No title

The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to the editor-in-chief, Sheldon Axler.

T h e F o u n d a t i o n s of Mathematics I would like to add a comment to the discussion bet w e e n S a u n d e r s Mac Lane and Craig Smoryfiski (Mathematical Intelligencer, Vol. 10 (1988), No. 3, 12-20) about the question of w h y mathematical logic got away from philosophy and the foundations of mathematics. The present writer has some experience in the area, having been active in the field of intuitionism and proof theory from 1965 to 1975. At that time the area was dominated by a few influential gurus, who, while not overly technically skilled, relentlessly gave out directives about what was the good and what was the wrong w a y of doing foundations, in much the same way as Saunders Mac Lane gives out directives on how to do mathematics. If one's work was not in the line of these gurus, one risked attracting edicts from one of these in the form of denigrating remarks at conferences or in survey articles. Some logicians avoided the area of foundations and preferred to work in purely technical areas where the gurus in question had no influence; I left the field of mathematical logic completely. The best way to guarantee the progress of mathematics in the long run is to let people do their research in peace. Bruno Scarpellini Mathematisches Institut der Universitdt Basel 4051 Basel, Switzerland

9Mathematics in Social Science In two recent Opinion columns Neal Koblitz and Herbert Simon disagreed about the merits of some work by Samuel Huntington. My purpose in writing is not to comment on the content of the debate per se, but rather to suggest that Koblitz draws the wrong lesson from this interchange. In particular, I disagree with his

statement that many social sciences are " . . . fields in which mathematical methods are rarely appropriate and are often misused." I w o n ' t d i s p u t e the p o i n t t h a t m a t h e m a t i c a l methods are "often misused," because I'm not sure what counts as often. My main quarrel is with the words "rarely appropriate." In my view mathematical methods are highly appropriate in the social sciences because they i m p o s e a discipline that is generally lacking in the traditional verbal discourse used in these subjects. The Huntington affair is an excellent case in point. It is because Huntington stated his theory using mathematical notation that its faults were evident. Anyone with a minimal degree of scientific training who saw Huntington's equations would be led to ask the same sort of questions that Koblitz raised. H o w are these variables measured? Is this simple correlation or is causation being implied? What is the empirical evidence for these relationships? Do you literally mean division or do you just mean an inverse relationship? The fact that the relationship was written as an equation made these questions both natural and obvious. Suppose, instead, that Huntington had simply said "political participation increases as social frustration increases," as Koblitz would apparently prefer. Exactly the same questions would be appropriate for this verbal spcificafion: h o w can these variables be measured, is this correlation or causation, and so on. However, I suggest that it is much less likely that these questions would be asked. The terms "political participation" and "social frustration" are m u s h y enough so that asserting a relationship between the two terms seems to have, on the face of it, little empirical content. I have no idea whether Huntington's theories provide useful insights about social behavior; perhaps, as

THE MATHEMATICAL INTELLIGENCER VOL. I1, NO. 1 9 1989 Springer-Veflag New York

3

Herbert Simon suggests, there are interpretations of his work that can be defended. But in my view, Huntington's attempt to state his position clearly enough so that it could be criticized should be applauded. That the attempt was inadequate is clear; but the lesson to be learned from this episode is not that mathematics is inappropriate in the social sciences, but rather that an attempt at a mathematical formulation is a useful w a y to expose ambiguity and avoid nonsense. In the words of Francis Bacon: "Truth emerges sooner from error than from confusion." Hal R. Varian Department of Economics University of Michigan Ann Arbor, M148109 USA

Neal Koblitz Replies Hal Varian has it backwards. H u n t i n g t o n ' s use of equations, far from making his theory easier to judge, evidently intimidated and mystified his readers. For over a decade, to the best of my knowledge, no one criticized those equations. Finally, it was not Huntington's social "science" colleagues, b u t rather a group of mathematicians and natural scientists who exposed the foolishness of Huntington's work. Huntington's "attempt at a mathematical formulation" should, as Varian suggests, "be a p p l a u d e d " only if one thinks that snowing one's colleagues deserves applause. Neal Koblitz Department of Mathematics University of Washington Seattle, WA 98195 USA

Eric Temple Bell. For a profile of Eric Temple Bell, which is scheduled to appear as a dedicatory chapter in the sequel to Mathematical People (Birkh/iuser Boston, 1985), I would welcome hearing from people who have recollections of Bell or possess letters from him. Constance Reid 70 Piedmont Street San Francisco, CA 94117 USA 4

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989

Herbert Simon suggests, there are interpretations of his work that can be defended. But in my view, Huntington's attempt to state his position clearly enough so that it could be criticized should be applauded. That the attempt was inadequate is clear; but the lesson to be learned from this episode is not that mathematics is inappropriate in the social sciences, but rather that an attempt at a mathematical formulation is a useful w a y to expose ambiguity and avoid nonsense. In the words of Francis Bacon: "Truth emerges sooner from error than from confusion." Hal R. Varian Department of Economics University of Michigan Ann Arbor, M148109 USA

Neal Koblitz Replies Hal Varian has it backwards. H u n t i n g t o n ' s use of equations, far from making his theory easier to judge, evidently intimidated and mystified his readers. For over a decade, to the best of my knowledge, no one criticized those equations. Finally, it was not Huntington's social "science" colleagues, b u t rather a group of mathematicians and natural scientists who exposed the foolishness of Huntington's work. Huntington's "attempt at a mathematical formulation" should, as Varian suggests, "be a p p l a u d e d " only if one thinks that snowing one's colleagues deserves applause. Neal Koblitz Department of Mathematics University of Washington Seattle, WA 98195 USA

Eric Temple Bell. For a profile of Eric Temple Bell, which is scheduled to appear as a dedicatory chapter in the sequel to Mathematical People (Birkh/iuser Boston, 1985), I would welcome hearing from people who have recollections of Bell or possess letters from him. Constance Reid 70 Piedmont Street San Francisco, CA 94117 USA 4

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989

Allen Shields*

Felix Hausdorff: Grundziige der Mengenlehre The first edition of Hausdorff's classic Principles of Set Theory appeared 75 years ago. Although there had been other treatments of the material somewhat earlier (see the Anhang, p. 449 for references), this is the book from which succeeding generations of mathematicians learned the elements of set theory and point set topology. The Introduction begins with the statement that the book is intended to be a textbook and not a m o n o g r a p h , and that complete proofs are given. Hausdorff states that W. Blaschke (Prague) drew all the figures. The first two chapters deal with sets and functions. The notations are not modern: If A and B are sets than their union and intersection are denoted (~(A,B) and ~(A,B). If the sets are disjoint, then the union is also denoted by A + B. The difference of two sets, denoted B - A, is defined only when A is a subset of B. On page 14 "rings" and "fields" of sets are introduced: a family of sets is called a ring if it is closed under finite unions and intersections, a family of sets is called a field if it is closed under differences and finite unions. A footnote states that the terminology is taken from the theory of algebraic numbers but that there is only a partial analogy which should not be pushed too far. A family of sets is called a c-family if it is dosed under countable unions; it is called a g-family if it is closed under countable intersections. Thus one has or-rings and or-fields. The third chapter deals with cardinal numbers and their arithmetic, including exponents. Here F. Bernstein's theorem is proved: if A and B are sets, if there is a one-to-one map of A onto a subset of B, and if there * Column editor's address: Department of Mathematics, University of Michigan, A n n Arbor, MI 48109-1003 USA

