, 1979.
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ANDRE WEIL 6 May 1906 — 6 August 1998
Biog. Mems Fell. R. Soc. Lond. 45, 519-529 (1999)
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ANDRE WEIL 6 May 1906 — 6 August 1998 Elected For.Mem.R.S. 1966 BY JEAN-PIERRE SERRE, F O R . M E M . R . S . College de France, 3 rue a" Ulm, 75231 Paris, France
INTRODUCTION Andre Weil died in Princeton in August 1998; he was ninety-two years old. His last years were saddened by the loss of his wife Eveline, and by the infirmities of old age; death was perhaps a relief for him.
THE MAN He was born in Paris in 1906, into a Jewish family. His father, a doctor, came from Alsace, his mother was of Austrian origin but was born in Russia. He had a sister, Simone, three years younger; the two children were very close, and remained so until Simone's death in 1943; thereafter, Andre Weil was much involved in publishing the many manuscripts that she had left. His book Souvenirs d'apprentissage (47)* contains a charming account of the unorthodox but scholarly education that he received; the outcome was a lively taste for ancient languages (Latin, Greek, Sanskrit) and a firm vocation to be a mathematician. This led him to enter the Ecole Normale Superieure in 1922, when he was only sixteen (he was said to walk there in short trousers). He left in 1925, first in the class despite a blank paper on rational mechanics, This memoir is a translation of one written for the Academie des Sciences, of which Weil was a Member, and is published with the permission of the Academie. The original text appeared in French in L'Enseignement Mathematique (Geneve) 45, 5-16 (1999). * Numbers in this form refer to the bibliography at the end of the text.
521
© 1999 Academie des Sciences
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a subject that did not seem to him to be part of mathematics. He went to Italy and then Germany, which was then home to some of the finest mathematicians of the time, including Hilbert, Artin, von Neumann and Siegel. After submitting his thesis in 1928, when he was 22, he spent two years on the staff of Aligarh University in India, a post found for him by the Sanskrit scholar Sylvain Levi, whose course he had attended at the College de France. Thereafter, it was Marseille, then Strasbourg from 1933 to 1939. It was during his stay in Strasbourg that he and some friends from the Ecole Normale (Henri Cartan, Jean Dieudonne, Jean Delsarte...) joined together to found the Bourbaki group. In 1939, he was in Finland when war was declared; after almost being shot as a Soviet spy, he returned to France and was imprisoned at Rouen, charged with 'insoumission'. He was soon freed, and after various adventures described in his Souvenirs he managed to leave for the USA in 1940. He stayed there for some years before spending two years in Brazil, but it was only in 1947 that he was given a post commensurate with his talents: he became professor at the University of Chicago and then in 1958 he moved to the Institute for Advanced Study at Princeton, where he remained for the last 40 years of his life. The Institute suited him very well, both in the freedom that he had to teach (or not to teach) as he pleased and in the high level of its professors and visitors. (His proper place, in his homeland, should have been the College de France; I often dreamed of a chair of mathematics that he would occupy, but alas it was not to be.) To finish this account of Weil's curriculum vitae, I list some honours that he received (or rather, that he agreed to receive). He was a member of the US National Academy of Sciences, a Foreign Member of the Royal Society and of course a member of the Paris Academy of Sciences; he received the Wolf Prize in 1979 (the same year as Jean Leray, and a year before Henri Cartan) and the Kyoto Prize in 1994; this last prize gave him particular pleasure because of the excellent relations he had always had with Japanese mathematicians. THE WORK Now I come to the essential part, that is, to his work. His first publication was a Comptes Rendus note in 1926 (1). In the 50 years that followed, he published a dozen books and more than a hundred papers, written in French, in English, and sometimes in German. His papers have been collected in the three volumes of his Oeuvres mathematiques published by Springer in 1979 (45). They include Weil's valuable Commentaires, which explain their motiviation. It is not possible to classify these works by subject, because so many themes are combined. It is true that one might play the American game of listing keywords: zeta, Siegel, rational points, Abelian varieties, and so on; but that hardly makes sense. It seems to me that the only possibility is to follow the chronological order, as is done in his Oeuvres. 1. Let us begin with his thesis (2). It is concerned with Number Theory and more particularly with Diophantine equations, that is to say, with rational points on algebraic varieties. At that time, the only known method was Fermat's descent; very often, the application of this method depended on explicit calculations, so that a different little miracle seemed to happen in each particular case. Weil was the first to see that behind these computations there was a general principle, which he called the theoreme de decomposition; this theorem allowed a sort of transfer between algebra (in principle easy) and arithmetic (harder). He deduced what we now call the Mordell-Weil theorem: given an Abelian variety A
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and a number field K, the group A(K) of points defined over K is finitely generated. The proof was far from easy: the algebraic geometry of the time had not yet developed the tools that were needed. Fortunately, Weil had read the works of Riemann at the Ecole Normale, and he was able to replace the missing algebra by analysis: theta functions. So he was able to reach his goal. However, this goal was by no means an end. As in almost all of Weil's works, it was much more a point of departure, from which one could attack other problems. In this instance, the problems were as follows. • Prove the fmiteness of the set of integer points of an affine curve of genus g > 0. This was done a year later by Siegel, who combined Weil's ideas with those from the theory of transcendental numbers. • Prove the fmiteness of the set of rational points of a curve of genus g > 1 (the Mordell conjecture). This was done 55 years later by Faltings. • Make effective (that is to say, explicitly computable) the qualitative results of Mordell-Weil, of Siegel and of Faltings. This question is still open, and of great interest to arithmeticians. 2. In the years after his thesis, Weil tried various paths that might lead to the Mordell conjecture. One of them led to his monograph Generalisation des fonctions abeliennes (8), a text presented as analysis, whose significance is ...asentialry algebraic, but whose motivation is arithmetic! (And, one asks oneself, who in the world beside Weil and Siegel could have understood this text in 1938?) The success of his thesis arose from the use of Abelian varieties, and in particular of Jacobians; for the Mordell conjecture, Weil felt that it would be necessary to go beyond the Abelian setting. The Jacobian parametrizes line bundles of rank 1 (and degree 0); one has to parametrize fibre bundles of any rank (that is to say, to pass from GL\ to GL„—this was to be one of his favourite themes). However, in 1938 no one, not even he, knew what an analytic vector bundle was—still less an algebraic vector bundle: it was not till a decade later that such a notion would be introduced (by Weil himself). This little detail did not stop him. He introduced an equivalent notion, that of 'classe de diviseurs matriciels', and by analytic methods (following Riemann and Poincare) he proved the Riemann-Roch formula and what we nowadays call the duality theorem (which he called the 'inhomogeneous Riemann-Roch theorem'). It was a real tour-de-force! Unhappily, to define vector bundles was not enough: one needed also their 'moduli varieties', to replace Jacobians. From the point of view of algebraic geometry, this is a very tough passage-to-the-quotient problem, which was solved only some 20 years later, by Grothendieck and Mumford. Weil had to be content with partial results, partly unproved but which would turn out to be essentially correct; a fortiori, he could make no arithmetic application. A set-back, perhaps? No, for his work on Riemann-Roch served as a model to others fifteen years later, and the moduli varieties that he tried to construct turned up again at the roots of other questions: in differential geometry, with Donaldson, and in characteristic p > 0, with Drinfeld. 3. During the period 1925^0, Weil was far from confining himself to Number Theory. Here are some of his achievements. • In the analysis of several complex variables, he introduced a generalization of the Cauchy integral, which is nowadays known as the Weil integral (3, 4). He deduced
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• • • •
a generalization of Runge's theorem: if D is a bounded domain defined by polynomial inequalities, then every holomorphic function on D is a limit of polynomials for the compact convergence topology. In the theory of compact Lie groups, he used topological methods (Lefschetz's formula) to prove the conjugacy of maximal tori (5). In non-Archimedean analysis, a subject then in its infancy, he defined />-adic elliptic functions (6). In topology, he gave the definition of uniform spaces (7). He published a book L'integration dans les groupes topologiques et ses applications (10) in which he expounded, in a style at once elegant, concise and Bourbaki-like, the two aspects of the theory that were accessible at that time: the case of compact groups (orthogonality relations of characters) and that of commutative groups (Pontryagin duality and Fourier transform).
4. We return to number theory and algebraic geometry, with the celebrated note of 1940 (9). Between 1925 and 1940, the German school, led by Artin and Hasse, had found remarkable analogies between algebraic number fields and fields of functions of one variable over finite fields (in geometric language: curves over finite fields). Each of them has zeta functions, for which a Riemann hypothesis can be stated. In the function field case, Hasse was able to prove this hypothesis for genus 1. What about genus > 2? During his stay in Rouen, Weil saw the answer: instead of working with curves, that is to say varieties of dimension 1, one should use varieties of higher dimension (surfaces, Abelian varieties) and adapt to them results proved (over the complex field) by topological or analytic methods. He sent to the Comptes Rendus a note (9) that begins as follows: 'Je vais resumer dans cette Note la solution des principaux problemes de la theorie des fonctions algebriques a corps de constantes fini This note contains a sketch of proof, no more; everything depends on a 'crucial lemma' culled from Italian geometry. How should one prove this lemma? Weil soon realized that it is possible only if one completely reworks the definitions and results that form the foundations of algebraic geometry, in particular those involving intersection theory (so that one has a calculus of cycles that replaces the missing homology). He was thus led to write his Foundations of algebraic geometry (12), a massive and rather dry book of 300 pages, which was only replaced twenty years later by the no less massive and dry Elements de geometrie algebrique of Grothendieck. Once the Foundations were in place, Weil could return to curves and their Riemann hypothesis. He published in quick succession two works: Sur les courbes algebriques et les varietes qui sen deduisent (15) and Varietes abeliennes et courbes algebriques (16). After eight years and more than 500 pages, his Note of 1940 was at last justified! What are the rewards? First, the Riemann hypothesis has down-to-earth applications. It gives upper bounds for trigonometric sums of one variable (17), for example the following (useful in the theory of modular forms): 2jt (x + x)
< 2-Jp for p prime,
where the sum is over integers x with 0 < x
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(23, 24, 41, 42), including notably the theory of complex multiplication (25, 26) that was done simultaneously and independently by Taniyama and Shimura. 5. Guided by the case of curves, and also by explicit computations for hypersurfaces defined by diagonal equations, Weil (18) formulated what were immediately known as the Weil conjectures. These conjectures were about (projective non-singular) varieties over a finite field. They amount to supposing that the topological methods of Riemann, Lefschetz, Hodge, and others, can be adapt ;ii to work in characteristic p > 0; from this point of view, the number of solutions of an equation (mod p) appears as a number of fixed points, and one can compute it by the Lefschetz trace formula. This truly revolutionary idea thrilled the mathematicians of the time, as I can testify at first hand; it has been the origin of a major part of the progress in algebraic geometry since that date. The objective was reached only after about twenty-five years, and then not by Weil himself but (principally) by Grothendieck and Deligne. The methods that they were led to develop remain among the most powerful of present-day algebraic geometry; they have had applications in varied fields including the theory of modular forms (as Weil had foreseen) and the determination of the characters of the 'algebraic' finite groups (Deligne-Lusztig). 6. Weil returned to arithmetic in 1951 with his work on class field theory (20). This theory had attained an apparently definitive form in 1927, when Artin had proved the general reciprocity law. In the language introduced by Chevalley, the main result states that the Galois group of the maximal Abelian extension of a number field K is isomorphic to the quotient CKIDK, where CK is the idele class group of K and DK is the connected component of CK. (Accordingly, one describes what happens above K in terms of objects taken from K itself, just as a topologist describes the coverings of a space in terms of its classes of loops.) However, a disagreeable feature of this statement is that it is not CK itself that is a Galois group, but only its quotient CKIDK. Weil started with the idea that CK itself should be a Galois group in a suitable sense (in what sense we still cannot say). It this were true, it would imply remarkable functorial properties of the groups CK (for instance, if LIK is a finite Galois extension there should be a canonical extension of Gal(L/AT) by CL). One could try to prove these properties directly, and that is what Weil did. Here again, there were important consequences, as follows. • One was led to the study of the cohomology groups of the groups CL\ it is the origin of the cohomological methods in class field theory as developed by Nakayama, Hochschild, Artin, Tate, and others. • The new 'Weil groups' so defined allow one to define new types of L-functions, containing as particular cases both Artin's non-Abelian L-functions and Hecke's L-functions with 'Grossencharacter'. As Weil said, one has thus wedded Artin to Hecke! 7. A little later, Weil published a study (22) (completed later (37)) on explicit formulae in the theory of numbers; these formulae (essentially known to specialists, it seems) related sums taken over prime numbers to sums taken over the zeros of zeta functions. Weil rewrote them in a very suggestive manner (for example, by highlighting the analogy between finite and infinite places—another of his favourite themes). The most interesting result is a translation of the Riemann hypothesis in terms of the positivity of a certain distribution. Might this translation help to prove the Riemann hypothesis? It is too soon to say.
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8. Various works of Weil between 1940 and 1965 were concerned with differential geometry. They include the following. • With Allendorfer, a Gauss-Bonnet formula for Riemannian polyhedra (11). • Proof of the theorems of de Rham in a letter to Cartan in 1947 (see (21)). This letter greatly influenced Cartan in his formulation of the theory of sheaves (due initially to Leray). • Harmonic forms and Kahler theory (14, 27); these were the basic tools for the application of analytic methods to algebraic geometry. • Theory of connections and introduction of the Weil algebra (19). • Deformations of locally homogeneous spaces and discrete groups (28, 31, 32); rigidity theorems are proved for co-compact discrete subgroups of simple Lie groups of rank > 1. 9. During the 1950s and 1960s, Weil devoted a series of articles to themes inspired by Siegel. He writes in his Oeuvres (vol 2, p. 544): 'To comment on the works of Siegel has always appeared to me to be one of the tasks that a present-day mathematician may most usefully undertake.' Note the lovely understatement of the word 'comment'! Weil did much more: • In (29) and (30) he developed systematically the adelic methods introduced by Kuga and Tamagawa. These not only give us again Siegel's theorems on quadratic forms, but suggest new problems, for instance to show that the Tamagawa number of any simply connected group is 1 (we now know that this is so, thanks to the work of Langlands, Lai and Kottwitz). • In two Acta Mathematica memoirs (33, 34) he returned to quadratic forms and Siegel's formula from an entirely different point of view. He introduced and studied a new group, the metaplectic group, as well as a representation of this group (nowadays called the Weil representation). The Siegel formula appeared as the equality of two distributions, one of them a sort of Eisenstein series, the other an average of theta functions. This result is not limited to quadratic forms: Weil showed that it applies to all classical groups, and it implies local-to-global theorems (Hasse principle) as well as the determination of Tamagawa numbers. 10. Hecke's work too was an inspiration for Weil. In 'L'avenir des mathematiques' (13) he was already writing about Euler products 'whose extreme importance for the theory of numbers and the theory of functions have only just been revealed to us by Hecke's work'. Twenty years later (35), he made a decisive contribution to Hecke's theory, by showing that the validity of certain functional equations for a Dirichlet series and its twists by characters is equivalent to the fact that the series comes from a modular form. One thus obtains something very precious in mathematics, namely a dictionary modular forms o
Dirichlet series.
The implication => was due to Hecke, who had also proved the reverse implication in the particular case of level 1; Weil's new idea was to use twisting. One of the more interesting aspects of his theory is the way in which the constants in the functional equations vary by twisting (i.e. by tensor product). This work led to several developments, including some by Weil himself (36). It fits now into the so-called 'Langlands philosophy'. One of its consequences was a precise formulation
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of a slightly vague conjecture made by Taniyama in 1955, according to which every elliptic curve over Q is 'modular'. The work of Weil suggested that the appropriate modular 'level' should be the same as the 'conductor' of the elliptic curve, that is to say, it is determined by the primes and types of bad reduction of the curve. This allowed a great deal of numerical verification, before the result was finally proved (subject to some technical restrictions) by Wiles in 1995. 11. The last publications of Weil were concerned with the history of mathematics. It is a subject that had interested him for a long time, as certain of the Notes historiques in Bourbaki show (in particular, those on the differential and integral calculus in Fonctions d'une variable reelle, chapters I—III). He began with a short booklet, at once mathematical and historical: Elliptic functions according to Eisenstein and Kronecker (40); he says he enjoyed writing it, and his enjoyment communicates itself to the reader. The following works are more clearly historical. Above all, one must cite his Number Theory—an approach through history from Hammurabi to Legendre (46) in which he describes the history of the theory of numbers up to 1800, stopping just before the Disquisitiones arithmeticae (his readers would have liked him to go further and to write on Gauss, Jacobi, Eisenstein, Riemann, and others—but he did not.) As expected from him, it is mathematics that is the subject of these books, and not the private lives or the social relationships of the mathematicians. Only the history of ideas matters; what a refreshing point of view! It is not easy to write such books. One needs linguistic and literary gifts, and Weil did not lack them. One must also, above all, be able to judge when an idea is truly new and when it is standard technique (he discussed this in 'History of mathematics: why and how' (44)); that is certainly the most difficult part for a historian of mathematics who is not a mathematician (see (38, 39, 43)). I end this description, too superficial I fear, of what Andre Weil did. What makes his work unique in the mathematics of the twentieth century is its prophetic aspect (Weil 'sees the future') combined with utmost classical precision. To read and study this work, and to discuss it with him, has been among the greatest joys of my mathematical life.
ACKNOWLEDGEMENT The frontispiece portrait was presented to the London Mathematical Society and is reproduced with permission.
BIBLIOGRAPHY The following publications are those referred to directly in the text. (1) (2) (3) (4) (5) (6)
1926 1928 1932 1935
Sur les surfaces a courbure negative. Compt. Rend. 182, 1069-1071. L'arithmetique sur les courbes algebriques. Acta Math. 52, 281-315. Sur les series de polynomes de deux variables complexes. Compt. Rend. 194, 1304-1305. L'integrale de Cauchy et les fonctions de plusieurs variables. Math. Ann. I l l , 178-182. Demonstration topologique d'un theoreme fondamental de Cartan. Compt. Rend. 200, 518520. 1936 Sur les fonctions elliptiques p-adiques. Compt. Rend. 203, 22-24.
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(7)
1937
(8) (9)
1938 1940
(10) (11)
1943
(12)
1946
(13)
1947
(14) (15) (16)
(17) (18) (19) (20) (21) (22)
1948
1949 1951 1952
(23) (24)
1954
(25)
1955
(26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) (38) (39) (40) (41)
1958 1960 1961 1962
1964 1965 1967
Memoirs
Sur les espaces a structure uniforme et sur la topologie generale. Acta Sc. et Ind. no. 551, pp. 3 40. Paris: Hermann. Generalisation des fonctions abeliennes. J. Math. Pur. Appl. 17 (9), 47-87. Sur les fonctions algebriques a corps de constantes fini. Compt. Rend. 210, 592-594. L'integration dans les groupes topologiques et ses applications. Paris: Hermann. (Second edn 1953.) (With C. Allendorfer) The Gauss-Bonnet theorem for Riemannian polyhedra. Trans. Am. Math. Soc. 53, 101-129. Foundations of Algebraic Geometry (Am. Math. Soc. Colloq., vol. 29). New York: American Mathematical Society. (Second edn 1962.) L'avenir des mathematiques. In Les grands courants de la pensee mathematique (ed. F. Le Lionnais), pp. 307-320. Paris: Cahiers du Sud. (Second edn, A. Blanchard, Paris, 1962.) Sur la theorie des formes differentielles attachees a une variete analytique complexe. Comment. Math. Helv. 20, 110-116. Sur les courbes algebriques et les varietes qui sen deduisent. Paris: Hermann. Varietes abeliennes et courbes algebriques. Paris: Hermann. (Second edition containing (15) and (16) together, published under the title Courbes algebriques et varietes abeliennes, Hermann, Paris, 1971.) On some exponential sums. Proc. Natl Acad. Sci. USA 34, 204-207. Numbers of solutions of equations in finite fields. Bull. Am. Math. Soc. 55, 497-508. Geometrie differentielle des espaces fibres (unpublished). Sur la theorie du corps de classes. J. Math. Soc. Japan 3, 1-35. Sur les theoremes de de Rham. Comment. Math. Helv. 26, 119-145. Sur les 'formules explicites' de la theorie des nombres premiers. Comm. Sem. Math. Univ. Lund (vol. dedicated to Marcel Riesz), pp. 252-265. On Picard varieties. Am. J. Math. 74, 865-894. On the projective embedding of abelian varieties. In Algebraic geometry and topology, a symposium in honor of S. Lefschetz, pp. 177-181. Princeton University Press. On a certain number of characters of the idele-class group of an algebraic number-field. In Proc. Int. Symp. on Algebraic Number Theory, Tokyo-Nikko, pp. 1-7. Science Council of Japan. On the theory of complex multiplication. In Proc. Int. Symp. on Algebraic Number Theory, Tokyo-Nikko, pp. 9-22. Science Council of Japan. Introduction a I'etude des varietes kahleriennes. Paris: Hermann. On discrete subgroups of Lie groups. Ann. Math. 72, 369-384. Adeles and algebraic groups. Princeton: Institute for Advanced Study. Sur la theorie des formes quadratiques. In Colloque sur la Theorie des Groupes Algebriques, pp. 9-22. Bruxelles: C.B.R.M. Paris: Gauthier-Villars. On discrete subgroups of Lie groups (II). Ann. Math. 75, 578-602. Remarks on the cohomology of groups. Ann. Math. 80, 149-157. Sur certains groupes d'operateurs unitaires. Acta Math. I l l , 143-211. Sur la formule de Siegel dans la theorie des groupes classiques. Acta Math. 113, 1-87. Uber die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. Math. Ann. 168, 149-156.
1971 Dirichlet series andautomorphic forms. (Lecture Notes no. 189.) Heidelberg: Springer-Verlag. 1972 Sur les formules explicites de la theorie des nombres, Izv. Mat. Nauk (Sen Mat.) 36, 3-18. 1973 Review of 'The mathematical career of Pierre de Fermat', by M.S. Mahoney. Bull. Am. Math. Soc. 79, 1138-1149. 1975 Review of 'Leibniz in Paris 1672-1676, his growth to mathematical maturity', by Joseph E. Hofmann. Bull. Am. Math. Soc. 81, 676-688. 1976 Elliptic functions according to Eisenstein and Kronecker. (Ergebnisse der Mathemalik, vol. 88.) Heidelberg: Springer-Verlag. Sur les periodes des integrates abeliennes. Commun. Pure Appl. Math. 29, 813-819.
779
Andre Weil (42) (43) (44) (45) (46) (47)
529
1977 Abelian varieties and the Hodge ring (unpublished). 1978 Who betrayed Euclid? Arch. Hist. Exact Sci. 19, 91-93. History of mathematics: why and how. In Proc. Intern. Math. Congress, Helsinki, vol. 1, pp. 227-236. Acad. Sci. Fennica. 1979 Oeuvres Scientifiques—Collected Papers, 3 vols. Heidelberg: Springer-Verlag. 1984 Number Theory—an approach through history from Hammurapi to Legendre. Basel: Birkhauser. 1991 Souvenirs d'apprentissage. Basel: Birkhauser.
Wolf Prize in Mathematics, Vol. 2 (pp. 781-843) eds. S. S. Chern and F. Hirzebruch © 2001 World Scientific Publishing Co.
Academic Appointments and Awards
Born March 23, 1907, in New York City Yale University, Ph. B., 1928; Mus. B., 1929; honorary Sc. D., 1947 Harvard University, Ph. D. 1932 National Research Council Fellow in Mathematics, 1931-33 Instructor to Professor of Mathematics, Harvard University, 1933-52 Professor of Mathematics, Institute for Advanced Study, 1952-77; Professor Emeritus, 1977-89 Member; American Mathematical Society, Mathematical Association of America, National Council of Teachers of Mathematics, National Academy of Sciences, American Philosophical Society, Swiss Mathematical Society (honorary) American Mathematical Society: Colloquium Lecturer, 1946 Vice President, 1948-50 Editor, American Journal of Mathematics, 1944^9 Editor, Mathematical Reviews, 1949-54 Committee on Visiting Lectureship (chairman), 1946-51 Committee for Summer Institutes (chairman), 1953-54 National Science Foundation: Chairman of Panel on Mathematics, 1953-56 Researcher for Applied Mathematics Panel of NDRC of OSRD, 1943-45 Exchange Professor to France from Harvard, 1951-52 Exchange Professor to College de France (Fulbright), 1957 Committee on Support of Research in the Mathematical Sciences of the National Research Council, 1966-67 L. R. Ford Award for paper, "The Mathematics of Physical Quantities," 1969 National Medal of Science, 1967 Wolf Prize in Mathematics (shared) for 1982 Steele Prize, American Mathematical Society, 1985 Consultant to School Mathematics Study Group, Cambridge Conference in School Mathematics, Education Development Center, and other groups International Commission on Mathematical Instruction, President, 1979-82
XI
Bibliography of Hassler Whitney
The coloring of graphs, NAS Proc. v. 17, 1931, 122-125 . Non-separable and planar graphs, NAS Proc. v. 17, 1931, 125-127 . A theorem on graphs, Annals of Math. (2) v. 32, 1931, 378-390 . Note on Perron's solution of the Dirichlet problem, NAS Proc., v. 18, 1932, 68-70. Non-separable and planar graphs, AMS Transac. v. 34, 1932, 339-362. Congruent graphs and the connectivity of graphs, Am. Jour. Math. v. 54, 1932, 150-168. Regular families of curves I, NAS Proc., v. 18, 1932, 275-278. Regular families of curves II, NAS Proc., v. 18, 1932, 340-342. A logical expansion in mathematics, AMS Bull., v. 38, 1932, 572-579. The coloring of graphs, Annals of Math., (2) v. 33, 1932, 688-718. A characterization of the closed 2-cell, AMS Transac, v. 35, 1933, 261-273. A set of topological invariants for graphs, Am. Jour. Math., v. 55, 1933, 231-235. On the classification of graphs, Am. Jour. Math., v. 55, 1933, 236-244. 2-Isomorphic graphs, Am. Jour. Math., v. 55, 1933, 245-254. Regular families of curves, Annals of Math. (2) v. 34, 1933, 244-270. Characteristic functions and the algebra of logic, Annals of Math. (2) v. 34, 1933, 405-414. Planar graphs, Fundamenta Math., v. 21, 1933, 73-84. Analytic extensions of differentiable functions defined in closed sets, AMS Transac., v. 36, 1934, 63-89. Derivatives, difference quotients and Taylor's formula, AMS Bull., v. 40, 1934, 89-94. Differentiable functions defined in closed sets I, AMS Transac, v. 36, 1934, 369-387. Derivatives, difference quotients and Taylor's formula II, Annals of Math. (2) v. 35, 1934,476-481. Functions differentiable on the boundaries of regions, Annals of Math. (2) v. 35, 1934, 482-485. On the abstract properties of linear dependence, Am. Jour. Math., v. 57, 1935, 509-533. Differentiable manifolds in Euclidean space, NAS Proc, v. 21, 1935, 462-464, reprinted in Receuil Mathematique. T. 1(43) N. 5 (1936) 783-786. Sphere-spaces, NAS Proc, v. 21, 1935, 464^468, reprinted in Receuil Mathematique. T. 1(43) N. 5 (1936) 787-791. A function not constant on a connected set of critical points, Duke Math. J., v. 1, 1935, 514-517. Differentiable functions defined in arbitrary subsets of Euclidean space, AMS Transac, v. 40, 1936, 309-317. Differentiable manifolds, Annals of Math. (2) v. 37, 1936, 645-680. The imbedding of manifolds in families of analytic manifolds, Annals of Math. (2) v. 37, 1936, 865-878. On regular closed curves in the plane, Compositio Math. 4, 1937, 276-284. On matrices of integers and combinatorial topology, Duke Math. J., 3, 1937, 35-45. On the maps of an n-sphere into another n-sphere, Duke Math. J., 3, 1937, 46-50. The maps of an n-complex into an n-sphere, Duke Math. J., 3, 1937, 51-55. On products in a complex, NAS Proc, v. 23, 1937, 285-291. xii
[35] Analytic coordinate systems and arcs in a manifold, Annals of Math. (2) 38, 1937, 809-818. [36; Topological properties of differentiable manifolds, AMS Bull. 43, 1937, 785-805. [37 A numerical equivalent of the four color problem, Monatshefte fur Math, und Phys. 3, 1937, 207-213. [38; Cross sections of curves in 3-space, Duke Math. J., 4, 1938, 222-226. [39 On products in a complex, Annals of Math. (2) 39, 1938, 397-432. [40; Tensor products of abelian groups, Duke Math. J., 4, 1938, 495-528. [41 Some combinatorial properties of complexes, NAS Proa, 26, 43-148. [42 On the theory of sphere-bundles, NAS Proa, 26, 1940, 148-153. [43 On regular families of curves, AMS Bull., 47, 1941, 145-147. [44; On the topology of differentiable manifolds, Lectures in Topology, U. of Michigan Press, 1941, 101-141. [45 Differentiability of the remainder term in Taylor's formula, Duke Math. J., 10, 1943, 153-158. [46; Differentiable even functions, Duke Math. J., 10, 1943, 159-160. [47; The general type of singularity of a set of 2n — 1 smooth functions of n-variables, Duke Math. J., 10, 1943, 161-172. [48; Topics in the theory of Abelian groups, I. Divisibility of Homomorphisms, AMS Bull., 50, 1944, 129-134. [49 On the extension of differentiable functions, AMS Bull., 50, 1944, 76-81. [50; The self-intersections of a smooth n-manifold in 2n-space, Annals of Math. (2), 45, 1944, 220-246. [51 The singularities of a smooth n-manifold in (2n-l)-space, Annals of Math. (2), 45, 1944, 247-293. [52; Algebraic topology and integration theory, NAS Proa, v. 33, 1947, 1-6. [53 Geometric methods in cohomology theory, NAS Proa, v. 33, 1947, 7-9. [54 Complexes of manifolds, NAS Proa, v. 33, 1947, 10-11. [55; On ideals of differentiable functions, Am. Jour. Math. 70, 1948, 635-658. Relations between the second and third homotopy groups of a simply-connected space, [56 Annals of Math. (2) 50, 1949, 180-202. Classification of the mappings of a 3-complex into simply-connected spaces, Annals [57; of Math. (2) 50, 1949, 270-284. An extension theorem for mappings into simply-connected spaces, Annals of Math. [58 (2) 50, 1949, 285-296. La topologie algebrique et la theorie de l'integration, Colloques Intemationaux du [59; CNRS XII, Topologie Algebrique, 1947, 107-113, published by CNRS, Paris, 1949. (With L. H. Loomis) An inequality related to the isoperimetric inequality, AMS Bull., [6o; 55, 1949, 961-962. [61 On totally differentiable and smooth functions, Pacific J. Math. 1, 1951, 143-159. [62 r-dimensional integration in n-space, Proc. Int. Cong. Math., 1950, vol. 1, 245-256, Amer. Math. Soc. 1952. [63 On singularities of mappings of Euclidean spaces, I. Mappings of the plabe into the plane, Annals of Math. (2) 62, 1955, 374^110. [64; On functions with bounded n-th differences, J. de Maths. Pres et Appl. 36, 1957, 67-95. [65; Geometric Integration Theory, Princeton University Press, 1957, Princeton, NJ, xv + 397 pages. (Princeton Math. Series 21) Translated into Russian [book]. [66 Elementary structure of real algebraic varieties, Annals of Math. (2) 66, 1957, 545-556. xiii
[67] Singualrities of mappings of Euclidean spaces, Symposium Intemacional de Topologia Algebraica, Mexico, 1956, 285-301, Mexico, La Universidad Nacional Autonoma. [68] (With F. Bruhat) Quelques proprietes fondamentales des ensembles analytiques-reels, Comm. Math. Helv. 33, 1959, 132-160. [69] (With A. Dold) Classification of oriented sphere bundles over a 4-complex, Annals of Math. (2) 69, 1959, 667-677. [70] On bounded functions with bounded n-th differences, AMS Proc. 10, 1959, 480-481. [71] (With A. M. Gleason) The extension of linear functionals defined on //-infinity, Pacific J. Math. 12, 1962, 163-182. [72] The work of John W. Milnor, Proceedings ICM 1962m Institut Mittag-Leffler Djursholm, Sweden, xlviii-1. [73] Local properties of analytic varieties, in: differential and combinatorial topology (Symposium in Honor of Marston Morse), Princeton, NJ., Princeton University Press. [74] Tangents to an analytic variety, Annals of Math (2) 81, 1965, 496-549. [75] The mathematics of physical quantities. Part I, Mathematical Models for measurement, Am. Math. Monthly 75 (1968), 115-138, Part II, Quantity structures and dimensional analysis, ibid., 237-256. [76] Logic Fad or tool? Nico 4, 1969, Revue per du Centre Beige de Pedagogie de la Mathematique, 2-14. [77] On reducibility in the four color problem, unpublished manuscript, 1971. [78] (With W. T. Tutte) Kempe chains and the four color problem, Utilitas Mathematica 2(1972), 241-281. [79] Complex Analytic Varieties, Addison-Wesley Pub. Co., Reading, MA 1972, xii +399 pp. [book], [80] Math Activities, multilithed, Institute for Adv. Study, 1974 [book]. [81] Comment on the division of the plane by lines, Am. Math. Monthly 86(1979), p. 700. [82] Moscow 1935: Topology moving toward America, in A Century of Mathematics in America, Part I, P. L. Duren Ed., 96-117, Amer. Math.
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Hassler Whitney was a student ofG. D. Birkhoff at Harvard University, where he earned a Ph.D. in 1932. After two years as a National Research Fellow (1931-1933), he served on the Harvard faculty until 1952. He has since been at the Institute for Advanced Study, where he is now Professor Emeritus of Mathematics. A recipient of the National Medal of Science, the Wolf Prize, and the AMS Steele Prize, Professor Whitney is also a member of the National Academy of Sciences. His research has been primarily in the areas of topology and analysis.
Moscow 1935: Topology Moving Toward America HASSLER WHITNEY The International Conference in Topology in Moscow, September 4-10, 1935, was notable in several ways. To start, it was the first truly international conference in a specialized part of mathematics, on a broad scale. Next, there were three major breakthroughs toward future methods in topology of great import for the future of the subject. And, more striking yet, in each of these the first presenter turned out not to be alone: At least one other had been working up the same material. At that time, volume I of P. Alexandroff / H. Hopf, Topologie, was about to appear. I refer to this volume as A-H. Its introduction gives a broad view of algebraic topology as then known; and the book itself, a careful treatment of its ramifications in its 636 pages. (It was my bible for some time.) Yet the conference was so explosive in character that the authors soon realized that their volume was already badly out of date; and with the impossibility of doing a very great revision, the last two volumes were abandoned. Yet a paper of Hopf still to come (1942) led to a new explosion, with a great expansion of domains, carried on especially in America. It is my purpose here to give a general description of the subject from early beginnings to the 1940s, choosing only those basic parts that would lead to later more complete theories, directly in the algebraic treatment of the subject. We can then take a look at some directions of development since the conference, in very brief form, with one or two references for those who wish a direct continuation.
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I also do not hesitate to draw a few conclusions on our difficulties with new research, with some comments on how research might be improved. What were early beginnings of "analysis situs"? Certainly a prime example is Euler's discovery and proof that for a polyhedron, topological^ a ball, if a0, a\ and a2 denote the numbers of "vertices," "edges" and "faces," then (1)
aQ-a\+a2
= 2.
How might one find something like this? Who might think of trying it out? These are questions looking directly for answers, rather than at situations to explore. For the latter, one might build up a picture:
yt^^t The first step here is to add a vertex, cutting one edge into two: this leaves QI — c*o unchanged. The second step is to add an edge joining two vertices; this leaves a2 - o.\ unchanged. Now it needs some playing to see that (1) contains both these facts. We might now say that we essentially know the formula (1); just the 2 is missing. That expression, generalizing to Q0 - a\ + a2 - 03, etc., is known as the Euler characteristic. (Also Descartes discovered it much earlier; see A-H, p. 1.) Can you be taught how to think? If you are in a particular subject, there may be tricks of the trade for that subject; Polya shows this for some standard parts of mathematical thinking. But trying to learn to carry out research by studying Polya is unlikely to get you far. It is the situation you are in which can lead to insights, and any particular thinking ways are quite unlikely to apply to different sorts of situations. "Sharpening your wits" on peculiar questions may keep your mind flexible so that new situations can let you think in new directions. Thus Lakatos, "Proofs and refutations" can give you ideas, samples, of thoughts; the usefulness is less in learning that in keeping your mind flexible. A popular pastime in Konigsberg, Germany, was to try to walk over each of its seven bridges once and only once. Euler showed how to organize the situation better and check on the possibility. Can we find a way to get naturally at this? If we started in the island, say at A, and crossed the upper left bridge, why not sit down at C and think it over instead of wandering around aimlessly?
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If we can find the desired path, we can certainly simplify it by using just the paths shown; and putting a gate in each bridge, we can check on which ones we have crossed. Thus we crossed gate 3, and must next cross either 4 or 5. But then we must cross the other of these gates later, and find ourselves back at C with no way to reach any uncrossed bridges. This is enough for us to start organizing. The final result, applying to any such situation, was given by Euler. A most famous question is of course the four color problem: Can one color any map on the plane or globe in at most four colors so any adjoining regions are of different colors? A first "proof was given by Kempe, in 1869, who introduced the important tool of "Kempe chains." The mistake was discovered by Heawood in 1890. A major step in advance was given by G. D. Birkhoff, in a paper in 1913 on "The reducibility of maps." In the early 1930s, when I was at Harvard, exploring the problem among other things, Birkhoff told me that every great mathematician had studied the problem, and thought at some time that he had proved the theorem (I took it that Birkhoff included himself here). In this period I was often asked when I thought the problem would be solved. My normal response became "not in the next half century." The proof by computer (W. Haken and K. Appel) began appearing in non-final form about 1977.