is also a one-to-one map of B onto a subset of A, then there is a one-to-one map of A onto B. Corollary: If a and b are cardinal numbers and if both a ~ b and also b ~ a, then a = b. Note, however, that at this point one does not yet know that any two cardinal numbers are comparable; this is proved in Chapter 5. Finally in this chapter it is shown that if c denotes the cardinal number of the continuum, then cc = 2 c. The fourth chapter deals with linearly ordered sets, the fifth with well-ordered sets, and the sixth with relations between ordered and well-ordered sets. Ordinal numbers and transfinite induction are introduced in Chapter 5, as is the first transfinite number 00. This chapter also contains Zermelo's proof that every set can be well ordered. The Axiom of Choice appears in the following form: if M is a nonempty set then to each nonempty subset A of M there is associated a distinguished element a = f(A) of A. Apparently H a u s d o r f f regards the existence of such a "choice function" f as obvious; at any rate the existence is merely asserted (p. 136) with no further discussion. The cardinal numbers associated with well ordered sets are called alephs; because of the well-ordering theorem every cardinal number is an aleph. It now follows that any two cardinals are comparable. Theorem: Every infinite aleph is equal to its square. The proof uses transfinite induction and is taken from Jourdain [1908]; the first (complete) proof is in Hessenberg [1906] p. 593. Chapter 6 (p. 141) contains the following result, sometimes called Hausdorff's Maximal Principle: If A is a partially ordered set, then A contains maximal linearly ordered subsets. This is proved using the well-ordering theorem of Zermelo. This is of course equivalent to Zorn's Lemma, which was only published 21 years later (see Zorn [1935]). J. L. Kelley in the appendix to Kelley [1957] emphasizes the Hausdorff

6 THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 1 9 1989Springer-VerlagNew York

Maximal Principle and uses it throughout the book. Hausdorff himself does not make as much use of it, often preferring to use transfinite induction. Nor does Hausdorff show that his principle implies the well-ordering theorem or the Axiom of Choice. Much of the material in Chapters 4, 5, and 6 is quite specialized and not of current interest. The last four chapters, 7, 8, 9, and 10, introduce topological ideas and related concepts. After defining metric spaces in Chapter 7, the author defines what are today called Hausdorff spaces (p. 213). This seems to be the place where abstract topological spaces were first defined. (Compare Weyl [1913], w where abstract two-dimensional Hausdorff manifolds are defined in terms of Euclidean neighborhoods satisfying certain axioms.) Hausdorff states in the notes at the end of the book (pp. 456-457) that he first developed this theory (Hausdorff spaces) in lectures at the University of Bonn in the summer semester of 1912. He goes on to discuss open sets, closed sets, limit points, boundary points, etc. Following Fr6chet a subset E of a topological space is said to be compact if every infinite subset of E has a limit point (which need not belong to E). Thus compact sets in this sense need not be closed. Hausdorff then proves (Borel's theorem) that every countable open cover of a closed compact set has a finite subcover. He also proves a partial converse: if every open cover (not necessarily countable) of E contains a finite subcover, then E is dosed and compact. In the notes (p. 457) he states: "'Many proofs of Borel's theorem have been given, some of them unimaginably complicated, compare Schoenflies [1913] p. 234." Finally, in w of Chapter 7 (p. 244) Hausdorff gives the m o d e m definition of connected space. He states in a footnote that many other definitions have been given, all different from that of the text, which seems to the author to be the most natural and the most general. Chapter 8, entitled "Point sets in special spaces," is by far the longest chapter in the book, almost 100 pages long. In w p. 263, the first and second axioms of countability (namely, each point has a countable neighborhood basis; and, the whole space has a countable basis) are introduced and discussed. In w a number of examples are gathered together with appropriate metrics: the space of all sequences (introduced by Fr6chet), (~2, C(0,1) with the s u p r e m u m norm, and C(0,1) with the L2 norm. In w the nonempty closed subsets of a closed compact set of a metric space are made into a metric space, with what is nowadays called the Hausdorff metric. The definition of this metric is equivalent to the following: d(A, B) = max{AB, BA}, where AB denotes the infimum of ~ such that B C A,, and A, denotes the set of points at distance less than e from A. Hausdorff states that the quantity AB occurs in Pomp6iu [1905]. In w of Chapter 8 Hausdorff shows (using a result of Liouville on the a p p r o x i m a t i o n of algebraic

numbers by rational numbers) how to construct a Gs set that contains all the rational numbers, but does not contain any irrational algebraic numbers. [Note by the column editor: LiouviUe's t h e o r e m is unnecessary here. It is easy to show directly that if D is any denumerable set of real numbers disjoint from the rationals, then there is a G8 set containing all the rationals but not containing any point of D.] In w the concept of a totally bounded subset of a metric space is introduced, and its relation to compactness is discussed. w discusses complete metric spaces. On p. 319 W. H. Young's theorem is proved: In a complete metric space an uncountable G~ set has at least the cardinality of the continuum (see Young [ 1906], Theorem 31, p. 64). (Earlier Cantor had proved this for dosed subsets of the real line.) This was a first step toward proving the corresponding result for general Borel sets, i.e., for verifying the continuum hypothesis for Borel sets; this was achieved two years later independently by Aleksandrov [1916] and by Hausdorff [1916]. w studies Euclidean spaces, while w studies topological properties of the Euclidean plane; in particular, the Jordan curve theorem is proved. The proof is taken from Brouwer [1910], but Hausdorff states that the method is very close to that used by Veblen [1905]. [Query by the column editor: Who gave the first complete proof of the Jordan curve theorem? Veblen's proof was from his dissertation, written at the University of Chicago under E. H. Moore.] Chapter 9 studies functions in topological spaces. Theorem II of w is the "Invariance of domain" for planar sets: If A and B are subsets of the Euclidean plane, with A an open set, and if B is a one-to-one continuous image of A, then B is an open set, and the inverse map from B to A is continuous. This is due to J(irgens [1879]. (The n-dimensional analogue is due to Brouwer.) The remaining sections of this chapter study functions and sequences of functions. The tenth and last chapter studies measure and content. The idea is to assign a nonnegative number f(A), the measure or content of A, to each set A in some collection of sets. The author states that the theory of measure developed in two stages: the first stage began with G. Cantor and H. Hankel, and was completed by G. Peano and C. Jordan. The principal requirement was finite additivity: f(A U B) = f(A) + fiB) when A and B are disjoint. The second stage, developed by E. Borel and H. Lebesgue, required the stronger condition of countable additivity. Hausdorff comments (p. 400): "'The transition from finite to countable additivity in the new measure and integration theory must be regarded as one of the greatest advances in mathematics." In the notes (p. 451) Hausdorff considers a ring ~ of sets, with a nonnegative function fdefined on ~ , such that f vanishes at the null set, and if X, Y E 9d?, with A = X U Y, B = X N Y, then fiX) + flY) = f(A) + fiB). THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989

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He proves that f extends uniquely to the smallest field of sets containing ~d~. By far the most famous part of this chapter occurs in the notes (p. 469). Here H a u s d o r f f presents his famous "paradox." Two subsets of the 2-sphere will be said to be congruent if there is a rotation of the sphere carrying one set onto the other. Hausdorff decomposes the sphere into four subsets, A, B, C, D, such that D is a countable set, A, B, and C are pairwise congruent, and finally, A is congruent to B U C! Thus there does not exist a finitely additive measure defined

This is the b o o k f r o m w h i c h succeeding generation o f m a t h e m a t i c i a n s learned the e l e m e n t s o f s e t theory a n d p o i n t set topology.

on all subsets of the 2-sphere, finite valued and not identically zero, such that congruent sets have the same measure. (Denumerable sets automatically have measure zero. Indeed, if n is a positive integer then one can choose n rotations of the sphere S such that the n images of D are all disjoint. Since these sets all have the same measure we have: nf(D) ~ f(S).) Hausdorff's proof depends on finding two rotations ~ and qJ such that ~2 = 1, & = 1, and there are no other relations between q~ and ~. Such measures do exist on the circle group (see, for example, Banach [1932], Chap. II, w However even on the circle such a measure cannot be countably additive. Indeed, on pp. 401-402 Hausdorff decomposes the circle into countably many pairwise congruent sets. The Hausdorff paradox was generalized in a famous paper by Banach and Tarski [1924]. For recent surveys see Wagon [1985] and French [1988]. Later in Chapter 10 Hausdorff develops the theory of Peano-Jordan content, then Lebesgue measure (on the line) and the Lebesgue integral (the integral of a positive function is defined as the measure of the set lying u n d e r the curve). Some of the convergence theorems are presented, as well as the basic differentiation theory. Thus Hausdorff's book served as an introduction to set theory, point set topology, as well as real analysis. We conclude with some information about Hausdorff's life, taken from a short article by W. Purkert (see Beckert-Purkert [1987], pp. 201-204). Additional information, including his publications u n d e r the name Paul Mongr6 (see below) may be found in DMV [1967]. Felix Hausdorff was born in Breslau on 8 November 1868, the son of a well-to-do merchant; the family 8

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moved to Leipzig in 1871. He graduated (promovierte) from Leipzig University in 1891 in astronomy, under H. Bruns. He completed the Habilitation in 1895 with a work entitled: " O n the absorbtion of light in the atmosphere," and became a Privatdozent, supported by his father. (Dozents were not paid a salary, they received only the small fees paid by students who attended the lectures.) During this period Hausdorff also wrote poems, and at least one successful play, under the pseudonym Paul Mongr6. In 1899 he married Charlotte Goldschmidt; they had one daughter. About 1900 Hausdorff became interested in Cantor's set theory. He lectured on it to three students in the summer semester of 1901. This may have been the first lecture course on set theory anywhere in Germany; Cantor himself, in his more than 40 years at Halle, never lectured on set theory. In a letter to Hilbert in 1907 Cantor indicated that he had suggested to Hausdorff some problems on ordered sets, and that he was pleased with Hausdorff's results. In 1901 Hausdorff was proposed for an associate (ausserordentliche) professorship at Leipzig. The faculty vote was 22 in favor and seven opposed. Before sending this on to the Minister, who had the final power to make the appointment, the Dean added a note stating that the minority had voted against the appointment because Hausdorff was of the "faith of Moses." He did receive the appointment. In 1910 Hausdorff went to Bonn as an associate professor, then in 1913 to Greifswald as a professor, and finally he returned to Bonn in 1921 as a professor. In 1935 the Nazis compelled him to retire. He was still permitted to publish until 1938. After that he prepared several manuscripts which went into storage; some were published after the War (see Hausdorff [1969]). On 26 January 1942, with deportation threatening, Hausdorff, his wife, and her sister committed suicide.

Bibliography Abbreviations: JFM = Jahrbuch~iberdie Fortschritte der Mathematik, MR = MathematicalReviews, ZBL = Zentralblattfar

Mathematik. P. S. Aleksandrov [1916], Sur la puissance des ensembles mesurables B, Compt. rend. Acad. Sci. Paris 162, 323-325. JFM 46, 301. S. Banach [1932], Th6orie des op6rations lin6aires, Warsaw. JFM 58, 420; ZBL 5, 209. S. Banach and A. Tarski [1924], Sur la d6composition des ensembles de points en parties respectivement congruentes, Fund. Math. 6, 244-277. JFM 50, 370-371. H. Beckert, W. Purkert [1987], Leipziger mathematische Antrittsvorlesungen, edited by H. Beckert and W. Purkert, Teubner Archiv zur Mathematik, Band 8, B. G. Teubner Verlagsgesellschaft, Leipzig. L. E. J. Brouwer [1910], Beweis des Jordanschen Kurvensatzes, Math. Ann. 69, 169-175. JFM 41, 544. DMV [1967], Felix Hausdorff zum Ged/ichtnis, Jahresb. Deutsch. Math. Verein. 69, 51-76. MR 34 #7330.

R. M. French [1988], The Banach-Tarski Theorem, Mathemat- A. Schoenflies [1913], Entwicklung der Mengenlehre und ical Intelligencer, 10(1988), No. 4, 21-28. ihrer Anwendungen, erste H/ilfte, B. G. Teubner, Leipzig and Berlin. JFM 44, 87. F. Hausdorff [1914], Grundz~ige der Mengenlehre, Veit & Co., Leipzig. Reprint: Chelsea Publishers, New York O. Veblen [1905], Theory of plane cur~es in non-metrical 1949. JFM 45, 123. analysis situs. Trans. Amer. Math. Soc. 6, 83-98. JFM 36, - [1916], Die M/ichtigkeit der Borelschen Mengen, 530. Math. Ann. 77, 430-437. JFM 46, 291. - [1969], Nachgelassene Schriften, edited by G. Berg- S. Wagon [1985], The Banach-Tarski paradox, Encyclop. Math. and Applic., vol. 24, Cambridge University Press. mann, Stuttgart, Teubner-Verlag. MR 39, 5306. MR 87e:04007. G. Hessenberg [1906], Grundbegriffe der Mengenlehre, Vandenhoeck & Ruprecht, GOttingen. JFM 37, 67-68. H. Weyl [1913], Die Idee der Riemannschen F1/iche, B. G. Ph. Jourdain [1908], On the multiplication of alephs, Math. Teubner, Leipzig; [1923] 2na ed.; [1955] 3'd ed. (StuttAnn. 65, 506-512, JFM 39, 101. gart). MR 16, 1097. English translation: [1964] The conE. Ji~rgens [1879], Allgemeine S/itze ~iber Systeme von zwei cept of a Riemann surface, Addison-Wesley, Reading, eindeutigen und stetigen reellen Funktionen von zwei Mass. MR 29 #3628. reellen Ver/inderlichen, B. G. Teubern, Leipzig. JFM I1, W. H. Young and Grace Chisholm Young [1906], The theory 269-270. of sets of points, Cambridge University Press. JFM 37, J. L. Kelley [1957], General topology, van Nostrand, 70. Princeton. MR 16, 1136. M. Zorn [1935], A remark on method in transfinite algebra, D. Pomp6iu [1905], Sur la continuit6 des fonctions de variBull. Amer. Math. Soc. 41, 667-670, JFM 61, 1028, ZBL 12, ables complexes, Ann. Fac. Toulouse 7, 264-315. JFM 36, 337. 454.