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A very important advance in mathematics took place in the mid nineteenth century, with the appearance of Riemann's thesis. Here he made an investigation of "Riemann surfaces," along with basic analytic considerations, in particular, moving from one "sheet" to another by going around branch points. This led to the general question of what a "surface" was, topologically, and the problem of classification. This culminated (Mobius, Jordan, Schafli, Dyck 1888) in the characterization of closed surfaces (without boundary); they are determined through their being orientable or not, and through their Euler characteristic. Let me note that H. WeyFs book Die Idee der Riemannsche Flache. Leipzig 1913, clarified many basic notions such as neighborhood, manifold, fundamental group, and covering space. A notable discovery was made by Gauss (who had made deep investigations in differential geometry, with special studies of the earth's surface). This was the expression as a double integral for the "looping coefficient" of two nonintersecting oriented curves C\ and C2 in 3-space /?3. Consider all pairs of points P in C\ and Q in C2, and
C2
the unit vector from P toward Q:
With P and Q as in the figure, if we let P' run over a short arc A in C\ about P and let Q' run similarly over B in C2, v{P', Q') will clearly run over a little square-like part of the unit 2-sphere S2 of directions in 3-space R2. The whole mapping is a little complex, since we are mapping C\ x C2, which is a torus, into S2. But we can see that it covers S2 an algebraic number 1 of times, as follows. For each P, look at the image of v(P,C2). From the figure we see that it is circle-like, down and to the left to start. When P is taken down to Pu the above circle has moved to the right and up, now going directly around Px. Continuing down and along C\ from Px to
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Pi, v(P, Ci) moves upward, to the left and down again. Thus that part of S2 directly to the right from Pi (see the arrow at that point) is swept over just once in the total sweep. We now use general theory (see A-H for instance) that says that S2 must be covered some integral number of times, hence once (algebraically). Gauss gave a numerical form to the double integral, in the general case of non-intersecting curves (see A-H, p. 497). If the result is not zero, the looping coefficient is not zero, and being invariant under deformations, one curve cannot be separated to a distance from the other without cutting through the other. Kronecker considered the common zeros of a set of functions fa fa. Equivalently, consider the vector field v(p) = (fa{p) fa{p)), and its zeros. This leads to the "Kronecker characteristic," generalizing the Gauss integral to higher dimensions. See A-H for some details. All this work was growing and expanding at the end of the last century. But I call this the end of the early period, since Poincare's studies, from 1895 on, gave a better general organization and important new directions of progress. The essentials of the early period were described in the article by Dehn and Heegaard, Enzyklopadie der Mathematischen Wissenschaften, III A B 3, 1907, and a very nice exposition of the analytical aspects was given by Hadamard in an appendix to Tannery, Introduction a la theorie des fonctions d'une variable, 2nd ed., 1910. Turning now to the middle period, Poincare set out to make a deep study of H-dimensional manifolds (locally like a part of «-space); these were basic in his work on dynamical systems. He cut them into "n-cells," each bounded by (n- l)-cells; and each of the latter is a face of two rt-cells. Each (n- l)-cell is bounded by (n - 2)-cells, and so on. Moreover "r-chains," written YLa>a\-< associating an integer a, with each r-cell o\, were defined. Now using d for boundary, each boundary do\ can be seen as an (r - l)-chain, and for a general r-chain A' as above, dAr = ^afio'. For any o-, with a given orientation, an orientation of each of its boundary cells ar~x is induced, and da[ is the sum of these with the induced orientations (see below). And since each ark~2 in the boundary of a[ is a face of just two o'~', with opposite orientations induced, we have dda[ = 0, and hence ddAr = 0 for all r-chains A'. A special case is the "simplicial complex," composed of "simplexes." In n-space R", an /--simplex is the convex hull PoP\-pr of a set of points Po, — pr lying in no (r - l)-plane. In barycentric coordinates, the points of Po-Pr are given by £ a,-p,-, each a, > 0, £ a, = 1. (This point is the center of mass of a set of masses, in amount a, at each p,.)
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Any Rr can be oriented in two ways. Choosing an ordered set of r independent vectors V[ vr determines an orientation. A continuous change to another set vj v'r gives the same orientation if independence was maintained. A simplex ar — Po-pr has a natural orientation, given by the ordered set p\ - po , Pr - Po of vectors. Note that the ordered set p\ - p0, p2 P\ pr - pr-1 is equivalent. The induced orientation of the face P\ • • • pr = ar~x of ar is denned by choosing v l f ..., vr to orient ar, with v 2 ,..., vr in ar~x (orienting it) and vt pointing from p0- • • pr out of <7r-1, as used just above. This holds true for the second set of vectors chosen above for Po- • pr\ and this shows also that p\- • pr has that orientation. In this way we may find the full expression for d (p0 • • • pr). Some instances of this relation are d (PoPi) = Pi- Po.
d {p0pi p2) = P1P2- P0P2 + P0P1 •
Later we will note that Kolmogoroff and Alexander might have found the correct products in cohomology by 1935 if they had kept such relations in mind, along with the relationship with differential geometry (typified by de Rham's theorem). In accordance with the influence of Emmy Noether in Gottingen in the mid twenties, we shift now to group concepts to simplify the work. If an r-chain Ar has no boundary, dAr = 0, we call it a cycle. Under addition, the cycles form a group Zr. Similarly we have the group of r-boundaries, Br, which is a subgroup of Zr since ddAr+l - 0 always. The factor, or difference, group, Hr — Zr mod Br, is the rth homology group of the complex. Any finite part of Hr (its elements of finite order) is the "torsion" Tr. For an example of the above ideas we look at the real projective plane P2. It can be described topologically as a closed disk a2, with opposite points
P
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on the boundary ("points at infinity") identified. The simplest possible cutting into cells is shown. The boundary relations are da1 = p-p
= Q,
da2 = 2a\
Thus the one-dimensional homology group has a single non-zero element (i.e., / / ' = r 1 ) , with a1 as the representative cycle. If we carry a pair of independent vectors (vj, V2) from a2 across a1, leading back into a2 on the other side, we see that the pair has shifted orientation: P2 is non-orientable. When a manifold is cut into cells of a reasonably simple nature, we may form the "dual subdivision" as follows. Put a new vertex in each «-cell. Let a new 1-cell cross each (n — l)-cell, joining two new vertices. Let a new (piecewise linear) 2-cell cross each (n - 2)-cell, finding a boundary waiting for it, and so on. The figure shows a portion of the construction for n = 2.
There is a one-one correspondence between r-cells of the original complex K (shown in heavy lines) and (n - r)-cells of the dual KD (shown in lighter lines), and incidence between cells of neighboring dimensions is preserved. If the manifold is orientable, this shows that homology in K is the same as cohomology in KD (except in extreme dimensions). From this we can see that the Betti numbers (ranks of the Hr) coincide in dimensions r and n-r, and the torsion numbers in dimensions r and n-r-I. This is the "Poincare duality" in a complex formed from an orientable manifold. Note also (see the dashed lines in the figure) that K and KD have a common simplicial subdivision, the "barycentric subdivision" K* of A". Also
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invariance of the homology groups under subdivisions is not hard to show; Alexander proved topological invariance of the ranks of the Hr in 1915. If we examine a cycle Ar of K and a cycle B5 of KD, with r + s > n, the intersection is seen to be expressible as a cycle Cr+S~" of A"*. This is a generalization of the intersection of submanifolds of M", of great importance in algebraic geometry for instance (Lefschetz, Hodge). It is quite clear (that is,- until 1935) that there is nothing of this sort in general complexes. Poincare applied considerations like these to his work in dynamical theory (for instance, the three body problem). But he could not prove a simply stated fact needed about area preserving transformations of a ring shaped surface. However, G. D. Birkhoff succeeded in proving this theorem in 1913. The fundamental group and covering spaces were also studied in detail by Poincare. In a space K, with a chosen point P, a curve C starting and ending at P defines an element of the fundamental group; any deformation of C, keeping the ends at P, defines the same element. One such curve followed by another gives the product of the two elements. The identity is defined by any curve which can be "shrunk to a point" (hence to P). The fundamental group is in general noncommutative. A space with vanishing fundamental group is called "simply connected." Great efforts were expended by Poincare to understand 3-dimensional manifolds. In particular, he conjectured that the 3-sphere was the only simply connected 3-manifold. This is as yet unproved. Alexander proved an entirely new kind of "duality theorem" in 1922: Given a complex K imbedded homeomorphically in an ^-sphere S", there is a strict relation between the homology groups of K and of S" - K. Alexander also gave in 1924 a remarkable example (using ideas of Antoine) of a simply connected surface S* (homeomorphic image of S2) in S \ cutting S 3 into two regions, one of which is not simply connected. We begin with the surface of a cylinder, stretched and bent around to have its two ends facing each other; the figure shows these facing ends, the gap G between them partly filled. We pull out, from each side of the gap, a piece, pulled into a cylindrical piece with a gap (like the original cylinder), these two pieces looped together, as shown; there are now two much smaller gaps, G\ and Gi. We next act in the same manner with each of these gaps, giving Gn and G\2 in G\ and Gj\ and Gn in Gj, and continue. The limiting surface S* has the stated properties. In fact, a loop going around each gap (?*,...*, gives an infinite set of independent generators in the fundamental group of the outside of S' in S 3 , as we see easily. (The inside of 5* is simply connected.) Going back to the early 1910s, Lebesgue discovered (1911) that a region of R", if cut into sufficiently small closed pieces, must contain at least n + 1
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of these pieces with a common point. This (when proved) gives a topological definition of that number n for R". L. E. J. Brouwer proved this in 1913. He was very active, with a general proof of invariance of dimension (a general definition of dimension was given by Menger and Urysohn), mappings of complexes and manifolds, studied through simplicial approximations, the Jordan separation theorem in n-space, coverings and fixed points of mappings, and other things. Alexandroff and Hopf were so inspired by all this that they dedicated their volume A-H to him. (If / is a mapping of a simplicial complex K into R", and f'{pj) is near f{Pi) for each vertex p, of K, the corresponding simplicial approximation is defined by letting / ' be linear over simplexes: f'{J2a'Pi) = J2aif'(Pi)- By subdividing into smaller simplexes, the approximation / ' can be made closer to/.) In the 1920s there was considerable rivalry between S. Lefschetz and W. V. D. Hodge in the applications of topology to algebraic geometry. In a Riemannian manifold M", a principal question was to find the "periods" of a differential form co, that is, the integrals fc, w over cycles C which would form a base for the homology in dimension r. In later work, the forms would be required to be harmonic.
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At one time I was visiting Hodge in Cambridge. In our taking a walk together, he said "Lefschetz claimed to have proved that theorem before I did; but I really did prove it first; besides which the theorem was false!" He liked intriguing questions, so I asked him one that was recently going around Princeton: A man walked south five miles, then east five miles, then north five miles, and ended up where he had started. What could you say about where he had started? (Or more popularly, what color was the bear?) He insisted it must be the north pole, and proceeded to give a careful proof; but I got the sense he did not really believe his proof was correct. (Try Antarctica.) A contrasting situation was less happy. With both Alexander and Lefschetz in Princeton, they naturally had many discussions on topology. But Alexander became increasingly wary of this; for Lefschetz would come out with results, not realizing they had come from Alexander. Alexander was a strict and careful worker, while Lefschetz's mind was always full of ideas swimming together, generating new ideas, of origin unknown. I saw this well in my year, 1931-1932, as a National Research Fellow in Princeton. I believe that Lefschetz never felt good about Veblen choosing Alexander, not him, as one of the first professors at the new Institute for Advanced Study. Let me mention here the famous Lefschetz formula for the algebraic number of fixed points of a self-mapping of a space, an example of Lefschetz's great power. The basic work on integration of differential forms in manifolds was given by G. de Rham in his thesis (1931, under E. Cartan). A complete identity was shown in the homology structure of Riemannian manifolds, as seen through the algebraic structure of a subdivision or through integrating differential forms; moreover, the intersections of submanifolds were related in the natural manner to the products of differential forms. There are three more recent books with fine accounts of this theory in extended form: W. V. D. Hodge, The theory and applications of harmonic integrals, Cambridge University Press, 1941; G. de Rham, Varieties Differentiates, Hermann, Paris, 1955; and H. Flanders, Differential Forms, Academic Press, 1963. The third is especially helpful to the untutored reader. I myself was greatly intrigued by de Rham's work, and studied his thesis assiduously when it appeared. Of course I looked forward to meeting him; I did not suspect the happy occasion in which this would take place. In the late twenties, Alexandroff and Hopf spent considerable time in Gottingen, especially influenced by E. Noether (as mentioned above). I was there for three weeks in early summer, 1928, after graduating from Yale, to get the sense of a great physics and mathematics center. I had physics notes to review, which I thought would go quickly; instead I found that I had forgotten most of it, in spite of much recent physics study. Seeing Hilbert-Ackermann, Grundziige der Theoretischen Logik, in a bookstore, I got it and started working on it, along with George Saute, a math student from Harvard. So I soon decided that since physics required learning and remembering facts, which I
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could not do, I would move into mathematics. I have always regretted my quandary, but never regretted my decision. Those weeks I was staying in the house of Dr. Cairo, along with some physicists, Paul Dirac in particular. We became quite friendly, and discussed many things together. One was the problem of expressing all possible natural numbers with at most four 2's, and common signs. For example we can write 7 = \/i(2/-2 - -2)/.2]. We finally discovered a simple formula, which uses a transcendental function taught in high school. (I'll let you look for it; it starts with a minus sign.) The authors of A-H speak also of a fruitful winter of 1931 in Princeton, influenced by Veblen, Alexander, and especially Lefschetz. The next autumn I found this also. At one time there were seven separate seminars going on together, one of them was devoted to my proof (just discovered) of a characterization of the closed 2-cell. One of my talks was to be on my cutting up process. But a few days before, I was horrified to find that there was a bad mistake in the proof. I worked desperately hard the next two days, and found a valid proof. Later, at the Moscow conference, Kuratowski told me that he especially liked that proof, for he had tried very hard to carry out such a process, but could not. Conversely, I had greatly appreciated his characterization of planar graphs through their containning neither of two graph types: five vertices, each pair joined, and two triples of vertices, each pair from opposite triples being joined. I did, however, find how to use my characterization of planar graphs through dual graphs to give his theorem. By the time of the conference, Heinz Hopf had become my favorite writer (and I later became a personal friend). I found his papers always very carefully written, with fine introductory sections, describing purposes and tools (and he made some similar comments on my writings; he told me he "learned cohomology" from my 1938 paper). I still want to speak of two of Hopfs theorems published before the conference. One was the classification of mappings of an ^-complex K" into the n-sphere S"; it required working separately with the Betti numbers and the torsion numbers. The other described a simple analytic mapping of S 3 onto S2 which could not be shrunk to a point; yet homology could not suggest its existence. The latter theorem was a basic step forward in studying the homotopy groups, to be presented at the conference. Also, it showed that formally the above-mentioned classification theorem could not easily be extended to higher dimensions Km, m> n. How did people learn topology at that time? For point set theory, HausdorfFs Mengenlehre was the bible. Menger's Dimensionstheorie was a help (superseded later by the Hurewicz-Wallman book). For "combinatorial" topology, Veblen's book Analysis situs was a very useful book in the 1920s. Kerekjarto's Topologie was a help (he disliked Bessel-Hagen; look up the reference to the latter in his index). Lefschetz's Topology (1930) became at one a basic reference; but it was very difficult to read. I failed completely
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to understand some broad sections. But soon Seifert-Threlfall Lehrbuch der Topologie appeared, a very fine book; it was admirable for students, and its chapters on the fundamental group and covering spaces remain a good source for these topics. Finally, the foreword to A-H was written soon after the Moscow conference. But, as mentioned earlier, one tragic result of the conference was the abandonment of later volumes. It is high time that we turned to the conference itself. Who was there? Most of the world leaders, that is, in the combinatorial direction. There was Heegaard, representing the old-timers. (Replying to his invitation, he wrote, "I could not resist coming and meeting the greats of present day topology.") Representing the great Polish school of point set theory were W. Sierpinski (but he could not come, I believe) and K. Kuratowski. Two great figures who could have added immeasurably to the conference had they been there, were Marston Morse (analysis in the large) and S. S. Chern (differential geometry, in the complex domain in particular). Apart from these (and Veblen, no longer active in this direction) there were, from America, Alexander, Lefschetz, J. von Neumann, M. H. Stone, and P. A. Smith; also W. Hurewicz and A. Weil (later to be U.S. residents). There were Hopf and de Rham from Switzerland, J. Nielson from Copenhagen, E. Cech from Czechoslovakia, and Alexandroff, Kolmogoroff (not usually thought of as a topologist) and Pontrjagin from U.S.S.R. Then there were younger people: Garrett Birkhoff, A. W. Tucker and myself from America; Borsuk, Cohn-Vossen, D. van Danzig, E. R. van Kampen (becoming a U.S. resident), G. Nobeling, J. Schauder, and others. The Proceedings of the conference came out as No. 5 of vol. 1 (43), of Recueil Math or Matematischiskii Sbornik, 1936. All papers were either published or listed here. There were about 40 members in all; a number of them missed being in the official photograph (see page 88). For many of us, coming to the conference was a very special event. And since I was one of three from America that met in Chamonix to climb together beforehand, I tell something about this. But to start, how did Alexander and de Rham first meet? Alexander told me (when he and I were at the Charpoua hut above Chamonix in 1933) how he and his guide Armand Charlet (the two already forming a famous team) were crossing the enormous rock tower, the Dru, from this same hut a few years before. They and another party crossed paths near the top; so since each had left a pair of ice axes at the glacier, they decided to pick up the other party's axes when they reached the glacier again. With all back at the hut, two of them discovered that they knew each other by name very well: Alexander and de Rham. I had had the great fortune to spend two years in school in Switzerland, in 1921-1923, including three summers. Besides learning French one year
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and German the next, I had essentially one subject of study: the Alps. The first of these years my next elder brother, Roger, was with me. We were very lucky in having an older boy, Boris Piccioni, quite experienced in climbing, in school with us; and in a neighboring school teacher, M. le Coultre, who was a professional guide also, inviting us all on three climbing trips, which included training in high alpine climbing. As a further consequence, nearly all my climbing has been without guides. In 1933 Alexander and I met for several fine climbs at Chamonix, then went on to Saas Fee for more climbing. We next went up to the Weisshaon hut, below the east side of the great Weisshorn, with the idea of trying an apparently unclimbed route, the E. ridge of the Schallihorn, a smaller peak just south of the Weisshorn. At the hut, there was Georges de Rham, with a friend Nicolet! They had just climbed the Weisshorn by the N. ridge and descended the E. ridge; tomorrow they would climb the E. ridge again, to descend the much more difficult S. ridge, the "Schalligrat." So we were all off early the next morning. Alexander and I found our ridge easier than expected, and never put on the rope during the ascent. (Near the top we found a bottle; it was apparently from a party traversing to the top part of the ridge in 1895.) The descent (now we were roped) was over the N. ridge and down to the Schallijoch (where we heard calls of greeting from the other party). The others watched our route going down the glacier, aiding their own descent, which was partly after dark. From this time on, de Rham and I often met during the summers, and did much fine climbing together. It seems that he was renowned in Switzerland as much for his climbing as for his mathematics. In the summer of 1939, my finest alpine climbing season, he and Daniel Bach and I crossed the Schallihorn by "our ridge" (now its third ascent), and went on to climb the "Rothorngrat" and Ober Gabelhorn (we having first climbed the Matterhorn). Georges' new "vibram"-soled boots were giving him trouble, so he stopped now, while Daniel and I returned to the Weisshorn hut and made a one-day traverse of the Weisshorn by the Schalligrat and N. ridge, closing the season. And imagine my surprise when, some years later, I bought a wonderful picture book "La Haute Route" of the high peaks, by Georges' friend Andre Roch, and saw the first picture in it: Daniel and I on the Schallihorn (taken by Georges)! To return to 1935: Alexander, Paul Smith and I met at Chamonix, climbed the Aiguille de Peigne together, then went on to further climbing; but the weather was turning bad, and we soon had to go on toward Moscow, (de Rham was already in Warsaw.) Alexander drove me to Berlin, and we took the night train from there. What was the main import of the conference? As I see it, it was threefold: 1. It marked the true birth of cohomology theory, along with the products among cocycles and cycles.
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2. The pair of seemingly diverse fields, homology and homotopy, took root and flourished together from then on. 3. An item of application, vector fields on manifolds, was replaced by an expansive theory, of vector bundles. Yet seven years later, a single paper of Hopf would cause a renewed bursting open of the subject in a still more general fashion. We now look at the remarkable way in which these matters developed at the conference. The first major surprise was from Kolmogoroff, an unlikely person at the conference, who presented a multiplication theory in a complex, applying it also to more general spaces. The essence of the definition lies in the expression (A) • • Pr) X (tfo • ' Qs) = (A) • • • PrQO ' ' '
provided that the right-hand side is a simplex; besides, an averaging over permutations is taken. (One obvious problem is that the product seems to be of dimension r + s + 1, one more than it should be.) When he had finished, Alexander announced that he, also, had essentially the same definition and results. (Both had papers in press.) From the reputations of these mathematicians, there must be something real going on; but it was hard to see what it might be. I digress for a moment to say what happened to this product. Within a few months, E. Cech and I both saw a way to rectify the definition. We each used a fixed ordering of the vertices of a simplicial complex K, and defined everything in terms of this ordering. The basic definition was simply (with the vertices in proper order) (p0 • • • pr) — (Pr--
Pr+s) = {Po ' ' ' Pr ' ' • Pr+s).
whenever the latter is a simplex of K. Alexander at once saw the advantage of this, and rewrote his paper from this point of view {Annals of Math., 1936). Another event at the conference was the defining of the homotopy groups in different dimensions of a space, with several simple but important applications, by Hurewicz. Alexander responded by saying he had considered that definition many years (twenty?) earlier, but had rejected it since it was too simple in character and hence could not lead to deep results. Perhaps one lesson is that even simple things may have some value, especially if pushed long distances. Both E. Cech and D. van Danzig also said that they had considered or actually used the definition of Hurewicz. Thus at the time of the conference, the homotopy groups were very much "in the air." I now turn to the paper that had the most intense personal interest for me. Hopf presented the results of E. Stiefel (written under Hopf s direction), "Richtungsfelder und Fernparallelismus in /z-dimensionalen Mannigfaltigkeiten.". It was concerned with the existence of several independent
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vector fields in a manifold. Both in generality, and (largely) in detail, this was just what I had come to Moscow in order to present myself! Stiefel had more complete results; in particular, that all orientable 3-dimensional manifolds were "parallelizable." On the other hand, I had given a much more general definition; for example, for submanifolds of Euclidean space (or of another manifold), I considered normal vector fields also. Moreover, I considered sphere (or vector, or fiber) bundles over a complex as base space, and found that results were best expressed in terms of cohomology, not homology, in the complex (for manifolds it did not matter). I spoke briefly of these things right after Hopfs talk; but still had to decide afresh how to talk about my own work. Moreover, on my way to the conference I had already become uncertain on how to talk; for I had realized that Hopfs classification of the mappings of K" into Sn could be presented much more simply in terms of cohomology than of homology. In fact, it seemed to me highly worthwhile to show this in detail, as the possibly first true use of cohomology, and the simplest possible example of its usefulness. I therefore gave two shorter talks, one giving a fuller account of my work on sphere bundles, and the other, a pretty complete proof of the Hopf theorem with cohomology. I want to speak briefly of two further presentations. Tucker spoke on "cell spaces," a thesis written under Lefschetz's direction, which gave certain specifications about what can usefully be considered a "complex." This cleared up some important matters which played a real role in both Cech's and my exposition of cohomology and products in our coming papers in the Annals of Mathematics. The other was Nobeling's presentation, which occupied the full last morning of the conference. (I was not there; I had left early for Leningrad, hoping to meet the composer Shostakovich (which did not happen), and to make the five-day boat trip from Leningrad through the Kiel Canal to London, which was quite interesting.) Nobeling's talk was to present, in outline, the proof that all topological manifolds can be triangulated, von Neumann reported on this conference as follows: Nobeling demonstrated amply that he had answers to every possible question that one might think of. (Within the year, van Kampen found the error in the proof. Disproving the theorem took much longer.) I give a brief description of Hopfs mapping theorem (about K" -* S") through cohomology; take n - 2 for ease of expression. (See my papers in the Duke Math Journal, 1937.) First, "coboundary" is dual to "boundary": If da = ! + ••-, then Sz = a + • • •. With the language of scalar products, or, better, considering a cochain as being a linear function of chains, we can write Sz-a = r-do. 5Xr • Ar+l = Xr dAr+i.
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Whereas the "boundary" of a cell makes good geometric sense, the "coboundary" does not. In the figure, dx stretches into three pieces; but why stretch so far? In the theorem, our first step is to deform any mapping / into a "normal" mapping. To this end, choose a definite point P of S2 (the south pole). For each vertex /?, of K, we may deform / into / ' so that f'(p,) = P. Of course
all cells of K with pi on the boundary must be pulled along some also. Do this for all vertices, so they are now all at P. Now each 1 -cell xj of K has its ends mapped into P; we may pull xy, along S2 down to P, keeping its ends at P, extending the deformation in any manner through the rest of K. This gives a normal mapping f0, in which the 1-dimensional part Kl of K lies at P. Now take any 2-cell a2 of K. It is a standard theorem (first proved by Brouwer) that (since its boundary is at P) it lies over S2 with the degree di = d{a2), this being an integer. (If it only partly covers S 2 , each piece of S2 is covered an algebraic number 0 of times; we may shrink the mapping to P, keeping do} at P.) We remark in passing that when x is pulled down to P, how far we choose to extend the deformation into 2-cells depends on how far those 2-cells reach beyong T; thus Sx plays a role in the proceedings. Let us write X(a2) — d{ for each a2. (Or we could write X = X^/°f-) This is the cochain X defined by the deformed mapping. We could deform a r, ;rmal mapping f0 into a different normal mapping /i as follows. Choose a 1 -cell x, and sweep it up and over S2, and down the
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other side back to P, keeping its ends always at P. (In the figure, we show two stages, fh and fh, 0 < t\ < t2 < 1, of this deformation of r.) Extend the deformation over the neighboring 2-cells of K. For each of of K which has T on its boundary (positively), df is increased by 1; thus the change in the corresponding cochain is simply to add St. In this manner we may add SY to the cochain X, for any 1-dimensional cochain Y; but the cohomology class of X remains unchanged. Since there is no 3-dimensional part in K, all 2-cochains are cocycles; thus there is a definite 2-dimensional cohomology class associated with the original mapping. We have one thing still to prove: Given any deformation of a normal mapping f0 into another one, f\, the same cohomology class is defined. We use a standard technique to do this. Let f,{0 < t < 1) denote the deformation. Set F(t, p) — fi{p). Now F is a mapping of the product space I x K into S2, where / is the unit interval (0, 1).
J /
/ <— .-(--. \ \
/
i
7
\ \ \
\
\ \
\
Ox a2
txa2
,1
\xa2
I xa2
If we alter F for 0 < t < 1, it will give a new deformation of f0 into f\. So look at any vertex p, of K, and the corresponding line segment / x p, of I x K. This segment is mapped by F into a curve in S2 starting and ending at P. We may alter F(t, /?,) by pulling this curve along S2 down to P, and extending the mapping to the rest of / x K. Doing this for each vertex of K, we have defined a new deformation of f0 into f\, in which each vertex of K remains at P. We have already seen that this implies that the cohomology classes of fa and fx differs by a coboundary, and the proof is complete.
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I have spoken of my great interest in the papers of Hopf and de Rham. In the mid thirties I saw my main job of coming years to be the extending of the general subject of sphere bundles. For this purpose, I was very anxious to have the basic foundations in as nice a state as possible; I could not work with concepts that were at all vague in my mind. As part of this, I wanted to work both with algebraic and with differential methods; hence I needed, as far as possible, a common foundation of both (and Hopf and de Rham were my best models). But these subjects had been quite separate, and hence the notations used were very different in character. So I tried to devise notations that could allow the fields to work more closely together. The use of "contravariant" and "covariant" vectors raised a quandary. A covariant tensor was one whose components (depending on the coordinate system) transform "like" or "with" the partial derivatives of a function. But for me, the basic object was a vector space, and its elements, vectors, should be the base of operations: They should be called "covariant" (if anything), not contravariant. Also homology dealt directly with geometric things, and should have the prefix (if any) "co" not "contra." In any case, I would not use prefixes differently in homology and in differential geometry. So I started publishing, using the term "cohomology" for the new topic, omitting any use at all of "contra," and disregarding the (for me) wrong use of "co." This was picked up quickly by others, and the inherent reverse of "co" and "contra" remains. There was a further block to my progress. I had to handle tensors; but how could I when I was not permitted to see them, being only allowed to learn about their changing costumes under changes of coordinates? I had somehow to grab the rascals, and look straight at them. I could look at a pair of vectors, "multiplied": u V v. And here, I needed MVV = - W M . So I managed to construct the rest of the beasts, in "tensor products of abelian groups." {Duke Math Journal, 1938). Before long I noticed that neat form, using less space, was the sine qua non of mathematical writing: the CORRECT definition of the tensor product of two vector spaces must use the linear functional over the linear functional over one of them. So this is the way in which later generations learned them. Only in 1988 did I make a further discovery (or rediscovery?): A typical "differential 2-form" is u v v; and this is already a product! Any simplex, say PoP\ Pi, is a bit of linear space, and writing v,7 = pj - pt, a natural associated 2-form is v0i V v^. Another such product in poPxPiPi is, for instance, VQI
V (V,2 V V23).
Then why not write {PoPl) — {P\PlPl) = PoP\PlPl>
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whenever the right hand side is a simplex? From the basic definition only of differential forms associated with simplexes (or a simplicial complex), nothing could be more natural. (This was soon called the "cup" product.) Why did not Kolmogoroff and Alexander (and lots of others) think of it? I think this is a real lesson to be learned: Keep wide contexts and broad relations in mind; new connections and extended methods may show up. I mention still the "cap" product, typified by PiPiPi — P0P1P2P3 = P0P1, and for cochains X and Y and chains A,
X~{Y~A)
= (X~
Y)-A.
What was the aftermath of the conference? To a large extent, the younger generation took over. In the U.S.S.R., L. Pontrjagin was coming into full flower (in particular, with topological groups and duality). J. Leray (France) and J. Schauder (Poland), collaborating in large part, and S. S. Chern (China, U.S.A. and elsewhere) were bringing powerful tools into play. New domains such as sheaves and spectral sequences were playing a big role. In the U.S.A., N. E. Steenrod, S. Eilenberg, and S. Mac Lane were playing an increasing role, especially in building edifices from extremely general principles (with categories and functors for the foundation). We are getting into the 1940s, with an astounding pair of papers by H. Hopf about to arrive on the scene. The first of these papers, "Fundamentalgruppe und zweite Bettische Gruppe," was communicated to the Commentarii Mathematicii Helvetia on September 12, 1941. In this paper Hopf gives an algebraic construction of a certain group G* from any given group G. Now let K be a complex, with second homology (Betti) group B1 and with fundamental group G. Let S2 be the subgroup of B2 formed from the "spherical cycles," continuous images of the 2-sphere. Then Hopf proves that the factor group B2/S2 is isomorphic with G*. Thus the fundamental group of a complex has a strong influence on the second homology group. For example, if G is a free abelian group of rank p, then G\ is a free abelian group of rank p(p - l)/2, so this is a lower bound for the rank of the second homology group. The construction of G* is as follows. Represent the fundamental group G as a factor group F/R, where F is a free group and R is a subgroup (of relations in F). Define the commutator subgroups [FR] (generated by all commutators frf~xr~x for / e F and r e R) and [FF] (using f\hf^x ffl). Then G\ = (R ~ [F,F])/[FR). (Actually, this group G* had been defined much earlier, by I. Schur, in Berlin.) At this point I turn over a description of happenings to S. Mac Lane, "Origins of the cohomology of groups", L 'Enseignement Mathimatique, vol. 24, fasc. 1-2, 1978. This is an admirable paper, full of descriptions of modern
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fields of work and their interrelations, all showing the enormous influence of the above paper of Hopf (and another to follow a few years later). Here is one quote from Mac Lane. Hopfs 1942 paper was the starting point for the cohomology and homology of groups; indeed this Hopf group G* is simply our present second homology group H2{G, Z). This idea and this paper were indirectly the starting point for several other developments: Invariants of group presentations; cohomology of other algebraic systems; functors and duality; transfer and Galois cohomology; spectral sequences; resolutions; Eilenberg-Mac Lane spaces; derived functors and homological algebra; and other ideas as we will indicate below. I had the great pleasure of reviewing Hopfs paper for Math Reviews (the review appeared already in November 1942). I also, many years later, wrote an informal paper whose purpose was to bring out the essential reasoning (commonly geometric in character) of various basic theorems. This paper ended with the quoted formula of Hopf. It was published as "Letting research come naturally," Math. Chronicle 14 (1985) (Auckland, New Zealand) 1-19. I mention just one crucial idea of Hopf: that of "homotopy boundary," k, from which everything flowed: From a fixed point P of A^, choose a (simplicial) path to any 2-simplex a2, go around its boundary, and back to P:
k = ydo2y~y
Now k (in R) is a relation in the fundamental group G of K, associated with the cell a2, considered as a 2-dimensional chain in K. (Write G = F/R, where the free group F is the fundamental group of the 1-dimensional part Kl ofK. Now keRcF.) I will say a few words still about my own work in the direction of topology. My one fairly full account of researches in sphere bundles appeared in Lectures in Topology, University of Michigan Press, 1941, under the title "On the topology of differentiable manifolds." This paper is certainly the basis of some awards I have received in later years. Largely to help prepare
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good foundations for my planned book on sphere bundles, I wrote the book Geometric integration theory, Princeton University Press, 1957. This volume received quite unexpected acclaim over many years. On the other hand, in this period, Warren Ambrose once said to me "I calculate that the publication date of your book on sphere bundles is receding at the rate of two years per year." He was quite right. I was having increasing difficulty in finding good geometric foundations for the topological aspects. Then I had an unexpected piece of good fortune; rather, two. Steenrod's book on fiber bundles appeared, doing a far better job than I could possibly have done, and I was invited to join the faculty at the Institute for Advanced Study. I also had new sources of inspiration: Henri Cartan (we played music together at times, piano and violin) was combining topology with analytic studies (several complex variables), which I studied with determination, getting preparation for my final book on complex analytic varieties (whose unfulfilled purpose was to help in the foundations of singularity theory). And R. Thorn and I had started working (independently) on singularities of mappings, my last major field of work. He had also given a very simple and general proof of my "duality theorem," really the formula for the characteristic classes of the product of two sphere bundles over the same base space, which I had carried out through a full examination of the geometric definitions. (And later, under the general definitions of Eilenberg-Steenrod, the theorem became part of the definition of sphere bundles.) I did still write some further papers, with which I was amply satisfied, about ideals of differentiable functions {Am. J. of Math, 1943), On totally differentiable and smooth functions {Pacific J. Math., 1951), On functions with bounded nth differences {J. de Math. Pures at appliquees, 1957), somewhat outside my normal fields of work. But younger workers, in America in particular, were taking over in a strong way; I mention John Mather especially, in singularities of mappings. So, to return to the title of this paper, I have seen general topological and algebraic methods flourishing all over the world increasingly, as the "center of mass" of such studies moves still nearer to the U.S.A. shores.
R e p r i n t e d from D U K E MATHEMATICAL J O U H N A L
Vol. 1, N o . 4, December, 1935
A FUNCTION NOT CONSTANT ON A CONNECTED SET OF CRITICAL POINTS B Y HASSLER WHITNEY
1. Introduction. Let /(x l ; • • • , xn) be a function of class Cm (i.e., with continuous partial derivatives through the with order) in a region R. Any point at which all its first partial derivatives vanish is called a critical point of / . Suppose every point of a connected set A of points in R is a critical point. It is natural to suspect then that / is a constant on A. But this need not be so. We construct below an example with n = 2, m = 1, A = a n arc. The example may be extended to the case n = n, m = n — 1, A = an arc. The arc and the function on the arc are easily defined. The extension of the function through the rest of the plane or space is given by a theorem of the author. 1 The question settled in this paper was raised implicitly in a paper of W. M. Whyburn.2 It is brought up by his definition of critical sets as the maximal connected subsets of the set of critical points on which the function takes a single critical value. Theorem 2 of Whyburn's paper shows that an example of the type given in the present paper can be constructed only by using critical sets which have points that cannot be joined in these sets by rectifiable arcs. It would be interesting to discover how far from rectifiable a closed set must be to be a set of critical points of some function but not a critical set of the function. It may be remarked that any closed set may be a critical set.3 For fixed n and m large enough, m 5: [(n — 3)2/16 -f- n], where [n] is the integral part of n, f must be constant on any connected critical set, as shown by M. Morse and A. Sard in an unpublished paper. The example shows that it is in general impossible to express the values of a function f{x\, • • • , xn) along a curve which is not rectifiable by means of an integral of a function of partial derivatives of / of order g n — 1 along the curve.4 2. The arc. Let Q be a square of side 1 in the plane. Let Q0, Qi, Qi, Qi be squares of side 1/3 lying interior to Q in cyclical order, each a distance 1/12 Received May 16, 1935; presented to the American Mathematical Society, October 26, 1935. The example given here was discovered in 1932 while the author was a National Research Fellow. 1 H. Whitney, Analytic extensions of differenliable functions defined in closed sets, Transactions of the American Mathematical Society, vol. 36 (1934), pp. 63-89, Lemma 2. We refer to this paper as AE. 2 W. M. Whyburn, Bull. Amer. Math. Soc, vol. 35 (1929), pp. 701-708. 3 See A. Ostrowski, Bull, des Sciences Math., Feb. (1934), pp. 64-72. * For such an expression (using partial derivatives of any desired order) along a rectifiable curve, see H. Whitney, Functions differenliable on the boundaries of regions, Annals of Mathematics, vol. 35 (1934), pp. 482-485, (1) and (3).
807 F U N C T I O N N O T CONSTANT ON S E T O F CRITICAL
POINTS
515
from the boundary of Q. Let q and q' be the centers of the sides of Q along Qo, Qi, and along Q3, Q0. Let qi and qt be centers of adjacent sides of Qi (i = 0, 1, 2, 3) so that ,_! and qi face each other (i = 1, 2, 3), and go is near q, q's is near q'. Let A0 be a line joining q and 50, let At join g,_i and qi (i = 1, 2, 3), and let 4 4 join qs and '. Suppose we have constructed squares Q^.-.i, , points ,,...<,, ,•,...<,, and lines i4,v..,-, (each iA = 0 , 1 , 2, 3; each j * = 0 , 1 , 2,3,4) fori < s. By taking a square Qn i,_,, shrinking it to a third its size, and turning it around and upside down if necessary, we may place it in Q;, • «,_, so that g,,. ••«,..» and ;,;,_, go into {,...<,_,
and <&,...*,_,, and thus construct squares Q^.-i,, etc. We continue this process indefinitely. Let Qi,,-,... be the point common to Q, Qilf Q^i,, • • • for each (n, *2, • • • )•
The line segments Atl...i, together with the points Qi,;,... form an arc A. It may be represented as the topological image of the segment (0, 1) by letting •Aj,...if correspond to the segment /2tHhl 2t._i + 1 2t. 2ii + 1 2t_i + 1 2i, + l \ \ 9 "•" " ' - "•" 9.-1 + 9. > 9 + •••+ Q,-! + 9. J1
516
HASSLER
WHITNEY
and letting Q,,i,... correspond to the number 2ii + 1 9 3. The function f(x, y).
2U + 1 , 92 + ' •- •
We first define f(x, y) along the arc A as follows: j.
A
at Q^...,
ii
,
,
i,
/ =| + g+....
/ increases from 0 to 1 as we run along A from q to ?'. Set foo(x,y) = f(x,y), fw(x,y) = foi(x,y) = 0 on .4. We shall show that/go is of class C1 on A in terms °f (/oo, /io, /oi) (see AE). It will follow from Lemma 2 of AE that the definition of f(x,y) may be extended over the plane (in particular, over Q) so that / is of class C1; also df/dx = fw = 0, df/dy = foi — 0 on A, and hence each point of A is a critical point of/. As /io and /oi are continuous in A, we need merely prove that for each < > 0 there is a 8 > 0 such that if (x,y) and (x',y') are points of A whose distance apart is r < o, then (1)
\f(x',y')
-f(x,y)\
< re,
(see AE (3.1) and (3.2)). The proof rests on the following two facts. (a) If (x,y) and (x',y') are points of A in Q„ ... ,•„ then (2)
\f(x',y')
-f{x,y)\
S 1/4".
(b) If (x,y) and {x',y') are points of A separated by some point Q,^..., and if Qi,....-, is the smallest square containing them both, then (3)
'>r23-^
Assume (a) and (b) are true.
(4)
Given e > 0, choose sQ and 5 so that
softy<«,
«
W
Now let (x,y) and (x',y') be any two points of A distant r < 8. If no point Qi.i,... separates them, then f(x',y') = f(x,y), and (1) holds. Otherwise, let Q,,... i, be the smallest square containing them both, (b) gives , .-. s > So .
< r < 8< +1
12 3«
12 3'°
+l
Hence \f{x',y')-f(x,y)\ r
< 12 • 3«+* = ~ 4'
36
/3V \4/
<
809 FUNCTION NOT CONSTANT ON SET OF CRITICAL POINTS
517
It remains to prove (a) and (b). (a) is obvious from the definition of/. To prove (b), we consider three cases: neither point is in any square Q,-,...;, ,-,+1; each point is in such a square; one point is in such a square, and the other is not. In the first two cases we see that r > 1/(2- 3*+1); in the third case, (3) holds. 4. Generalization to higher dimensions. We shall indicate how the corresponding example is constructed for n = 3, m = 2; the generalization to higher dimensions is obvious. Let Q be a cube of side 1. Let Q0, • • • , Q1 be cubes of side 2/5 arranged in Q so that Qi is adjacent to Q;_i. Let q and q' be the centers of faces of Q which are adjacent and adjacent respectively to Qo and Q7- Define qi and qi (i = 0, • • • , 7) as before, and similarly for A$, • • • , .48. The process is continued indefinitely, as before. A is again an arc. We set on Aix...i,,
/ = - + . . . + _ ,
at Q*,..., / = i + | + . . . . Set /ooo = /, /B/ST = 0 (a + /S + T = 1 or 2) on A. To prove that / = /ooo is of class C2 in terms of these functions we need merely show that (5)
A = ! / ( * ' , y', z') - f{x, y,z)\
< r*e on A
(r < 5) .
Note that / varies by at most 1/8* in any Qi, ...i,. Now if the smallest cube containing (x,y,z) and (x',y',z') is Qi,...,-,, r is of the order (2/5)*, while A g 1/8*; hence A/r2 is of the order (25/32)', which is < t for r small enough and hence s large enough. Note that all partial derivatives off (of order ^ n — 1 = 2) vanish on A. HARVARD
UNIVERSITY.
SINGULARITIES OF MAPPINGS OF EUCLIDEAN SPACES B Y HASSLER WHITNEY
1. Introduction We shall describe here some results and methods pertaining to the following general problem (details will appear elsewhere). Suppose a m a p p i n g / 0 of an open set R in n-space En into w-space Em is given (we shall w r i t e / : Rn —*• Em). How can we alter / 0 slightly, obtaining a mapping / with nicer and simpler properties? By the Weierstrass approximation theorem (generalized), we may require t h a t / be analytic in R; if/ 0 was r- smooth (had continuous partial derivatives through the r t h order), we may require the partial derivatives o f / through the r t h order to approximate those of f0 (we then c a l l / an r-approximation). Now take any regular point p of/, t h a t is, a point p such t h a t / is of maximum rank v = inf (n, m) a t p. (Equivalently, using coordinate systems in En and in Em, the Jacobian matrix of/ at p is of rank v.) Then, by the implicit function theorem, we may choose coordinates so that / has the form (1.1)
yl = xl
(i = 1, • • • , v),
y' = 0
(i > v, if m > n).
Hence at any regular point p, / h a s the structure of the particular mapping (1.1); we shall be satisfied w i t h / h e r e . Any non-regular point we call a singular point of/. I n general (unless m > 2M; see §4) we cannot avoid the presence of singular points. We would then like to reduce them as much as possible, making them lie in small and simple point sets, and requiring the structure of/ to be as simple as possible in the neighborhood of a singular point. We shall show t h a t the set >S" of singular points may be made to form a smooth manifold plus boundary. There will be subsets of S', also manifolds with boundaries, consisting of points in the neighborhood of which/is more complicated. This gives a splitting of S' into sets, in each of which / satisfies specified conditions. Thus the singularities o f / are divided into various types. We give the geometric basis of this splitting in P a r t I. Examples of the types of singularities will be found in P a r t I I . The manner of defining the singularities is as follows. Let Lr be the space of possible values of the differentials of a m a p p i n g / : Rn —• Em through the order r at a p o i n t y (see §2). Each Lr contains a certain set of manifolds, say L',^, L[ 2) , • • • , of various dimensions. G i v e n / , t h e values at p of the differentials of/ give a mapp i n g / ' ' : i2n—>- Lr, for each r. (We m a y keep r 5^ 2M; see §11.) By a slight change in / , we may r e q u i r e / ' to be "crosswise" to the L't, (see §5); then fr(R) does not intersect any Lf-, of small dimension, and it intersects other i ^ , in as simple a manner as possible. Using a suitable r, we then say / is locally generic. The sets in i2 which map into the LT(i) are the singular sets of/; at a point of one of these sets, we say / has a generic singularity.
286
HASSLER WHITNEY
Suppose p is a generic singularity; say q = fr(p) e L'iy T h e n / r is crosswise t o •^'i) a * ?• I* follows that for a n y good (r + 1 ^ a p p r o x i m a t i o n / j to f,f{ will also be crosswise to U(i) at some point qt = / [ ( i ^ ) ; thus a generic singularity cannot be removed in this manner. (It m a y be removable undera£-approximation for smaller k; see for instance §22 of [13].) We say the above fx has a singularity at px of the same type t h a t / h a d a t p. A basic question now is, to what extent is fx near px like / near pi If the division into singular sets is sufficiently complete, then we would like fx t o be obtainable f r o m / b y "changes of coordinates." Explicitly, there is then a mapping F defined by y i = (f><(xl, • • • , X n )
(i — 1, • • • , m),
such t h a t b o t h / a n d / j m a y be p u t into this form by proper choices of coordinate systems. We then choose a particular mapping F, and call it a normal form for the type of singularity, and we say t h a t / is stable at p. Our principal conjecture is t h a t the division of singularities into types satisfies the above condition. The general program m a y be described as follows. (a) Carry out the definition of types of singularities, as proposed below. This will be seen to run into questions about the relation of planes to certain algebraic cones. (The basic theoretical considerations are not difficult, but carrying out t h e description of the singularities in high dimensional cases seems very complicated.) I t is then easy to show t h a t arbitrarily near any / 0 there is a locally generic / . (b) Show t h a t the division into types is complete, in t h a t any locally generic mapping is stable at each point. This is the most difficult part of the program. (The choice of a normal form for a given type of singularity is relatively easy.) A further study should include the following: (c) Find topological properties relating to the singularities, both locally and in t h e large (with En and Em replaced by smooth manifolds). We shall not discuss this problem here. See [11] and [13]; also Thom, [4], [5] and [6]. The program has been carried through in certain cases, as follows: For m = 1, we have a real function / i n En; the singular points are the critical points of/. The theory of Marston Morse, in [1] and [2], covers this case (see §16). For m ^> 2n, there are no singularities; see §4. For m = 2n — 1, we can have singularities a t isolated points; see [9] and §20. For n = m — 2, S' consists of smooth curves, and there are isolated points of other t y p e on the curves; see [13] and §17. Suggestions have been made about the possible types of singularities in more general cases. F . Roger [3] described the types we call SfK R. Thom found explicitly the types in low dimensional cases in [4] (the entry IS 2 (5J) appears first for (n, m) = (5, 4) (see §25), not (4, 3)). B u t no proofs t h a t the division into types is complete in these cases has been given. Singularities which can be removed by small deformations (for instance branch points; see §7 of [13]) of course may nevertheless be of importance. We shall not consider these here.
SINGULARITIES OF MAPPINGS OP EUCLIDEAN SPACES
287
I. GEOMETRIC STRUCTURE OF SINGULARITIES
2. The differentials of a mapping Let V(En) = V denote the space of vectors in En. The differential df(p) of/ at p is the linear transformation of Vn into Vm defined by (2.1)
df(p)-v=\un^+l-[j(p + tv)-f(p)}eVm,
veV\
The second differential d?f(p) is the bilinear transformation of Vn X Vn into Vm defined by (2.2) ffif(p) • (v, w) = lim, t ^ 0 + - \J(P + «, + tw) - f(p + w) St
-f(p
+ tw)+f(p)].
It is symmetric: (2.3) dj(p) • (v, w) = d*f(p) • (w, v). Higher differentials are defined similarly, or by induction. If coordinate systems in the spaces are given, then naming the first r differentials of/ is equivalent to naming the partial derivatives of orders up through r. Relation (2.3) corresponds to the symmetry of cross partial derivatives. Let U- = £} denote the space of linear transformations of V" into Vm; it is a linear space, of dimension nm. More generally, let £ r denote the space of multilinear symmetric transformations of the Cartesian product V" X • • • X V {r factors) into Vm. Thus an element of £ r is a multilinear symmetric function F(vlt • • • , vr), whose values are vectors of Vm. Set U = fi1 © • • • © £'.
(2.4)
Now with/given, dkf(p) is an element of 2k, for each k. Let/ r (p) denote the set of d f(p), k
(2.5)
d"f(p)e2k,
f'(p)=(df(p),d*f(p),---,d'f(p))eL'. 3. The structure of H
Each point T £ L1 is a transformation of Vn into Vm. The rank of T is the dimension of the image space T( Vn). Let Lxp denote the set of points of L1 of rank p. Now (3.1)
L1 = L\ U L\ U • • • U L\,
v=M(n,
m).
l
It is easy to see that each L f is a manifold, of dimension (3.2)
dim (Lxp) = (n + in)p — p2 = nm — (n — p)(m — p).
The codimension of a manifold in a fixed containing space is the difference of the two dimensions. Hence (3.3)
codim {L\) = (n — p)(m — p).
Clearly any limit points of Lxp not in L* lie in the sets Lp,, p' > p. We express this by saying that the L1 form a manifold collection.
HASSLER WHITNEY
288
4. The rank of/ n
m
G i v e n / 0 : R -+E , (4.1)
there corresponds a m a p p i n g / J = df0 : R" —»- L1. Set
p(n, m) — sup {r + 1: (» — r)(m ~ r) > n);
this is the largest number r + 1 such t h a t the condition shown holds. Set p = p{n, m) — 1; then codim (L\) = (n — p')(m — p') > n. I t follows [13, Theorem 11A] t h a t a small deformation of/ 0 will give a m a p p i n g / such t h a t no point df(p) lies in Z*,. We s a y / i s of rank p atpifdf(p) is of rank p. The rank o f / i s the smallest rank a t any point. T h e result above is t h a t by a small deformation, we may make f of rank ^_p(n, m). The deficiency of/ is (with v — inf (n, m)) (4.2)
dfc (/, p)=v-
rank (/, p),
dfc (/) = v - rank (/).
Set (4.3)
d(n, m) = v — p{n, m).
Then we may make f of deficiency <1 d(n, m). Some special cases of this are the following: (a) If m ^> 2n, t h e n p(n, m) = n, d(n, m)= 0; hence we can remove all singularities in this case. Take any smooth manifold M, and set/ 0 (j>) = q0 e E2n (p e M). Applying the result above gives an "immersion" / of M in Ein. (There may be self-intersections of/(.3f).) (b) Suppose m > f ( n — 1 ) . Using r = n — 2 in (4.1) shows t h a t p(n, m) I> n— 1. Thus all singular points m a y be made of deficiency <^ 1. (c) For n ^ 2m — 3, we find p(n, m)~^>_m — 1, and d(n, m) <1 1 again. (d) If n = m < (k + l ) 2 , we find 5(n, m) <£ k. The results given are the best possible, in t h a t there exist mappings such t h a t any sufficiently nearby mapping has t h e rank p{n, m) and deficiency <5(ra, m). Some values of 5(n, m) are given in the table. (This was found in large part by Wolfsohn, [14].)
0 0 1 2 1 1
0 0 1 1 2 2
0 0 0 1 1 2
Table of 6(n, m).
0 0 0 1 1 2
0 0 0 0 1 1
10
11
12
0 0 0 0 0 1
0 0 0 0 0 1
0 0 0 0 0 0
SINOULABITIES OF MAPPINGS OF EUCLIDEAN SPACES
289
5. Crosswise mappings Consider F : R" —* 2?A, and let M be a smooth manifold in EN. Suppose F(p) = q e M. At } we have the tangent vector space P(M, q) of M, and the image dF(p){Vn) of Vn under dF a t p. We say F is crosswise to M if, for all such p, these two vector spaces span VN: dF{p)(Vn)
(5.1)
+ P(M, F(p)) =
F*.
Note t h a t if n — dim (M) < A\
i.e.,
n < codhn (Jf),
then F(R) n i f m u s t be void. LEMMA (Thorn, [6]). Let M be a manifold collection in LT. Then arbitrarily near any f0 : R" —*• Em there is an ffor which fr is crosswise to each manifold of M. S u p p o s e / ' : Rn —*• Lr is crosswise to the smooth manifold M, and codim(M) = 3
6. The singular sets Sk m
L e t / : R" -+E L\. Set (6.1)
be locally generic (§1). Then (§5) f1 = df is crosswise to the
Sk = (ft-H^U),
v = inf (n, m).
Then (§5) the <Sfc are smooth manifolds in En; they form a manifold collection. Now S0 is the set of regular points of/, and S' = Sl U S2 U • • • is the set of singular points: / is of deficiency k in Sk. By (3.3) and §5, (6.2)
codim (S,_p) = (n -
provided this number is <^n: otherwise, S (6.3)
p)(m -
p),
is void. This gives
codim (Sk) = k(\n — m — k).
As special cases (provided the numbers shown are between 0 and n),
{ (
codim (Sx) = n — m + 1,
dim (#j) = m — 1,
codim (S2) = 2(w - m + 2), codim (ASX) = m - » J- 1.
dim (S2) = 2m — n - 4, dim (S^ = 2n — m — 1,
codim (5 2 ) = 2(m - n ^ 2),
dim (5,) = 3» - 2» - 4.
815 290
HASSLEK WHITNEY
7. Structure of the Sk Take any point p where / i s of rank p\ then p eSr_ ,q = f1(p) e L J. I t is easy to • , ym) m a y be chosen near p see t h a t coordinate systems (x1, • • • , xn) and (y1 and f(p) respectively so t h a t / h a s the form (7.1)
yl = x* i
k
Let y\ denote dy jdx .
(i£
p),
yj = ftix1, • • • ,;
(J > P)-
The Jacobian matrix of (7.1) is /
0
J"
J'
Vp+x
<+1
J'
J =
ii
m
Vn
I is the unit matrix of p rows and columns, and 0 is a matrix of zeros. Moreover, t h e elements of J " and of J ' vanish at p. The yk occurring in J' are a set of hf = (n — p)(m — p) real functions in E" near p. Now assume t h a t / i s locally generic. Then f1 is crosswise to L*, and as a result, it is easy t o see t h a t the gradients of these functions are independent a t p. We may choose functions z' ( ^ p < j ^ n ) which vanish a t p and whose gradients are independent of the above gradients; now these y\ and z} may be chosen as new coordinates in En near p, with origin at p. Note t h a t (7.2)
rank (J) = rank (J') + p.
Hence the set <S,_ ( p + i ) is given (near p) by setting all (k + 1 )-rowed determinants of J' equal to zero. This is a set of algebraic equations of degree k + 1 in h of the new coordinates. For an example, see §23 below.
8. The null spaces N(p) The null space N(p) of / a t p is the set of all vectors v mapped into 0 by df(p). Clearly [n — m + k ifn^m, forp eSk, dim (N(p)) •• (8.1) ifn< For p € 8k, N(p) can be related in various ways to Sk near p, and more generally to the sets S,*-i> , S1 near p. This will be used to split Sk into several sets.
9. The differentials dfr Given / , we have f{p)
e LT, and since ddkf = dk+1f,
df' = d(df, •••,d'f)=
(d*f, •••,
d'+Y).
r
Thus knowing the value of f a t p, the possible values for the tangent vectors dfr(p) • v are severely restricted; only the last term d'^fip) • v is arbitrary. There are also restrictions due to the symmetry of higher differentials. Thus, P(p) = df{p) e V-, hence df\p) • v e i 1 , [dP(p) • v] • u e Vm, and [dp{p) -v]-u=
d2f(p) • («, v) = d*f(p) • (v, u) = [dfi(p) • u] • v.
SINGULARITIES OF MAPPINGS OF EUCLIDEAN SPACES
291
Suppose we know the point fr+l{p) e LT+1. Then we also know the dkf(p) (k
10. The mapping/2 into L2 In the expression (2.4) for Lr, if we drop out the last terms, we obtain any L , k < r. Hence to each point of LT corresponds a definite point of each Lk, k
codim (Lrp) = codim {L\),
(10.1)
a n d / r is transversal to the IS if f1 is crosswise to the Lxp. Supposing t h a t / 1 is crosswise to the L*, we now consider, for each point p, the relation of the null space N(p) to the sets Sk which contain p. To give the relation of a vector v to Sk (p e Sk) is to give the relation of dp^(p) • v to L\_h; hence a given position of N(p) relative to the Sk corresponds to a property of dfa)(p), i.e., a property in L2 (see §9). By §§6 and 8, (10.2)
dim (Sj) + dim (N(p)) = n
if n ^ w;
hence we may expect, at most of the points p eSv that N(p) and the tangent plane P(Slt p) have only the zero vector in common. More generally, (10.3)
dim (N(p)) = codim (Sk) + (k — l)(n — m + k)
if n ^ m.
Then h=6im[N(p)nP(Sk,p)] might be 0, or > 0 . To say that/!(^) = q e Lxp and/ 1 is crosswise to i j at q is to say that P(p) e L2f but/ 2 (p) avoids a certain subset of L%p (of codimension >w). Let L*2 be the part of L2 in no such subset; set L*2 = L*2 n L2. The different values of h above correspond to different subsets of L*2k, and give a splitting (10.4)
e = e , . U ^ " ' ' ' .
Set (10.5)
flw
= (P)-\L:\h);
then Sk = Sk0 U Skl U • • • .
Now Sk h is the subset of Sk where, if we consider / in Sk alone, / has deficiency h. The Skh are called Sh(Sk) in Thom, [4] (provided that n I> m). We have similar splittings for n < in.
HASSLEB. W H I T N E Y
292
Now consider p e S2, assuming n ^> m. We must consider the relation of N(p) not only t o P(S2, p), but also in relation to Bx near p. This is a more complex situation; see §23 below. The possible relationships of N{p) to the structure of S1 near p is reflected in a splitting of the L*22 h near/(p), and this gives a corresponding splitting of the S2 h- We shall not give names t o the new sets here. I n a similar manner, each Sk h is split into subsets, on considering t h e possible relationships of N{p) to the St (I <1 k) near p e Sk. Note t h a t S0 (the set of regular points) is not split. Also, since (using coordinate systems) the possible relationships of the N(p) to the Sk are expressible by algebraic equations, and hence the new sets in L*2 are algebraic varieties a n d hence manifold collections, we see t h a t the new sets in E" are manifold collections. There is an open subset of S' where N(p) is in the most general position; t h e rest of S' forms a manifold collection of dimension less t h a n t h a t of Sv 11. Further splitting of the Sk I n L , we have the subset L*3 corresponding to mappings / such t h a t f2 is crosswise to the sets in L*2. We have also sets L*3 c: L*3, corresponding to the Lxf (see §10); they also correspond to the L*2 in L2. B u t these sets L*2 have been split u p : this gives a splitting of the L*3. We shall split these further. The Sk in E" have been split into subsets, forming a manifold collection. At any point p, we have considered the relation of N(p) t o t h e Sk; we now consider it also in relation t o the new sets. The various possible relationships correspond to facts about df2, and hence to subsets of L*3 (see §9); this gives t h e desired further splitting in L*3. Through (J 3 )" 1 , we find a further splitting of the Sk. As in the last section, the first new manifolds we obtain are of dimension one less t h a n t h a t of the previous new ones at most, and hence of dimension two less t h a n that of S1 a t most. REMARK. Though we have described the manner of splitting in L*2 and in L*3 through a discussion of the function / , t h e definitions of t h e sets in L2 and L3 are clearly intrinsic; they depend on the properties of the £ r alone. We next consider the relation of the N(p) to the new sets, giving a splitting in L** and hence a new splitting of the Sk, the largest dimension of a new manifold being three less t h a n t h a t ofS1 at most, etc. The process must stop after t h e new manifolds are of dimension at most zero. Hence the l a r g e s t / r and Lr we need use are those for which r — 1 = dim (Sj). I n L*r, we can make fr crosswise to all manifolds obtained, and this gives us the desired locally generic / ; see also §13. The final requirements on / employ dr+1f. Thus (see §6) the requirements on / to be locally generic involve derivatives through the order a t most 3
{
m -hi
(n ^
TO),
2» — m + 1
(n <^
TO).
12. The singular sets Sf Suppose first t h a t » ^ m. In a neighborhood U <= ^ of a point p of S j , we m a y choose a set Vjip'), • • • . vm_l(p') of independent vectors orienting P(SV p'); choose
SINGULARITIES OF MAPPINGS OF EUCLIDEAN SPACES
293
also v^ip'), • • • , vn(p') orienting N(p') (p' e U); see (8.1). For most points p' e U, the whole set vy(p'), • • • , vn(p') is independent, and an orientation of En is determined. I n some parts of U', one orientation of En m a y be determined, and in other parts, t h e opposite orientation: these parts are divided by the set Sl 1 U »Sj 2 U Hence (12.1)
dim (,?!!) = dim (S,) -
1 == m - 2
(n ^ TO).
We shall write S'p for t h e manifold S j x. For p e Sf\ N(p) U P(Slt p) = Q(^) is of dimension 1; it m a y point from p into P(SV p) in either side of P(Sf\ p); or at exceptional points, it may lie in P(S(^, p). This exceptional set is a manifold in Sf\ of dimension m — 3; it is the inverse image u n d e r / 3 of a certain set in L3. At these points p, N(p) fl P(<S'12), p) = Q'(p) is of dimension 1 instead of 0; we call this set S l l v o r '^i3)- At a point of S^\ Q'{p) may point into PfSj 2 *, p) in either side of P{Sf\ p), or be tangent t o t h e latter; thus we find S(^\ etc. W e have (12.2)
dim (S^) = m — i
(1
These singular sets (inverse images of sets in L ) were found by Roger [3], and appear i n T h o m [4] with t h e notation S^Sj), <S1(<S'1(<S'1)), etc. Note t h a t the last one corresponds to a set in Lm; this gives actual occurrences of y(n,TO)in (11.1), for n ^> TO. This gives t h e total splitting of Sv if n <^ 4. For m ^> n, codim (Sj) = TO — n + 1 a n d dim (N(p)) = l ( y e S j ; hence iV(p) will he in P(SV p) (for locally generic/) in a manifold Sl x = i S ^ of codimension m — n -\- 1 in /S^, and hence of dimension 3n — 2m — 2. For p e Sj 2 ', iV^(j>) lies in P{SV p)\ it will he in P(S^\ p) at each point of a manifold S111 = Sf* of codimension m — re + 1 in S^\ etc. Thus (12.3)
codim (Sf) = i(m — n + 1) if present (1 <1 i fg m 2> n).
The definition of the (Sr<11> shows t h a t / is locally one-one with nonvanishing Jacobian in each S^ — S^+1); b u t t h e image has a cusp manifold in t h e next set 5<; +1) .
13. On the classification of singularities As before, if we cut out a certain subset of IS, r = y(n,TO),of codimension >n, corresponding t o mappings not crosswise t o t h e sets we have found in Z * r _ 1 , we have left L*r; a n y mapping / 0 : Rn -^-Em is arbitrarily near a mapping / with f(p) e L*r (p 6 R), and any s u c h / i s locally generic. Morevoer, fT is automatically crosswise to the sets of the splitting of L*r. The various sets of t h e splitting in L*r define t h e different types of singularities for locally generic / . Note t h a t each set A of the splitting in L* r _ 1 (r = y(n,TO))corresponds to a set B in LT which is not split further; b u t p a r t of B m a y be cut out when we reduce IS to L*r, and what is left m a y be composed of several connected pieces, which m a y correspond to different types of singularities. This happens for instance in t h e case m = 1, w > 1; see §16.
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WHITNEY
14. Encasing of singularities m
Given/ : R" —> E , the s-dimensional encasing of/ is a mapping F : Rn+' —*• E"^' denned as follows. Write En+> = En X E', Em+' = Em X E>, and (14.1)
F(p, q) = (f(p), q)
(p eE",qe
E>).
To each generic singular point of / corresponds a generic singular point of F; the latter is the s-dimensional encasing of the former. Examples will be given below. Clearly (14.2)
rank {F, (p, q)) = rank (/, p) + s;
hence, for each q e E", (14.3)
Sk(F) n (En xq) = Sk(f) X q, n
oodim (Sk(F)) = codim (£,(/)). n
Clearly NF(p, q) lies in the space V of E always; hence the relation of null spaces to the Sk is the same for F as for/, and we see step by step that the Sk are split into subsets in the same way for each. Finally, at any p, lying in a certain singular set, f (for some r) is crosswise to the corresponding subset in LT{n, m)\ hence Fr is crosswise to the corresponding set in LT(n + s, m -j- s), showing that F is locally generic. 15. Increasing n Given / : Rn -+ Em, with n ^ m, define F : R n + 1 -> Em near a point p e Sj as follows. Since dim [df(p) • Vn] = m — 1, we may choose a vector w0 in Vm not in df{p) • V". (There are essentially two choices.) Set (15.1)
F(p,t)=ffr)
+ t*w0.
n
With a coordinate system in E , dF df
dF
For p near p and t ^ 0, the last vector is independent of the former; hence rank (F, (p', t)) = rank (/, p) -f 1 = m = v, and (p', t) is not a singular point of F. For t = 0, NF(p', 0) = Nf(p') + V1; the relation of these spaces to the S^ for F is the same as that of the N(p') to the S^l) for/. Thus 5 1 ( i ) for/becomes part of Stli) for F at < = 0. The above discussion fails for the Sk,k^>. 2. For instance, S2 may be present for mappings E* —>- 2J4, but not for mappings Eb -> ^J4 (see §4). II. EXAMPLES OF SINGXTLAMTIES
16. Critical points of real functions (See M. Morse, [1] and [2]). For En-^-E1, we let E1 be the set of real numbers. Here, L1 is simply the set of real linear functions (i.e., covectors) in Vn; it is of dimension n, and L\ consists of 0 only. Since codim (S^ = codim (Zj) = n, iS^ consists of isolated points.
SINGHLABITIES OF MAPPINGS OF EUCLIDEAN SPACES
295
With a Cartesian coordinate system in E" and hence in £A, the components of fl(p) = df(p) are (16.1)
J KP = ( — , • • • , —n ) . P(p)
'
'
W
dx J
Now (compare §9)
dp/dx* = (dyid&dx*, •••, d2fldx"dx*), and if / is locally generic, these vectors in L1 are independent for i = 1, • • • , n. That is, the matrix of second partial derivatives is of rank n. With proper choice of coordinates,/near p may be written in the form (for some k) (16.2)
y = (x1)2 + • • • + (x*)2 — (xk+1)2
(xn)2.
For n = 1, y = ±(a^) 2 . (If we allow coordinate systems reversing orientation, we may write y = (a;1)2.) If we now increase n as in §15, choosing w0 e E1 sometimes positive and sometimes negative, we obtain the above critical points. Note that i j 2 is L2 except for all points where the determinant \d2fjdxidxi\ = 0. The different parts of L*z correspond to critical points of different index, i.e., with different numbers of minus signs in (16.2). 17. The case n = m = 2 For E1 —> E1, a typical singularity is given by y = x2. If we encase this (§14), we obtain (17.1)
y1 = x\
y2 = (x2)2;
this has a singular set St on the line x2 = 0. Consider the mapping defined by (17.2)
yl = x1,
y2 = xxx2 — (x2f.
The Jacobian is J = dy2Jdx2 = x1 — 3(x2)2, which vanishes on the curve Sv defined by x1 = 3(x2)2. With unit vectors ev e2 along the axes, the image of any vector v under df is
w*. *•> • <*x+*,)=(*!£+»° g^ 1 %+v2 £ ) • Along St, this equals (v1, v1x2); hence N(p) consists of all vectors oe2 for p e Sv Except for p = (0, 0), N(p) is not tangent to S^, hence we are not in »S^2) here. But we have tangency at (0, 0). (The image u n d e r / of Sx is a curve with a cusp point at the origin.) Also, N(p) is crossing Sx here (the mapping p is crosswise to the corresponding set), as is easily seen; hence the origin is in Sf\ For fuller details, see [13].
18. The case m = 2, n ^ 3 If we increase n as in §15, we find singular curves Sv with isolated points <S(12) on them, as in §17. We find typical mappings, first for a point p of St — yS^2), with n = 3. By (17.1), the images of all vectors at p He in the ^-direction; hence we may
HASSLER
296
WHITNEY
choose w0 = te2 in (15.1); t is replaced by x3. Inserting the sign ^ in (17.1), this gives (18.1)
y1 = x\
y2=
± (z*)« ± (a;3)2,
in St ~
Sf\
Similarly, using (17.2), (18.2)
y1 = x1,
y2 = xW - (a;2)3 ± (ar»)2
in Sf>.
Continuing this process gives, for typical singularities in the general case, (18.3)
yl = x\
y2=±
(18.4)
1
2
1
y = a; ,
(a-2)2 ± • • • ± (xn)2 1 2
2 3
mS1-
3 2
Sf\
n 2
y = a; * — (a; ) ± (x ) ± • • • ± (z )
in SfK
19. The case n ]> m = 3 Here, we have singularities of types Sj, Sf\ S^. Typical examples are, for n = 3, (19.1) (19.2) (19.3)
y1 = a;1, 1
y2 = a;2,
1
2
y = x, t/1 = x1,
2
y = a; , y2 = x2,
y3 = (x3)2 3
in ^ -
5
3 3
y = a^ar — (x )
S?\ in S f - S[3\
y 3 = a^x3 + a;2(x3)2 — (x3)*4
inSf'.
3
In the last example, the Jacobian is dif/dx , and St is the surface Sx :
20. The case m = 2n — l Here, Sr consists of isolated points. By [9], a typical singularity is given by (20.1)
(y<=xi \y' =
(i= 1, . . . , n - 1), 1
rt'-^
(j = n, n + 1, • • • , 2ra — 1).
For instance, for n = 2, (20.2)
y1 = x1,
y2 = a^a;2,
y> = (x2)2.
21. The case m = 2w — 2 If we take the singularity above for £">_1 —*-E2n~3, and encase it as in §14, we find a typical singularity in Sx for E" —»- E2n~2. We have extra variables, say x° and y°, and the extra equation y° = x°; S1 is the a^-axis. For n ^> 3, (12.3) shows that 5<12) is void; for n = 2, we are back to the case n = m = 2.
SINGULARITIES OF MAPPINGS OF EUCLIDEAN SPACES
297
22. Some other cases with m> n The lowest case not considered is the case (n, m) = (4, 5). Here, (22.1)
dim (SJ = 2,
dim {S^)
= 0
(n = 4, m = 5).
In general, by (12.3), 5(x2> appears only if 2w appears only if 3»i n is also for the case n = 6,TO= 7.
23. The case w = w = 4 In this case, Sx m a y contain sets (S^21, S^3) and S^\ but no £ j 2- We may now also have isolated points belonging to <S2. We examine these more closely. As in §7, we m a y choose coordinates in the two spaces so that, near the origin j)0, we have y1 = x1, y2 = x2. Now the matrix J' in §7 is
yl
yU
4
4 !
2/3
y\\
(23.1)
We may also (§7) use these y\ as new coordinates near p0: (23.2)
X2 = y\,
Xi = yl
X 3 = y\,
X4 =
y\.
The parts of S1 and of S2 near p0 are
(23.3)
(23.4)
St:
; X1 \
X21 | = X X X 4 - X 2 X 3 = 0.
| X3
X4 '
S 2 : X 1 = X 2 = X 3 = X 4 = 0.
Thus, in these coordinates, S j is a quadratic cone, with vertex a t pQ, j)0 being the single point of S2. Let T be the tangent cone oiS1 at p0; it consists of all vectors v such t h a t p0 -r v e Si (and hence p0 4- on e j ^ , all a). We now ask what relation N(p0) (of dimension 2) can have to Sv If we look for vectors in both N(p0) and T, we find a quadratic relation; there m a y be no common vectors ^ 0 , or there may be common vectors in one or in two distinct directions. There can also be more special relations (for instance, if N(p0) is the (X 1 , X 2 )-plane, then N{p0) c T). We give two examples: (23.5)
y1 = x1,
y2 = y2,
y3 = XVJ? 4- x2x* 4- x 3 * 4 ,
and ((a) y* = xW - i ¥ (23.6)
4- i{x3)2 -
((b) yi = x 1 * 4 4- x2x3 -
i(x 3 ) 2 -
i(x*)2, £(x 4 ) 2 .
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HASSLER
WHITNEY
We find a t once t h a t S1 is given in the two cases by (a)
(x 1 ) 2 + (x 2 ) 2 -
(x 3 ) 2 -
(x 4 ) 2 = 0,
(b)
(x 1 ) 2 -
(x 2 ) 2 + (x 3 ) 2 -
(a;4)2 = 0.
I n both cases, N(p0) consists of the vectors in the (x3, x 4 )-plane. Jn (a), N(p0) n T has no vector ^ 0 ; in (b), it contains those vectors with Vs = i'i> 4 . We thus have two distinct types of singularities, which might be called S2a, S2b. 24. The case n = m = 5 Here, S2 consists of curves. W e shall illustrate a special singularity a t an isolated point of a curve of S2. Define/ by y* = x* (i = 1,2, 3),
y* = x 2 x 4 -j- (a:4)2 -f x 4 ^ 5 ,
(24.1) y5 = x V - x 4 * 5 + K* 4 ) 2 + K* 5 ) 2 + fx^x 4 ) 2 . TheniS 2 is the x x -axis (near the origin p0), and forp e *S2, 2V(^)) consists of the vectors in the (x 4 , x^-plane. Note t h a t S2 is not tangent to N(p); hence / maps S2 into a smooth curve. The singular points are given by y\yl — y\y\ = 0, i.e., (x2 + 2x 4 + ar^x 3 + x 4 + x 5 ) — x 4 (x 4 + x 5 + x*x4) = 0. Let ^'(x 1 ) be t h a t part of S' with x 1 fixed; it is a quadratic cone. Let T(x1) be the corresponding cone of vectors, as in §23, and Nix1), the null space at (x1, 0, • • • , 0). A simple computation shows t h a t N(x1)T n (x1) has no non-zero vectors if x 1 < 0, and has two independent directions if x 1 > 0; for x 1 = 0, it is the 1-dimensional space of all vectors a(e 4 — e 5 ). The origin is a new type of singular point. 25. The case n = 5, m = 4 Here we have no S2, but there can be points of S.^ 2 . Note t h a t dim (<Sj) = 3, dim {N(p)) = 2(p eSj. Consider the m a p p i n g / defined by y' = x* (i = 1, 2, 3), and (25.1)
y* = x*x4 -j- x 2 * 5 + x V x 5 + i ( x 4 ) 3 + ^(x 5 ) 3 .
Then JSJ is given by y\ = y\ = 0, t h a t is, by x 1 + x V + i ( x 4 ) 2 = x 2 + x 3 x 4 + ^(x 5 ) 2 = 0, and N(p) consists of all vectors in t h e (x 4 , x 5 )-plane (p eS^). Note t h a t a t the origin p0, the tangent 3-plane to S1 is t h e (x3, x 4 , x^-plane, which contains N(p0). This is a phenomenon which cannot be removed by arbitrarily small alterations of/. To see this, note t h a t the components of the differentials of y\ and y\ are
dyiW
y*»
2/24
2/34
yU y«
dyi,
vU
2/25
2/35
y\h
2/55
824 SINGULARITIES OF MAPPINGS OF EUCLIDEAN SPACES
299
To have the (x*, a^-plane tangent to Slt we must have dy\(P) • e< = dy\(p) • e,- = 0 , i = 4, 5. These equations give y\i = y\s = y\b = o> occurring at the origin p0 in the present example. Note that y\i = **> y « = &> yU = ^> and rfy*4 J*
^45
d
yU
=
0 0 0 1 0 0 0 1 0 0 0 0 0 0 1
Letting p move in Sv this shows that the mapping (y| 4 , y\h, y| 5 ) is crosswise to (0, 0, 0) in 3-space. Now take any sufficiently good 3-approximation / ' to / . I t is easy to see that new coordinates £', y* close to the old may be found so that yi = £{(i = 1, 2, 3). We will have Sx f o r / ' close to Sx for/, and J* for/' close to J* for/. Hence there will be a solution of y\± — y| 5 = y| 5 = 0, with crosswiseness, as required. (The crosswiseness can of course be expressed in L3.) Note that the fact that the second differential is symmetric (y| 4 = y\b) plays an essential role here.
III.
GENERAL DEFINITIONS
26. Local equivalence of mappings Take/j : B"—>-E™ (i = 1, 2), and t a k e ^ e Rv We say/ 2 a,tp2 is equivalent tofx at p j if we can write (26.1)
/ 2 = rpf^ near p2, an
where <j> : £7!J —>- JSJ & V '• &T ~*• E™ a r e smooth homeomorphisms of rank n and m respectively. Let 91* denote arithmetic s-space (of all ordered sets of s real numbers). If we take E\ = 91", E% = %m, we have introduced coordinates into E\ and E? through
2/'= ^(x\ ••• ,xn)
(i = 1, • • • , m).
We noted in §1 that if/ : i?" —>- i?™ is regular at p, then / is equivalent at p to the mapping (1.1). If p is a singular point of/, we would like to choose coordinates in some way to give a particularly simple representation (26.2) for/. Such a choice we call a normal form for the singular point. For instance, for/ : E1 -+E1, if/ has a minimum at p and d2f(p) =fi 0, an equivalent mapping is given by y = x2. Various normal forms were given in Part II.
27. Local stability We say / : R" —>• Em is stable at p if there is a neighborhood U of p with the following property. Take any neighborhood V of p, U' <= U. Then for any sufficiently good s-approximation/' to / in U, for some s, there is a point p e V such t h a t / ' at p' is equivalent to / at p.
300
HASSLER WHITNEY
Clearly the mapping (1.1) is stable at all points (use good 1-approximations). So is the mapping y = x2 at x = 0, using 2-approximations; but if we used only 1-approximations, we could obtain a new mapping with y constant in some small interval, and local equivalence would then fail. One form of the principal conjecture is that any locally generic mapping (denned through the discussion of Part I) is stable at all points. We give a strengthened form in the next section.