Comments o n p a s t c o l u m n s . Vol. 9, No. 2. At the end of the column we quoted from P. S. Aleksandrov's introduction to the Russian edition of Kelley [1957] where he refers to Moore-Smith convergence. The usual reference is to Moore-Smith [1922]. Aleksandrov (p. 7) states that "this same concept was discovered two decades earlier by the talented Odessa mathematician S. O. ~atunovski." However Aleksandrov does not give a reference. Since then V. I. Arnol'd has kindly lent me a book, ~atunovski [1923], which partly clarifies the matter. Arnol'd had obtained the book from his father, I. V. Arnol'd, who was also a mathematician and who had attended classes taught by ~atunovski. We quote from the introduction.

The appearance in print of this book I owe to my pupil I. V. Arnol'd who took on himself the labor of preparing for print my lectures on Introduction to Analysis, given at the Novorossisk University and then at the Odessa Institute of Popular Instruction [Narodnogo Obrazovaniya]. The primary purpose of the course was to establish the concept of real number. He goes on to say that he will introduce convergence along certain sets (he calls them manifolds), but he will not assume that these sets are linearly ordered. Instead, he only needs to require that to each two elements in the set there is a third element greater than both of them. He states that one can use this concept to prove the existence of the definite integral. In the text itself he does not discuss integrals, though he does point out that the set of all partitions of a closed interval, partially ordered by mesh size, has the above property. Thus we can conclude that he had the idea of directed sets, at least for some years prior to 1923. There

S. O . ~ a t u n o v s k i

is no indication that he applied the concept to general topology. As regards E. H. Moore, it is quite possible that he too had the idea long before 1922. He spent much of his life working Oh a theory of "general analysis," only part of which was ever published (see Moore [1935]). I remember meeting a former student of his some thirty years ago, who was supposed to be working on Moore's manuscripts; Moore died in 1932. For a brief biography of ~atunovski see Bogolyubov [1983], p. 29. Vol. 10, No. 2. In the course of discussing the origin of the modern concepts of differentiable manifold and Riemann surface we referred to the reproduction of Felix Klein's handwritten lecture notes. Such reproductions were called "Autographien." In this connection we have received an interesting letter from B. G. Teubner Verlag (701 Leipzig, Goldschmidt Str. 28, THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989 9

German Democratic Republic), the original publishers of these handwritten notes. The letter is signed by Dr. Sc. Genschorek (Cheflektor), and by Weiss (Lektor). They write that for his "Riemann Surfaces" notes, Klein himself wrote them out by hand; they were then hand-copied by collaborators for the printing process. As to whose handwriting we actually have in the published copy, they say that Klein wrote at various times mentioning people who had helped him with different sets of notes. For example concerning his lectures in Leipzig 1880/81 on geometric function theory, he writes that in 1892 the notes for the first semester were written in an Autographie by Paul Epstein (earlier the lecture notes had been reworked by Ernst Lange). [Note: Epstein is remembered, among other things, for the Epstein zeta function. He was dismissed from Frankfurt University in 1935 by the Nazis, and committed suicide in 1938. See Siegel [1966] pp. 470 for more details.] Another time Klein states that his assistant, Dr. Ernst Hellinger, "excellently qualified," prepared the Autographie. [Hellinger later worked in integral equations and operator theory. He was dismissed from Frankfurt University in 1935, emigrated to the USA, and finished his career as a professor at Northwestern University.] Still another time it was Ft. Schilling who prepared the Autographie. Teubner Verlag has recently (1986) republished Klein's lectures on Riemann Surfaces, with commentaries by G. Eisenreich and W. Purkert (Teubner-Archiv zur Mathematik, vol.5). They have also published (vol. 7) Klein's Funktionentheorie in geometrischer Behandlungsweise (the Leipzig lectures, 1880/81), with commentaries by F. K6nig and an introduction by F. Hirzebruch. Two additional books of mathematical historical interest are: Weierstrass's lectures on function theory 1886, and Berlin mathematicians of the nineteenth century. These books are available outside Eastern Europe from Springer-Verlag. We wish to thank Herren Genschorek and Weiss for this information. Bibliography A. N. Bogolyubov [1983], Mathematicians and Mechanicians, a biographical reference book, lzdat. Naukova Dumka, Kiev. MR 85f:01003. J. L. Kelley [1957], General topology, van Nostrand, Princeton. MR 16, 1136. Russian edition: Ob~aya topologiya, Izdat. "Nauka", Moscow (1968). MR 39 #907. E. H. Moore [1935], General analysis, written with the collaboration of R. W. Barnard, The American Philosophical Society, Philadelphia. ]FM 61, 433. E. H. Moore and H. L. Smith [1922], A general theory of limits, Amer. J. Math. 44 (1922), 102-121. JFM 48, 1254. S. O. ~atunovski [1923], Introduction to analysis, Mathess, Odessa. C. L. Siegel [1966], Gesammelte Abhandlungen III, SpringerVerlag, Berlin and Heidelberg, MR 33, 5441. 10

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A Nice Little Earner Once every decade the UK government produces a survey of the career achievements of university graduates. The latest such exercise has just appeared, the National Survey of 1980 Graduates and Diplomates, Department of Employment. (A detailed excerpt is given in Nature 334, 4 August 1988, pp. 393-4.) And guess what subject comes out with the highest salaries? For women as well as for men? Mathematics. Well, in combination with computing, but then we can all turn our hands to that, can't we? Here is an excerpt of the figures, including a few non-scientific subjects. It's very instructive, and could scotch a few myths.

Average Salaries in 1986 (s sterling p.a.) Average Salary Degree Subject

Men

Women

Mathematics, Computing Electrical Engineering Physics, Maths/Physics Biochemistry Business Studies, Economics, Law, and Accountancy Architecture, Town Planning, and other vocational subjects

16,610 15,250 14,330 11,470

13,520 10,480 11,540 10,850

16,480

13,420

12,650

9,950

We've always known we're like gold dust. Apparently, the rest of society agrees! Ian Stewart

The Opinion column offers mathematicians the opportunity to write about any issue of interest to the international mathematical community. Disagreements and controversy are welcome. An Opinion should be submitted to the editor-in-chief, Sheldon Axler.