28. Topologies in function space. Let S be the set of all pairs (/, R), where / : R" —> Em is co-smooth. We sha.ll give rough descriptions of two topologies i n S , the normal topology and the stability topology, turnings into topological spacesS n a n d S , respectively. InS„, we say that (/', R') is near (/, R) if R' is near R (with a simple definition), and for some large r, / ' is a good r-approximation to / in R n R'. In S,, we say (/', R') is near (/, R) if we can make (26.1) hold, with the following requirements. There are open sets Rv R[ near R, R' respectively,
29. General problems For each R c E", we have a subset
and
f2:R%-+Em
be mappings such that, with y defined by (11.1), Wn-m)(Pi) =
ft.
flin-m)(P2) = 9z-
SINGULARITIES
OF MAPPINGS OF EUCLIDEAN
SPACES
301
Then
1. M. MORSE, Relations between the critical points of a real function of n independent variables, Trans. Amer. Math. S o c , 27 (1925), pp. 345-396. 2. M. MORSE, The critical points of a function of n variables, ibid., 33 (1931), pp. 72-91. 3. F . ROGER, Sur les varietes critiques . . . , C. R. Acad. Sci., Paris. 208 (1939), pp. 29-31. 4. R. THOM, Les singularites des applications diffirentiables, Seminaire Bourbaki, Paris, May, 1956. 5. R. THOM, Les singularites des applications diffirentiables, Annates de l'lnstitut Fourier, VI (1956), pp. 43-87. 6. R. THOM, Vn lemme sur les applications differentiables, Boletin de la Sociedad Matematica Mexicana, 1, ser. 2 (1956), pp. 59-71. 7. A. \V. TUCKER, Branched and folded coverings. Bull. Amer. Math. S o c , 42 (1930), pp. 859-862. 8. H. W H I T N E Y , Differentiate manifolds, Ann. of Math. 37 (1936), pp. 645-680. 9. H. W H I T N E Y , The general type of singularity of a set of In — 1 smooth functions of n variables, Duke Math. J., 10 (1943), pp. 161-172. 10. H . W H I T N E Y , The self-intersections of a smooth n-manifold in 2n-space, Ann. of Math., 45 (1944), pp. 220-246. 11. H. W H I T N E Y , The singularities of a smooth n-manifold in (2n — l)-space, ibid., pp. 247293. 12. H . W H I T N E Y , On the extension of differentiable functions. Bull. Amer. Math. S o c , 50 (1944), pp. 76-81. 13. H . W H I T N E Y , On singularities of mappings of Euclidean spaces, I, Mappings of the plane into the plane, Ann. of Math., 62 (1955), pp. 374-410. 14. N. Z. WOLFSOHN, On differentiable maps of Euclidean n-space into Euclidean m-space, Harvard thesis, 1952. See the abstract in Bull. Amer. Math. S o c , 81 (1955), p. 171.
827 ANNALS OF MATHEMATICS
Vol. 66, No. 3, November, 1957 © 1957 The Johns Hopkins University Press. Reprinted with permission.
ELEMENTARY STRUCTURE OF REAL ALGEBRAIC VARIETIES B Y HASSLER
WHITNEY
(Received April 4, 1957)
1. Introduction A real (or complex) algebraic variety V is a point set in real n-space R" (or complex n-spaee C") which is the set of common zeros of a set of polynomials. The general properties of a real V as a point set have not been the subject of much study recently (but see for instance [2], [3] and [4]); attention has turned more to the complex case, the complex projective case, and especially the abstract algebraic theory. Facts about the real case are sometimes needed in the applications; proofs are commonly very difficult to locate. Our first object is to give a proof involving no algebraic theory that through a certain splitting process, V may be expressed as a union of "partial algebraic manifolds". (Part of the proof is standard.) To prove the finiteness of the process, the Hilbert basis theorem must be used. The principal concept used is the "rank" of an ideal at a point; it has immediate geometric-analytic meaning. The splitting procedure works equally well in the complex case; in fact, in the case of real or complex analytic varieties. Each partial manifold Mi may have limit points in later Mj ; this is unavoidable. Also, Mi need not be connected; but it can have at most a finite number of topological components. We prove this by showing that for any subvariety V of V (V may be void), V — V has at most a finite number of topological components. (This conjecture was communicated to me by D. Mostow.) The methods used in the splitting process are used here also, together with some standard algebraic material. Use is made of the relation of V to the smallest complex variety V* containing V; the irreducible components of V and of V* correspond, as does the notion of rank at points of V. The structure of the splitting of Theorem 1 can be seen with the help of the irreducible components of V or V*. The more obvious splitting, by dimension, is given also; it is not of the same elementary character as the other. Examples are given to show some of the peculiar properties of real algebraic varieties. We do not enter into the more difficult and important questions about local structure; the possibility of triangulation of V is a theorem in this direction. 2. Definitions The following definitions hold both for the real and the complex case. The 1 See [6], Appendix t o C h a p t e r IV. Some revision in t h e proof is necessary. For instance lemma on p . 125 is used in the real domain on p . 128, where it does not apply. T h u s a circle in t h e plane is a real algebraic v a r i e t y , while its projection into a line is a closed line segm e n t , which is not an algebraic v a r i e t y .
545
828 546
HASSLER WHITNEY
(polynomial) ideal 1(Q) of a point set Q d Rn (or Q c C") is the set of poly. nomials which vanish in Q; if / and g are in I(Q), so i s / + g, and so is <j>f for any polynomial #. Any set S of polynomials defines a variety V — V(S) (§1)- s generates an ideal / , consisting of the linear combinations of elements of S with polynomial coefficients, and / (Z I ( V ( J S ) ) . The differential df(p) of a function / at a point p is a covector, that is, a linear function of vectors v. The definition, geometric or through the use of a coordinate system, is
df(p)-v = l i m ^ I [f(p + to) - f(p)] = *ZiVidJM.
(2.1)
t
dXi
The rank rnkp(S) of a set S of functions at a point p is the number of independent differentials dfi{p), • • • , df,{p), the /< being in S; if S is an ideal, the set of all covectors df(p) (/ e S) forms a vector space, whose dimension is rnk(jS). Clearly rnkp((S) is also the number of independent differentials from any set of functions generating S; see (10.6). The rank rnk p (Q) of a point set Q at p e Q is rnk p (I(Q)); the rank rnk(Q) of Q is the largest value of rnk p (Q) for p e Q. An algebraic partial manifold M is a point set, associated with a number p, with the following property. Take any p e M. Then there exists a set of polynomials / i , • • • ,f?,of rank p at p, and a neighborhood U of p, such that M n U is the set of zeros in U of these fi. The number n — p is the dimension of the partial manifold. Note that M need not be closed or connected, and that any open subset of M is also a partial algebraic manifold. In the complex case, if M is of (complex) dimension r, it may be considered, topologically, as a real manifold, of real dimension 2r. 3. The theorems The first two theorems are stated for the real case, but will be proved first in the complex case. n THEOHEM 1. Let V d R bea real algebraic variety, and let Mi be the set of points p e V where the rank of V is its maximum. Then Mi is an algebraic partial manifold of dimension n — mk(V), and Vi = V — Mi is void oris a proper algebraic subvariely of V. We may now apply this splitting process to Vi if it is not void, giving Vi = V2UM2,
etc.
The first part of the following theorem is an immediate consequence of the Hilbert basis theorem. THEOHEM 2. This process comes to an end after a finite number of steps, giving (3.1)
V = MiuM2u
•••uM,,
each Mi being an algebraic partial manifold in V, and the Mt being disjoint. We have (3.2)
s ^ 2 n - 1.
Note that each point set (3.3)
Vi = Mt+1 u • • • u M.
829 REAL ALGEBRAIC VARIETIES
547
is an algebraic variety, and hence is a closed set. We say that V is expressed in this manner as a manifold collection. REMARK. Using some algebraic theory, we will give another such splitting in §§9 and 11, with the property that the manifolds are of decreasing dimension; hence there will be at most n terms. THEOREM 3. A real algebraic variety has at most a finite number of topological components. THEOREM 4. Let V be a subvariety of the real variety V; then V — V has at most a finite number of topological components. As a corollary, we have THEOREM 5. In the expression (3.1) (real case), each Mi has but a finite number of topological components. 4. The variety Vi Given the functions / i , • • • , fT, their differentials are independent at p if and only if some Jacobian determinant (using the natural coordinate system of 22") (4.1)
a(/i,---,/r) d(x X l , • • • , X\r)
dfi/dxXl • • • d/i/da*, d/V/dxx, • • • dfr/dx\r
is ;^0 at p. Set p = rnk(F). By definition of M\ and F : , / = 1(F) is of rank p at all points of Mi and of rank < p at all points of Fj ; that is, all Jacobian determinants of elements ft , • • • , fp oi I are 0 at all points of Vi, while this is not true for any point of Mi . Let S be this set of Jacobian determinants; these are polynomials. Now Vi is just the set of points where all polynomials of / and all polynomials of S vanish; hence Ft is an algebraic variety. 5. A special coordinate system We prove an elementary lemma. LEMMA 1. Let fi, • • • , f„ have independent differentials at p. Then there is a coordinate system (x\, • • • , x'n) in a neighborhood U of p such that (5.1)
x'i(q) = Uq) - Up) in U
(i = 1, • • • , p).
Choose covectors £p_..i , • • • , £ „ which, with the dft(p), form an independent set. Set Xi{q) = ^-(q — p) for i > p; then (5.2)
dx'i(q) = dfi(q)
(i ^ p),
Jx'i(q) = fc (i > p).
Hence the Jacobian of the transformation (xi, • • • , xn) —> (:r'i . • • , i'„) is ^ 0 at p, and (xi, • • • , x'n) is a coordinate system. 6. The operations $„ Take any p « M i . We may choose polynomials / i , • • • , f„ in 1(F) with inde-
830 548
HASSLBR WHITNEY
pendent differentials at p; we keep these/, fixed in this section. Given any polynomial g and any ft = (MI , • • • , nP+i), define the polynomial (6.1;
$M0 =
Choose a coordinate system (x[, • • • , x'n) in a neighborhood £7 of p by Lemma 1. Now dfi/dx'j = 53* (dfi/dxh)(dxh/dx'j), and applying the general Lagrange identity to the determinant of such elements gives d(/i, ••• ,f,,g) Hfi ,--••, f„, g) d f a i , • • • , a:,, , ay>+1) = y> fl(*i, • • • , x,, x'k) "i<"-<"/.+i d(Xnt ... ,Xlit>,x^p+l) d(x'i, ••• ,x'f,x[) Because of (5.2), the left hand side is simply dg/dxl ,ilk > p. Let T* denote the last term on the right. Then this relation and (6.1) give (6.2)
^ 7 = ^<-<^
T^g
inU
(k> p).
Let M* be the (n — p)-dimensional manifold in U defined by the vanishing of the fi ; it is the part of the (x p + i , • • • , xn)-coordinate plane in U. Given a function g and a point p' e M*, dg(p')-v = 0 for all vectors v tangent to M* at p if and only if dg(p') is dependent on the dfi(p'), that is, if and only if all ^g are 0 at p; or again, dg/dx'k = 0 at p' for k > p. We may iterate the operation, forming &&$, <&*$>$?$, etc. Let H(g) be the set of all such polynomials, and let Jv(g) be the ideal of analytic functions in U generated by H(g), consisting of all functions ^ v*(p)hi(p) w-ith hi e H(g) and 4>i analytic in U. We show that each partial derivative of g of any order with respect to the variables xf+i, • • • , xn is an element of Jv{g)By (6.2), this holds for first partial derivatives. Using induction, it is sufficient to show that if <j> is analytic in U and ft =<$„•• • $>„g, then d(<j>h)/ dxk is in Jv(g) for k > p. Differentiating <j>h and applying (6.2) with h in place of g shows this to be true. 7. The algebraic partial manifold Mi We shall complete the proof of Theorem 1. Take any p e Mi, and let / i , • • • , / , be as in §6. Choose U, the x't and M* again; we may suppose M* is connected. We need merely show that Mi n U = M*. Since Vi is closed, we may suppose Vi n U = 0; hence MinU=VnUcM* (since /< e 1(7)), and there remains to prove M* C V. Take any polynomial g e 1(F); we must prove that g = 0 in M*. Since rnk(F) = p, the differentials dfi, • • • , df„, dg are dependent throughout V; hence all
831 REAL ALGEBRAIC VARIETIES
549
8. Dimension and rank of a complex variety We give some general facts about a complex algebraic variety V, which will show what the splitting V = Vx u Mx is in the complex case; (3.2) will also follow in this case. The facts will be used to study the real case in §10. Lemmas 2 and 3 will be used in the proofs of the last theorems. Consider a complex variety V C C, with corresponding ideal of complex polynomials, which we call / = I*(F). Suppose that V is irreducible; then / is prime. Adjoint elements £i, • • • , £„ to the field C of complex numbers, with the following algebraic relations: For any polynomial /, (8.1)
/(&, •••,*«) = 0
if and only if / ( * , , • • • , ,r„) <= I.
By definition, the dimension of F is the transcendence degree of the set £i, • • • , £„ ; let this be r. We may suppose £i , • • • , %r are algebraically independent (after a change of coordinates); then each f, (i > r) satisfies an irreducible equation P< = 0 with coefficients in the field C ( | i , • • • , £;-i), and these give the full algebraic properties of the £i and hence of J. LEMMA 2. A complex irreducible variety V of dimension r is of rank ;£n — r at all its points; it has points of rank n — r. Any proper subvariety of V is of dimension
m k p ( F u V) = mk p (F)
if
ptV
-
V.
This clearly holds in the real domain also. Clearly 2= holds; we prove ^ . Choose/ e l*(V) such that/(p) = 1. If rnk p (F) = p, choose fi, • • • , fP f I*(F) so that dfi (p), • • • , dfjp) are independent. Set f'i = //, ; then /'< « I*(F u V), and df'dp) = dfi(p), so that the df'i(p) are independent. A complex algebraic variety F (also a real one) has a unique minimal expression as a union V\ u • • • u Vh of irreducible varieties; by definition, dim(F) = sup dim (F,). If the F» are all of the same dimension, we say F is of constant dimension. LEMMA 5. Lei V be of constant dimension r. Then in the splitting V = Vx u Mi , Mi is of (complex) dimension r, and dim (Vi)
M'[ = M'a-U^V-,
M*i=UM'I.
550
HASSLER WHITNEY
Each Mi is non void; for if M'[ is void, for instance, then V = V'n u V'2 u • •. VH , which would clearly contradict uniqueness. By Lemmas 2 and 4, rnk p (F) = n — r for p e Mt, and hence Mt C Mi. Since Mi is a manifold, it is easy to see that Mi = Mt ; also dim(Mi) = r. Now
u
V1^(UiVii)u(Ui<jVinVi); since V'u and Vi n F,- are of dimension
V = Wi u • • • u Wa ,
dim(Wi) = r<,
n < • • •
the Wi being of constant dimension. The Wi are clearly uniquely determined (in a minimal representation); let us set (9.2)
T(V)
= 2ri +
(- 2r«.
Now Mi is the set of simple points of Wi which are in no other W,-. That is, if Wi = W\ u M\ as in Lemma 5, then (9.3)
Mi = M\ - W2 u • • • u Wa .
For points of Wi u • • • u Wa are of rank ^ n — r2, and (see the proof of Lemma 5) the only points of Wi of rank n — rx in V are those shown. We prove (3.2) in the complex case. If dim(V) = n, then V = C", and V = Mi. Supposing dim(F) < n, (9.2) shows that (9.4)
r ( 7 ) ^ E « 2 * = 2" - 1.
Express Wi as the union of irreducible varieties; omitting any that lie in any Wj (j' > 1) gives a minimal representation (9.5)
Vi = W'{ u • • • u Wi u PF2 u • • • u Wa ,
with W" of constant dimension r" , r" < • • • < r% < n (see Lemma 5). Since Y, 2r* < 2 r i , we find r ( 7 i ) < r ( 7 ) . Therefore, with F,- as in (3.3), (9.6)
2 n - 1 ^ r(V) > T(VI) > •
> r(y._!) > 0,
and (3.2) follows. (b) Using the facts of §8, an obvious splitting procedure is as follows. Write V as in (9.1), and write Wa = W'a u M'a as in Lemma 5. Set (9.7)
Mt = M'a - Wi u • • • u Wa-i,
Vt = V - Mt .
Then Mt is an algebraic partial manifold of dimension ra = dim(F), and Vi is an algebraic variety of dimension
833 REAL ALGEBRAIC VARIETIES
551
We thus express F a s a manifold collection, using at most n — 1 algebraic partial manifolds, of decreasing dimension. 10. Real varieties as parts of complex ones Let 9te and Ex denote the real part and complex conjugate of the complex number x respectively. Now EEx = x, E(x + y) = Ex + E?/, £(xy) = (Ex) (£?/), and Six = —i Ex. For any p = (xi, • • • , x„) e C, set Ep = (Exi, • • • , &r„). For any polynomial / = 2^4x,...n„:ril ' ' • z»",
(IO.I)
/(son, • • •, x0) = e Z eAx.-.x^ 1 • • • *»n.
Define the operation T on polynomials / :
(io.2)
r/(P) = M P ) + e/(Sp)].
Then (10.1) gives (10.3)
r £ AM • • -lA
• • • ari" = E 9Mx,• • x ^ 1 • • • xx„";
either relation shows that T / is real (i.e. it takes on real values at real points), and Tf = / i f and only i f / i s real. Also, (10.4)
/ = Tf - iTiif). n
For a point set Q a C , let 6Q denote the set of points Ep, p e Q. Say Q is real-symmetric if EQ = Q. LEMMA 6. Given the real variety V CZ R", there is a unique smallest complex variety V* CZ C" containing V; it is defined by the set of polynomials l(V). Now V* is real-symmetric, and V is the real part of V*. If I = I(V) and I* = I*(V*) are the associated ideals, then I CZ I*, and I is the set of real polynomials in I*; it also consists of all Tf with f « /*, and I* is the complex ideal generated by I. The polynomials I(V) define a complex variety V*, whose real part is V; because of (10.1), V* is real-symmetric. To prove that V* is the smallest, we show that any complex polynomial / that vanishes in V also vanishes in V*. S i n c e / = 0 in V, Tf and T(if) are 0 in V and hence in V*; therefore, by (10.4), / = 0 in V*. By definition of V*, I CZ I*; since V c: V*, 1 contains all real polynomials in /*, and hence is just this set of polynomials. For any / e I*, Tf el CZ I*. This set of functions Tf is the set of real polynomials in /*, and hence is / . Because of (10.4), / generates /*. Let V*(V) denote the complex variety associated with the real variety V by Lemma 6. Let 9t(F*) denote the real part of the complex variety V*. LEMMA 7. Let V* = V*(V), and let V* , • • •, F * be the irreducible components of V*. Then the F , = SR(Fif) are the irreducible components of V, and V* = V*(F,)Define the V* and V( as above, and set F-* = V*(F,); then V'>* c Vl. Since V C V'i* u • • • uV'a* and V* is minimal, V* = V\* u • • • uV'*; hence Vi* — V* . Suppose Vn u Va were a splitting of Vi into proper sub varieties; then if V*j = V*(Fjj), we have V* = V*i u V*2 (since V* is minimal), and V*j is a
552
HASSLER WHITNEY
proper subvariety of V* , contrary to the irreducibility of 7 * . Finally, no Vi i s contained in the union of the other 7^ ; for if it were, 7 * , being minimal, would be contained in the union of the other 7 * . Let rnk* denote the rank in the complex case. LEMMA 8. With V and V* as in Lemma 6, (10.5)
rnk p (7) = rnk*(7*),
p
e V.
This is immediate from the fact that / generates /*, and the relation (10.6)
dZ*.(p)/.(p)
=
X>,(p)#.(p)
if all /,(p)
are
0.
11. The splitting processes, real case (a) Given the real variety V, set 7 * = V*(7). Split both 7 and 7 * according to Theorem 1 (for the real and complex cases respectively): (11.1) LEMMA
7 = 7,uM1(
V*=VtvMt.
9. With these notations,
(11.2)
Jf, = mMt),
V1 = « ( F ? ) .
need not equal VX ; see §12, (f). First we prove rnk(7) = rnk*(F*). By Lemma 8, ^ is true. Suppose < were true; then the points of M% would be of rank (in V*) greater than rnk(F), and hence, by (10.5), MX would contain no points of V; but then V C VX , contrary to the minimal property of V*. The first relation now follows from (10.5) and the fact that V = 3J(F*). The second relation now follows also. We now prove (3.2) in the real case. Define T(V) to be T ( V * ( V ) ) (see §9). We prove T ( 7 I ) < T(V); then (9.6) will hold, and (3.2) will follow. In (9.5) we found VX in the first splitting for V*(V) = V* = WX u • • • u Wt: REMARK. V * ( 7 I )
(11.3)
VX = W[* u Wt u • • • u Wt ,
W[* = W"* u • • • u Wi*t
the WX and W1* being of constant dimension as given there, and WX = W\* u Mi* being the first splitting of WX • By Lemma 9, Vi = ?R.(VX) c VX ; hence (11.4)
V*(F,) C VX .
We show now that (11.5)
W* C V*(7i),
* ^ 2.
Let U jWn be the expression of W* in terms of irreducible components (i ^ 2). By (11.3), W?t
Y*(Fi) = 7** u Wt u • • • u Wt,
as in §9, T ( 7 I ) < r ( 7 ) follows, and (3.2) is proved.
7** c W{*;
835 REAL ALGEBRAIC VARIETIES
553
(b) Given V, the splitting of V* = V*(F) of §9 (6) gives a corresponding splitting of V into at most n — 1 real algebraic partial manifolds of decreasing dimension, as we now show. With W* = U jWa and Wi,- = 1R(W*j) as above, set
then, as seen above, W% — V*(TF,,), and hence W* = V*(TF;). If IF* = W* u M'* is the splitting of Theorem 1, Lemma 9 shows that the splitting of Wa is Wa = W u il/',
IT' = SR(ir'*),
3 f = 9?(M'*).
Set V[ = TT, u • • • u W„_, u W,
M[ = AT' -
TF, u • • • u TF„_i ;
then V = F i u itfi , rnk (TF„) = rnk* (IF*) (see the proof of Lemma 9), and, dim (Mi) = dim (AT) = n - rnk* (TT*) = r a . Set V[* = V*(7i) = IT? u • • • u T^_! u V* ( W ) ; now V*(W) c: W'*, and hence (applying Lemma 5 to W%) dim (Fi*) < r« . We now repeat the process with Vi and F i * , splitting a manifold of dimension
554
HASSLEK WHITNEY
(d) The variety defined by the equation x2 + 2xyz + y2 = 0 contains the z-axis and two separate surfaces. Dropping out one of the latter would leave a point set which is locally like an algebraic variety but is not one in the large (e) If W is the curve x2 — y3 = 0 in the plane, then Mi(W) is the curve minus the origin, and M2(W) is the origin. If X is the cone z2 — (x2 + y2) = 0 in R* then Mi(X) is the cone minus the vertex and M2{X) is the vertex. If Y is the surface x — {y — z2)2 = 0, then Mi(Y) is the part of Y with x ?* 0, and we are left with a curve like W; hence M 2 (F) is a curve and M3(Y) is a point. Now let V be the union in R3 of a point po, and varieties like W, X and Y; then Mi = po, M2 is formed by Mi{W) (M2 = Mi(W) - X u F), and the later Mt are formed by Mt(W), Mt(X) u Mi(Y), MX{X), Ma(Y), and M3(Y) respectively. Thus 7 = 23 — 1 steps are necessary, showing that (3.2) cannot be improved for n — 3. (It clearly cannot be for any n.) (f) The irreducible polynomial f = (x2 + y2)2 — £ defines a real variety V and a complex variety V*; clearly V* = V*(F), a n d / generates both 1(F) and I*(F*). With the notations of (11.1), Vi contains (0, 0, 0) only; but V* contains (1, i, 0) for instance, so that Vt * V*(F,). 13. Proof of Theorem 3 If the theorem is false, we may suppose V is a smallest variety with an infinite number of topological components. For otherwise, we could choose a proper subvariety Vi with this property, then a proper subvariety V2 of Vi, etc., contrary to the Hilbert basis theorem. Consider the splitting V = Vi u Mi of Theorem 1; then Vi has but a finite number of topological components, and hence Mi has an infinite number which are closed sets; they are of the same dimension r. We show first that r > 0. For suppose r = 0. Then the components pi, p 2 , • • • of Mi are single points, and V is of rank n — r = n at each; hence, by Lemma 8, the pt are also topological components of V* = V*(V). Let V\{ be an irreducible component of V* containing pi ; then rnk Pi (F*j) ^ rnk Pj (F*) = n, and by Lemma 2, dim (Fj^) = 0. By Lemma 3, V\( contains p< only. But the number of irreducible components of V* is finite, a contradiction. Since r > 0, the closed topological components Mi, M'% , • • • of Mi are not single points, and we may choose a point q<> = (ai, • • • , a„) which is not equidistant from all points of M[ ; set (13.1)
g=(xi-
aif + . . . + ( * . _ an)2.
For any polynomials / i , • • • , / p (p = n — r) and any n = (m, ••• , /tp+i), let $V(/i> • • •, jQ denote the right hand side of (6.1). Let V be the variety defined by the set of all these polynomials with the /* in / = 1(F), together with the polynomials of / ; then V C V. Let Pi be a point of M< nearest qo • Then dg(j>i) vanishes for tangent vectors to M[ at go, hence dfi, • • • ,df„,dg are dependent at pf if the /* are in I (see §6), hence the$ M (/i, • • • , / p ) are 0 at p<, and thus p,- e V. Therefore V contains an infinite number of topological components.
837 REAL ALGEBRAIC
555
VARIETIES
Also g is not constant in Mi ; hence (since M[ is connected) there is a point p e Mi such that dg(p)-v 9^ 0 for some tangent vector v; there are polynomials / 1 , • • • , f„ in / with dfi(p), • • • , df„(p) independent, and now this set, with dg(p), is independent (see §6); hence some $„(fi, • • • , /„) is ^0 at p, and p is not in V. Therefore V is a proper subvariety of V, a contradiction, completing the proof. 14. Proof of Theorem 4 As above, if the theorem is not true, we may suppose V is a smallest variety such that for some subvariety V", V — V" has an infinite number of topological components. Write V = Vj u Mi as in Theorem 1, and set V = V" u Vi . Then M' = V — V has an infinite number of components; for otherwise, V would contain an infinite number of the components of V — V", and hence V — V" would have an infinite number of components, contrary to the minimality of V. Since M' C Mi, the components of M' are manifolds of some fixed dimension r. (Because of Theorem 3, all but a finite number of these have limit points in V.) Set I = 1(F), / ' = I(F'). Let h[, •••, h\ be a basis for / ' , and set (14.1)
h = ih'if
(h'y)2
+ ••• +
then h = 0 in V and h > 0 in Rn — V. We show that h is bounded in all but a finite number of the topological components of M'. For if not, then the polynomial h — 1 would vanish throughout a proper subset of each of an infinite number of components of M', and hence / , together with h — 1, would define a variety with an infinite number of components, contrary to Theorem 3. Let Mi, M2, • • • be a set of components of M', in each of which h is bounded. Choose go = (ai, • • •, a„) not in V, define g by (13.1), and set (14.2)
hk = g
T—
dxk
(k = 1, • • • , n).
— 2(xk — a,k)h
Then u = h/g is analytic inffi" — 50, and (14.3)
du 1 , . D„ — = -51 hk in R — q0. dxk
g
Since h is bounded in M,• and h = 0 in y , there is a compact subset Wi of ikf,- where w attains its maximum. Set dfi/dx^ *i(/i, •••>/,)
=
dfp/dx^ fvHi
••• ••• *
dfi/dx^^ dfjdx^ ILL
and let ^ ( / i , • • • , /„) be the same, with h^ replaced by du/dx^ ; that is, ^ is given by (6.1), with u in place of g. Then by (14.3), SvCfi, • • • > /P) =
0**,(A , • • • , / , )
in
Rn
-
go.
556
HASSLEK W H I T N E Y
Let W be defined by the set of all the polynomials*i(/i >"••>/*) with t h e / in / , together with / itself. Then, as in the last theorem, W, (Z W for i = l 2 • • • . Also for any point p in any M\ such that du(p) • v ^ 0 for some tangent vector v, we can find fi, ••• ,f„ in / with ^ ( / t , • • • , / „ ) s* 0 and hence 3V(/i, • " >/P) ^ O a t p , showing that p is not in W. Take any topological component W\ of Wj ; we show that it is a topological component of W. If W" is the topological component of W containing W'i, we show that u is constant in W" ; this will prove W" = W\ . Let JVi u • • • u Nt be the splitting of W as in (3.1). Take any topological component X of any Xi} = Wi n JVy ; we show that u is locally constant in X. It will follow that u is constant in X; hence u takes on at most a denumerable number of values in W" , and W" being connected, u is constant in W" . Take any p e l . Since If" is a topological component of W and N, is a manifold, W" contains all points of N, in some neighborhood Ui of p. Take U a Ui so that JVj n [/ is connected; then N, n U" = X.-y n [/ = X n [/. By Lemma 1, for some a, we may suppose U small enough so that there is a coordinate system (xi, • • • , xn) in U with Nj n U being the part of the (x'„+i, • •-• , x'n)-plane in U; we may also suppose that N, n U C M't. If u is not constant in iVy n U, then some du/dxt is j£ 0 at some point p' e JVy n £7, A; > a. Hence there is a vector v tangent to Nj and therefore tangent to Mi at p', with du(p')-v 9-^ 0. But as seen above, this means that p' is not in W, a contradiction. We have now proved W" = W\ . We have found a topological component W\ of W in each Mi ; but this contradicts Theorem 3, and the proof is complete. THE
I N S T I T U T E F O B ADVANCED S T U D Y BIBLIOGRAPHY
[1] W. V. D . HODGE AND D . P E D O E , Methods of algebraic geometry, volume I I , University Press, Cambridge, 1952. [2] S. LEPSCHETZ, L'Analysis situs et la g6om£trie algebrique, P a r i s , 1924 and 1950. [31 J . N A S H , Real algebraic manifolds, Ann. of M a t h . , 56 (1952), p p . 405-421. [4] O. A. O L E I N I K , Estimates of the Betti numbers of real algebraic hypersurfaces, Rec. Math. (Mat. Sbornik) N . S . , 28 (70), 1951, p p . 635-640. [5] B . L. VAN D E B W A E K D E N , Algebra, volume I I (third e d . ) , Springer, Berlin, 1955. [6] , Einfiihrung in die algebraische Geometrie, Springer, Berlin, 1939 (Dover, New York, 1945).
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HASSLER WHITNEY (23 March 1907-10 May 1989)
840
"1 he Institute foi Advanced Study
HASSLER WHITNEY (23 March 1907-10 May 1989) Hassler Whitney was born in New York City to a family with a tradition of contributions to world knowledge. His father was a state supreme court judge; his mother was an artist, who was also active in politics. A grandfather, William D. Whitney, was a linguist and Sanskrit scholar and another grandfather, Simon Newcomb, was an astronomer. A greatgrandfather surveyed the Atlantic coastline for Thomas Jefferson and a great uncle was the first to survey Mount Whitney. Whitney attended Yale University and received a baccalaureate degree in physics in 1928 and in music in 1929. He earned a Ph.D. in mathematics at Harvard University in 1932. From 1932 to 1952 he taught at Harvard. He moved to the Institute for Advanced Study in 1952, where he retired in 1977. Whitney was a topologist of great originality. His contributions were broad and could be roughly divided into the following areas: 1) Differential topology. Nineteenth-century mathematics was mainly mathematical analysis. At the turn of the century the importance of topology began to be recognized. It proceeded in two directions: (1) pointset topology, where the spaces are very general; (2) combinatorial topology, where the spaces are locally polyhedral. Differentiation does not seem to play a role. In fact, there was, as there is now, a sentiment against the calculus. A common interesting saying was: "Whenever I see a derivative it gives me nausea." 1 In such an atmosphere Whitney created differential topology, which became a most active mathematical area in recent times. 2) Cup products. A fundamental problem in combinatorial topology is to define topological invariants of manifolds. The most elementary among these are the Betti numbers, based on the boundary relation. It was Emmy Noether who observed that these can be built into an algebraic structure; the homology groups of all dimensions. A much deeper concept is the introduction of a multiplication based on intersection theory. It is remarkable that the dual of the homology groups, the cohomology groups, has most beautiful properties. Using the cup product introduced by Whitney, the direct sum of cohomology groups has a ring
1 Edgar R. Lorch, "Mathematics at Columbia during adolescence," A Century of Mathematics in America, 3: 153, American Mathematical Society, 1989. I also heard similar statements from other sources.
PROCEEDINGS OF THE AMERICAN PHILOSOPHICAL SOCIETY, VOL. 138, N O . 3, 1 9 9 4
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BIOGRAPHICAL MEMOIRS
structure and becomes the cohomology ring. This can be defined for spaces more general than manifolds. If there is a continuous mapping / : X — Y, it induces a ring homomorphism /* : H*(Y) — H*(X), where H*(X) and H*(Y) are the cohomology rings of the spaces X and Y respectively. Cohomology groups are useful in the applications of topology. They are closely related to multiple integration. Whitney wrote a book on integration which is full of original geometrical ideas but unappreciated by the analysts. From Betti numbers to the cohomology ring was a great development in topology. Whitney's cup products made a crucial contribution. 3) Sphere bundles. This is perhaps the contribution for which Whitney is best known, even the name originating from him. It is a remarkable fact that a family of spheres which is locally a topological product may not be a global product. The first invariants of such a phenomenon are the so-called Stiefel-Whitney characteristic classes. They were discovered almost simultaneously by Whitney and E. Stiefel, the latter in his thesis written under the supervision of Heinz Hopf. Stiefel restricted himself to the tangent bundle of a manifold and drew beautiful conclusions, including the theorem that a closed orientable 3-dimensional manifold is parallelizable. Whitney saw the merit of the general notion of a sphere bundle over any space. In particular, this leads to algebraic operations on bundles. The description of the characteristic classes of the sum of two sphere bundles is the important Whitney duality theorem. Whitney's original proof covers the general case of local coefficients. It was very long and was never published. (I can still remember when he showed me the proof on a snowy Sunday when I visited his home near Watertown; it was like a book. The first proof, for an important special case, was given by Wu Wen-Tsun; cf. Annals of Mathematics 49 [1948]: 641-653.) Fiber bundles have since become a fundamental notion in topology. 4) Stratified manifolds and singularities. Whitney realized that the notion of a smooth manifold is not broad enough. For example, a cube is not a smooth manifold and algebraic varieties generally allow singularities. He introduced the notion of a stratification of a manifold. Personally it is my feeling that stratified manifolds will be the main object in differential geometry. They already play an important role in McPherson's theory of preverse sheaves. 5) Miscellaneous. Whitney liked mathematical problems and was able to come up with ingenious solutions. An example was his characterization of the closed 2-cell. I think it is not unfair to say that he has worked on the four-color problem. He realized its difficulty. He was happy with the computer proof by W. Haken and K. Appel. Another result is the Graustein-Whitney theorem on the regular homotopy of closed curves in the plane, an elementary result leading to much development. In his last years he devoted much time to the mathematical education of children, arousing their interest by asking challenging questions.
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As mentioned in the beginning he had a degree in music. He was an accomplished violin and viola player. Whitney received the National Medal of Science in 1976, the Wolf Prize in 1982, and the Steele Prize of the American Mathematical Society in 1985. In an article2 on the occasion of the centennial of the American Mathematical Society Whitney gave a report on the Topology Conference in 1935 in Moscow. The article is both personal and mathematical and is full of interesting facts and anecdotes. Being an avid mountaineer himself, he recounted how he met J. Alexander and G. de Rham in the Swiss mountains. He was a member of the Geneva section of the Swiss Alpine Club. On 20 August 1989, his ashes were placed at the summit of the Swiss mountain Dent Blanche by Oscar Burlet, a mathematician and member of the Swiss Alpine Club. ELECTED
1947 SHIING-SHEN CHERN
Professor of Mathematics University of California Berkeley
2 Hassler Whitney, Moscow 1935, "Topology moving toward America," A Century of Mathematics in America, 1: 97-117. There is an error on page 117: the simple proof of the duality theorem was given by Wu Wen-Tsun, cf. text.
Wolf Prize in Mathematics, Vol. 2 (pp. 845-863) eds. S. S. Chern and F. Hirzebruch © 2001 World Scientific Publishing Co.
Bibliography
1977 1. (with J. Coates) Rummer's criterion for Hurwitz numbers, Algebraic Number Theory, Kyoto, 9-23. 2. (with J. Coates) Explicit reciprocity laws, Societe Math. France, Asterisque, 41-42, 7-17. 3. (with J. Coates) On the conjecture of Birch and Swinnerton-Dyer, Invent. Math. 39, 223-251. 1978 4. Higher explicit reciprocity laws, Ann. Math. 107, 235-254. 5. (with J. Coates) On p-adic L-functions and elliptic units, J. Aus. Math. Soc. 26, 1-26. 1980 6. On modular curves and the class group of Q((p), Invent. Math. 58, 1-35. 1982 7. (with K. Rubin) Mordell-Weil groups of elliptic curves over cyclotomic fields, in Proceedings of a Conference on Number Theory related to Fermat's Last Theorem (Birkhauser), pp. 237-254. 1983 8. (with B. Mazur) Analogies between function fields and number fields, Amer. J. Math., June, 507-521. 1984 9. (with B. Mazur) Class fields of abelian extensions of Q, Invent. Math. 76, 179-330. 1986 10. On p-adic representations of totally real fields, Ann. Math. 123, 407-456. 11. (withB. Mazur) Onp-adic analytic families of Galois representations, Compos. Math. 59, 231-261. 1988 12. On ordinary X-adic representations associated to modular forms, Invent. Math. 94, 529-573.
846
1990 13. The Iwasawa conjecture for totally real fields, Ann. Math. 131, 493-540. 14. On a conjecture of Brumer, Ann. Math. 131, 555-565. 1995 15. Modular elliptic curves and Fermat's last theorem, Ann. Math. 141, 443-551. 16. (with R. Taylor) Ring-theoretic properties of Hecke algebras, Ann. Math. 141, 553-572. 1996 17. Modular forms, elliptic curves, and Fermat's last theorem, in Proc. of Int. Congress of Mathematicians, Zurich (1994), Vol. I, pp. 243-245. 1997 18. (with C. Skinner) Ordinary representations and modular forms, in Proc. of the National Academy of Sciences, 94, 10520-10527. 1998 19. (with C. Skinner) Nearly ordinary deformations representations, preprint.
of irreducible
residual
1999 20. (with C. Skinner) Residually reducible representations and modular forms, Pub. I.H.E.S., 89, 5-126. 2000
21. Twenty years of number theory, in Mathematics: Frontiers and Perspectives, eds. V. Arnold, M. Atiyah, P. Lax and B. Mazur (AMS), pp. 329-342. 2001 22. (with C. Skinner) Buse change and a problem of Serre, Duke Math. J. 107, 15-25.