An International Language for Mathematics Robert S. Strichartz Mathematics has long been an international discipline. Euler, a German-speaking Swiss, spent much of his career in Russia, but by writing primarily in Latin he could be assured that all his contemporaries could read his works. It has been a long time since Latin was the universal language of scholarly writing. Some would s a y - - a n d many more would h o p e - - t h a t English has taken its place. If so, this has been a short-

12

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term p h e n o m e n o n , a result of successful American cultural imperialism; we certainly cannot rely on it continuing indefinitely. And there are still many exceptions to the rule that all important mathematical writing is in English (or rapidly translated into English). The French persist in writing in their language, and the Russians produce a body of Russian language mathematics of large volume and high quality. The Germans, Italians, Hispanics, and Japanese have by and large capitulated; we do not know what the Chinese will do as they become a mathematical superpower. Therefore, to accept English as the international language of mathematics strikes me as shortsighted, presumptuous, and dangerous. Given that mathematicians will be writing in a number of different languages, what are the prospects for rapid and intelligible translations? Recently, the mathematics library at Cornell suggested the possibility of dropping its subscriptions to some translated Russian journals in order to save money. The reasoning is that these journals are not frequently used, and we already receive the Russian-language originals, so our collection would remain complete. Alas, only the Russian-born among us can read them. The reason we do not drop the subscriptions to the originals is that their cost is negligible in comparison to the cost of the translations. The translation process is a time-consuming and difficult task, and the results are sometimes unsatisfactory when the translators do not understand the texts as mathematics. To translate an entire journal cover to cover certainly seems inefficient, when only a few of the translations will actually

Springer-Verlag New York

be read at a particular library. On the other hand, the meaning, and so can be translated into an Intermath cost of translating articles on demand may also be pro- sentence. Undoubtedly there are many ways of realizing the meaning of a mathematical sentence in a nathibitively expensive. The computer would seem the natural solution to ural language, so an Intermath translated text may our translating problem. Of course, attempts so far to lack the spontaneity and stylistic elegance of a wellhave automatic computer translation of general texts written text. Also, a portion of the text, especially in have not been successful. I would guess that some the introduction and in motivating remarks, may be modification of G6del's Incompleteness Theorem beyond the reach of the Intermath language. But on would show that automatic translation, if not impos- balance, aren't we better off having a translation that sible, is at least an extremely difficult task. However, is accurate and intelligible, even if it is stylistically tesuch problems would not necessarily arise for transla- dious and missing a few paragraphs? H o w m u c h w o u l d mathematical writers have to tion of texts that were designed in advance for easy translation. Mathematical texts use a limited vocabulary in a change their practices in order to produce a text that very stylized way, and thus suitably prepared mathe- would be suitable for automatic translation into Intermatical texts would seem the first plausible candidates math? This is probably the key issue that will deterfor an automatic translation system based on an inter- mine whether an Intermath system can be used. If we national language that would serve as an intermediary c a n n o t write, " t h e r e are an infinite n u m b e r of between different natural languages. Say we call this primes," but are forced to substitute, "the cardinality language Intermath; a mathematician writing in En- of the set of prime numbers is infinite," I am afraid we glish would have her text automatically translated into will all curse the system and do what we can to saboIntermath. No one would read the text in Intermath, tage its adoption. On the other hand, if all we are rebut a Chinese mathematician could have the Inter- quired to do is answer a few queries about ambiguous math text automatically translated into Chinese. To sentences, I don't think very many of us will object. ensure the correctness of the translation, the author Probably we cannot know how such a system would would simply have the Intermath translation of the work until one is constructed and tested. But even in text automatically translated back into English (pre- the worst case, if such a system proved impractical, I sumably the computer would not be allowed to cheat still think we would learn a lot about the way matheby peeking at the original). If the sense of the text is matics is written. preserved, then the translation into Intermath is corWho would construct the system, and who would rect; if not, the author will have to correct the errors. finance it? I think the project is challenging enough W h a t w o u l d be i n v o l v e d in setting up such a that talented mathematicians, computer scientists, system? We already have a simple model of what a a n d linguists w o u l d work on it, p r o v i d e d it was mathematical language might be in the formal lan- funded. Therefore I appeal to the professional matheguages of mathematical logic. In principle these lan- matical societies to consider organizing and funding guages are capable of expressing any mathematical ar- such a project. gument, although in practice such expressions would The benefits of such a project would extend beyond be overly complicated. This is not surprising if we re- the mathematical community. No doubt other sciences member that these formal languages were created for and technical disciplines might imitate the idea sucthe purpose of studying the algebra of mathematical cessfully. Also, by strengthening the tradition of interproof, so a premium was put on limiting the number national cooperation in mathematics (and other culof symbols and the different ways they can be com- tural disciplines), the creation of an international bined. Intermath, as a formal language, would be cre- m a t h e m a t i c a l l a n g u a g e m a y help to b u i l d some ated to embody all the different ways of expression bridges between the peoples of this planet. Our govused by mathematicians in their normal modes of ernments may never be able, on their own, to break writing. The inventors of Intermath will have to ex- out of the quagmire of hostility and warfare that has amine actual mathematical texts to see how language characterized international relations since time immeis used, and then capture these uses in Intermath. A morial. But we, the mathematicians of the world, can medium-sized vocabulary of technical terms will be maintain a system of international relations based on needed; they will have to be updated from time to cooperation, friendship, and mutual respect that can serve as a model for others to follow. time to include newly coined terms. An automatic translation program back and forth between all the natural languages and Intermath will have to be written, along with a program for modifying Intermath text by a user who does not know In- Mathematics Department termath. The underlying assumption is that every sen- Cornell University tence of a mathematical text has an u n a m b i g u o u s Ithaca, NY 14853 USA

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Interview With Dirk Jan Struik David E. Rowe

Dirk J. Struik was born in Rotterdam in 1894, where he attended the Hogere Burger School from 1906-1911 before e n t e r i n g L e i d e n University. At Leiden he studied algebra and analysis with J. C. Kluyver, geometry with P. Zeeman, and physics under Paul Ehrenfest. After a brief stint as a high school teacher at Alkmaar, he spent seven years at Delft as the assistant to J. A. Schouten, one of the founders of tensor analysis. Their collaboration led to Struik's dissertation, Grund-

the founding editors of the journal Science and Society, Professor Struik has been one of the foremost exponents of a Marxist approach to the historical analysis of mathematics and science. At the present time he is completing a study on the history of tensor analysis while working on his autobiography. He is a passionate devotee of Sherlock Holmes. This interview is excerpted from a December 1987 conversation.

ziige der mehrdimensionalen Differentialgeometrie in direkter Darstellung, published by Springer in 1922, and numerous other works in the years to follow. From 1923 to 1925 Struik was on a Rockefeller Fell o w s h i p while s t u d y i n g in Rome a n d G6ttingen. During these years he and his wife Ruth, who took her degree under Gerhard Kowalewski at Prague, met m a n y of the leading mathematicians of the era--LeviCivita, Volterra, Hilbert, Landau, et al. After befriending Norbert Wiener in G6ttingen, Struik was invited to become his colleague at M.I.T., an offer he accepted in 1926. He taught at M.I.T. until his retirem e n t , except for a f i v e - y e a r p e r i o d d u r i n g the McCarthy era w h e n he was accused of having engaged in subversive activities. He has also been a guest professor at universities in Mexico, Costa Rica, Puerto Rico, and Brazil. Beyond his work in differential geometry and tensor analysis, Professor Struik is widely known for his acc o m p l i s h m e n t s as a historian of mathematics and science. His Concise History of Mathematics (recently reissued with a new chapter on 20th century mathematics) has gone through several printings and has been translated into at least sixteen languages. His Yankee Science in the Making, a classic account of science and technology in colonial New England, is considered by many to be a model study of the economic and social underpinnings of a scientific culture. As one of 14