847
Modular Forms, Elliptic Curves, and Fermatrs Last Theorem ANDREW W I L E S
Mathematics Department, Princeton University Princeton. NJ 08544, USA
The equation of Fermat has undoubtedly had a far greater influence on the development of mathematics than anyone could have imagined. After 1847 most serious mathematical approaches to the problem followed the line introduced by Kummer. This approach involved a detailed analysis of the ideal class groups of cyclotomic fields. The class number formulas used in Rummer's theory are refinements of the well-known class number formula of Dirichlet of 1838. To recall a special case, if q is a prime with q = 3 mod 4 and q / 3, then the class number of Q(\/~9) is (B - A)/q where B = £ quadratic nonresidues mod q A = T. quadratic residues mod q. Such formulas can be recast in the language of Galois modules as follows. If M ~ (Qp/Zp) (x) with a Gal(Q/Q)-action we define h(M) by h(M) = # k e r : # 1 ( Q . A / ) —• II H1 {(finr, e
M).
For a general p-divisible M we need to modify the condition at p, but for the example given above we can just take Mp — (Q p /Z p ) (x) where x is the quadratic character of Q(v/—?)• Then knowing the class number of Q(y/—?) is equivalent to determining h(Mp) for all p. However, despite considerable progress on such problems, no convincing conjectures appeared that were strong enough to imply Fermat's Last Theorem. Ultimately. the solution came from a quite different source, although it did also rely in part on a generalized class number formula of the above type. We begin with a rather special but influential example from the work of Eichler (1954). Let E be the elliptic curve: y2 + y = x3 - x2 - lftr - 20. Let Np denote the number of solutions of this (affine) equation mod p. Consider the modular form f
(l-qn)2(l-qUnf
=
qU
=
V(z)2 vCUz)2 = q - 2q2 - q3 + 2qA + q5 + q6 - 2q7 + • • • Proceedings of the International Congress of Mathematicians, Zurich, Switzerland 1994 © Birkhauser Verlag, Basel, Switzerland 1995
848 244
Andrew Wiles
Then / is a modular form on T0 (11) and if we write / = Hanqn then Eichler showed that av = p — Np for each prime p ^ 11. The L-function of E is thus given by the Mellin transform of / , L(f.s) = E a „ n~". We call an elliptic curve modular if there is a modular form with this property. In the 1960s Shimura. building on the work of Eichler. showed how to associate an elliptic curve over Q to any newform of weight 2 on Ty (.V) that has rational Fourier coefficients. One therefore has a triangle Newforms of weight 2 on TQ(N) with rational Fourier coefficients
Elliptic curves over Q up to isogeny
Shimura —> r
V
'/
Compatible systems of £-adic representations where s is simply defined as the composite. Serre in special cases and Faltings in general later proved that t was injectivc. However, the fundamental question posed in [Wc], which was apparently first raised in an imprecise form by Taniyama and in a precise form by Shimura. was whether r was surjective. In other words, is every elliptic curve over Q modular? During the next 25 years, the triangle was enormously developed. The map r was studied in great generality under the name of Shimura varieties and the map s was also studied in great generality as part of the Langlands programme. One significant advance was in the analogue of s for weight one. Here the image is more naturally replaced by 2-dimensional complex representations. The construction of these was given in [D-S] and the crucial converse theorem for representations with soluble image was proved by Langlands in [L] and completed by Tunnell in [T]. However, the original problem on the surjectivity of r remained untouched. In 1985. Frey suggested a completely new approach to Fermat's Last Theorem. If p > 5 is an odd prime and ap + \P = cp were a solution to Fermat's equation then he proposed showing that the elliptic curve E : y2 = x[x — ap) (x + If) could not be modular. To exploit this idea Serre formulated a conjecture on Galois representations which applied to the Galois module E[p] of p-division points implied that this curve could not be modular. Ribet then proved Serre's conjecture in the summer of 1986 (see [R]). It remained to prove that elliptic curves over Q arc modular, or less generally that all semistablc elliptic curves over Q are modular, as those considered by Frey would necessarily be of this form. The main theorems of [Wi] are: THEOREM 1 Every semistable elliptic curve over Q is modular. THEOREM 2 (Fermat's Last Theorem) abc = 0.
If ap + hP = cp with a.b.c in Q. then
To set the stage for the proof in [Wi] one begins by replacing the problem on elliptic curves with a problem on Galois representations. Thus instead of considering the map r we consider the map .s-. Wc consider the extension of the map
849 Modular Forms. Elliptic Curves, and Fermt's Last Theorem
245
s covering forms on Li(A r ) and restrict its image to consider only £-adic representations for a single choice of i. T h e proof itself begins with the crucial observation t h a t for any elliptic curve E over Q. PE.Z is modular where p^.3 is the representation on t h e 3-division points of E. This is immediate from the theorem of Langlands and Tunnell referred to above, although we actually want to know t h a t it is also the reduction of the representation associated to a form of weight 2. T h e proof then proceeds by showing t h a t , under the hypothesis that pE_3 is irreducible, every suitable lifting to a G L T ( Z 3 ) representation is modular; in other words, it is in the image of the extended m a p s. This is part of a more general theory describing conditions under which 2-dimensional ^-adic representations that are liftings of modular mod i representations should themselves be modular. The key ingredient in proving these results is the forging of a surprising link with a certain generalized class number formula. T h e link is made using some new arguments from commutative algebra as well as an elaborate study of the properties of modular forms. The commutative algebra enters in trying to relate two rings, one arising from the theory of deformations of Galois representations and the other from the theory of modular forms. T h e theorem of Langlands and Tunnell permits one to choose a modular lifting of PE.3 and it is the adjoint of this representation to which the class number formula is attached in the manner described earlier. T h e solution to this class number problem is based on duality theorems in Galois cohomology and on a construction using Hecke rings, which was inspired by a variant of Iwasawa theory (see [Wi] and [T-W]). These arguments only work at the moment when PE.3 is irreducible. To include the other semistable curves we use a different argument involving families of elliptic curves with the same representation on their 5-division points. At the time of the congress, one step in the argument was not complete, but it was completed a few weeks afterwards. For a fuller account of the proof we refer to [Wi] and [T-W]. References [D-S]
Deligne. P. and Serre. J-P., Formes modulaires de poids 1. Ser 7. Ann. Sci. Ecole Norm. Sup. 4 (1974). 507-530. [L] Langlands. R.. Base change for GL(2). Ann. of Math Stud.. 96 (1980). Princeton University Press. Princeton, NJ. [R] Ribet, K.. On modular representations of Ga/(Q/Q) arising from modular forms. Invent. Math. 100 (1990). 431-476. [T] Tunnell. J.. Artin's conjecture for representations of octahedral type. Bull. Amer. Math. Soc. 5 (1981). 173-175. [T-W] Taylor. R. and Wiles, A.. Ring-theoretic properties of certain Hecke algebras. Ann. of Math. 142 (1995). 553-572. [We] Wreil, A., Uber die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. Math. Ann. 168 (1967). 149-156. [Wi] Wiles, A.. Modular elliptic curves and Fermat's Last Theorem. Ann. of Math. 142 (1995), 443-551.
Mathematics: Frontiers and Perspectives 2000
Twenty Years of Number Theory Andrew Wiles
We begin with three problems considered by Fermat: (1) Which prime numbers can be written as the sum of two integer squares? (2) Is there a right-angled triangle with rational length sides and area 1? (3) Do there exist solutions to the equation: xn + yn = zn for n > 3? The answer to the first question is: these are precisely the primes p which are congruent to l m o d 4 . The answer to the second question is: no. The solutions to these two problems were found by Fermat himself. The third problem of course needs no introduction. The kind of mathematics needed to solve the first question, the theory of quadratic forms, quadratic fields, quadratic residues, ... occupied number theorists until the early years of the nineteenth century. The mathematics used originally to attack the third problem, cyclotomic fields, class groups, factorization in number fields, ... occupied number theorists for another hundred years after that. Moreover, as is well-known, the mathematics eventually used to solve this problem incorporated much more from the twentieth century, modular forms, class field theory, arithmetic geometry, ... The second problem is actually older than Fermat's time and in its most general form, where 1 is replaced by an arbitrary integer n, it is still unsolved. It is in fact equivalent to a special case of one of the most challenging problems in modern number theory. One of the great attractions of number theory is precisely the way that simply stated problems such as these can generate the most profound developments in the subject. In this article we give a highly selective review of some of the principal developments in number theory in the last twenty years. At the same time we highlight some of the outstanding problems that we believe will be at center stage in years to come. Here we have chosen, from the many fascinating conjectures current in the field, those that seem to us to be both easy to state and impossible to avoid. There is nothing new or radical in our choices, This is the text of a lecture of the same title given at The International Congress at Berlin on August 19, 1998. ©2000 International Mathematical Union
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ANDREW WILES
but we hope that this personal selection conveys the beauty and excitement of this field of mathematics. In order to help towards this goal we have consciously written in a nontechnical style. The price for this is that we do not always state results in their most general or precise form. However this precision can easily be obtained from the references cited. Most of the problems we have selected fall into two categories; problems about solving equations, usually in the rational numbers, and problems about Galois extensions of the rationals. Advances in each of these areas has affected the development of the other, so we will not present them separately. We begin with polynomial equations in one variable which are related to the most basic structures of algebraic number theory. Let f(x) £ Z[x] be an irreducible polynomial with integer coefficients. A root a of such a polynomial generates a number field F — Q(a). The factorization of this polynomial, modulo primes, determines much of the arithmetic of this number field. In particular the Dedekind zeta function can be defined as the product, up to finitely many factors,
CF(S) ~ n n a - p~{des s where f(x) = Y[ fi(x)modp
/,)s _i
(s e c )
)
is a factorization into irreducible factors. This
i=i
generalizes the Riemann zeta function for the number field Q. The principal analytic properties of these functions (with one well-known exception!) were proved for the field Q by Riemann in his celebrated 1859 paper and completed by Hecke in about 1918 for the general case. These include the analytic continuation to the whole complex plane with the exception of a simple pole at s = 1, and the functional equation. A further fundamental result on these functions is the formula for the residue at s = 1 in terms of the units of F , the class number, and certain other invariants. This is known as Dirichlet's class number formula. If F is Galois over Q, i.e., all the roots of f(x) = 0 are polynomials in a, then one obtains a factorization of the zeta function analogous to the decomposition of the regular representation of a group. If I is the set of irreducible representations of Gal(F/Q) then the factorization has the form
(F(s) = Y[as,p)desppel
The fundamental problems now are to extend the analytic properties of the zeta function to these factors £(s, p). The functional equation is easily verified but the holomorphy and the analogue of the formula of Dirichlet are not. The first of these is known as the Artin conjecture (1923). (See [1]). CONJECTURE
complex plane.
(Artin). If p =£ 1 then ((s,p)
is holomorphic in the whole
TWENTY YEARS OF NUMBER THEORY
331
The second unsolved problem, giving an explicit formula for C(l)P) is the conjecture of Stark (1970), see [2]. The first major progress on the Artin conjecture which goes beyond elementary considerations from group theory was a theorem of Langlands (1980). This proved the Artin conjecture in many cases where p is two dimensional and the image is a solvable group. The result was extended by Tunnell (1981) to cover the remaining cases where the image of p is solvable and two dimensional. THEOREM ([3], [4]). If p is two dimensional, irreducible and has solvable image then ((s,p) is holomorphic. Moreover these results are actually valid with any ground field in place of Q. Langlands' proof depends on the theory of base change which was initiated by Saito and Shintani and further developed by Langlands. Behind it lies the whole edifice of the Langlands programme. Unfortunately no comparable progress has been made on Stark's conjecture. A polynomial in two variables defines a curve, and in 1983 Faltings proved a celebrated conjecture of Mordell, which states the following. T H E O R E M ([5]). Any curve of genus > 1 defined over a number field has only finitely many points in the given number field.
Faltings' proof also solved a number of other well-known problems in number theory, some of which are more technical but more useful than the final theorem. The final step in the proof was the Shafarevich conjecture, namely that there are finitely many isomorphism classes of curves with good reduction outside a given finite set S. That this implied the Mordell conjecture was known from earlier work of Parshin, the idea being to associate curves to rational points via certain controlled coverings. Via Torelli's theorem, this translates into a finiteness statement about abelian varieties. Faltings approaches such finiteness results by using the moduli space of principally polarized abelian varieties together with classical results about Galois extensions. (One result which he proves on the way, and which we will refer to again later, is the isogeny conjecture which states that an abelian variety is determined up to isogeny by its associated Galois representations). The proof of Mordell's conjecture was a masterpiece of technique and ingenuity. Note however that it is not effective and no effective proof has yet been given. So even theoretically we cannot determine all the points on a general curve over a given number field, as the proof gives no bound on their size. Subsequently Vojta [6], (1989) gave a new proof of the theorem using more classical methods, in particular using diophantine approximation. Faltings [7], [8] (1990) then saw how to extend this for the first time to prove finiteness theorems for certain higher dimensional varieties. What are the proper generalizations of this? A lot of work has already gone into it. For curves, the condition that the genus is > 1 is equivalent to the curve being "hyperbolic". So this motivated the following conjecture of Lang.
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ANDREW WILES
C O N J E C T U R E (Lang). / / X is hyperbolic and defined over a number field, then X should have finitely many points in the given number field.
We will not try to specify which definition of hyperbolic is most appropriate here but refer to [9] for more details. Some varieties do have infinitely many points even over Q. In particular this can occur for abelian varieties and does occur for F 1 . So a second more algebraic way of formulating a conjecture extending Faltings' theorem is the following. (Lang). If {ft : A\ —> X} is the set of nonconstant maps to X from abelian varieties or P 1 , then X — Ufi(Ai) should have finitely many points in any given number field. CONJECTURE
Faltings' higher dimensional theorem gives a proof of this second conjecture for subvarieties of abelian varieties ([8]). In principle then this conjecture suggests that there are three basic problems, first to solve this conjecture, second to determine the nonconstant maps from abelian varieties or projective lines to a given variety X and third to determine when an abelian variety has infinitely many points over a given number field. As an example of the subtlety of the second problem, as well as of a recent counterexample to a famous conjecture of Euler from 1769, we have the following result of Elkies [10], (1988) (the result was also found independently by Zagier but not published). THEOREM.
There are infinitely many protectively distinct integer solu-
tions of xi + y4 + z4= For example: 26824404
+
153656394
uA. +
187967604
=
206156734.
The theorem is proved by finding an elliptic curve on this surface with infinitely many rational points on it. This is a particular instance of the third problem which we mentioned of determining when an abelian variety has infinitely many rational points over a given number field. This problem is addressed by the Birch and Swinnerton-Dyer conjecture and its generalizations. The original conjecture concerned only elliptic curves, and we will now restrict to this case and also restrict our number field to be the rationals. Here too there has been significant progress in the last twenty years. First let us recall some of the arithmetic of elliptic curves. An elliptic curve over Q can be thought of as an equation of the form l 2 y = cubic' where the cubic is assumed to have rational coefficients and nonzero discriminant A. An example is given here of the curve: y2 = x 3 + 17.
TWENTY YEARS OF NUMBER THEORY
333
Group S t r u c t u r e P + Q + R = 0
A straight line will meet the elliptic curve in three points, at least if we include the point at oc. As Newton observed, if the line has rational coefficients and if two of these points are rational, then so is the third. More profound is the fact that if we make the sum of the points on the line equal to zero and (say) make the point at oc into the 'zero', then the rational points form an abelian group. It is not hard to see that by taking the limiting case of a tangent at a point we can often get infinitely many points this way by starting with a single rational solution. In 1922, Mordell proved however that this group is finitely generated. The question is: how can we decide whether this group is finite or infinite? This is the question answered conjecturally by the Birch and Swinnerton-Dyer conjecture. To describe this we first need the L-function of an elliptic curve E : y2 = 3 x + ax + b. This is defined in a similar way to the zeta function of a number field. Let Np and ap be given by Np = # { ( z , y ) : y2 = x3 + ax + 6(modp)},
ap = p - Np.
Then the L-function of E is, up to finitely many factors, the infinite product,
UE.s)
app
+ pl - 2 s \ - l
pfA
The theorem of Mordell states that the group of rational points on E has the form E(Q) = Z 9 ® (finite group). The conjecture states the following: CONJECTURE (Birch and Swinnerton-Dyer, [11], 1962). L(E,s) has a zero at s = 1 of order exactly g. In particular E has infinitely many points over Q if and only if L(E, 1) = 0.
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ANDREW WILES
There's one immediate problem with this conjecture; the Euler product does not converge at s = 1. However it had been conjectured already in the 1930s by Hasse that the L-series should indeed have an analytic continuation. More usefully we have the following modularity conjecture. C O N J E C T U R E (Taniyama-Shimura; Weil [12]. 1967). There is a holomorphic modular form f of weight 2 such that L(E,s) = L(f,s).
This conjecture would imply the analytic continuation and functional equation for L(E,s). What it says precisely is the following. If L(E,s) = T,ann~s, then writing / = T,anqn with q = e2iriz we know a priori that / is a holomorphic function on the upper half plane. To be a modular form of weight 2 it must also satisfy, for some integer N, the transformation property (1)
f(^L£\=(cz
+ d)kf(z)
for all
|Y°
£) G SL2(Z) : c = Omodivj.
This should be true with TV equal to the conductor of E. There is also a technical condition that / is holomorphic as z approaches a cusp. To see this condition of modularity in a more striking way, we give an example of Eichler (1954) which was the first nonspecial instance of this conjecture. (Some rather special examples had been given by Hecke much earlier but these all corresponded to a very special class of elliptic curves, namely those with complex multiplication). Consider the elliptic curve given by y2 + y = x 3 — x2. We can list the number Np of solutions mod p of this equation, and also ap = p — Np. (
solutions of
~j
p
Np = I # y2 + y = x 3 - x 2 modp >
2 3 5 7
4 4 4 9
ap = p-
Np
-2 -1 1 -2
On the other hand consider the following infinite product. oc
qU(l-qn)\l-qnn)2
f= n=l
- q - 2q2 - q3 + 2q4 + q5 + q6 - 2q7 + • • • .
This formal expansion is, when we set q = e2v2Z, the Fourier expansion of a modular form of weight 2. Now when we consider the Fourier coefficients we see that the coefficient of qp is the same as the ap in the table above. This is true for all primes other than 11, as was proved by Eichler. Thus the coefficients of this form determine the number of solutions of the original
TWENTY YEARS OF NUMBER THEORY
335
equation modp for all primes p ^ 11. The modularity conjecture asserts that to every elliptic curve there exists such a form which counts the modp solutions in this way. There is another way of writing the conjecture due to Shimura which we also need. There is a curve defined over the rationals denoted Xo(N) which over C is given as the quotient of the upper half plane by the group of matrices in (1). The conjecture asserts that every elliptic curve over Q should be a quotient of an XQ(N) for some choice of N (TV depending on the elliptic curve). Now let us return to the conjecture of Birch and Swinnerton-Dyer. If we assume the modularity conjecture for the elliptic curve E, then the Birch Swinnerton-Dyer conjecture makes sense. In 1952. Heegner showed that one could sometimes hope to construct rational points on E by taking certain canonical points in XQ(N) defined over number fields, taking their images on E, and then taking a trace so that the resulting points are defined over Q. The problem with this is that the trace might only give points of finite order. In 1983, Gross and Zagier proved the following theorem. T H E O R E M (Gross-Zagier [13] 1983). Assume that E is modular. If L(E,l) = 0 and L'(E,1) ^ 0 then Heegner's construction gives a point of infinite order on E over the rationals.
This is a remarkable result. It means that one can prove the existence of infinitely many rational solutions to an equation by testing the vanishing of a certain analytic function. It even yields an effective solution to Gauss' class number problem by an earlier result of Goldfeld. In 1988, Kolyvagin built on this work of Gross and Zagier. (Kolyvagin [14]). Assume that E is modular. Then (i) If L(E, 1 ) ^ 0 then #£?(Q) is finite. (ii) If L(E, 1) = 0 but L'(E, 1 ) ^ 0 then g = 1, i.e., E(Q) = Z® ('finite group;. THEOREM
The first part had been known in the special cases alluded to before by a different method due to Coates and Wiles since 1976. However, in view of the modularity conjecture, Kolyvagin's method should give the most general case. Kolyvagin's method involved introducing some new algebraic techniques, in particular the notion of an Euler system. It also required some analytic results on the existence of twists of the L-function of E which either have no zero or a zero to the first order. Such problems have been widely studied in the last ten years. The key remaining problem then is to prove unconditionally that if L(E, 1) = 0, then there exists a point of infinite order. The result of Gross and Zagier makes the additional hypothesis that the derivative does not vanish. And of course all this presupposes the modularity conjecture. Meanwhile, in 1985, Gerhard Frey produced an extraordinary surprise by suggesting that the modularity conjecture should imply Fermat's Last Theorem.
336
ANDREW WILES
The idea was this. Suppose that a? + b" = ,
p >5
were a solution to Fermat's equation. Then consider the elliptic curve E:y
= x(x-ap)
(x + P).
This elliptic curve has discriminant (apbpcp)2. a perfect p-th power and this fact affects the arithmetic of £ as follows. Consider the p-torsion points on E, p times
(2)
E\p\ = {Pe
E(C) :> + P+---
+ P = 0}.
It is known from results of Serre and Mazur that the coordinates (x, y) of these P's would generate an extension L of Q with Gal(L/Q) ~ GL2(Z/pZ) and, because the discriminant is a p-th power, L is almost unramified. Experience shows that such extensions should be hard to find, but why does the modularity conjecture intervene? To understand this we need to recall a conjecture of Serre. CONJECTURE (Serre [15]). (i) If p : Gal{K/Q) ~ GL2(Z/pZ), — — 1. then there exists a modular form f — YlanQn such that ae = tracep(Frobl)
for primes
£»
detp(c)
0.
(ii) Predicts the minimum level one can choose for an f in (i). Here c denotes the restriction of any complex conjugation to K C C, and Frob£ is a certain element in the Galois group called the Frobenius at I. In 1986 ([16]) Ribet proved that if (i) is true for a given p then (ii) is also true for p. The consequence of this was that Frey's suggestion was vindicated: the modularity conjecture did indeed imply Fermat's Last Theorem. The proof is that the p constructed from the Frey curve satisfies (i) by the modularity hypothesis, and hence also (ii) by Ribet's theorem. Since p was almost unramified, we can compute the minimum level for / and it is 2 - but there are no modular forms of level 2! The modularity conjecture, at least for many elliptic curves, is now proved. Indeed we have the theorems (Wiles, see [17], [18], 1994) THEOREM.
Every semistable elliptic curve is modular.
THEOREM.
Fermat's Last Theorem is true.
The class of elliptic curves covered by the first result is enough to include all Frey curves and so implies F.L.T. The proof of the modularity conjecture depends on a study of the relationship between Galois representations and modular forms. Consider the triangle,
TWENTY YEARS OF NUMBER THEORY
Modular forms
s
(an <E Z and with Euler product)
337
Elliptic Curves ^
(up to isogeny)
Galois Representations E[m] for all m
It is known that to certain modular forms, namely those with Fourier coefficients in Z and associated to Euler products, we can associate an elliptic curve. This goes back to Eichler and Shimura. The conjecture says that this association is bijective. So the naive approach is to try to count these sets. This we cannot do, even today. Instead we associate Galois representations to the elliptic curves. As in (2) we consider E[m] and the Galois extension K =
338
ANDREW WILES
Here is a good point to record also some of the many outstanding number-theoretic problems about Galois extensions. Two of the most important have been mentioned already, the Artin conjecture and Serre's conjecture. There is more generally the problem of nonsoluble base change which is embedded in Langlands programme. To these we could add the inverse Galois problem, which is the problem of showing that any finite group is a Galois group, as well as the modularity of more general Galois representations. We turn now to a small selection of other results in number theory of the last twenty years that are not directly related to these topics. The first is a theorem of Iwaniec (1987) and Duke (1988). A breakthrough in technique by Iwaniec, breaking the convexity bound in the estimation of certain Lfunctions, led to the solution of an old problem on ternary quadratic forms by Duke, see [20]. The question is: when can we solve (for x,y.z) the equation 0
,
2
2
ax + by + cz = n, where a, b and c are arbitrary fixed positive integers? (Duke originally considered the case a ~ b = c = 1. This was extended with the help of Schulze-Pillot.) T H E O R E M (Iwaniec, Duke, Schulze-Pillot). For sufficiently large squarefree integers n, the equation above has a solution provided it has a solution mod(8abc)2.
The proof is not effective as we do not know how large n has to be for the result to hold. However it would be effective with GRH (the generalized Riemann hypothesis). The idea of the proof, as often in the theory of quadratic forms, is to consider the theta function Q(z\
=
Y^
e2m(an\+bnl+cnl)z
_
This is a modular form of weight 3/2 and the proof depends on a powerful study of the analytic properties of such forms. A different result on quadratic forms from a quite different source is the following theorem of Margulis, conjectured by Oppenheim (1929) for the case n > 5 and by Davenport for the case n > 3. T H E O R E M (Margulis [21], 1987). Let Q = axx\ + h anx\\ at G M, and Q indefinite. Suppose some cn/dj ^ Q, n > 3. Then for any E > 0, 3 x e Z " - {0} such that |Q(x)| < e.
The methods here are drawn from ergodic theory and are quite different from the techniques used for the other results that I have described. Note that the form does not have to be diagonal for this theorem to hold. One change in number theory over the last twenty years is that it has become an applied subject. (Perhaps one should say it has gone back to being an applied subject as it was more than two thousand years ago.)
TWENTY YEARS OF NUMBER THEORY
339
Public key cryptography has changed the way we look at secrecy and codes. The RSA system of Rivest, Shamir and Adleman ([22], 1978) depends on the practical difficulty of factoring a number. A base or central agency picks two large primes p and q and forms their product n = pq. It also picks c. and then computes d. such that cd = 1 mod(p— 1) (q — 1). Then d and n are made public but not c, p or q. A field agent can now send a message to base as follows. First the message is made numerical, say (3571...). (Strictly speaking the message should be broken up into pieces, each of which is small compared with n.) The message is encoded as x = (3571... )d mod n. To decode the message n we just compute x c mod n. However c is known only to the base, and it seems that the knowledge of c.p or q is extremely hard to get from the knowledge of d and n. This has not however been rigorously proved, and it is a central problem in the field to determine whether it is or is not hard to obtain c. The use of such codes has rejuvenated elementary number theory. The seventeenth century problems of generating large primes, primality testing and factoring now pose new and precise problems. How fast can algorithms for answering these questions be? The question of primality testing is almost solved (and in practice is solved). Remarkably a complete theoretical solution would follow from GRH. The other two are not theoretically solved, although the first of the problems seems much easier in practice than the third. This brings us again to the Riemann hypothesis, undoubtedly the most important problem in number theory and perhaps in all of mathematics. We will not discuss it here. Instead we will discuss one development that was. at least in part, inspired by the Riemann hypothesis. In 1941, WTeil proved an analogue of the Riemann hypothesis for curves over finite fields. He achieved this in part by describing the zeta function as the characteristic polynomial of Frobenius in a certain representation. Already in 1942, Weil was suggesting an analogous construction for number fields that might be related to zeroes of the zeta function. It turned out that it was not related to the zeroes of the classical zeta function but instead to the zeroes of the p-adic zeta function introduced in 1964 by Kubota and Leopoldt. The construction is as follows. Let An be the p-part of the class group of the field Q(Cpn-t-0Then one defines a vector space V by V = (limAi) ®QP. One picks a generator 7 of Gal(Q(£p^)/(Cp)) and observes that 7 acts on V. This 7 is the analogue of the Frobenius. The following theorem was originally conjectured by Iwasawa. T H E O R E M (Mazur-Wiles, [23], 1982). The characteristic polynomial of 7 — I on V~ is given (after a change of variable) by the p-adic zeta function
ofQ(Q)+-
340
ANDREW WILES
Here V~ is the piece of V on which complex conjugation acts by —1. This p-adic zeta function is related to the classical zeta function in that the values at the negative integers are essentially the same. Although this theorem has no obvious bearing on the Riemann hypothesis, it does give a deeper understanding of the class number formula of Dirichlet and the Birch-Tate conjecture. There has been a great deal of work on p-adic Lfunctions and p-adic methods generally in the last twenty years, and we should cite the results of Rubin [24] and Kato (see [25]) giving partial generalizations of this theorem to elliptic curves, thus yielding new results on the Tate-Shafarevich group. (The proofs use the technique of Euler systems introduced by Kolyvagin who, together with Rubin, obtained the first results on this problem.) Obviously whole areas of number theory, such as transcendence theory, have been omitted from this survey. Rather than try to be exhaustive we prefer to go back to an earlier result. As we observed before, one of the great charms of number theory is that one often finds that classical problems represent deep and sophisticated challenges to modern mathematics. We give an example of this by returning to the second of the problems mentioned in the introduction and asking more generally whether, given an integer n, one can find a right-angled triangle with rational length sides and area n. This problem goes back a thousand years, and even more if one considers special cases. The case n — 1 was, as we indicated, solved by Fermat. However the general case is unsolved. One can prove easily that n is the area of a right-angled triangle < = >
En(Q)
is infinite
where En is the elliptic curve y2 = x3 — n2x. But the Birch and SwinnertonDyer conjecture gives a criterion for the right-hand side to hold. Indeed we should have En{Q) is infinite
<= >
L(En, 1) = 0.
The forward direction is known by a theorem of Coates and Wiles (1976). an earlier special case of the theorem of Kolyvagin. The converse is unknown and represents one of the great challenges of modern number theory. A particularly striking way of stating this conjecture was given by Tunnell ([26]) using results of Waldspurger and Shimura. He observed that (for odd square-free integers n) L(En. 1) = 0 < = > # { x , y,z£Z:
2x2 + y2 + 8z 2 = n}
= 2 x # { x , y, z G Z : 2x2 + y2 + 32z 2 = n}. This gives a completely elementary, but as yet unproved, criterion for n to be the area of a right-angled triangle. (A similar criterion holds if n is even). Even if the criterion does hold, one still has the problem of finding such a triangle. In the case where L(E, 1) = 0 and L'(E, 1) ^ 0, the theorem of
TWENTY YEARS OF NUMBER THEORY
341
Gross and Zagier gives a viable method. Using this, Zagier showed that the simplest right-angled triangle with rational sides and with area 157 is 224403517704336969924557513090674863160948472041 8912332268928859588025535178967163570016480830
6803298487826435051217540 411340519227716149383203
411340519227716149383203 21666555693714761309610
References 1. Heilbronn, H., Zeta functions and L-functions in "Algebraic Number Theory". Ed. Cassels, J. and Frohlich, A., Academic Press, London (1967). 2. Tate, J., Les conjectures de Stark sur les fonctions L d'Artin en s = 0. Progress in Mathematics, vol. 4 7 , Birkhauser, Boston (1984). 3. Langlands, R. Base change for GL(2). Ann. of Math. Studies, 96, Princeton University Press, Princeton (1985). 4. Tunnell, J., Artin's conjecture for representations of octahedral type. Bull. A.M.S. 5, (1981) pp. 173-175. 5. Faltings, G. Endlichkeitssatze fur abelsche Varietaten iiber Zahlkdrpern. Invent. Math. 73, No.3, (1983) pp. 549-576. 6. Vojta, P., Siegel's theorem in the compact case. Ann. M a t h 133, (1991) pp. 509-548. 7. Faltings, G., Diophantine approximation on abelian varieties. Ann. M a t h 1 3 3 , (1991) pp. 549-576. 8. Faltings, G., The general case of S. Lang's conjecture. Perspect. M a t h . vol. 1 5 , Academic Press, Boston (1994). 9. Lang, S., Number Theory HI. Encyclopaedia of Mathematical Sciences, vol.60, Springer-Verlag, Heidelberg (1991). 10. Elkies, N., On A4 + B4 + C4 = D4. Math. Comput. 5 1 , No. 184, (1988) pp. 825-835. 11. Birch, B., Swinnerton-Dyer, H., Notes on elliptic curves I, II. J. Reine Angew. Math 212, (1963) pp. 7-25. 12. Weil, A., Uber die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. Math Ann. 168, (1967) pp. 149-156. 13. Gross, B., Zagier. D., Heegner points and derivatives of L-series. Invent. Math. 84, (1986) pp. 225-320. 14. Kolyvagin, V.A., Finiteness of E(Q) and \\[(E,Q) for a class of Weil curves. Math. USSR, Izv. 32. (1989) pp. 523-541. _ 15. Serre, J.-P., Sur les representations modulaires de degre 2 de Gal(Q/Q). Duke Math. J. 54, No. 1, (1987) pp. 179-230. _ 16. Ribet, K., On modular representations of Gal(Q/Q) arising from modular forms. Inv. Math 100, (1990) pp. 431-476. 17. Wiles, A., Modular elliptic curves and Fermat's last theorem. Ann. M a t h 142, (1995) pp. 443-551. 18. Taylor, R., Wiles, A., Ring-Theoretic properties of certain Hecke algebras. Ann. Math. 1 4 1 , (1995) pp. 553-572.
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19. Darmon, H., Hyperelliptic curves, Hilbert modular forms and Fermat's last theorem. Preprint. 20. Duke, W., Some old problems and new results about quadratic forms. Notices of A.M.S., 44(2), (1997) pp. 191-197. 21. Margulis, G., Discrete subgroups and ergodic theory in Number Theory, Trace Formulas and Discrete groups. Ed. Albert, Bombieri, Goldfeld, Academic Press, San Diego (1988). 22. Adleman, L., Rivest, R., Shamir, A., A method for obtaining digital signatures and public-key crypto systems. Commun. ACM 2 1 , (1978) pp. 120-126. 23. Mazur, B., Wiles, A., Class fields of abelian extensions ofQ. Invent. Math. 76, (1984) pp. 179-330. 24. Rubin, K., The "main" conjectures of Iwasawa theory for imaginary quadratic fields. Invent. Math. 103, (1991) pp. 25-68. 25. Scholl, A., An introduction to Kato's Euler systems, In Galois representations in arithmetic algebraic geometry. L.M.S. lecture notes, vol. 254, Cambridge University Press, Cambridge, UK (1998). 26. Tunnell, J., A classical diophantine problem and modular forms of weight 3/2. Invent. Math. 72, (1983) pp. 323-334. D E P A R T M E N T O F M A T H E M A T I C S , P R I N C E T O N U N I V E R S I T Y AND T H E , I N S T I T U T E F O R ADVANCED STUDY, PRINCETON,
NJ
Wolf Prize in Mathematics, Vol. 2 (pp. 865-918) eds. S. S. Chem and F. Hirzebruch © 2001 World Scientific Publishing Co.
Curriculum Vitae
Born
Kobryn (USSR), 24 April 1899
Education 1918-1920 University of Kiev, Russia 1921-1924 University of Rome, Italy
Degrees 1923 Dottore in Matematica, Univ. of Rome, Italy Positions held 1925-27 Fellow International Education Board ("Rockefeller Fellow") 1927-29 Johnston Scholar 1929-32 Associate in Mathematics, Johns Hopkins University 1932-37 Associate Professor, Johns Hopkins University 1937-45 Professor, Johns Hopkins University 1945-47 Research Professor, University of Illinois 1947-59 Professor of Mathematics, Harvard University 1959-69 Dwight Parkes Robinson Professor of Math., Harvard University 1969-86 Professor Emeritus, Harvard University
1931 (summer) Visiting lecturer, University of Chicago 1935 Visiting lecturer, University of Moscow 1934-35 and 1960 Visiting member, Institute for Advanced Study, Princeton 1940-41 Visiting lecturer, Harvard University 1945 Visiting professor, University of Sao Paulo, Brazil 1948, 1968, 1969 Visiting professor, University of California at Berkeley 1953, 1963, 1967, 1970, 1972 Visiting professor, Rome, Italy 1961, 1967, 1970 Member — Institute des Hautes Etudes Scientifiques (Paris — Bures-sur-Yvette) 1963 Visiting professor, University of Mexico 1965 Visiting professor, University of Pisa 1966, 1968, 1970, 1971 Visiting professor, Purdue University 1969 Visiting professor, University of Tel-Aviv 1972 Visiting professor, University of Cambridge 1973 Visiting professor, Ecole Polytechnique, Paris
1943 Elected member of the National Academy of Sciences 1944 Cole prize in Algebra, Amer. Math. Soc.
866 1947 Colloquium lecturer of the Amer. Math. Soc. 1965 National Medal of Science 1969, 1970 President of the American Mathematical Society
Member Academia Scientias Brazil Academia Scientias Peru Academia Nazionale dei Lincei (Italy) London Mathematical Society (Honorary) Istituto Lombardo delle Scienze (Honorary) American Mathematical Society National Academy of Sciences American Philosophical Society American Academy of Arts and Sciences
Honorary Degrees 1959 Dr. Sc, College of Holy Cross, Worcester, Massachusetts 1965 Dr. Sc, Brandeis University 1973 Dr. S c , Purdue University 1981 Dr. Sc, Harvard University Died
4 July 1986
867
Preface T h e series "Mathematicians of Our Time" embraces, at least in principle, the works of living mathematicians; therefore, the term "collected works," as applied to this series stands of necessity for an open-ended entity, because the author— contrary to the old cliche that "mathematics is a young man's game"—may still be actively engaged in research and therefore continue to produce papers while the "collected" works is being printed. Thus, in my case, the bibliography of papers that appeared in the first volume did not include two papers, one of which was in course of publication and another of which was in preparation. (These two papers form, together with the last paper [89] of the list printed in the first volume, a sequence of three papers under the common title "General theory of saturation and of saturated local rings.") The paper that was then "in course of publication" appears in the second volume as the last published paper [90] in the bibliography, while the third paper of that series, which was in course of publication at the time of publication of the second volume, appears in the third volume as paper [93] of the bibliography. The bibliography in the present (final) volume includes three papers that have been written since the appearance of the third volume. These are [96], published in 1978, [97] and [97a], published in 1979. Paper [96] is the only one in this volume that does not deal with equisingularity. T h e complete set of four volumes includes all my published works, with the following exceptions (the numbers in brackets refer to the bibliography as printed in this volume): 1. Books [6,25,72,75]. 2. Lecture notes [87,92]. 3. Expository articles in fields to which I have made no original contribution myself [1,3,5,11]. All these articles deal with the foundations of set theory, and I wrote them in my early postgraduate years in Rome at the urging and with the encouragement of my teacher F. Enriques, whose primary interest at the time was in the philosophy and history of science and who was editor of a series of books entitled "Per la Storia e la Filosofia delle Matematiche." As the reader can see, the book listed under [6] was published in that series. T h e editorial preparation and the writing of introductions to each volume is entrusted to the capable hands of younger men, who are experts in the field of algebraic geometry and who at one time or another have been either my students at Harvard or have been closely associated with me in some capacity at Harvard or elsewhere. Thus, the editors of the first volume, H. Hironaka and D. Mumford, as well as M. Artin, who joined D. Mumford as editor of the xi Zariski, Oscar/Collected Works (Vol. IV, pp. xi-xviii) © The MIT Press. Reprinted with permission.
PREFACE
868 second volume, are truly leaders in the field of algebraic geometry and have studied at Harvard. B. Mazur, who joined M. Artin as editor of the third volume, is now my colleague at Harvard and as an expert topologist made an important contribution in his joint introduction with M. Artin. Of the two editors of the present fourth volume, J. Lipman studied at Harvard and is now professor at Purdue University, very active in the field of algebraic geometry, while B. Teissier, professor at the Ecole Polytechnique in Paris, has visited Harvard on various occasions. I had very close scientific contact with Teissier when I was a Visiting Professor at the Ecole Polytechnique in 1973. While all the papers printed in these collected works belong, without exception, to algebraic geometry, the reader will undoubtedly notice that beginning with the year 1937 the nature of my work underwent a radical change. It became strongly algebraic in character, both as to methods used and as to the very formulation of the problem studied (these problems, nevertheless, always have had, and never ceased to have in my mind, their origin and motivation in algebraic geometry). A few words on how this change came about may be of some interest to the reader. When I was nearing the age of 40, the circumstances that led me to this radical change of direction in my research (a change that marked the beginning of what was destined to become my chief contribution to algebraic geometry) were in part personal in character, but chiefly they had to do with the objective situation that prevailed in algebraic geometry in the 1930s. In my early studies as a student at the University of Kiev in the Ukraine, I was interested in algebra and also in number theory (by tradition, the latter subject is strongly cultivated in Russia). When I became a student of the University of Rome in 1921, algebraic geometry reigned supreme in that university. I had the great fortune of finding there on the faculty three great mathematicians, whose very names now symbolize and are identified with classical algebraic geometry: G. Castelnuovo, F. Enriques, and F. Severi. Since even within the classical framework of algebraic geometry the algebraic background was clearly in evidence, it was inevitable that I should be attracted to that field. For a long time, and in fact for almost ten years after I left Rome in 1927 for a position at the Johns Hopkins University in Baltimore, I felt quite happy with the kind of "synthetic" (an adjective dear to my Italian teachers) geometric proofs that constituted the very life stream of classical algebraic geometry (Italian style). However, even during my Roman period, my algebraic tendencies were showing and were clearly perceived by Castelnuovo, who once told me: "You are here with us but are "not one of us." This was said not in reproach but good-naturedly, for Castelnuovo himself told me time and time again that the methods of the Italian geometric school had done all they could do, had reached a dead end,
xii
PREFACE
869 and were inadequate for further progress in the field of algebraic geometry. It was with this perception of my algebraic inclination that Castelnuovo suggested to me a problem for my doctoral dissertation, which was closely related to Galois theory (see [2,12] and section 2 of the introduction by M. Artin and B. Mazur to volume III). Both Castelnuovo and Severi always spoke to me in the highest possible terms of S. Lefschetz's work in algebraic geometry, based on topology; they both were of the opinion that topological methods would play an increasingly important role in the development of algebraic geometry. Their views, very amply justified by future developments, have strongly influenced my own work for some time. This explains the topological trend in my work during the period 1929 to 1937 (see [15,16,17,20,22,27,28,29,31] and sections 5 and 7 of the introduction by M. Artin and B. Mazur to volume III). During that period I made frequent trips from Baltimore to Princeton to talk to and consult with Lefschetz, and I owe a great deal to him for his inspiring guidance and encouragement. The breakdown (or the breakthrough, depending on how one looks at it) came when I wrote my Ergebnisse monograph Algebraic Surfaces [25]. At that time (1935) modern algebra had already come to life (through the work of Emmy Noether and the important treatise of B. L. van der Waerden), but while it was being applied to some aspects of the foundations of algebraic geometry by van der Waerden, in his series of papers "Zur algebraischen Geometrie," the deeper aspects oibirational algebraic geometry (such as the problem of reduction of singularities, the properties of fundamental loci and exceptional varieties of birational transformations, questions pertaining to complete linear systems and complete "continuous" systems of curves on surfaces, and so forth) were largely, or even entirely, virgin territory as far as algebraic exploration was concerned. In my Ergebnisse monograph I tried my best to present the underlying ideas of the ingenious geometric methods and proofs with which the Italian geometers were handling these deeper aspects of the whole theory of surfaces, and in all probability I succeeded, but at a price. T h e price was my own personal loss of the geometric paradise in which I so happily had been living. I began to feel distinctly unhappy about the rigor of the original proofs I was trying to sketch (without losing in the least my admiration for the imaginative geometric spirit that permeated these proofs); I became convinced that the whole structure must be done over again by purely algebraic methods. After spending a couple of years just studying modern algebra, I had to begin somewhere, and it was not by accident that I began with the problem of local uniformization and reduction of singularities. At that time there appeared the Ergebnisse monograph Ideal-
xiii
PREFACE
870
theorie of W. Krull, emphasizing valuation theory and the concept of integral dependence and integral closure. Krull said somewhere in his monograph that the general concept of valuation (including, therefore, nondiscrete valuations and valuations of rank > 1) was not likely to have applications in algebraic geometry. On the contrary, after some trial tests (such as the valuation-theoretic analysis of the notion of infinitely near base points; see title [35]), I felt that this concept could be extremely useful for the analysis of singularities and for the problem of reduction of singularities. At the same time I noticed some promising connections between integral closure and complete linear systems; a systematic study of these latter connections later led me to the notions of normal varieties and normalization. However, I also concluded that this program could be successful only provided that much of the preparatory work be done for ground fields that are not algebraically closed. I restricted myself to characteristic zero: for a short time, the quantum j u m p top =£ 0 was beyond the range of either my intellectual curiosity or my newly acquired skills in algebra; but it did not take me too long to make that j u m p ; see for instance [48,49,50] published in 1943 to 1947. I carried out this initial program of work primarily in the four papers [37,39,40,41] published in 1939 and 1940. From then on, for more than 30 years, my work ranged over a wide variety of topics in algebraic geometry. It is not my intention here, nor is it the purpose of this preface, to brief the reader on the nature of these topics and the results obtained or the manner in which my papers can be grouped together in various categories, according to the principal topics treated. This is the task of the editors of the various volumes. I will say only a few words about the four volumes. T h e papers collected in the first volume are divided in two groups: (1) foundations, meaning primarily properties of normal varieties, linear systems, birational transformations, and so on, and (2) local uniformization and resolution of singularities. These two subdivisions correspond precisely to the twofold aim I set to myself in my first concerted attack on algebraic geometry by purely algebraic methods—an undertaking and a state of mind about which I have already said a few words earlier. As a matter of fact, of the four main papers that I mentioned earlier as being the chief fruit of my first huddle with modern algebra and its applications to algebraic geometry, exactly two [37,40] belong to "foundations," while the other two [39,41] belong to the category "resolution of singularities and local uniformization." The papers collected in the second volume are also divided in two groups: (1) theory offormal holomorphic functions on algebraic varieties (in any characteristic), meaning primarily analytic properties of an algebraic variety V, either in the
PREFACE
871 neighborhood of a point (strictly local theory) or—and this is the deeper aspect of the theory—in the neighborhood of an algebraic subvariety of V (semi-global theory); (2) linear systems, the Riemann-Roch theorem and applications (again in any characteristic), the applications being primarily to algebraic surfaces (minimal models, characterization of rational or ruled surfaces, etc.). My work on formal holomorphic functions was a natural outgrowth of my previous work on the local theory of singularities and their resolution. In the course of this previous work I developed an absorbing interest in the formal aspects of Krull's theory of local rings and their completions. In particular, I gave much thought to the possibility of extending to varieties V over arbitrary ground fields the classical notion of analytic continuation of a holomorphic function defined in the neighborhood of a point P of V. I sensed the probable existence of such an extension provided the analytic continuation were carried out along an algebraic subvariety W of V passing through P. It was wartime, and my heavy teaching load at Johns Hopkins University (18 hours a week) left me with little time for developing these ideas. Fortunately I was invited in January 1945 to spend at least one year at the University of Sao Paolo, Brazil, as exchange professor under the auspices of our Department of State. My light teaching schedule at Sao Paolo gave me the necessary leisure time to concentrate on an abstract theory of holomorphic functions. The year spent at Sao Paolo also presented me with a superlative audience consisting of one person—Andre Weil (who spent two or three years in Sao Paolo)—to whom I could speak about these ideas of mine during our frequent walks. The full theory of holomorphic functions—in the difficult case of complete (projective) varieties—was developed by me in my 1951 Transactions memoir [58]. However, the germ of this theory, in the easier case of affine varieties, appears already in my 1946 paper [49] written and published in Brazil. The key ingredient of the theory developed in this earlier paper is the concept of certain special rings, which later were named "Zariski rings," and properties of the completion of these rings. It is also this earlier Brazilian paper that led me to the discovery of a connection between the general theory of holomorphic functions and the connectedness theorem on algebraic varieties (and, in particular, the so-called principle of degeneration of Enriques). This connection was fully developed in my memoir [58] mentioned above. To a more strictly local frame of reference belong such papers as [52], [53], and [59] which deal with analytic properties of normal points of a variety. As to the third volume, it includes all my papers that have not been included in the first two volumes (other than books, lecture notes, and certain expository articles mentioned at the beginning of the preface) with one general exception: the entire set of papers that deal either with the theory of equisingularity or with
xv
PREFACE
872 the theory of saturation is included in volume IV. T h e set of these papers (all published since 1964) consists of fifteen titles in the bibliography, namely the titles [80]-[97a], excluding the titles [83], [84], [87], [92], and [96] (paper [83] appears in the third volume, while paper [84] appears in the first volume since it deals with the reduction of singularities; the titles [87] and [92] are lecture notes; paper [96] deals with the resolution of singularities of an algebraic surface). In particular, all the papers I wrote before 1937 (except the paper [26], which belongs to the theory of algebraic surfaces and was therefore included in the second volume), whether in Rome (before 1927) or at the Johns Hopkins University (on or after 1928), appear in the third volume. T h e bulk of these papers, published during the period 1928-1937, is topological in nature, as I mentioned earlier in the preface. T h e reader will find in the introduction by M. Artin and B. Mazur an illuminating discussion of these papers and of their impact on later work by other mathematicians. Their discussion includes, in particular, my papers dealing with the following three topics: (1) solvability in radicals of equations of certain plane curves; (2) the fundamental group of the residual space of plane algebraic curves; (3) the topology of the singularities of plane algebraic curves. In one of my papers on the latter topic, namely, paper [22] ("On the topology of algebroid singularities"), I have found a number of misprints and (in section 5) a proof that is incomplete. These are dealt with in an addendum immediately following the paper, which contains a list of the errata and a complete proof of the result stated in the last five lines. As I pointed out earlier, all the papers in the present volume IV (except title [96]) deal with the theory of equisingularity and also with a special case of equisingularity: the theory of saturation. T h e reader will find in the introduction of J. Lipman and B. Teissier an excellent exposition of (and I quote from that introduction), "the vision toward which the papers in this volume point." Suffice it to say that the theory of equisingularity, which I have initiated with two papers in 1965 and a third paper in 1968, under the common title "Studies in equisingularity" (titles [81], [82] and [85]) and on which I and other mathematicians continued to work up to the present time, is still largely an open field, and a definitive and convincing complete general theory of equisingularity is still not available, at least not in print. The last paper in this volume [97] develops a general theory of equisingularity. By a "general" theory, I mean one which is based on a "satisfactory" definition of equisingularity of a given r-dimensional variety V (of embedding dimension ^ r + 1, locally at each of its points; in particular, of a hypersurface V) along any irreducible subvariety W of V, of codimension > 1, i.e., such that dim W < r — 1. (The case of codimension 1 has been completely settled in my paper [82].) By a "convincing" or "satisfactory"
xvi
PREFACE
873 general theory of equisingularity, I mean one which, in the first place, is not contradicted, sooner or later, by counter-examples, and, in the second place, agrees with what one would expect from equisingularity when tested in examples against the behavior of V under a monoidal transformation centered in W. In my student days in Rome algebraic geometry was almost synonymous with the theory of algebraic surfaces. This was the topic on which my Italian teachers lectured most frequently and in which arguments and controversy were also most frequent. Old proofs were questioned, corrections were offered, and these corrections were—rightly so—questioned in their turn. At any rate, the general theory of algebraic surfaces was very much on my mind in subsequent years, as witnessed—on a expository level—by my monograph [25] on algebraic surfaces, and—on a more significant research level—by the connection which I have found exists for varieties V of any dimension between normal (respect., arithmetically normal) varieties and the property that the hypersurfaces of a sufficiently high order (respect., of all orders) cut out on V complete linear systems. With this result as a starting point and with the conviction, indelibly impressed in my mind by my Italian teachers, that the theory of algebraic surfaces is the apex of algebraic geometry, it is no wonder that as soon as I realized that further progress in the problem of resolution of singularities would probably take years and years of further effort on my part, I decided that it was time for me to come to grips with the theory of algebraic surfaces. I felt that this would be the real testing ground for the algebraic methods which I had developed earlier. In his introduction to the second part of volume II Mumford says that he believes that my research on linear systems was to me "something like a dessert" (after the arduous efforts of the previous phase of my work). Objectively, Mumford may be right, but to me, subjectively, the proposed new work on linear systems felt more like the "main course." This work was also, in part, an answer to the following challenge sounded by Castelnuovo in his 1949 introduction to the treatise "Le superficie algebriche" of Enriques: "Verra presto il continuatore dell'opera delle scuole italiana e francese il quale riesca a dare alia teoria delle superficie algebriche la perfezione che ha raggiunto la teoria delle curve algebriche?" (Note Castelnuovo's answer: "Lo spero ma ne dubito".) With this challenging question of Castelnuovo in mind, the reader will read with particular interest Mumford's analysis of my papers on linear systems and of later work done by others in the theory of algebraic surfaces. T h e reader will then realize that the theory of surfaces is still a very lively topic of research and that everything points to the likelihood that this theory will reach the degree of perfection dreamed of by Castelnuovo, except that this will not be the work of one "continuatore," but of many.
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874 In 1950 I gave a lecture at the International Congress of Mathematicians at Harvard; the title of that lecture was "The fundamental ideas of abstract algebraic geometry" [60]. This is a good illustration of how relative in nature is what we call "abstract" at a given time. Certainly that lecture was very "abstract" for that time when compared with the reality of the Italian geometric school. Because it dealt only with projective varieties, that lecture, viewed at the present time, however, after the great generalization of the subject due to Grothendieck, appears to be a very, very concrete brand of mathematics. There is no doubt that the concept of "schemes" due to Grothendieck was a sound and inevitable generalization of the older concept of "variety" and that this generalization has introduced a new dimension into the conceptual content of algebraic geometry. What is more important is that this generalization has met what seems to me to be the true test of any generalization, that is, its effectiveness in solving, or throwing new light on, old problems by generalizing the terms of the problem (for example: the Riemann-Roch theorem for varieties of any dimension; the problem of the completeness of the characteristic linear series of a complete algebraic system of curves on a surface, both in characteristic zero and especially in characteristic/? # 0; the computation of the fundamental group of an algebraic curve in characteristic p 41 0). But a mathematical theory cannot thrive indefinitely on greater and greater generality. A proper balance must ultimately be maintained between the generality and the concreteness of the structure studied, and usually this balance is restored after a period in which it was temporarily (and understandably) lost. There are signs at the present moment of the pendulum swinging back from "schemes," "motives," and so on toward concrete but difficult unsolved questions concerning the old pedestrian concept of a projective variety (and even of algebraic surfaces). T h e r e is no lack of such problems. It suffices to mention such questions as (1) criteria of rationality of higher varieties; (2) the study of cycles of codimension > 1 on any given variety; (3) even for divisors D on a variety there is the question of the behavior of the numerical function of n: dim \nD\; and finally (4) problems, such as reduction of singularities or the behavior of the zeta function, which are still unsolved when the ground field is of characteristic p =t 0 (and is respectively algebraically closed or a finite field). These are new tasks that face the younger generation; I wholeheartedly wish that generation good speed and success. Oscar Zariski Cambridge, Massachusetts
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Bibliography of Oscar Zanski
[1] I fondamenti della teoria degli insiemi di Cantor, Period. Mat., serie 4, vol. 4 (1924) pp. 408-437. [2] Sulle equazioni algebriche contenenti linearmente un parametro e risolubili per radicali, Atti Accad. Naz. Lincei Rend., CI. Sci. Fis. Mat. Natur., serie V, vol. 33 (1924) pp. 80-82. [3] Gli sviluppi piu recenti della teoria degli insiemi e il principio di Zermelo, Period. Mat., serie 4, vol. 5 (1925) pp. 57-80. [4] Sur le developpement d'une fonction algebroide dans un domaine contenant plusieurs points critiques, C. R. Acad. Sci., Par.is, vol. 180 (1925) pp. 1153-1156. [5] // principio de Zermelo e la funzione transfinita di Hilbert, Rend. Sem. Mat. Roma, serie 2, vol. 2 (1925) pp. 24-26. [6] R. Dedekind, Essenza e Significato dei Numeri. Continuita e Numeri Irrazionali, Traduzione dal tedesco e note storico-critiche di Oscar Zariski ("Per la Storia e la Filosofia delle Matematiche" series), Stock, Rome, 1926, 306 pp. The notes fill pp. 157-300. [7] Sugli sviluppi in serie delle funzioni algebroidi in campi contenenti piu punti critici, Atti Accad. Naz. Lincei Mem., CI. Sci. Fis. Mat. Natur., serie VI, vol. 1 (1926) pp. 481-495. [8] Sull'impossibilitd di risolvere parametricamente per radicali un'equazione algebrica f{x,y) = 0 di genere p > 6 a moduli generali, Atti Accad. Naz. Lincei Rend., CI. Sci. Fis. Mat. Natur., serie VI, vol. 3 (1926) pp. 660-666. [9] Sulla rappresentazione conforme dell'area limitata da una lemniscata sopra un cerchio, Atti Accad. Naz. Lincei Rend., CI. Sci. Fis. Mat. Natur., serie VI, vol. 4 (1926) pp. 22-25. [10] Sullo sviluppo di una funzione algebrica in un cerchio contenente piu punti critici, Atti Accad. Naz. Lincei Rend., CI. Sci. Fis. Mat. Natur., serie VI, vol. 4 (1926) pp. 109-112. xix BIBLIOGRAPHY Zariski, Oscar/Collected Works (Vol. IV, pp. xix-xxvi) © The MIT Press. Reprinted with permission.
[11] El principio de la continuidad en su desarrolo historico, Rev. Mat. Hisp.Amer., serie 2, vol. 1 (1926) pp. 161-166, 193-200, 233-240, 257260. [12] Sopra una classe di equazioni algebriche contenenti linearmente un parametro e risolubili per radicali, Rend. Circolo Mat. Palermo, vol. 50 (1926) pp. 196-218. [13] On a theorem of Severi, Amer. J. Math., vol. 50 (1928) pp. 87-92. [14] On hyperelliptic d-functions with rational characteristics, Amer. J. Math., vol. 50 (1928) pp. 315-344. [15] Sopra il teorema d'esistenza per le funzioni algebriche di due variabili, Atti Congr. Internaz. Mat. 2, Bologna, vol. 4 (1928) pp. 133-138. [16] On the problem of existence of algebraic functions of two variables possessing a given branch curve, Amer. J. Math., vol. 51 (1929) pp. 305-328. [17] On the linear connection index of the algebraic surfaces z" =f(x,y), Nat. Acad. Sci. U.S.A., vol. 15 (1929) pp. 494-501.
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[18] On the moduli of algebraic functions possessing a given monodromie group, Amer. J. Math., vol. 52 (1930) pp. 150-170. [19] On the non-existence of curves of order 8 with 16 cusps, Amer. J. Math., vol. 53 (1931) pp. 309-318. [20] On the irregularity of cyclic multiple planes, Ann. of Math., vol. 32 (1931) pp. 485-511. [21] On quadrangular 3-webs of straight lines in space, Abh. Math. Sem. Univ. Hamburg, vol. 9 (1932) pp. 79-83. [22] On the topology of algebroid singularities, Amer. J. Math., vol. 54 (1932) pp. 453-465. [23] On a theorem of Eddington, Amer. J. Math., vol. 54 (1932) pp. 466-470. [24] Parametric representation of an algebraic variety, Symposium on Algebraic Geometry, Princeton University, 1934-1935, mimeographed lectures, Princeton, 1935, pp. 1-10. [25] Algebraic Surfaces, Ergebnisse der Mathematik, vol. 3, no. 5., SpringerVerlag, Berlin, 1935, 198 pp.; second supplemented edition, with
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BIBLIOGRAPHY
appendices by S. S. Abhyankar, J. Lipman, and D. Mumford, Ergebnisse der Mathematik, vol. 61, Springer-Verlag, Berlin-HeidelbergNew York, 1971, 270 pp. [26] (with S. F. Barber) Reducible exceptional curves of the first kind, Amer. J. Math., vol. 57 (1935) pp. 119-141. [27] A topological proof of the Riemann-Roch theorem on an algebraic curve, Amer. J. Math., vol. 58 (1936) pp. 1-14. [28] On the Poincare group of rational plane curves, Amer. J. Math., vol. 58 (1936) pp. 607-619. [29] A theorem on the Poincare group of an algebraic hypersurface, Ann. of Math., vol. 38 (1937) pp. 131-141. [30] Generalized weight properties of the resultant of n + 1 polynomials in n indeterminates, Trans. Amer. Math. S o c , vol. 41 (1937) pp. 249-265. [31] The topological discriminant group of a Riemann surface of genus p, Amer. J. Math., vol. 59 (1937) pp. 335-358. [32] A remark concerning the parametric representation of an algebraic variety, Amer. J. Math., vol. 59 (1937) pp. 363-364. [33] (In Russian) Linear and continuous systems of curves on an algebraic surface, Progress of Mathematical Sciences, Moscow, vol. 3 (1937). [34] Some results in the arithmetic theory of algebraic functions of several variables, Proc. Nat. Acad. Sci. U.S.A., vol. 23 (1937) pp. 410-414. [35] Polynominal ideals defined by infinitely near base points, Amer. J. Math., vol. 60 (1938) pp. 151-204. [36] (with O. F. G. Schilling) On the linearity of pencils- of curves on algebraic surfaces, Amer. J. Math., vol. 60 (1938) pp. 320-324. [37] Some results in the arithmetic theory of algebraic varieties, Amer. J. Math., vol. 61 (1939) pp. 249-294. [38] (with H. T. Muhly) The resolution of singularities of an algebraic curve, Amer. J. Math., vol. 61 (1939) pp. 107-114. [39] The reduction of the singularities of an algebraic surface, Ann. of Math., vol. 40 (1939) pp. 639-689.
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BIBLIOGRAPHY
[40] Algebraic varieties over ground fields of characteristic zero, A m e r . J. Math., vol. 62 (1940) p p . 1 8 7 - 2 2 1 . [41] Local unifortnization
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pp. 852-896. [42] Pencils on an algebraic variety and a new proof of a theorem of
Bertini,
T r a n s . A m e r . M a t h . S o c , vol. 50 (1941) p p . 4 8 - 7 0 . [43] Normal varieties and birational correspondences,
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A n n . of Math., vol. 4 3 (1942) p p . 5 8 3 - 5 9 3 . [45] Foundations of a general theory of birational correspondences, T r a n s . A m e r . M a t h . S o c , vol. 5 3 (1943) p p . 4 9 0 - 5 4 2 . [46] The compactness of the Riemann manifold of an abstract field of algebraic functions, Bull. A m e r . M a t h . S o c , vol. 4 5 (1944) p p . 6 8 3 - 6 9 1 . [47] Reduction
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of Math., vol. 4 5 (1944) p p . 4 7 2 - 5 4 2 . [48] The theorem of Bertini on the variable singular points of a linear system of varieties, T r a n s . A m e r . M a t h . S o c , vol. 56 (1944) p p . 1 3 0 - 1 4 0 . [49] Generalized semi-local rings, S u m m a Brasiliensis M a t h e m a t i c a e , vol. 1, fasc. 8 (1946) p p . 1 6 9 - 1 9 5 . [50] The concept of a simple point of an abstract algebraic variety, T r a n s . A m e r . M a t h . S o c , vol. 62 (1947) p p . 1-52. [51] A new proof of Hilbert's Nullstellensatz,
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(1947) p p . 3 6 2 - 3 6 8 . [52] Analytical irreducibility of normal varieties, A n n . of M a t h . , vol. 4 9 (1948) pp. 352-361. [53] A simple analytical proof of a fundamental
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mations, Proc. Nat. A c a d . Sci. U.S.A., vol. 35 (1949) p p . 6 2 - 6 6 . [54] A fundamental variety,.Ann.
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la theorie des functions
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holomorphes sur une
vanete algebrique, Colloque d'Algebre et Theorie des Nombres, Paris, 1949, pp. 129-134. [56] Postulation et genre arithmetique, Colloque d'Algebre et Theorie des Nombres, Paris, 1949, pp. 115-116. [57] (with H. T. Muhly) Hilbert's characteristic function and the arithmetic genus of an algebraic variety, Trans. Amer. Math. Soc, vol. 69 (1950) pp. 78-88. [58] Theory and applications of holomorphic functions on algebraic varieties over arbitrary ground fields, Mem. Amer. Math. Soc, no. 5 (1951) pp. 1-90. [59] Sur la normalite analytique des varietes normales, Ann. Inst. Fourier (Grenoble), vol. 2 (1950) pp. 161-164. [60] The fundamental ideas of abstract algebraic geometry, Proc. Internat. Cong. Math., Cambridge, Massachusetts, 1950, pp. 77-89. [61] Complete linear systems on normal varieties and a generalization of a lemma of Enriques-Severi, Ann. of Math., vol. 55 (1952) pp. 552-592. [62] Le probleme de la reduction des singularites d'une variete algebrique, Bull. Sci. Mathematiques, vol. 78 (January-February 1954) pp. 1-10. [63] Interpretations algebrico-geometriques du quatorzieme probleme de Hilbert, Bull. Sci. Math., vol. 78 (July- August 1954) pp. 1-14. [64] Applicazioni geometriche della teoria delle valutazioni, Rend. Mat. e Appl., vol. 13, fasc. 1-2, Roma (1954) pp. 1-38. [65] (with S. Abhyankar) Splitting of valuations in extensions of local domains, Proc. Nat. Acad. Sci. U.S.A., vol. 41 (1955) pp. 84-90. [66] The connectedness theorem for birational transformations, Algebraic Geometry ar.d Topology (Symposium in honor of S. Lefschetz), edited by R. H. Fox, D. C. Spencer, and A. W. Tucker, Princeton University Press, 1955, pp. 182-188. [67] Algebraic sheaf theory (Scientific report on the second Summer Institute), Bull. Amer. Math. Soc, vol. 62 (1956) pp. 117-141. [68] (with I. S. Cohen) A fundamental inequality in the theory of extensions of valuations, Illinois J. Math., vol. 1 (1957) pp. 1-8.
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[69] Introduction to the problem of minimal models in the theory of algebraic surfaces, PubJ. Math. Soc. Japan, no. 4 (1958) pp. 1-89. [70] The problem of minimal models in the theory of algebraic surfaces, Amer. J. Math., vol. 80 (1958) pp. 146-184. [71] On Castelnuovo's criterion of rationality pa=P2 — 0ofan algebraic surface, Illinois J. Math., vol. 2 (1958) pp. 303-315. [72] (with Pierre Samuel and cooperation of I. S. Cohen) Commutative Algebra, vol. I, D. Van Nostrand Company, Princeton, N.J., 1958. [73] On the purity of the branch locus of algebraic functions, Proc. Nat. Acad. Sci. U.S.A., vol. 44 (1958) pp. 791-796. [74] Proof that any birational class of non-singular surfaces satisfies the descending chain condition, Mem. Coll. Sci., Kyoto Univ., series A, vol. 32, Mathematics no. 1 (1959) pp. 21-31. [75] (with Pierre Samuel) Commutative Algebra, vol. II, D. Van Nostrand Company, Princeton, N.J., 1960. [76] (with Peter Falb) On differentials in function fields, Amer. J. Math., vol. 83 (1961) pp. 542-556. [77] On the superabundance of the complete linear systems \nD\ (n-large) for an arbitrary divisor D on an algebraic surface, Atti del Convegno Internazionale di Geometria Algebrica tenuto a Torino, Maggio 1961, pp. 105-120. [78] La risoluzione delle singolarita delle superficie algebriche immerse, Nota I e II, Atti Accad. Naz. Lincei Rend., CI. Sci. Fis. Mat. Natur., serie VIII, vol. 31, fasc. 3-4 (Settembre-Ottobre 1961) pp. 97-102; e fasc. 5 (Novembre 1961) pp. 177-180. [79] The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface, Ann. Math., vol. 76 (1962) pp. 560-615. [80] Equisingular points on algebraic varieties, Seminari dell'Istituto Nazionale di Alta Matematica, 1962-1963, Edizioni Cremonese, Roma, 1964, pp. 164-177. [81] Studies in equisingularity I. Equivalent singularities of plane algebroid curves, Amer. J. Math., vol. 87 (1965) pp. 507-536.
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[82] Studies in equisingularity II. Equisingularity in co-dimension 1 (and characteristic zero), Amer. J. Math., vol. 87 (1965) pp. 972-1006. [83] Characterization of plane algebroid curves whose module of differentials has maximum torsion, Proc. Nat. Acad. Sci. U.S.A., vol. 56 (1966) pp. 781-786. [84] Exceptional singularities of an algebroid surface and their reduction, Atti Accad. Naz. Lincei Rend., CI. Sci. Fis. Mat. Natur., serie VIII, vol. 43, fasc. 3-4 (Settembre-Ottobre 1967) pp. 135-146. [85] Studies in equisingularity HI. Saturation of local rings and equisingularity, Amer. J. Math., vol. 90 (1968) pp. 961-1023. [86] Contributions to the problem of equisingularity, Centro Internazionale Matematico Estivo (C.I.M.E.), Questions on Algebraic varieties. I l l ciclo, Varenna, 7-17 Settembre 1969, Edizioni Cremonese, Roma, 1970, pp. 261-343. [87] An Introduction to the Theory of Algebraic Surfaces, Lecture Notes in Mathematics, No. 83, Springer-Verlag, Berlin, 1969. [88] Some open questions in the theory of singularities, Bull. Amer. Math. Soc, vol. 77 (1971) pp. 481-491. [89] General theory of saturation and of saturated local rings, I. Saturation of complete local domains of dimension one having arbitrary coefficient fields (of characteristic zero), Amer. J. Math., vol. 93 (1971) pp. 573-648. [90] General theory of saturation and of saturated local rings, II. Saturated local rings of dimension 1, Amer. J. Math., vol. 93 (1971) pp. 872-964. [91] Quatre exposes sur la saturation, Notes prises par J. J. Risler, Asterisque, 7 et 8 (1973) pp. 21-29. [92] Le probleme des modules pour les branches planes, Cours donne au Centre de Mathematiques de l'Ecole Polytechnique, octobre-novembre (1973). Redige par Francois Kmety et Michele Merle, pp. 1-144. Avec un Appendice de Bernard Teissier. [93] General theory of saturation and of saturated local rings, III. Saturation in arbitrary dimension and, in particular, saturation of algebroid hypersurfaces, Amer. J. Math., vol. 97 (1975) pp. 415-502.
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[94] On equimultiple subvarieties of algebroid hypersurfaces, Proc. Nat. Acad. Sci. U.S.A., vol. 72 (1975) pp. 1425-1426. [95] The elusive concept of equisingularity and related questions, Algebraic geometry; T h e Johns Hopkins Centennial Lectures (supplement to the American Journal of Mathematics) (1977) pp. 9-22. [96] A new proof of the total embedded resolution theorem for algebraic surfaces (based on the theory of quasi-ordinary singularities), Amer. J. Math., vol. 100 (1978) pp. 411-442. [97] Foundations of a general theory of equisingularity on r-dimensional algebroid and algebraic varieties, of embedding dimension r + 1, Amer. J. Math., vol. 101 (1979) pp. 453-514. [97a] Abstract of the paper "Foundations of a general theory of equisingularity on r-dimensional algebroid and algebraic varieties, of embedding dimension r + 1," Symposia Matematica, vol. XXIV (Instituto Nazionale di Alta Matematica Francesco Severi, Rome, 1979).
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BIBLIOGRAPHY
883
Oscar Zariski 1899-1986 and 2) given any two solutions xl and i 2 , all other solutions x are rational functions of i l and xl. Already in his first work he strongly showed his ability to combine algebraic ideas (the Galois group), topological ideas (the fundamental group), and the "synthetic" ideas of classical geometry. The interplay of these different tools was to characterize his life's work. He pursued these ideas with the support of a Rockefeller fellowship in Rome during the years 1925-1927. His son Raphael was born there on July 18, 1925. In 1927, he accepted a position at Johns Hopkins and in 1928 his family moved to the U.S.A. to join him. Here his daughter Vera was born on September 14, 1932. A crucial paper in this phase of his career is his analysis [3] of an incomplete proof by Severi that the Jacobian of a generic curve of genus g has no nontrivial endomorphisms. Seven's paper reads as though the proof were complete. Zariski discovered the problem and found a very ingenious argument to remedy it, but neither were well received by Severi who published his own correction independently. The effect of this discovery seems to have been to turn Zariski's interests to the study of the topology of algebraic varieties, especially of the fundamental group, where the rigor of the techniques was beyond question and the tools were clean and new. He travelled frequently to Princeton to discuss his ideas with Lefschetz. In this phase of his career, roughly from 1927 to 1935, he studied the fundamental group of a variety through the fundamental group of projective nspace minus a divisor. This work is characterized by the spirit of exploration and discovery and, in spite of much recent interest, it remains a largely uncharted area. One result will give the flavor of the new things he turned up: according to another incomplete paper of Severi, it was widely believed that all plane curves of fixed degree with a fixed number of nodes (ordinary double points) belonged to a single algebraic family. What Zariski found was that curves with a fixed degree and a fixed number of cusps (the next most complicated type of double point) could belong to several families. He exhibited curves CI and C2 of degree 6 with 6 cusps such that the fundamental groups of their complements were not isomorphic! In 1935, however, Zariski completed his monumental review of the central results of the Italian
Oscar Zariski was born on April 24, 1899, in the town of Kobryn, which lies on the border of Poland and the U.S.S.R. It was Russian at the time of Zariski's birth, was Polish between the two world wars, and is now Russian again. He was the son of Bezalel and Chana Zaristky, and was given the name of Asher Zaristky, which he changed to Oscar Zariski when he came to Italy. Kobryn was a small town where his mother ran a general store, his father having died when he was two. In 1918, he went to the University of Kiev in the midst of the revolutionary struggle. He was seriously wounded in one leg when caught in a crowd that was fired at by troops, but recovered after two months in the hospital. As a student, he was attracted to the fields of algebra and number theory as well as to the revolutionary political ideas of the day. He supported himself partly by writing for a local Communist paper. This is most surprising for those of us who only knew him much later, but calls to mind the quip—A man has no heart if he is not a radical in his youth and no mind if he is not a conservative in his mature years. Because of the limitations of the education available in the U.S.S.R. at the time, in 1921 Zariski went first to the University of Pisa and six months later to the University of Rome where the famous Italian school of algebraic geometry, Castelnuovo, Enriques, and Severi, was flourishing. He had no money and the fact that universities in Italy were free to foreign students was an important consideration. Zariski was especially attracted to Castelnuovo, who immediately recognized his talent. Castelnuovo took him on a three hour walk around Rome after which Zariski realized that he had been given an oral exam in every area of mathematics! Castelnuovo saw in Zariski a man who would not only push their subject further and deeper, but would find radically new ways to overcome its present limitations. Zariski was fond of quoting Castelnuovo as saying "Oscar, you are here with us, but are not one of us," referring to Zariski's doubts even then of how rigorous their proofs were. Zariski met his wife, Yole Cagli, while a student in Rome and they were married on September 11, 1924, in Kobryn. He received his doctorate in the same year. His thesis ([1],[2]) classified all rational functions y = P(x)/Q(x) of i such that 1) x can be solved for in terms of radicals starting with y, 891
884 school, his Ergebnisse monograph Algebraic Surfaces [4], His goal had been to disseminate more widely the ideas and results of their research, but the result for him was "the loss of the geometric paradise in which I so happily had been living"'. He saw only too clearly that the lack of rigor he had touched on was not a few isolated sores but a widespread disease. His goal now became the problem of restoring the main body of algebraic geometry to proper health. Algebra had been his early love and algebra was blooming, full of beautiful new ideas in the hands of Noether and Krull, and various applications to algebraic geometry had already been proposed by van der Waerden. Zariski threw himself into this new discipline. He spent the year 1935-1936 at the Institute for Advanced Study in Princeton, and met regularly with Noether, then at Bryn Mawr, learning the new field through first hand contact with the master. The fifteen years or so that followed, 19381951, if you take the years between his paper [5] recasting the theory of plane curve singularities in terms of valuation theory and his monumental treatise [6] on his so-called "holomorphic functions" (sections of sheaves formed from completions of rings in I-adic topologies), saw the most incredible outpouring of original and creative ideas in which tool after tool was taken from the kit of algebra and applied to elucidate basic geometric ideas. Many mathematicians in their forties reap the benefits of their earlier more original work; but Zariski undoubtedly was at his most daring exactly in this decade. He corresponded extensively in this period with Andre Weil, who was also interested in rebuilding algebraic geometry and extending it to characteristic p with a view to its number-theoretic applications. Although they only rarely agreed, they found each other very stimulating, Weil saying later that Zariski was the only algebraic geometer whose work he trusted. They managed to get together in 1945 while both were visiting the University of Sao Paulo in Brazil.
The first theme is the study of birational maps which lead him to the famous result universally known as "Zariski's Main Theorem". This was the final result in a foundational analysis of birational maps between varieties, "maps" which are one to one and onto outside of a finite set of subvarieties of the range and domain, but which "blow up" or "blow down" special points. Zariski showed that if there are points P and Q in the range and domain which are isolated corresponding points, i.e. the set of points corresponding to P contains Q but no curve through Q. and the set of points corresponding to Q contains P but no curve through P, and if. further. P and Q satisfy an algebraic restriction—they are normal points—then in fact Q is the only point corresponding to P and vice versa (slightly stronger: the map is biregular between P and Q). Zariski's proof of this was astonishingly subtle, yet short. The second theme from this period is the resolution of singularities of algebraic varieties, which culminated in his proof that all algebraic varieties of dimension at most 3 (in characteristic zero) have "nonsingular models," i.e., are birational to nonsingular projective varieties. In dimension 3. this was a problem that had totally eluded the easy-going Italian approach. Even in dimension 2, although some classical proofs were essentially correct, many of the published treatments definitely were not. Zariski attacked this problem with a whole battery of techniques, pursuing it relentlessly over 6 papers and 200 pages. Perhaps the most striking new tool was the application of the theory of general valuations in function fields to give a birationally invariant way to describe the set of all places which must be desingtilarized. The result proved to the mathematical world the power of the new ideas. For many years, this work was also considered by everyone in the field to be technically the most difficult proof in all algebraic geometry. Only when the result was proven for surfaces in characteristic p by Abhyankar and later for varieties of arbitrary dimension in characteristic 0 by Hironaka2 was this benchmark surpassed!