Rowe: You entered the University of Leiden in 1912 with

the intention of becoming a high school mathematics teacher. What made you change your mind and how did you manage to break into the academic world? Struik: The man who enabled me to enter academic life was Paul Ehrenfest. Ehrenfest was born in Vienna and studied in St. Petersburg and G6ttingen. He and his Russian wife, Tatiana (Tanja), had made a name for themselves with their book on statistical mechanics. It was the first work to take into account the achievements of Boltzmann and Gibbs, a great step forward at the time. In 1912 Ehrenfest was appointed professor of mathematical physics at Leiden, succeeding the great H. A. Lorentz. Ehrenfest felt greatly honored to serve as Lorentz's successor, but he was dismayed by the stiff formality of the Leiden academic world where students only saw their professors in class and half the student body disappeared by train before s u n d o w n . Having come from G6ttingen, he was greatly influenced by the atmosphere there, and so he implemented some of the same reforms that Felix Klein h a d introduced. One of these was the mathematical-physical library, the Leeskammer, which on Ehrenfest's instigation was housed in Kamerlingh Onnes's laboratory. There students could browse

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1 9 1989 Springer-VerlagNew York

Curie, Rutherford, Einstein--so it is little wonder that the most talented students were attracted to physics. I never felt quite at home in this field; I was always more adept at thinking in terms of mathematical-and especially geometrical concepts. Rowe: Did Ehrenfest's ideas influence you in any definite way beyond the impact of his personality?

Dirk Struik during his student days.

through a wide variety of books rather than being confined to picking out a few at a time from the dusty university library. Just like in G6ttingen, the Leeskammer proved to be a central meeting spot for students and faculty alike, and before long there was considerable intermingling between them. Rowe: Were physics and mathematics closely allied fields at

Leiden ? Struik: To a considerable degree, although perhaps no more than elsewhere at this time. This was of course a period in which revolutionary changes were taking place in physics, and Leiden was one of the leading centers in the world with Lorentz, Ehrenfest, and Kamerlingh Onnes. Lorentz and Kamerlingh Onnes were Nobel Prize winners. The latter presided over his cryogenic lab where he ran experiments on the liquification of gases under low temperatures; only shortly before this he had discovered superconductivity. Lorentz was by n o w curator of Teyler's M u s e u m in Haarlem, but he came to Leiden once a week to lecture on a variety of subjects from statistical mechanics to electrodynamics, all in his serene and masterful way. It was often said that his lectures were full of pitfalls for the unwary, as he had a way of making even the most difficult things look easy. We heard other est e e m e d visitors from a r o u n d the w o r l d - - M a d a m e

Struik: Oh indeed, he himself had been influenced by Felix Klein's views, which stressed the underlying unity of ideas that were historically unrelated, like group theory, relativity theory, and projective and non-Euclidean geometry. The way he taught statistical mechanics and electromagnetic theory, you got the feeling of a growing science that emerged out of conflict and debate. It was alive, like his lectures, which were full of personal references to men like Boltzmann, Klein, Ritz, Abraham, and Einstein. He told us at the beginning that we should teach ourselves vector analysis in a f o r t n i g h t - - n o babying. Ehrenfest's students all acknowledge how much his method of exposition has influenced their own teaching. I remember a digression he once entered into on integral equations that I later used in my own course. He also recommended extracurricular studies; in my case he advised me to study group theory (again Klein's influence) together with a fellow pupil. I once asked Ehrenfest what was then one of the difficult questions of that day: whether or not matter exists. He proceeded to explain not only the status of matter as of 1915 (when E = mc 2 had just been put on the map), but also how the facts of sound and electricity tie in with the three dimensions of space, noting that if the Battle of Waterloo had been fought in a two-dimensional space we would be able to detect the sound of its cannon fire even today. Rowe: Who were some of the other students you got to know in Leiden? Struik: There were several whom I got to know quite well, especially through our scientific circle "Christiaan H u y g e n s . " One of the most outstanding was Hans Kramers. He, too, came from Rotterdam, but he attended the Gymnasium, so we did not know one another in high school. He later took his Ph.D. under Niels Bohr at Copenhagen and eventually succeeded Ehrenfest at Leiden. Another was Dirk Coster who also studied under Bohr and co-discovered a new elem e n t ( H a f n i u m - - H a f n i a e is the Latin for Copenhagen). He later returned to the Netherlands and became professor of physics in Groningen. Rowe: What did you do after graduation? Struik: My stipend had run out so I had to look for THE MATHEMATICAL INTELL1GENCER VOL. 11, NO. 1, 1989 1 5

work, which was not difficult to find in the summer of 1917 with so many young fellows tucked away in garrison towns. I took a job as a teacher of mathematics at the H.B.S. (high school) in Alkmaar, twenty miles north of Amsterdam. But in November I received a letter from Professor J. A. Schouten in Delft inviting me to join him as his assistant there. Schouten was by training an engineer, but he eventually succumbed to his love of mathematics. His doctoral dissertation, which was published by Teubner, dealt with the construction and classification of vector and affinor (tensor) systems on the basis of Felix Klein's Erlangen Program. After some soul-searching, I decided to accept his offer, and I ended up spending the next seven years in Delft. The salary was less than at Alkmaar, but it gave me a wide-open window on the academic world. Rowe: It must have been an exciting period to work on tensor calculus. Struik: It surely was. Schouten had shown that an application of the ideas in Klein's Erlangen Program could lead to an enumeration not only of the rotational groups underlying ordinary vector analysis, but others like the projective and conformal groups for any number of dimensions. Schouten's formal apparatus was algebraic, but it was accompanied by suggestive geometric constructs. We n o w know, of course, that Elie Cartan was working on related problems from a different point of view. With his great mastery of Lie group theory and Darboux's tri~dre mobile, Cartan was able to dig deeper and obtain his own results with an almost deceptive elegance. But none of us knew of Cartan's w o r k in 1918; his fame came much later. Schouten's work appealed to me first because of its close ties with Klein's Program, which was already familiar to me through Ehrenfest, and secondly because of its close connection with Einstein's general theory of relativity. It was not just the formal apparatus of tensors that interested me, it was the dialectics involved. For Klein, these were the interplay between complex functions, Euclidean and non-Euclidean geometry, continuous and discontinuous groups, Galois theory and the properties of the Platonic solids, et al. For Einstein, his field theory established connections between geometry, gravitation, and electrodynamics. Rowe: To what extent was Schouten's mathematics related to recent developments in Einstein's theory? Struik: At the time I joined him in Delft he was busy applying his ideas to general relativity theory, i.e., the direct analysis of a Riemannian space of four dimensions. The algebra involved was fairly simple, but the differentiation required new concepts because the curvature is non-zero. Schouten was able to introduce co-

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variant differentiation on such a space by considering w h a t he called g e o d e s i c a l l y m o v i n g c o o r d i n a t e systems. This enabled him to introduce new structure into the already existing tensor calculus utilized by Einstein. It was top-heavy with formalism, but Lorentz took an interest in it and helped to see that it was published by the Dutch Academy of Sciences. One day in 1918 Schouten came bursting into my office waving a paper he had just received from Levi-Civita in Rome. "He also has my geodesically moving systems," he said, "only he calls them parallel." This paper had in fact already been published in 1917, but the war had prevented it from arriving sooner. As it turned out, Levi-Civita's approach was much easier to read, and of course he had priority of publication. But few people realize that Schouten barely missed getting credit for the most important discovery in tensor calculus since its invention by Ricci in the 1880s. Rowe: You must have had a good working relationship with him. Struik: Yes, though Schouten was neither an easy chap to work with nor to work for, but we had few difficulties, especially after I outgrew the position of being merely his assistant and became his collaborator and friend. I certainly learned a great deal from him; especially the combination of algebraic and geometric thinking typical of Klein and Darboux. Our first common publication appeared in 1918; it investigated the connection between geometry and mechanics in static problems of general relativity. Thus it accounted for the perihelion movement of Mercury, then a crucial test for Einstein's theory, by a change of the metric corresponding to a corrective force.

Rowe: When did you complete your doctoral thesis? Struik: Originally, I planned to write my dissertation with Kluyver at Leiden on a subject in algebraic geometry, either on the application of elliptic functions to curves and surfaces, or a topic related to the RiemannRoch theorem in the spirit of the Italian and German schools. De Rham's work appeared shortly afterward, revealing that there was a future in this field of research, especially since he showed h o w one could utilize concepts from tensor analysis. But in 1919 I was not aware of these possibilities, and anyway I had become increasingly occupied with t e n s o r calculus through Schouten. I therefore arranged to have W. van der Woude, the Leiden geometer, as my thesis advisor, although the actual work grew out of my collaboration with Schouten on the application of tensor methods to Riemannian manifolds. I finally completed my thesis in 1922 and received my Ph.D. in July of that year. It was written in German and published by Springer in Berlin. The title was Grundzi~ge der mehrdi-

Rowe: Did you learn a lot from him about tensor calculus?

Jan Arnoldus Schouten

Struik: No, not really. In Rome he suggested that I should take up a new field. He showed me a paper he had recently published on the shape of irrational periodic waves in a canal of infinite depth and asked if I would like to tackle the same problem for canals of finite depth. It involved complex mapping in connection with a non-linear integro-differential equation to be solved by a series expansion and a proof of its convergence. Even t h o u g h Levi-Civita's m e t h o d s appeared applicable to this case, the problem was far from trivial. It also appealed to me, as I liked to test my strength in an unfamiliar field. Rowe: Did he give you any further guidance with this

problem, or was he too busy with his own affairs?

mensionalen Differentialgeometrie in direkter Darstellung. Following a time-honored tradition, I paid for the book myself, which was an easy proposition in 1922. The inflation in Germany was such that it is entirely possible that the little party I threw afterwards for family and friends cost me more in guilders than the whole dissertation of 192 pages. Rowe: I believe it was around this time that you and your wife first met. Struik: Yes, Ruth and I met at a German mathematical congress in 1922 and were married in the ancient Town Hall of Prague in July of the following year. She had a Ph.D. from the University of Prague, where she had studied under Georg Pick and Gerhard Kowalewski. Her thesis was a demonstration of the use of affine reflections in building the structure of affine geometry, a new subject at the time. After our marriage we settled in Delft for a brief time before travelling to Rome on a Rockefeller Fellowship. We spent nine months there while I worked with Tullio Levi-Civita. Rowe: What sort of a man was Levi-Civita? Struik: He was short and vivacious; his manner combined great personal gentleness and charm with tremendous will power and self-discipline. He was then about 50 years old and at the height of his fame as a pure and applied mathematician. His internationalist outlook derived from the ideals of the Risorgimento. His wife was a tall blonde woman of the Lombard type who was equally charming and graceful. She had been a pupil of his and was now his faithful friend and devoted companion; they had no children.

Struik: I had the benefit of seeing him often, either at his apartment in the Via Sardegna or at the University near the Church of San Pietro in Vincoli where I often had a look at Michelangelo's Moses, which I greatly admired. The Leiden philosopher Bolland once said that the Moses remains gigantic even in the smallest r e p r o d u c t i o n . Yes, Levi-Civita was one of those persons who in spite of a busy and creative career always seemed to find time for other people. He was remarkably well organized. I can still hear him saying, after I asked him to write a letter for me, "Scriver6 i m m e d i a t e m e n t e " - - a n d he did. Rowe: How did your work on canal waves come out? Struik: It went well, and I was able to bring it to a successful conclusion. Abstracts of it were published in the Atti of the Accademia dei Lincei, and later the full text came out in Mathematische Annalen. It was evidently read and studied, and later the theoretical results were experimentally verified by a physicist in California. Rowe: Who were some of the other interesting figures you

met during your year in Italy? Struik: On the floor above Levi-Civita's apartment lived Federigo Enriques, who was known for his research in algebraic geometry and the philosophy of science. When he heard that Ruth had graduated with a thesis in geometry, he invited her to prepare an Italian edition of the tenth book of Euclid's Elements. She accepted, and spent much of her time preparing the text with modern commentary. Maria Zapelloni, a pupil of Enriques, corrected her Italian. It was published along with the other books in the Italian edition of the Elements. Besides Enriques, there were a number of other prominent mathematicians whom I THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989

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met either at the university or at small dinner parties thrown by Levi-Civita and his wife. There was gentle Hugo Amaldi, who was then writing a book on rational mechanics with Levi-Civita. Then there was Guido Castelnuovo with his strong Venetian accent, and (but only at the university) the grand old man, Senatore Vito Volterra, President of the Accademia dei Lincei. I followed his lectures on functional analysis, which were largely based on his own researches. His delivery was impeccable, a style that reminded me of Lorentz's" lectures. Volterra was a senator, as was Luigi Bianchi, who came to Rome from Pisa for sessions of the Senate, and whose books and papers had been among my principal guides in differential geometry. On a day excursion with the Levi-Civitas we also met Enrico Fermi, but since he was a physicist we saw little of him thereafter. Little did we imagine that he would one day be a man of destiny, a real one, not like the fascist braggart known as "I1 Duce."

Rowe: It seems that Italian mathematicians took a fairly ac-

tive role in politics from the time of Cremona and Brioschi.