At the same time, these were years of terrible personal tragedy. During the war, all his relatives in Poland were killed by the Nazis. Only his immediate family and those of two of his siblings who had moved to Israel escaped the holocaust. He told the story of how he and Yole were halfway across the U.S., driving back to the East Coast, the day Poland was invaded. They listened each hour to the news broadcasts on their car radio, their only link to the nightmare half a world away. There was nothing they could do. In this period of his work, Zariski solved many problems with his algebraic ideas. Three themes in his work are particularly beautiful and deep and I want to describe them in some detail. 1
The third theme is his theory of abstract "holomorphic functions." The idea was to use the notion of formal completion of rings with respect to powers of an ideal as a substitute for the idea of convergent power series, and to put elements of the resulting complete rings to some of the same uses as classical holomorphic functions. The most striking application was to a stronger version of the "Main Theorem." known as the connectedness theorem. The connectedness theorem states that if a birational map from X to 2 S.S. Abhyankar, Local uniformization on algebraic surfaces of characteristic p # 0, Annals of Math.. 63 (1956); H. Hironaka, Resolution of singularities of an algebraic variety of characteristic 0. AnnalB of Math.. 79 (1964).
Preface by Zariski to his Collected Works, MIT Press.
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885 Y is single-valued and if a point Q of Y is normal, then the inverse image of Q on X is connected (we are assuming X and Y are complete, e.g. projective). This result was later one of the inspirations for Grothendieck's immense work in rebuilding with yet newer tools the foundations of algebraic geometry 3 . This phenomenal string of papers caught the attention of the mathematical world. Zariski received the Cole Prize from the American Mathematical Society in 1944. In 1945, he moved to a Research Professorship at the University of Illinois. Early in the forties, his work had caught the attention of G.D. Birkhoff who decided he must come to Harvard 4 and, indeed, in 1947 he received and accepted an offer to come to Harvard University, where he remained for the rest of his life. He was a very strong influence on the mathematical environment/at Harvard and he enjoyed the opportunity of luring the best people he could to Harvard and bringing out the best in each of his students. While he was chairman, the Dean, McGeorge Bundy, used to refer to him as that "Italian pirate," so shrewd was he in getting his way, inside or outside the usual channels. Whenever Harvard's baroque appointment rules, known as the Graustein Plan (after the earlier mathematician who invented it), jibed with his plans, he used them; but whenever they did not, he feigned ignorance of all that nonsense and insisted the case be considered on its own merits. Over the next thirty years, he made Harvard into the world center of algebraic geometry. His seminar welcomed Weil, Hodge, Nagata, Kodaira, Serre, Grothendieck, and many others. The stimulating evenings at his home and the warm welcome extended by Oscar and Yole were not easily forgotten.
birational equivalence class of varieties [8], and on the classification of varieties following Enriques [9] (now known as the classification by Kodaira dimension). In each of these areas he spread before his students the vision of many possible areas to explore, many exciting prospects. Although he himself had developed a fully worked out theory of the foundations of algebraic geometry, he welcomed the prospect of yet newer definitions and techniques being introduced because they would make the subject itself stronger. He embraced the new language of sheaf theory and cohomology, working through the basic ideas methodically as was his custom in the Summer Institute in Colorado in 1953 [10], although he never adopted this language as his own. When Grothendieck appeared in the field, he immediately invited him to Harvard. Grothendieck, for his part, welcomed the prospect of working with Zariski. Because Grothendieck's political beliefs did not allow him to swear the oaths of loyalty required in those unfortunate days, he even asked Zariski to investigate the feasibility of continuing his mathematical research from a Cambridge jail cell, i.e., how many books and visitors would be allowed! The final phase of Zariski's mathematical career was a return to the problems of singularities. Zariski had absolutely no use for the concept of retirement and he dedicated his sixties and seventies and as much of his eighties as he could to a broadbased attack on the problem of "equisingularity". The goal was to find a natural decomposition of an arbitrary variety X into pieces Y„ each one made up of a subvariety of X from which a finite set of lower dimensional subvarieties have been removed, such that along each subvariety Vi, the big variety X had essentially the same type of singularity at each point. Zariski made major strides towards the achievement of this goal, but the problem has turned out to be quite difficult and is still unsolved. Zariski's last years were disturbed by his fight with his hearing problem. Zariski was always very lively both in mathematical and in social interactions with his friends and colleagues, picking up every nuance. He was struck with tinnitus, which produced a steady ringing in his ears, a greater sensitivity to noise, and a gradual loss of hearing. This forced him into himself, into his research and kept him close to home. Only the boundless love of his family sustained him in his last years. He died at home on July 4, 1986. Many honors flowed to Zariski in welldeserved appreciation of the truly extraordinary contribution he had made to the field of algebraic geometry. He received honorary degrees from Holy Cross in 1959, Brandeis in 1965, Purdue in 1974, and from Harvard in 1981. He received the National Medal of Science in 1965, and the Wolf Prize, awarded by the government of Israel, in 1982. His friends, his students, and his colleagues
His work of reconstruction of algebraic geometry had started with the writing of the monograph Algebraic Surfaces, and now that Zariski felt he had reliable and powerful general tools, it was natural for him to see if he could put all the main results of the theory of surfaces in order. He initiated the modern work on the duality theorems for cohomology (called by him the "lemma of Enriques-Severi" [7], before the topic was taken up by Serre and Grothendieck), the questions of the existence-ef minimal nonsingular models in each 3 Grothendieck's style was the opposite of Zariski's. Whereas Zariski's proofs always had a punch-line, a subtle twist in the middle, Grothendieck would not rest until every step looked trivial. In the case of holomorphic functions, Grothendieck liked to claim that the result was so deep for Zariski because he was just proving it for the Oth cohomology group. The easy way, he said, was to prove it firBt for the top cohomology group, then use descending induction! 4 The story, which I have from reliable sources, is that Birkhoff approached Zariski and said in his magisterial way: "Oscar, you will probably be at Harvard within the next five years."
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886 will remember not only the beautiful theorems he found, but the forcefulness and the warmth of the man they knew and loved.
[5] Polynominal ideals defined by infinitely near base points, Amer. J. Math., vol. 60 (1938) pp. 151-204. [6] Theory and applications of holomorphic functions on algebraic varieties over arbitrary ground fields. Mem. Amer. Math. Soc. no. 5 (1951) pp. 1-90. [7] Complete linear systems on normal varieties and a generalization of a lemma of EnriquesSeveri, Ann. of Math., vol. 55 (1952) pp. 552-592. [8] Introduction to the problem of minimal models in the theory of algebraic surfaces. Publ. Math. Soc. Japan, no. 4 (1958) pp. 1-89. [9] On Castelnuovo s criterion of rationality pa = P2 = 0 of an algebraic surface, Illinois J. Math., vol. 2 (1958) pp. 303-315. [10] Algebraic sheaf theory (Scientific report on the second Summer Institute), Bull. Amer. Math. Soc, vol. 62 (1956) pp. 117-141.
REFERENCES
[lj Sulle equazioni algebriche contenenti linearmente un parametro e risolubili per radicali, Atti Accad. Naz. Lincei Rend., CI. Sci. Fis. Mat. Natur., serie V. vol. 33 (1924) pp. 80-82. [2] Sopra una classe di equazioni algebriche contenenti linearmente un parametro e risolubili per radicali. Rend. Circolo Mat. Palermo, vol. 50 (1926) pp. 196-218. [3] On a theorem of Seven, Amer. J. Math.. vol. 50 (1928) pp. 87-92. [4] Algebraic Surfaces, Ergebnisse der'Mathematik, vol. 3, no. 5., Springer-Verlag, Beilin, 1935, 198 pp.; second supplemented edition, with appendices by S.S. Abhyankar, J. Lipman, and D. Mumford, Ergebnisse der Mathematik, vol. 61, Springer-Verlag, Berlin-Heidelberg-New York, 1971, 270 pp.
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887 Reprinted from Vol. II, PROCEEDINGS OF THE INTERNATIONAL CONGRESS OF MATHEMATICIANS,
1950
THE FUNDAMENTAL IDEAS OF ABSTRACT ALGEBRAIC GEOMETRY OSCAR ZARISKI
1. Introductory remarks. The past 25 years have witnessed a remarkable change in the field of algebraic geometry, a change due to the impact of the ideas and methods of modern algebra. What has happened is that this old and venerable sector of pure geometry underwent (and is still undergoing) a process of arithmetization. This new trend has caused consternation in some quarters. It was criticized either as a desertion of geometry or as a subordination of discovery to rigor. I submit that this criticism is unjustified and arises from some misunderstanding of the object of modern algebraic geometry. This object is not to banish geometry or geometric intuition, but to equip the geometer with the sharpest possible tools and effective controls. It is true that the lack of rigor in algebraic geometry has created a state of affairs that could not be tolerated indefinitely. Effective controls over the free flight of geometric imagination were badly needed, and a complete overhauling and arithmetization of the foundations of algebraic geometiy was the only possible solution. This preliminary foundational task of modern algebraic geometry can now be regarded as accomplished in all its essentials. But there was, and still is, something else and more important to be accomplished. It is a fact that the synthetic geometric methods of classical algebraic geometry, operating from a narrow and meager algebraic basis and faced by the extreme complexity of the problems of the theory of higher varieties, were gradually losing their power and in the end became victims to the law of diminishing returns, as witnessed by the relative standstill to which algebraic geometry came in the beginning of this century. I am speaking now not of the foundations but of the superstructure which rests on these foundations. It is here that there was a distinct need of sharper and more powerful tools. Modern algebra, with its precise formalism and abstract concepts, provided these tools. An arithmetic approach to the geometric theories which we were fortunate to inherit from the Italian school could not be undertaken without a simultaneous process of generalization; for an arithmetic theory of algebraic varieties cannot but be a theory over arbitrary ground fields, and not merely over the field of complex numbers. For this reason, the modern developments in algebraic geometry are characterized by great generality. They mark the transition from classical algebraic geometry, rooted in the complex domain, to what we may now properly designate as abstract algebraic geometry, where the emphasis is on abstract ground fields. 2. Revision of the concept of a variety. My object is to present some of the fundamental ideas of abstract algebraic geometry. I must begin with the very
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concept of a variety, since the arithmetic point of view led to a subtle revision of this concept and revealed some of its aspects that were not visible in the classical case. What I want to discuss in this connection concerns the following two topics: (1) the set-theoretic modifications in our conception of a variety as a set of points, modifications which were made methodologically necessary by the introduction of the well-known notion of a general point of an irreducible variety, due to Emmy Noether and van der Waerden; (2) the distinction between absolute and relative properties of a variety, a distinction which was made possible only by the admission of arbitrary ground fields. If we wish to arrange matters so that the general point of an irreducible variety be an actual point of the variety, we must allow point coordinates which are elements of some transcendental extension of the ground field. Furthermore, in the theory of algebraic correspondences it is essential to operate simultaneously with any finite number of independent general points of one and the same variety. It follows that we must have a reservoir of- infinitely many transcendental for the point coordinates in our geometry. For these reasons, it was found convenient, following Busemann and Andre1 Weil, to fix once and for always a universal coordinate domain; this is to be an algebraically closed field having infinite transcendence degree over the particular ground field A; in which we happen to be interested. Once this universal domain has been fixed, only such ground fields will be allowable which are subfields of the universal domain and over which the universal domain has infinite transcendence degree. We deal then with projective spaces over the universal domain, and all our varieties will be immersed in these spaces. This being so, if fc is any allowable ground field and if a variety V admits a system of defining equations with coefficients in fc, then fc is said to be a field of definition of V. Naturally, any variety V has infinitely many fields of definition. A property of V is relative or absolute according as it does or does not depend on the choice of the field of definition of V. For example, irreducibility of a variety is a relative property. But we also have the so-called absolutely irreducible varieties which are irreducible over each one of their fields of definition. The concept of the general point (x) of an irreducible variety V/k is a relative concept. On the other hand, the dimension of that irreducible variety V/k, i.e., the transcendence degree of the function field k(x) of V/k, is an absolute concept. A necessary and sufficient condition that a variety V be absolutely irreducible is that it be irreducible over some algebraically closed field of definition. An equivalent condition is the following: an irreducible variety V/k is absolutely irreducible if the ground field k is quasi-maximally algebraic in the function field k(x) of V/k, i.e., if every element of k(x) which is separably algebraic over A; belongs to fc. In theory, it would be sufficient to restrict the study of varieties to absolutely irreducible varieties, since any variety has a unique representation as a sum of absolutely irreducible varieties. However, in practice, and especially in the foundations, such a restriction introduces unnecessary complications.
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It may be advisable to give a special name to those varieties which admit every (allowable) ground field as field of definition. Obviously, these are the varieties which are defined over the prime field of the given characteristic p. I propose to call them universal varieties. The projective space and the Grassmannian varieties are examples of universal varieties. Another important class of universal varieties is obtained by considering the set of all algebraic varieties, of a given order and dimension, in the n-dimensional universal projective space and introducing in that set an algebro-geometric structure based on the Chow coordinates of a cycle. The study of these varieties (of which the Grassmannian varieties are special cases) is closely connected with the outstanding problem of developing a theory of algebraic equivalence of cycles on a given variety, and will no doubt be a fundamental object of future research. The definition of a variety as a set of points having coordinates in the universal domain has some startling, and perhaps unpleasant, set-theoretic implications. We have populated our varieties with points having coordinates which are transcendental over k. Thus, if x and y are independent variables the pair (x, y) is a legitimate point of the plane; and—what is worse—if a;'and y' are other independent variables, then (x', y') is another point of the plane, quite distinct from the point (x, y). This is shocking, especially if we recall that our universal domain has infinite transcendence degree and that consequently we have created infinitely many replica of that ghostlike point (x, y). However, we are dealing here with a methodological fiction which is extremely useful in proving very real theorems. For instance, the entire theory of specializations is based on this set-theoretic conception of a variety, and the entire elementary theory of algebraic correspondences can be developed on that basis in the most effortless and simple fashion. Furthermore, most results concerning irreducible subvarieties of a given variety can be best expressed and derived as results concerning the general points of these subvarieties. Nevertheless fiction remains fiction even if it is useful, and I feel that perhaps our varieties have altogether too many points to be good geometric objects. As the theory progresses beyond its foundational stage, some cuts and reductions may become necessary. Thus, one may begin first of all by eliminating isomorphic replica of points, by identifying points which are isomorphic over the given ground field k. Or one may restrict the coordinate domain to the algebraic closure of k. Or one may do both of these things at the same time. I have no strong convictions on these issues, and I am quite content in leaving their settlement to the future development of the theory of algebraic varieties. But to round up this discussion, let me indicate briefly some topological aspects of these issues. Given a variety V and given any field k (not necessarily a field of definition of V), there is a natural topology on V, relative to k: it is the topology in which the closed sets are intersections of V with varieties which are defined over k. In particular, if V itself is defined over k, then the closed sets on V are the subvarieties of V which are also defined over k, and in terms of this topology
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general points and specialization of points are easily denned. Thus, a general point of an irreducible variety V/k is a point whose closure is the entire variety V; a point Q is the specialization of a point P, over k, if Q belongs to the closure of P. It is clear that in this topology even the weakest separation axioms are not valid. The only points of V which constitute closed sets are the points having coordinates which are pure inseparable over k. Hence V is not a 7V space. It is not even a T0-space, for if P and Q are fc-isomorphic points, then each belongs to the closure of the other. However, if we identify fc-isomorphic points of V, we restore the separation axiom To. If, moreover, we restrict the coordinate domain to the algebraic closure of the ground field k, then V becomes a 7\-space. An even more radical revision of the concept of a variety has been offered by Andre" Weil. His so-called abstract varieties are not defined as subsets of the projective space, but are built out of pieces of ordinary varieties, pieces that must hang together in some well-defined fashion. It is still an open question Avhether the varieties of Weil can be embedded in the projective space. In all that precedes I have used deliberately the term "general point" rather than that of "generic point". When the Italian geometers speak of a property enjoyed by the generic point of an irreducible variety, they mean a property that is enjoyed by all points of V, except perhaps those which belong to some proper subvariety of V. It is clear that this is not equivalent to saying that the general point of V has that given property. There is equivalence if and only if we are dealing with a property of points that can be expressed by equations and inequalities (with coefficients in k) connecting the point coordinates. But not every algebro-geometric property is of this category. For instance, it is possible to define algebraically the notion of analytical irreducibility of a variety V at a point. Now if W is, say, an algebraic curve on V, V may be analytically reducible at the generic algebraic point of the curve and analytically irreducible at all the general points of the curve. From our point of view, according to which W consists of both algebraic and transcendental points, either one of the following statements is false: (1) V is analytically irreducible at the generic point of W; (2) V is analytically reducible at the generic point of W. In the complex domain this corresponds to the following state of affairs: at the generic (complex) point of the curve W the variety V decomposes into several analytical branches, but these branches are permuted transitively along closed paths traced on the Riemann surface of the curve W. This is a good example of the difference between the meanings of general and generic. I shall pass now, without delay, to more concrete topics dealing with the major developments in abstract algebraic geometry. Roughly speaking, these major developments come under the following headings: (1) theory of specializations; (2) normal varieties; (3) analytical methods; (4) theory of valuations; (5) Abelian varieties. Time will not allow me to discuss the very general and elegant theory of Abelian varieties which we now possess and for which Andr6
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Weil is entirely responsible. Let me, then, first make a few remarks about the theory of specializations. 3. The theory of specializations. Specialization arguments in abstract algebraic geometry are the arithmetic substitute, or analogue, of continuity arguments of classical algebraic geometry, and have been largely developed by van der Waerden. The theory of specializations centers around one basic fact, concerning extensions of specializations: if an algebraic function / is defined at the general point of an irreducible variety V/k, then it is possible to extend the domain of the function / to the entire variety V, including therefore also those points of V at which the explicit expression of / is indeterminate. In this statement, the function / need not be a numerical function; the values of / may be points of another variety V, and when that is so, we are dealing with an algebraic correspondence between V and V. It is well-known that the theorem on extensions of specializations is equivalent to the existence of resultant systems in elimination theory. Without advocating the elimination of elimination theory, it may be pointed out that also the Hilbert Nullstellensatz, in its homogeneous form, can be used as a foundation for the theory of specializations (and hence also of algebraic correspondences). In fact, the Nullstellensatz provides a key to the whole of the elementary theory of algebraic varieties, including such topics as the dimension theory, the principal ideal theorem (in its geometric formulation), the decomposition of a variety under ground field extensions, etc. In my forthcoming Colloquium book, this part of the theory of varieties is built entirely around the Hilbert Nullstellensatz. One of the most important applications of the theory of specializations was the development of the general intersection theory. This application is due to van der Waerden and Andre Weil, with the work of Severi serving as a geometric background for the general plan of this undertaking. At present, then, we have a complete intersection theory which is valid for any nonsingular variety over an arbitrary algebraically closed ground field. A parallel development is the intersection theory for algebraic manifolds due to Chevalley. Chevalley's theory is an outstanding example of the arithmetization of some of the concepts and methods of the theory of analytic functions which are used in algebraic geometry. As far as the local analytical treatment is concerned, modern algebra has provided us with the necessary tools. I refer to the theory of local rings and their completions, due to Krull and Chevalley and further enriched by important contributions by I. S. Cohen and P. Samuel. The present intersection theories all have an absolute character, since they refer to an algebraically closed ground field. It is still an open question whether there exists a consistent relative intersection theory, i.e., an intersection theory relative to a given ground field. The fact that there exists such a thing as the relative order of a variety seems to indicate the possible existence of a relative intersection theory. Another unsolved question is whether there exists a rea-
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sonable intersection theory on varieties which have singularities, for instance—• and above all—on normal varieties. The example of algebraic cones shows that in the case of singular varieties one may have to use fractional intersection numbers and—more generally—fractional cycles. 4. Normal varieties. I shall now discuss briefly the concept of a normal variety, especially from the standpoint of the theory of specializations. This concept, which is purely arithmetic in character, turned out to be a useful contribution even to classical algebraic geometry. An irreducible variety V/k is said to be normal at a -point Q if the local ring of V at that point is integrally closed. The variety V is normal if it is locally normal at every point. Normality is a relative property. The normalization of a variety V consists in passing from V to a birationally equivalent variety V such that: (1) V is normal; (2) the birational transformation between V and V has no fundamental points on V, i.e., to every point of V there corresponds on V at most a finite number of points. By these two conditions the normal variety V is uniquely determined by V, to within a regular birational transformation (a birational transformation is regular if it is (1,1) without exceptions and if corresponding points have the same local ring). Normal varieties were originally introduced in connection with the problem of the resolution of singularities, for a normal variety of dimension r has the property that its singular locus is of dimension at most r — 2. This property of normal varieties is connected with the well-known fact that as far as the minimal prime ideals are concerned, the ideal theory of integrally closed Noetherian domains does not differ essentially from the classical ideal theory of Dedekind domains. At any rate, the process of normalization does have the effect of resolving all the singular loci of V, of dimension r — 1. But there are other properties of normal varieties which are of particular interest for the theory of specializations, and hence also for the general theory of algebraic correspondences. Suppose that V is a normal variety and that T is a birational transformation of V into some other variety V. Then the following theorem holds: if to a given point Q of V there corresponds on V more than one point, then (1) the point Q is fundamental for the birational transformation T, i.e., to Q there corresponds on V an infinite set of points, and (2) the set of points of V which correspond to Q is a variety, all irreducible components of which are of positive dimension. The really significant and nontrivial part of this theorem is the second part. It is this that I have given in a Transactions paper as the "main theorem" on birational transformations and for which I gave a short and simple proof, based on valuation theory and the theory of local rings, in a recent note in Proc. Nat. Acad. Sci. U. S. A. In terms of specializations, the above "main theorem" signifies, roughly speaking, that, in the case of normal varieties, the presence of an isolated specialization implies the uniqueness of that specialization. As was pointed out by Andre Weil, this property of normal varieties leads at once to the proof of the uniqueness of the intersection mul-
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tiplicity of two varieties at a common isolated intersection. The uniqueness of intersection multiplicity is, on the other hand, the crucial point of the whole of intersection theory, and the fact that this point can be disposed of in such a casual manner by the use of a general theorem on normal varieties illustrates the usefulness of the concept of normality. The treatment of the intersection theory would be further simplified, in fact the whole theory would become almost trivial, if one could prove the normality of any complete algebraic system of cycles in the projective space, i.e., the normality of the Chow-van der Wacrden representative variety of such a system. It would even be sufficient to prove that this variety is analytically irreducible everywhere, for in that case the normalization process would lead to another representative variety, whose points are still in (1,1) correspondence with the cycles of the system (without exceptions) but which is normal. Another important aspect of normal varieties has to do with the theory of complete linear systems. Any normal variety V has the following characteristic property: the hypersurfaces of a sufficiently high order n cut out on V a complete linear system | nC \ , where C is any hyperplane section of V. This result, in conjunction with Hilbert's postulation formula and the existence of derived normal models, leads at once to an expression of the dimension of the complete system | nC | (n large), whether V is normal or not. In a joint paper of Muhly and myself, now in course of publication in Trans. Amer. Math. Soc, we define the virtual arithmetic genus p(V) as ± the constant term in Hilbert's postulation formula of V. In the case of algebraic surfaces, we prove that, in any birational class [V] of normal surfaces V, the numerical character p(V) is a monotone nonincreasing function of V, with respect to the following partial ordering of the class: V < V if the birational transformation from V to V is single-valued without exceptions. An essential ingredient of the proof is the remark that, since a normal surface V has only a finite number of singular points, the generic hyperplane section of V is normal (since it is a curve free from singularities). Now the normality of a generic hyperplane section of any normal variety has recently been established by Seidenberg. In virtue of this interesting result of Seidenberg, the monotone character of the virtual arithmetic genus p(V) can now be regarded as established for normal varieties of any dimension. In particular, p{V) is invariant under regular birational transformations. Added in proof: In view of the technical difficulties of the proof of Seidenberg's theorem, we point out that the results of our joint paper with Muhly do not actually require Seidenberg's theorem. All that is needed is the following statement: the general hyperplane-section of a normal variety V/k (k algebraically closed) is absolutely normal. This statement is an immediate consequence of the theorem of Bertini, of the classical Jacobian criterion for simple points, and of Weil's characterization of absolutely normal varieties. In the joint "paper of Muhly and myself it is proved that (a) the function p(F), defined in a given birational class of varieties of dimension ^ 3 , has a
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minimum; (b) this minimum is reached for the nonsingular varieties of the class; (c) this minimum is equal to the effective arithmetic genus of the field of algebraic functions determined by the given birational class. Each of these statements represents an unsolved problem for varieties of dimension greater than three. 5. Holomorphic functions and the principle of degeneration. The "main theorem" on birational transformations is a special case of a much more general "connectedness theorem" on algebraic correspondences, a theorem which in its turn contains as a special case a principle of degeneration for varieties over arbitrary ground fields. The proof of this theorem is based on a theory of abstract holomorphic functions which I have developed in a paper now in course of publication in the Memoirs of the American Mathematical Society, and which represents an extension of the analytical methods of abstract algebraic geometry from a local theory to a theory in the large. The use of normal varieties is essential in this theory. I shall now briefly outline the geometric background and the underlying ideas of this work. In the classical case, the principle of degeneration (first formulated by Enriques) asserts that if an irreducible variety V varies continuously and degenerates in the limit into a reducible variety V0, then this limit variety is connected. This principle is almost self-evident, since F 0 i s a continuous image of the irreducible—and therefore connected—variety V. I say "almost evident", because in order to assert that F 0 is a continuous map of V it would be necessary to show that the continuous variation of V can be accompanied by a continuous deformation of V into V0. The existence of such a deformation has always been taken more or less for granted. Now, the principle of degeneration can easily be transformed into an equivalent statement in which no reference is made to continuity or limits and which therefore makes sense also in the abstract case. First of all, if V is a variety and k is any ground field (not necessarily a field of definition of V), then the expression "V is connected over k" has a meaning, since V has a natural topology over k. In particular, if k is an algebraically closed field of definition of V and if V is connected over k, then it is easy to see that V is connected over every one of its fields of definition. We say then that V is absolutely connected. This definition can then be extended in an obvious fashion to effective algebraic cycles, i.e., to formal linear combinations, with positive integral coefficients, of absolutely irreducible varieties of the same dimension r. Now let M be an irreducible algebraic system of r-dimensional cycles, and suppose that M is defined and irreducible over a given ground field k. Then I prove the principle of degeneration under the following form: / / the general cycle of M/k is absolutely irreducible, then every cycle in M is absolutely connected. It is easy to transform this principle into a statement concerning the incidence correspondence associated with the system M, i.e., the correspondence
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in which to every cycle of M correspond all the points which belong to that cycle. The transformed statement can be itself incorporated into the following more general connectedness theorem on algebraic correspondences: THEOREM. Let T/k be an irreducible algebraic correspondence between two varieties V and V and let (P, P') be a general point pair of T/k. We make the following assumptions: (1) T~' is rational [i.e., k{P) CZ k(P')] and semi-regular. (2) The field k(P) is maximally algebraic in k(P'). Let W/k be any connected siibvariety of V. Then if V is analytically irreducible (in particular, locally normal) at each point of W, the total transform of W under T is a k-connected subvariety of V.
This last theorem covers a good deal more ground than does the principle of degeneration. The incidence correspondence of an algebraic system has the special property (not shared by arbitrary correspondences) that it has no fundamental points: to every point of the representative variety V of the system M there corresponds, on the carrier variety V, a variety of the same fixed dimension r, r = the dimension of the cycles of the system M. If we deal, however, with an arbitrary correspondence T and if a point Q of V happens to be a fundamental point of T, then the variety T{Q], whose connectedness is being claimed, may very well have, dimension higher than that of the total transform of the general point of V/k. The connectedness of that variety T{Q] is, in that case, not at all trivial even in the classical case (and—to our knowledge —has never been proved in the classical case, even for birational transformations 7'). It is in the proof of this theorem that the holomorphic functions come directly into play. Wilh the given subvariety W of V we associate "functions" on V which "are defined and holomorphic" along W. These functions are, by definition, certain specified elements of the direct product of the completions of the local rings of V at the various points of W. They are those elements of this direct product which can be represented by a finite number of sequences of elements of the function field of V, in such a manner that (1) each sequence converges uniformly on some open subset i \ of W and (2) the sets I \ cover W. These functions form a ring which we denote by ow . As a first tangible evidence of the nonartificiality of this new concept, we have the following: CONNECTEDNESS CRITERION. If V is analytically irreducible at each point of W, then W/k is connected if and only if the ring ow of holomorphic functions along W is an integral domain.
With this theorem, we are still very far from the proof of the connectedness theorem for algebraic correspondences. The high point, and also the most difficult part, of the whole theory is still to come. It is represented by a theorem of
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invariance of rings of holomorphic functions under rational transformations. It is noteworthy that while the proof of the principle of degeneration in the classical case is essentially a simple exercise in topology, our proof of this principle in the abstract case can be given only after a long and difficult journey. The end of this journey is as follows: Let T be an algebraic correspondence between two irreducible varieties V/k and V/k such that T~l is a rational transformation, semi-regular at each point of V. Let W be the total transform r { W } of W, where W is any subvariety of V. We have the ring o%r of functions on V, defined and holomorphic along W. We first show that there always exists a natural isomorphism Hw.w of ow into ow> . The fundamental theorem of invariance asserts the following: THEOREM. Let (P, P') be a general -point pair of T/k. If (a) k(P) is maximally algebraic in k(P') and (b) if V is locally normal at each point of W, then Hw, w is an isomorphism of ow onto ow> • [Note that condition (a) is automatically satisfied if T is a birational transformation.]
This theorem, together with the above connectedness criterion, gives immediately the connectedness theorem for algebraic correspondences. There is a number of very difficult problems suggested by the theory of holomorphic functions and which are still open even in the classical case. One of them is to prove that: I. The ring o% is Noetherian. The elements of o% which belong to the field of rational functions on V form themselves a ring, denoted by ow . This ring is the intersection of all the local rings of the points of W. The second problem is to prove the following: II. The ring o w is Noetherian. The non units in ow form an ideal, say'm. Whether or not Ow is Noetherian, one may consider the completion of o v with respect to the powers of this ideal. The next problem is to prove the following: III. If W is connected, then ow is the completion of ow • A special case of III is the following conjecture: if o w consists only of constants, then Ow — ow • The existence of nonconstant holomorphic functions on V, defined along W, is closely connected with the existence of a rational transformation T of V into some other variety V, such that W is the total transform of a point of V. It is obvious that the latter implies the former, but I have no proof of the converse. At any rate, if W can be transformed into a point Q' of some variety V by a rational transformation, then it is easily seen that ow — OQ> , and from the theorem of invariance of rings of holomorphic functions it follows that o% = o% . Hence in this case the conjectures I, II, III are true. 6. Transcendental theory of specializations. My report is very incomplete as it stands, but it would be glaringly incomplete if I had not said anything about the role of valuation theory in abstract algebraic geometry. The ordinary theory
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of specializations applies to finite sets of quantities; it is a finite theory of specializations. This theory does not do everything that continuity does in classical geometry, for it contains nothing that corresponds to the notion of a branch, whether algebraic or transcendental. It does not tell us anything about the different modes of approach to a point on a given variety. For this reason, the finer differential aspects of the local geometry of a variety, in particular the analysis of the neighborhood of a singular point, are outside the province of the finite theory of specializations. What was needed here was a theory which deals with the simultaneous specialization of all the rational functions on a given variety, therefore a transcendental theory of specializations. Valuation theory meets precisely this requirement. It is to be observed that it was precisely the general valuation theory, as developed by Krull, i.e., the theory of valuations having arbitrary value groups (nondiscrete as well as discrete, nonArchimedian value groups as well as Archimedian), that turned out to be the necessary tool for the solution of such a concrete algebro-geometric problem as the local uniformization of algebraic varieties and for partial progress in the problem of the resolution of singularities. Nothing less than the general valuation theory would have served that purpose, and that is so for the following two reasons: (1) Without the general concept of a valuation it is not possible even to formulate the problem of local uniformization in pure algebraic terms. The nearest algebraic substitute for the neighborhood of a point Q of a variety V is the set of all modes of approach to the point Q, therefore essentially, the set of all valuations of the function field of V which have center at Q. Here, by the center of a valuation v we mean the point whose coordinates are the y-residues of the coordinates of the general point of the variety V. And the only statement which can reproduce in algebraic terms, and without loss of power, that what has been a classical conjecture, namely that the complete neighborhood of the point Q can be represented by a finite number of power series expansions, is the statement that any valuation v of center Q can be uniformized with respect to V. By this I mean that there exists a birational transform V of V such that (a) the center of v on V is a simple point Q' of V and (b) the local ring of V at Q is contained in the local ring of V at Q' (this second condition implies that every element of the function field of V which is holomorphic at Q is also holomorphic at Q'). (2) The set of all valuations of the function field of V is a compact space in a suitable natural topology of that set. This space is called the Riemann surface of the function field of V. Now it is the compactness of the Riemann surface that makes it possible to apply the theorem of local uniformization of abstract algebraic geometry (i.e., the uniformization of a single valuation) to the classical problem of local uniformization of the complete neighborhood of a point of a variety. No reasonable proper subset of the set of all valuations has that compactness property. Thus, the set of all discrete valuations is not compact (ex cept in the case of curves, in which case all valuations are, of course, discrete),
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and neither is the set of all algebraic valuations (represented by algebraic branches). The problem of local uniformization has been settled, so far, only in the case of characteristic zero. The extension of the present proof to the case of nonzero characteristic will call for considerable algebraic skill and ingenuity. It is not a problem for geometers; it is a problem for algebraists with a feeling and intuition for all the unpleasant things that can happen in the case of nonzero characteristic. The essential difficulties of this problem are already apparent in the case of algebraic surfaces over an algebraically closed ground field. In this case I can prove the theorem of local uniformization for every valuation except those which are nondiscrete and have rational rank 1 (i.e., those whose value group is a dense set of rational numbers). The case which is particularly difficult is the one in which the value group of the valuation contains rational numbers having in the denominator arbitrarily high powers of the characteristic p. In this case, already such a simple surface as zp = fix, y) becomes untractable. Also the problem of the resolution of singularities is still unsolved; or—to put it into more cautious terms—no solution of this problem has ever come to my direct attention. For characteristic zero, this problem has been solved, so far, only for varieties of dimension :£3. 7. Concluding remarks. The Italian geometers have erected, on somewhat shaky foundations, a stupendous edifice: the theory of algebraic surfaces. It is the main object of modern algebraic geometry to strengthen, preserve, and further embellish this edifice, while at the same time building up also the theory of algebraic varieties of higher dimension. The bitter complaint that Poincare has directed, in his time, against the modern theory of functions of a real variable cannot be deservedly directed against modern algebraic geometry. We are not intent on proving that our fathers were wrong. On the contrary, our whole purpose is to prove that our fathers were right. The arithmetic trend in algebraic geometry is not in itself a radical departure from the past. This trend goes back to Dedekind and Weber who have developed, in their classical memoir, an arithmetic theory of fields of algebraic functions of one variable. Abstract algebraic geometry is a direct continuation of the work of Dedekind and Weber, except that our chief object is the study of fields of algebraic functions of more than one variable. The work of Dedekind and Weber has been greatly facilitated by the previous development of classical ideal theory. Similarly, modern algebraic geometry has become a reality partly because of the previous development of the general theory of ideals. But here the similarity ends. Classical ideal theory strikes at the very core of the theory of algebraic functions of one variable, and there is in fact a striking parallelism between this theory and the theory of algebraic numbers. On the other hand, the general theory of ideals strikes at most of the foundations of algebraic geometry and falls short of the deeper problems which we face in the postfoundational stage. Furthermore, there is nothing in modern commutative
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algebra that can be regarded even remotely as a development parallel to the theory of algebraic function fields of more than one variable. This theory is after all itself a chapter of algebra, but it is a chapter about which modern algebraists knew very little. All our knowledge here comes from geometry. For all these reasons, it is undeniably true that the arithmetization of algebraic geometry represents a substantial advance of algebra itself. In helping geometry, modern algebra is helping itself above all. We maintain that abstract algebraic geometry is one of the best things that happened to commutative algebra in a long time. HARVARD UNIVERSITY, CAMBRIDGE, MASS., U. S.
A.
THE COMPACTNESS OF THE RIEMANN MANIFOLD OF AN ABSTRACT FIELD OF ALGEBRAIC FUNCTIONS OSCAR ZARISKI
1. The existence of finite resolving systems. In an earlier paper 1 we have announced the result that the existence of a resolving system of the Riemann manifold of an abstract field of algebraic functions (in any number of variables) or—what is the same—the local uniformization theorem2 implies the existence of finite resolving systems of the Riemann manifold. We have proved this result for algebraic surfaces by arithmetic considerations.l The proof for the general case of varieties, which at that time was in our possession,8 and which we have promised to publish in a subsequent paper, was of similar nature, that is, it was based upon considerations involving the structure of certain infinite sequences of quotient rings. However, we have succeeded lately in finding a much simpler proof which is based on topological considerations. Let 2 be a field of algebraic functions of several variables, over an arbitrary ground field k. By the Riemann manifold M of 2 we mean the totality of places of S, that is, the totality of zero-dimensional valuations v of 2/&. If V is a projective model of 2/&, and if H is any subset of V, we denote by N(H) the subset of M consisting of those valuations b which have center in H. By a resolving system of M we mean a collection 25 = { Va} of projective models (finite or infinite in number) with the property that for any v in M there exists a F a in 25 such that the center of v on Va is a simple point. The topology which we introduce in M is simply this: we choose as a basis for the closed sets of M the sets N(W), where W is any algebraic subvariety of any projective model of 2. We prove that if topologized in this fashion, the set M is a compact* topological space. From this the result announced above follows immediately. For if { Va\ is a resolving system, and if we denote by Sa the singular locus of Va, then N( Va—Sa) is an open set and {N( Va — S„)} is an open covering of M. Received by the editors April 10, 1944. A simplified proof for the resolution of singularities of an algebraic surface, Ann. of Math. vol. 43 (1942) p. 583. * See loc. cit. footnote 1. * That proof was presented by us at a seminar in algebraic geometry at Johns Hopkins in 1942. 4 We use the term compact in the same sense as it is used by S. Lefschetz in his Algebraic topology (Amer. Math. Soc. Colloquium Publications, vol. 27, 1942). The old term is bicompact. 1
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Hence this covering contains a finite subcovering {N( Vi — Si)}, t = l, 2, • • • , m, and this means t h a t { V\, V2, • • • , Vm) is a finite resolving system of M. The proof of compactness of M given in the next section is based in p a r t on some simple algebro-geometric considerations, and in part on a theorem of Steenrod 6 on the compactness of the limit space of an inverse system of compact TYspaces. 2. T h e Riemann manifold as the limit space of an inverse system. Let S3 = { Va} be the collection of all projective models of X/k. By a point of Va we mean a zero-dimensional prime ideal in a suitable coordinate ring of Va, or, in other terms, a point is a prime onedimensional homogeneous ideal in the ring of homogeneous coordinates of the general point of Va. This defines Va set-theoretically as a set of points. We topologize Va by choosing as closed sets the algebraic subarieties of Va- I t is obvious t h a t Va then becomes a compact topological space in which points are closed sets (whence Va is a 7Vspace; however, it is not a Hausdorff space). If Va and Vb are two projective models of S//fe, we denote by irl the transformation of Vb onto Va defined by the birational correspondence between Va and Vb. We define a partial ordering < of the collection 23 as follows:V a
There is a (1, 1) correspondence between the points P*
• N. E. Steenrod, Universal homology groups, Amer. J. Math. vol. 58 (1936) p. 666. * See our paper Foundations of a general theory of birational correspondences, Trans. Amer. Math. Soc. vol. 53 (1943) p. 516.
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of M and the zero-dimensional valuations v of the field 2/&. If P* and v are corresponding elements, and if Va is any projective model of 2//fe, then TT*P* is the center of v on Va. P R O O F . Let s b e a zero-dimensional valuation of 2/& and let Pa,v be the center of v on any given projective model Va of 2//fe. For any two projective models Va, Vb it is then true t h a t Pa,v and Pb,„ are corresponding points in t h e birational correspondence irba. Hence P* = {Pa.v} is a point of M. Thus every zero-dimensional valuation v determines uniquely a point P* of M. If Vi and »2 are two distinct zero-dimensional valuations, then there exists a t least one projective model Va such that Pa.v^Pa.v,Hence
ifv^vtthanP^P*. Now let P* be an arbitrary point of M, P* = {P0}. We denote by 33 the least ring containing the quotient rings Q(Pa). Let Vb be a fixed projective model of "L/k and let Pb=irb*P*. We assert that if oi is a non-unit in Q(Pb) then 1/wSfSB. For assume t h a t l / w £ 9 3 . Then 1/w will belong to the ring generated b y a finite number of quotient rings Q(Pa), say Q(Pai), Q(Pai), • • • , Q(PaJ. Let Vc be the join of the varieties Vb, 7 B l , Va„ • • • , Va„ and let Pe=ir*P*. Since ira?P* = Pai and Trb*P* = Pb, we have irca.Pe = Poi and irbPe = Pb, and hence Q(Pai)QQ{Pc), Q(Pb)QQ(Pe). Therefore l/uEQ(Pc). This is a contradiction since any non-unit of Q(Pb) is obviously also a non-unit in We have therefore shown t h a t S3 is a proper ring (not a field). We now show t h a t S3 is a valuation ring. For this it is sufficient to show t h a t if £ is any element of 2 then either £ £ $ 3 or 1/£E33. We consider again a fixed projective model Vb of 2 / £ . We select a system of nonhomogeneous coordinates xi, xit • • • , xn of the general point of Vb in such a fashion t h a t the point Pb (=irb*P*) is a t finite distance with respect to these coordinates. Let Va be the projective model whose general point has as nonhomogeneous coordinates the elements x\, xit • • • , xn, £. If the point Pa (=ir$33 we have shown incidentally the following: if Va and Vb are any two projective models of 2 / £ and if 7r 0 *P*=P 0 and Tb*P* = Pb, then Pa and Pb are corresponding points of the birational correspondence between Va and Vb. For on the join Vc of Va and Vb we have the point P e = ire*P* and the relations Q(Pc)12Q(Pa), Q(Pc)^>Q(Pb)- These relations show t h a t if v is a n y zero-dimensional valuation whose center on Ve is the point Pe, then the center of v on Va is P a and its center on Vb is Pb. Hence P„ and Pb are indeed
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corresponding points. 7 With this observation in mind, let v be a zerodimensional valuation whose center on Vb is the point Pi and whose center on Vd is the point Pd. Since Pb is a t finite distance, we have v(xi)^0, i = l, 2, • • • , n. Since Pd is a t infinity, we must have n(£) < 0 . Hence v(l/£) > 0 , i>(x,/£) > 0 , and this shows that if we take V£) *i/£. • • • » *»>/£ as nonhomogeneous coordinates of the general point of Vd, then Pd is a t finite distance. Hence l/£GQ(Pd) Q%- This completes the proof of our assertion t h a t 53 is a valuation ring. Let v be the valuation defined by the valuation ring 93. We assert that v is zero-dimensional. For let v be of dimension s. We can find a projective model Vb on which the center of v is an s-dimensional variety W. If Pb=Tb*P*, then Q(Pb)Q% and this implies t h a t PhE.W* If s > 0 , then we can find a non-unit u in Q(Pb) such t h a t u^O on W, whence l/u(EQ(W)QS8, a contradiction. Hence s — 0, as asserted. The above relation Pb(E.W implies now Pb = W. This is true for any projective model Vb, t h a t is, the center of i; on any projective model Vb is the point Pb = irb*P*. This completes the proof of the theorem. 3. A generalization. Infinite direct products of projective lines. The idea of topologizing an algebraic variety V by choosing as closed sets the algebraic subvarieties of F c a n be used with good effect in order to topologize the set M* of all homomorphic mappings of any abstract field A into another abstract field K. In this general case we are dealing essentially with a generalization of the concept of the Riemann manifold of a field of algebraic functions (see the Remark a t the end of the paper). We begin with some topological preliminaries. Let {Ra be a system of compact topological spaces indexed by a set A = {a}. We assume t h a t each Ra is a TVspace; t h a t is, t h a t the points of Ra are closed sets. Elements of A shall be denoted by small Latin letters, a, b, c, • • • ; subsets of A shall be denoted by small Greek letters, a, )3, y, • • • . If a is a subset of A we shall denote by Ra the direct product PaGa Ra- If a Q 8 we denote by IT* the projection of Rp onto Ra. Finally, elements of Ra and Ra shall be denoted by xa, ya, z„, • • • and by xa, ya, z„, • • • respectively. If aE.cc and if ir"xa =xa, then xa shall be referred to as the a-component of xa. We assume t h a t for each finite subset a of A a topology has been assigned to Ra and t h a t the following three conditions are satisfied: (1) Ra is a compact topological space; (2) i / a Q then IT* is a closed 7
See our definition of corresponding points of a birational transformation, loc. cit. footnote 6, p. SOS. • See loc. cit. footnote 6, Theorem 3, p. 497.
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mapping (mapping = single-valued continuous transformation); (3) if a is a set with one element a then the topology assigned to Ra is exactly the topology of Ra. I t is clear t h a t in virtue of these two conditions Ra is a TVspace. For if xa is the a-component of xa, then (wZ)~lxa is closed and xa is the intersection of the closed sets (ir")-1^0« a £ « If we consider only finite subsets a of A and if we define a partial ordering in the collection {Ra} by setting Ra
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of the given family g, it follows t h a t every finite intersection of sets in 5 is again in the family %. For any element a in A and for any member F a * in JJj let iraF* = Fa.aIf a(£a then it is clear t h a t Fa,a=Ra, for then the a-component of the points of F* is not restricted. If a£.a and if F* = ira~1Fa, then Fa- Then ira~1xa is a basic closed set (since Ra is a 2Vspace) which meets every set F* in 2r. Consequently ir^XatEiS, *a<Ei5a. a n d the intersection n a F a , 0 consists only of the point xa. Let then x= \xa}, where xa= C\aFa,a. We show t h a t x is a common point of the sets F in JJ. Since ir^Xatizft > for any a, it follows that r\aQaiTalxa^:%, t h a t is, i r . r ^ G g - , where xa=irax. Therefore ir^lxa meets F*, t h a t is, irZlFa; hence xaE.Fa and 3cGir
where a= \ai, a*, • • • , aT) and where each «/ can take the values 1 or 2. I t is well known t h a t Va is a Segre variety, of dimension n, immersed in a projective space of dimension 2 n — 1 . The points of Va are in (1, 1) correspondence with w-tuples of ratios {a: 0 2/x 0 i}, a £ a , t h a t is, with the points of the direct product Ra^R^XR^X • •• XRaH- I t should be noted t h a t here we only consider points X" whose homogeneous coordinates are in K. We topologize Va by choosing as closed sets the algebraic subvarieties of Va- Then it is clear t h a t each Va becomes a compact topological 7\-space. If a = { f l i , a2, • • • , a , } and if P is a subset of a, say if j8= {oi, at, • • • , am\, m
t/S)
_
(«)
_
(a)
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where each e, S and y can take independently the values 1 or 2. T h u s 7r]i| is a single-valued rational transformation of Va onto Vp, and therefore -n% is closed and (ir^) - 1 is open. I t is clear t h a t the closed sets in the infinite direct product R*, as defined above, are the sets defined by (finite or infinite) systems of homogeneous equations, each equation involving the variables Xia) relative to some finite subset a of A. 4. The space of homomorphic mappings of one abstract field into another. We now further specialize our application b y assuming t h a t the set A is a field. The space R* is then the space of all single-valued transformations x*:a—»#„ =xa\/xai, of the field A into the set consisting of the elements of the field K and of the symbol °o. We shall now express in an appropriate homogeneous form the conditions t h a t a given mapping x* be a homomorphism. Let a be a subset of A consisting of three elements, ot= \alt at, C3}. On the corresponding variety Va let Fai,ai,a, be the algebraic subvariety obtained by imposing on the 6 parameters x», x&, i=ait a2, a 3 , the following condition: \2)
XailXa22Xa32
"T Xai2Xa2lXaa2
—
XafiXatfXatl-
Similarly we define another algebraic subvariety Gai,at,at of Va by the equation (3)
Xa\\Xa
2l#aj2 —
Xai2Xatf.XaiV
hetxaji/xafi = Xj,j = l, 2, 3, where Xj may be » . Suppose t h a t equation (2) holds true. Then if Xi and #2 are both different from °o we find Xi=xi+X2. If X\= 00 and a ; 2 ^ » , then #0,2 = 0, x^i-x^^O, whence (2) yields xa# = 0, t h a t is, x3= » . Assume now t h a t equation (3) holds. Again we find t h a t X3=#i*2i if both Xi and #2 are different from » . If xx= 00 and # 2 ^ 0 , then xHi = Q, x^i-x^^O, and (3) yields xaii = 0, t h a t is, x3= 00. Thus the equations (2) and (3) are the homogeneous counterparts of the equations # 3 = * i + # 2 and x3=XiXt respectively, and they include the conventions which are usually made for the symbol °o. We can therefore assert t h a t x* represents a homomorphic mapping of A into (K, «>) if and only if the following conditions are satisfied: for any three elements a%, a2, 03 of A such t h a t respectively 03 = ^1+02 or a3 = aia2, the projection ir^x* (where a = { o i , a2, 03}) must lie respectively on F^^.o, or on G^.oj.o,. Therefore, if we denote by M the set of all homomorphic mappings of A into (K, 00), we see t h a t
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OSCAR ZARISKI
[October
where the index a ranges over all sets a=\ai, a%, a%\ such t h a t a3 = ai-\-az, and the index j3 ranges over the sets /3= {bx, b2, b3} such t h a t b3 = bib2. We see thus t h a t M is an intersection of basic closed subsets of R*. Hence M is closed, and since R* is compact M is also compact. T h e case which is of special interest to us is t h a t in which if is a subfield of A. In this case we are interested in the relative homomorphisms of A into (K, oo ), that is, in the homomorphisms x* which leave each element of K invariant. If M* is the set of all these relative homomorphisms, then it is clear that M* is the intersection of M with the closed set Oa^Kir^a. Here, according to our notations, T a _1 a denotes t h a t subset of R* which consists of the points x* whose a-component xa is a itself (a(EK). Hence also M* is a compact space. It is convenient to describe in algebro-geometric terms the relative topology induced in M* by the topology of M. Let xi, xi} • • • , xn be a finite set of elements of A. For each x, we introduce a pair of homogeneous parameters xn, xa such that xa/xa=Xi. We consider the algebraic variety Z over K whose general point has as homogeneous coordinates the quantities X(t) defined by the parametric equations
where each e,- can take the values 1 or 2. If the quantities x, are algebraically independent, then the variety Z coincides with the variety Va defined by the equations (1), a being the subset {*i, x2, • • • , xn\ of A. B u t in general Z is a subvariety of Va. If x * £ M * , then the m a p p i n g * * of A into (K, a>) must preserve all the algebraic relations between X\, xt, • • • , xn over K, since x* is a homomorphism. It follows t h a t the point irax* of Va must lie on Z. Now we observe t h a t the homomorphism x* defines a unique valuation of A/K whose residue field is K itself and whose center on Z is the point irax*. Conversely, every valuation of A/K whose residue field is K defines a relative homomorphic mapping of A/K onto (K, oo). We conclude t h a t if W is any algebraic subvariety of Z, then the set of all valuations of A/K having K as residue field and having center on Wis a closed subset of M*. By taking different finite subsets {xi, * » , - • • , * » } of A and different subvarieties W of Z we obtain a family of closed subsets of M* which form a basis for the closed subsets of M*. R E M A R K . Suppose t h a t A is a field of algebraic functions in any number of variables, over a given ground field k. We identify the field K with the algebraically closed field determined by k. The Riemann manifold M of A is the set of all zero-dimensional valuations v
i944l
THE COMPACTNESS OF THE RIEMANN MANIFOLD
691
of A. By the ground field extension k—*K we can embed A in a field A'*=*KA. Every relative homomorphic mapping of A' onto (K, <») determines uniquely a zero-dimensional valuation of A'/K, and vice versa. Every zero-dimensional valuation of A '/K induces a unique zero-dimensional valuation of A/K, but a given zero-dimensional valuation of A /K may be extendable in more than one way to a zerodimensional valuation olA'/K. It follows that the Riemann manifold M' of A '/K coincides with the space M* of all relative homomorphic mappings of A' onto (K, ») and is therefore a compact space. The Riemann manifold M of A/K is obtainable from M' by topological identification and therefore can also be converted into a compact topological space. That is precisely what we have proved in §2. THE JOHNS HOPKINS UNIVERSITY
909 ANNALS OF MATHEMATICS
Vol. 49, No. 2, April, 1948 © 1948 The Johns Hopkins University Press. Reprinted with permission.
ANALYTICAL IRREDUCIBILITY OF NORMAL VARIETIES Br OSCAR ZARISKI
(Received August 11, 1947)
1. Introductory concepts By a local domain we mean an integral domain which is at the same time a local ring in the sense of Krull [4]. If m is the ideal of non-units in a local domain o and if o* denotes the completion of o (with respect to the powers of m), we say that o is analytically unramified if the zero ideal in o* is an intersection of prime ideals. In other words: o is analytically unramified if o* has no nilpotent elements. If p is a prime ideal in an arbitrary local ring o, we say that p is analytically unramified if the local domain o/p is analytically unramified. I t is well known that if o* is the completion of o then o*/o*p is the completion of o/p (Chevalley [1], Proposition 5). I t follows that a prime ideal p in a local ring o is analytically unramified if and only if the extended ideal o*p in the completion o* of o is an intersection of prime ideals. The following theorem has been conjectured by the author and proved by Chevalley ([2], Lemma 9 on p. 9, last sentence, and Theorem 1 on p. 11): The local ring of a point P of an irreducible algebraic variety V is analytically unramified. I t follow ^ that any prime ideal p in such a ring is also analytically unramified, because p defines an irreducible subvariety W of V, and the residue class ring o/p is the local ring of the point P, this point now being regarded as a point of W. Note the following special case: V is the affine n-space over k, and P is the origin. In this case the completion of the local ring of the point P is the ring k (x) of formal power series in n independent variables X\,x%, • • • , xn, with coefficients in k, and therefore it follows that every prime ideal in the polynomial ring k[x] splits into prime ideals in the power series ring k (x). In informal geometric language this result signifies that an irreducible algebraic variety V can decompose in the neighborhood of a point P only into "simple" analytical branches (i.e., none of the branches has to be "counted" more than once). At any rate, it is true in the complex domain that the analytical reducibility of V in the neighborhood of a point can be no worse ideal-theoretically than it is set-theoretically. We say that a local domain is analytically irreducible if its completion has no zero divisors, and that a variety V is analytically irreducible at a point P of V if the local ring of P is analytically irreducible. We recall that V is said to be locally normal at P if the local ring of P is integrally closed. The object of this paper is to prove the following theorem: / / an irreducible algebraic variety V is locally normal at a point P, then it is analytically irreducible at P. In the course of the proof of this Theorem we shall arrive incidentally at another proof of Chevalley's result. 352
910 NORMAL VARIETIES
353
Our theorem is to be compared with another result concerning normal varieties and proved by the author elsewhere ([7], Definition 4, Theorem 8(A) and Theorem 10, pp. 512-514). We have shown namely that if V is locally normal at P and if a birational transformation of V into another variety V sends P into a finite set of points of the variety V, then this set consists necessarily of a single point. We saw in this result a strong indication of the analytical irreducibility of normal varieties. For if a variety V consists of s branches in the neighborhood of a point P, then one would expect that these s branches cquld be separated by a suitable birational transformation. Such a transformation would then replace P by s distinct points. 2. Some auxiliary lemmas The first two of the following lemmas refer to an arbitrary local ring o and its completion o*. The ideals of non-units in these two rings are denoted by m and nt* respectively. LEMMA 1. If 21 is an ideal in o and b is any element of o, then o*2l : 0*6 = o*(H :b). PROOF. It is sufficient to prove the inclusion o*2l : o*b C o*(2I : b). Let u be any element of o*2I : o*b, u = lim Ui, w, e o. We have Uieu + m* ,+1 ,
tub eub + 6m**+1,
i.e.,
Uib
e o* 21 + o*6m i+1 .
Therefore utb t 21 + 6m i + V u< e 21 : b + m'+1, and hence u « o* (21 : b) + m* , + \ for all i. This implies that u e o*(2l : b), as asserted. LEMMA 2. If 21 and 93 are ideals in o and if 21 : 93 = 21, then o*2l : 0*93 = o*2I. PROOF. Let px , p 2 , • • • , })„ be those prime ideals of 21 which are not contained in any other prime ideal of 21 (i.e., they are maximal with respect to the property of being prime ideals of 21), and let o,-y be an element of- p,- which is not in py {i, j = 1, 2, • • • , g, i j* j). From 21 :93 = 21 it follows that 93 is not contained in any of the prime ideals of 21. Hence we can find an element an which is in 93 but not in p<(i = 1, 2, • • • , g). We set bj = a\jO%j • • • agj, 6 = 6i + 62 +
••• +
b„.
Then b e 93, b
CD
o*p = pf n p* n • • • n p,*,
each p* being a prime ideal in o*.
The lemmas 3-7 concern certain properties
1 Here we are making use of the relation o*S8 D » = S3 which holds for any ideal SB in o (Krull [4], Theorem 15; also Chevalley [1J, Proposition 5).
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OSCAR ZARISKI
of these h prime ideals p ; . Let p* be one these prime ideals, say p* = pi. Let s #* be one of the prime ideals of the zero ideal in o* such that ty* C p*. We set (2)
n = o*/?*,
? = P7$* •
LEMMA 3. The quotient ring R = Qv is a discrete valuation ring of rank 1. / / to is art element of o such that to « p, to <* p ' and if to is the ^-residue of to, then Ru is the ideal of non-units in R. PROOF. Let a* be an element of p2 D p* f~l • • • ("I pfc which does not belong to Pi, and let b be an element of oco : p, not in p. If we set c* = a*b, then c* 4 p! since b is not in pi.' We have p*a*
P T D o = p.
912 NORMAL VARIETIES
355
it follows from this same lemma that ai e r\hi=ipi (n~1))3 a n d hence aj) = a\ w, i.e. a*b = a m . In this fashion we find, after n steps, the relation a*bn = a„wn, where a„ is some element of o*. Hence a* e 0*0" : 0*6" C o*p(n) : 0*6". Since b i p, it follows from Lemma 2 that a* e o*p n) , and this completes the proof of the lemma. LEMMA 8. Let 21 = DLiqi be an ideal in 0, w/&ere qi, q 2 , • • • , q„ are -primary ideals belonging to distinct minimal prime ideals pi, p 2 , • • • , p„ in 0. If each of the n ideals pi is analytically unramified, then o*2l = flLiO*qi. PROOF. I t is sufficient to prove the inclusion o*2I D n,"_iO*q;. Let a* be any element of the ideal on the right. For each i we fix an element «,• which is in pi but not in p[2) and not in pj, for j ^ i. Moreover, let bi be an element of Don : pi, not in p};, j — 1, 2, • • • , n. We know from the proof of Lemma 7 that if pi is the exponent of q,- (so that qi is then necessarily the symbolic power pvP,)), then a*bil is a multiple of-on1, say a*6P1 = ami1, at e 0*. Since OH 4 pj, j y£ 1, it follows that «i e n,"_20*q.-, and hence, by a similar argument we have a?b? = a*o>f, where at e n,"=3o*qi. Ultimately we find o^&J1^2 • • • &'„" = atai^-u'n e 0*21. Hence if we set b = bllb? • • • b^, then a* e o*2l : 0*6, whence, by Lemma 1, a* e o*(2I : b), i.e., a* e o*2l, since b 4 p,-., 1' = 1, 2, • • • ,n. This completes the proof. LEMMA 9. If there exists an element t 5^ 0 in 0 such that all the prime ideals of the principal ideal o-t are analytically unramified, then 0* has no nilpotent elements. PROOF. Let pi , P2 , • • • , pn be the prime ideals of 0 • t and let us assume that all these ideals are analytically unramified. They are also minimal primes in 0 since 0 is integrally closed. Let o*p,- = p,i f~l p i2 f"l • • • fl pi„,-, i = 1,2, • • • , n, and let $ 1 , ^3a, • • • , $ , be the prime ideals of the zero ideal in 0* which contain a t least one of the prime ideals p,;- (i = 1, 2, • • • , n; j = 1, 2, • • • , v>). We know from Lemma 6 that each of the prime ideals p,-,- contains one and only one of the s ideals ^ j , and from the proof of that lemma it follows that 1
tinyt n • • • ntf = ntin?^ \pTnpV n - n ^ i ,
or, in view of Lemma 7,
yt n
??n%*n • • • n?.* = n?=1o* {p{y) n p2'> n - . - n ^ i c 0 * e w , where lim v(j) = + 00. Hence ty* D $ 2 D • • • fl $ ,
(3) 3
y? n $i n - • - n $.* = (o),
q . e .d.
We have aio> e pf(n), hence passing to the ^-residues fii, « we find &i <5 e '?<">. By Lemma 3 it then follows that 61 « ^J("~1>, whence c« c pj (n_1) . Similarly it is shown that o i f t f " ^ , * - 1 , 2 , ••-,*.
913 356
OSCAR ZARISKI
3. Application to algebraic varieties As a first application we shall show that from Lemma 9 it is possible to derive Chevalley's theorem stated in section 1 of this paper. We first observe that it is sufficient to prove that theorem under the assumption that V is locally normal at P. For suppose that the theorem has already been proved under this assumption and suppose that we are dealing with a variety V and a point P of V such that V is not locally normal at P. Then we pass to a derived normal model V of V, and we consider the points P[, P[, • • • , P[ which correspond on V to the point P. Let o denote the local ring of P and let 5 denote the intersection of the local rings of the points P,-. Then 5 is a semilocal ring in the sense of Chevalley [1] and is a finite o-module (o is in fact the integral closure of o in the function field CJ{V) of V; see [7], p. 511). The completion of o contains the completion o* of o as a subring, in fact is a finite o*-module ([1], Proposition 7, p. 699). To show that o is analytically unramified, it is therefore sufficient to show that the completion of o has no nilpotent elements. Now the completion of o is the direct sum of the completions of the s local rings of the points P't ([1], Proposition 8, p. 700), and since V is locally normal at each of the points Pi it follows, from our hypothesis, that the local rings of points Pi are all analytically unramified. Hence the completion of 15 has no nilpotent elements, since the rings of which it is a direct sum have no nilpotent elements. We shall now proceed by induction with respect to the dimension r of V. If r = 1 and if the curve V is locally normal at P , then P is a simple point of V, the local ring of P is a regular ring ([8], p. 19), in fact a valuation ring, and the completion of this ring is itself a regular ring ([4]) which therefore has no zero divisors at all. Having shown this for normal curves, it follows from the preceding observation that Chevalley's theorem is true for algebraic curves. Now let us assume that this theorem is true for algebraic varieties of dimension less than r, and let V be an r-dimensional irreducible variety which is locally normal at a given point P . We have to show that the local ring o of the point P is analytically unramified. By our induction hypothesis we have that every prime ideal in o is analytically unramified (compare with section 1), and in particular every minimal prime ideal in o is analytically unramified. Hence the assumption in Lemma 9 is automatically satisfied for any element t in o, t ^ 0, and since o is integrally closed, it follows by that lemma that o* has no nilpotent elements. Before proceeding to the proof of the analytical irreducibility of normal varieties, we shall make a few geometric comments about some of the lemmas proved in the preceding section. The local ring o is now the local ring of the point P of an r-dimensional irreducible variety V, and V is locally normal at P. Let (3) represent the decomposition of the zero ideal of o*. In that case the variety V consists, locally at P , of s*analytical branches Mi, M2, • • • , M,, each Mi being an analytical manifold. For each prime ideal *EJ3i of the zero ideal we define as in (2) the domain Qt = o*/^< • Then tt{ is the ring of holomorphic functions on the analytical manifold M,•. Now let p be a minimal
NORMAL
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357
prime in o and let (1) be its decomposition in o*. Then p represents an (r — 1)dimensional subvariety W of V which contains the point P and which decomposes, locally at P, into h analytical manifolds, JVi, N2, • • • , Nh . Lemma 6 signifies that each of the h analytical (r — 1)-dimensional branches N,- of W lies on (one and) only one of the analytical branches Mi of V. This result was to be expected from an intuitive geometric viewpoint, since the intersection of two distinct analytical branches M, and My of V is part of the singular manifold of V and since, on the other hand, the singular manifold of V cannot possess an (r — l)-dimensional component at the point P where V is locally normal. Naturally this entire geometric picture which we are painting has real significance only in the classical case, because in the abstract case an analytical manifold is not a point set at all (there is no conceivable incidence relation between Mi and points of the affine ambient space of V, except that we may say that the origin P of the branch Mi is on Mi). At any rate is is now becoming clear why the hypothesis that o is integrally closed was a priori necessary in the proof of Lemma 6. As to Lemma 3, we shall only make the following observation: the fact that the quotient ring R is a valuation ring is to be interpreted geometrically in the sense that if N,- lies on M{ then N, is not singular for M,. This interpretation is again in connection with the fact that the singular manifold of V must be, locally at P, of dimension < r. The remarks just made disregard our final result (which we shall now proceed to prove) that actually V is analytically irreducible at P, whence s = 1, and o* is an integral domain. I t is still an open question whether o* is integrally closed. In other words: if V is locally normal at P, is V also normal as an analytical manifold? We now proceed to the proof of the theorem on the analytical irreducibility of normal varieties stated in section 1. Let (3) be the decomposition of the zero ideal of o* into prime ideals. We shall prove in the next section the following relations:
(4)
op* +
i*j.
Assuming for the moment relation (4), we now show that the assumption that s > 1 leads to a contradiction. Let us then assume that s > 1 and let Uj be an element of (^3i -f- $ , ) D o, uj ^ 0, j" = 2, 3, • • • , s. If we set u = W2MS • • • « . , we can write u in the form u = v* + w*, where v* e $ ? and w* « « * = %* fl $ ? fl ••• fl $f. Let OM = pi' 1 ' ft p2M) D • • • D pi*"', where the p,- are minimal prime ideals in o' and let o*p< = pa D p% D • • • D p , ^ . (Since we are dealing with the local ring of a point of an algebraic variety V, we know that every prime ideal of o is analytically unramified.) All the lemmas of section 2 are applicable. In particular, we find by lemmas 7 and 8 that 0*u
= n t i riyii p * / " ' .
Each of the ideals p,-, contains one of the ideals $ „ .
If p,, 3 ty,,, then by Lemma
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OSCAR ZARISKI
4 any symbolic power of p,-,- contains %,, and in particular p,,"' Z) $ „ . If ^ i = l then y* e "$„
R*q = qf fl q* 0 • • • 0 q*.
The ring R* is a complete local domain of dimension n. Each of the prime ideals q,- is of dimension r, the same as the dimension of the prime fl-ideal R- q ([2], Theorem 1). Relation (4) is equivalent to the following relation: (6)
(t|? + q*) 0 R * Rq.
Let Q * be any prime ideal of q< + q,-. any such prime ideal O * we have: (7)
To prove (5) we have to show that for
Q * fl R * Rq.
* Note that u is not a zero divisor in o*, since u t o. Therefore the relation (v* + w*)u = u implies v* + w* = 1.
916 NORMAL VARIETIES
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Let p be the dimension of Q*. Since O* is a proper divisor of q< and q* , it follows that p < r. We pass to the quotient ring O = RD- • This ring is a local ring of dimension n — p, and since the ring R* is a regular complete ring, it follows that also D is a regular ring (Cohen [3], Theorem 20, p. 97). We consider the ideal D q . By well known properties of quotient rings, the decomposition of the ideal O q is obtained from the decomposition (5) of R*q by replacing the prime ideals q, by their extensions £)-q<. These extensions are prime ideals. Moreover O • q ; is the unit ideal in O if and only if Q * does not contain q<, and two distinct prime ideals which are contained in O * give rise to distinct extensions. Since q, and q, are both contained in O*, it follows that Oq is not a prime ideal. Another property of the ideal Oq which we shall have to use is that its prime ideals are all of dimension r — p. This follows from the fact that the ideals q< are of dimension r and the ideal O * of which O is the quotient ring is of dimension p. Now let j/i(x), / 2 (x), • • • , JN(X) } be a basis of the prime polynomial ideal q. These N polynomials will also form a basis of the ideal £>q. We denote by 3K the ideal of non-units of the regular (n — p)-dimensional local ring £). The additive group 3K/50J can be regarded as a vector space 9H*, of dimension n — p, over the field £>/2ft ([8], p. 6). We now use the two properties of the ideal Dq which have been derived above. Since this ideal is pure (r — p)-dimensional, it follows that at most n — r of the polynomials/<(x) map, modulo 9J22, on independent vectors of 9H*. Moreover, since the ideal £)q is not prime, it follows that the set of vectors of'OIL* which correspond to the polynomials /,(x) contains actually less than n-r independent vectors ([3], corollary on p. 87). On the other hand, if we consider the local vector space 911 = 9R(F/<S) of V in S (see [8], p. 6) we find that the set of vectors of 9R which correspond to the polynomials fi(x) contains exactly n-r independent vectors, since any variety V in iS is simple for S. We shall now show that relation (7), and hence also relation (4), follows from this discrepancy between the dimensionalities of the vector spaces spanned by the polynomials fi(x) in the vector spaces 911 and 9U* respectively. We first assume that the function field ZF(V) is separably generated over k. We use the results of our paper [8] concerning the vector space 9R = 9R(F/<S) and the vector space 2>{V) of local F-differentials in S ([8], p. 25). Let £, be the q-residue of x< and let £,• be the Q-residue of x(. Since the function field of V is separably generated over k, it follows that the Jacobian matrix d(/i ,fi,---, fs)/ d(xi, x2 , • • - , x„) is of rank n-r at x = £ ([8], Theorem 7', p. 31). On the other hand, since the polynomials /,(x) span in 9R* a space of dimension less than n-r, it follows a fortiori that any n-r rows in the Jacobian matrix d(fi(x), fi{%), • • • ,JV(z))/d(£i, x2, • • • , x„) are linearly dependent over the field 9D?/33Z'. 1
A few words will suffice to explain this assertion. The partial derivations d/dxi have obviously the property of transforming into itself the quotient ring of any prime ideal in k[x\. In particular, these derivations transform R into itself. Moreover, if m is the ideal of non-units in R, then the partial derivatives of any element of tn' are elements of m' - 1 .
360
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ZARISKI
Hence this m a t r i x is of rank less t h a n n-r. We conclude therefore t h a t while all the (n-r)-rowed minors of t h e m a t r i x d(f1{x),f2{x), • • • , fN(x))/d(xx, x2 , • • • , x„) belong t o O , a t least one of these minors is n o t in q. This establishes (7) a n d henoe also (4). If 7{V) is not separably generated over k, t h e proof is the same except t h a t instead of the ordinary Jacobian m a t r i x we m u s t use the mixed J a c o b i a n m a t r i x introduced in our p a p e r [8] on p . 38. I n conclusion we observe t h a t in our proof of the analytical irreducibility of normal varieties we h a v e m a d e use only of two special properties of local, integrally closed, domains which are t r u e for local rings of points of normal varieties and which are not k n o w n t o be true in general. These t w o properties are t h e following: 1) the local domain o, and every prime ideal in o, is analytically unramified; 2) if *53» a n d tyj are a n y two distinct prime ideals of the zero ideal in o*, t h e n relation (4) holds. T h e first p r o p e r t y is certainly false for general local domains, if we drop t h e condition t h a t o is integrally closed ([6]; also [8], p . 24, where t h e ring o defined in (8) is easily seen t o be analytically ramified), b u t it is possible t h a t it holds for all integrally closed local domains. Also t h e e x t e n t t o which relation (4) is valid in the non-geometric case, is an unsolved question. On the answer t o these questions depends t h e answer t o the following general question: if o is an integrally closed local domain, is it true that the completion of 0 is also an integral domain? I t is trivial t h a t the answer is affirmative if o is of dimension 1 (for o is then a discrete rank valuation ring). ADDED IN P R O O F . A related question is the following: if o is a local domain such t h a t (a) its integral closure o is a finite o-module, is it true then t h a t (b) o is analytically unramified? I t is known (Krull [5]) t h a t if o is of dimension 1 t h e n (a) and (b) are equivalent. An affirmative answer to the first question would imply an affirmative answer t o this second question, since it can be easily shown t h a t u n d e r assumption (a) o is analytically unramified if and only if o is analytically unramified. HARVARD UNIVERSITY REFERENCES
[1] C. CHEVALLEY, On the theory of local rings, Ann. of Math., vol. 44 (1943), pp. 690-708. [2] C. CHEVALLEY, Intersections of algebraic and algebroid varieties, Trans., Amer. Math. Soc, vol. 57 (1945), pp. 1-85. From this it follows that each of the derivations d/dn has a unique extension in the complete ring R*. This extension will be denoted by the same symbol d/dn . By the same argument, the extended derivation d/3x,- in R* can be further extended to the quotient ring O, and moreover, the partial derivatives of any element of 3JJ" are elements of 2K'-1. Now consider any n — r of the polynomials fi(x), say fi(x), / 2 (x), • •• , /„_ r (x). Since the corresponding vectors of 3JI/W2 are linearly dependent, there is a relation of the form: A*fi(x) + Atf,(x) + ••• + AJ_r/„_r(x) e W2, where the A* are elements of O, not all in 2K. Applying the derivation 9/dx,- and observing that the polynomials /,(x) belong to 9J2, we find that Aldf^/Bn + A*dfs(x)/dXi + • • • + • A*-r dfn_r(x)/dXi is in 5W, for t = 1, 2, • • • , n. Since the A* are not all zero modTO, the assertion in the text follows.
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[3] I. S. COHEN, On the structure and ideal theory of complete local rings, Trans., Amer. Math. Soc, vol. 59 (1946), pp. 54-106. [4] W. KBTJLL, Dimensionstheorie in Stellenringen, J. Reine Angew. Math., vol. 179 (1938), pp. 204-226. [5] W. KRULL, Ein Satz uber prim&re Integritdtsbereiche, Math. Ann., vol. 103 (1930), pp. 540-565. [6] F. K. SCHMIDT, Uber die Erhaltung der KeUe.nsa.tze der Idealtheorie bei beliebigen endlichen Korpererweiterungen, Math. Zeit., vol. 41 (1936), pp. 443-450. [7] O. ZARISKI, Foundations of a general theory of birational correspondences, Trans., Amer. Math. Soc, vol. 53 (1943), pp. 490-542. [8] A. ZARISKI, The concept of a simple point of an abstract algebraic variety, Trans., Amer. Math. Soc, vol. 61 (1947), pp. 1-52.
Wolf Prize in Mathematics, Vol. 2 (p. 919) eds. S. S. Chern and F. Hirzebruch © 2001 World Scientific Publishing Co.
Correction
As the reader of the two volumes of "Wolf Prize in Mathematics" will have realized, there was a m a x i m u m of approximately 50 pages for each Wolf Prize winner. In t h e case of Professor Hormander (Vol. 1) the number of pages exceeded 100. Professor Hormander had clearly stated t h a t of his dissertation "On the theory of general partial differential operators" only Chapter IV was to be printed. T h e total number of pages of Professor Hormander's contribution would then have been approximately 50. By some error, however, the entire paper was printed. T h e editors a n d the publisher regret t h a t this mistake happened. It is a pleasure for the editors to give Professor Hormander the opportunity to make the following comment on his dissertation. "As stated on page 408 in Volume 1, my intention was to include from my thesis just the five pages in Chapter IV, which point forward to later work on solvability of differential equations with variable coefficients and give a background for another included paper. (The rest of my thesis concerns differential operators with constant coefficients.) Moreover, t h e entire thesis should not have been reproduced anywhere without a correction of Lemma 3.2. It is obviously false as stated which is particularly embarrassing since in a footnote I a t t r i b u t e d the proof to a personal communication from Professor B. L. van der Waerden! T h e error was of course entirely my own. T h e correct statement should be: L e m m a 3 . 2 . If R is a ring such that C C R C C[z], then one can find a •d in the quotient field of R such that R
polynomial
As Professor van der Waerden pointed out, this is an easy consequence of Liiroth's theorem. However, my a t t e m p t to give a direct proof for t h e benefit of analysts was incorrect and m a d e me claim t h a t $ can be chosen in R, which is false. Fortunately the rest of the paper is not affected by this correction of Lemma 3.2, a p a r t from an obvious modification in the proof of Lemma 3.3."
Photos of Wolf Prize Winners
1978
1979
1978
Izrail M. Gelfand
1979
Carl L. Siegel
1980
AecM Weil
1981
Lars V. Ahlfors
Jean Leray
1980
Henri Cartan
1981
Andrei N. Kolmogorov
1982
Oscar ZarisM
Hassler Whitney
1982
1983/4
1983/4 / «
Mark Grigor'evich Krein
SMieg S. Chern
Paul Erdds
1986
1984/5
1984/5
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3 *A
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Hans Lewy
Kueihiko Kodaira
1987
1986
Atle Selberg
t
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Samuel Eilenberg
1987
Kiyosi ltd
Peter D. Lax
1989
1988
1988
i. FnecMch Hirzebrach 1989
Lars Hormander 1990
John W. Milnor 1992
1990
Ennio De Giorgi 1992
Lennart A. E. Carleson
Ilya Piatetski-Shapiro 1993
John G. Thompson
J993
Alberto P. Caklcron
MIkhael Gromov
1994/5
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Jacques Tits
Jilrgen K. Moser
199S/6
199S/6
Robert Langlands
1996/7
Andrew J. Wiles
1996/7
Joseph B. Keller
1999
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Yakov G. Sinai 1999
Elias M. Stein
2000
2000
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i ! Liszlo" Lovisz
Raoul Bott
2001
Saharon Shelah
^ ^ > Jean-Pierre Serre
2001
Vladimir I. Arnold