Struik: Indeed, political involvement was not uncommon among Italian scientists ever since the Risorgimento. The ones I knew were all anti-fascists with the sole exception of Francesco Severi, another outstanding algebraic geometer. On the other hand, their antipathy towards the Mussolini regime was not a militant one, so far as I could see. Volterra was an exception in this regard. He and Benedetto Croce actively attacked the regime from their seats in the Senate. After 1930 Volterra was dismissed from the University and stripped of his membership in all Italian scientific societies. The same thing later happened to Levi-Civita. To the honor of the Santa Sede, he and Volterra (both of w h o m were Jews) were soon thereafter appointed by Pope Pius XI to his Pontifical Academy. Rowe: What was the political atmosphere like during your stay in Rome?

Struik: One could not help be aware of fascism with all the blackshirts strutting through the streets of the city, but the political climate was relatively mild in those days, at any rate compared with what came later. The murder of Giacomo Matteotti, the Socialist opposition leader in the parliament, was still fresh in everyone's mind, and the resulting crisis in the government was very much unresolved. Mussolini tried to disavow the murder and tighten police control, but his dictatorship was off to a shaky start. Opposition papers could still appear, even the Communist Unit~. I was able to establish a contact with one of their contributors, and I used the information he passed on to 18

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me to write an occasional article for the Dutch party paper, De Tribune. I do not believe that I ever saw Mussolini in person, despite his high visibility. He used to parade a r o u n d on horseback in the Villa Borghese, but I had too much contempt for the sawdust Caesar to go out of my way to see this spectacle. Rowe: I guess Rome had plenty to offer mathematically in

those days. Were there many other foreigners who came to study or visit? Struik: Yes, there were other Rockefeller fellows in

Rome, and we struck up an amiable acquaintance with Mandelbrojt and Zariski, both of whom went on to become famous in their respective fields of research, Mandelbrojt at the Sorbonne and Zariski at Harvard. Mandelbrojt had worked on problems in analysis with Hadamard, and Zariski was studying algebraic geometry. Paul Aleksandrov, the Russian topologist, also spent some time in Rome. At that time he especially enjoyed the relative luxury of Italy after enduring the many privations in his homeland, which was just recovering from hunger and civil war. He told us that to do topology in Russia at that time you had to convince the authorities that it was useful for economic recovery. So the topologists told them that their field could be of service to the textile industry. Aleksandrov admired my winter coat, and when he learned that I had bought it with money from my stipend he dubbed it the "paletot Rockefeller." Rowe: It sounds as though all in all you had a splendid time

in Italy. Struik: Yes, we grew very fond of life there and enjoyed many memorable experiences. I remember visiting the Vatican on Christmas Eve to witness the opening of the Anno Santo, the Holy Year 1925, in which one could receive special indulgences. The e n o r m o u s basilica was filled with throngs of worshipers who had come to see the pope. He entered through a special door, the Porta Santa, in an ornately decorated chair carried high on the shoulders of selected members of the papal nobility, while the crowd shouted: "Viva il Papa!" Another occasion I recall occurred at a meeting of the Accademia dei Lincei. LeviCivita often took us to these sessions held in its ancient palace on the left bank of the Tiber. On this particularly ceremonious occasion the A c a d e m y was visited by II Re Vittorio Emmanuele and his still handsome Queen, Helena of Montenegro, once a famous b e a u t y . He h a d short l e g s - - e i n e Sitzgri~sse as the Germans would s a y - - b u t both majesties did very well on their decorous chairs, he with a bored face listening to the speeches. After the ceremony one was permitted to go up and shake hands with the monarchs, and I was amused to see how the Americans in the

audience crowded around them. I preferred to sample the pastry and sherry instead, and I was pleased to see that Levi-Civita also kept his distance from the Presence. It is said that when someone once asked Einstein what he liked about Italy, he answered, "Spaghetti and Levi-Civita." I felt pretty much the same way. I also l e a r n e d to like, with some a m u s e m e n t , that curious blend of Catholicism and anticlericalism found among many intellectuals and socialistically-inclined workers. I had not encountered this attitude in the Netherlands, where Catholics (at any rate in public) faithfully followed their clergy in matters of morals and politics. Catholic anticlericalism in Italy, on the other hand, dates back at least as far as the Risorgimento, when the Pope was an obstacle to reform and unity, but may in fact have had its roots in the Renaissance. Galileo is a good example. As my colleague Giorgio De Santillana once told me, Galileo's attitude can only be understood if one is aware of the phenomenon of anticlericalism among Italian Catholics. Giordano Bruno's statue on the Campo di Fiore in Rome is a typical example of this challenge to the papacy.

Struik: My fellowship from the Rockefeller Foundation was renewed for another year, but on the condition that we continue our studies in G6ttingen. In and of itself this was fine: G6ttingen was after all the mecca of mathematicians. But we had grown fond of Italy, its people (except for the blackshirts), its history, art, and science. And we had come to take its atmosphere of courtesy among mathematicians somewhat for granted.

died in the summer of 1925. Ruth and I attended his funeral, which was attended by most of the academic c o m m u n i t y in G6ttingen. There were a few short speeches, one by Hilbert, and I joined the group of those who threw a spade of earth over the grave. I felt as though I had lost one of my teachers. Ehrenfest had always emphasized the importance of Klein's lectures to his students, and we read many of those that circulated in lithograph form. They are full of sweeping insights that reveal the interconnections between different mathematical fields: geometry, function theory, number theory, mechanics, and the internal dialectics of mathematics that manifest themselves through the concept of a group. During my stay in G6ttingen, Courant invited me to help prepare Klein's lectures on the history of nineteenth and early twentieth century mathematics for publication, which I did. These first appeared in Springer's well-known "yellow series," and they remain, with all their personal recollections, the most vivid account of the mathematics of this period.

Rowe: What was the atmosphere in G6ttingen like?

Rowe: What about Hilbert?

Struik: Mathematically it was very stimulating, of course, but you had to have a thick skin to survive; the G6ttingen mathematicians were known for their sarcastic h u m o r . E m m y N o e t h e r , w h o was shy and rather clumsy, was often the butt of some joke, as was the good-natured Erich Bessel-Hagen. In von Kerekjarto's topology book there is a reference to BesselHagen in the index, but when you turn the page cited there is no reference to him in the text, only a topological figure that looks like a funny face with two big ears. That was the way they could treat you at G6ttingen, where ironical jokes about one's colleagues were always in vogue. It was a world apart from the courteous atmosphere in Italy.

Struik: I saw a fair amount of him in those days, although he was quite old by then. His main interest was foundations questions, as he was still in the thick of his famous controversy with L. E. J. Brouwer. Hilbert was very good at reinforcing his own enormous power and authority by making use of clever assistants whose time and brains he ruthlessly exploited, but not w i t h h o l d i n g credit where credit was due. Emmy Noether had been his assistant during the war years when he worked on general relativity. Hilbert was an East Prussian, and there was a distinctly Prussian quality about him that was reflected in his relationships with his assistants. Ruth and I once asked Hilbert's assistant, Paul Bernays, to join us on a Sunday morning walk. Bernays was then in his midthirties and already a well-known mathematician, but he actually had to ask the Herr Geheimrat (which was the tire one used in addressing Hilbert) whether he could spare him for a few hours.

Rowe: Where did your ventures take you after Italy?

Rowe: Did you have any contact with the older generation of mathematicians in G6ttingen? Struik: Yes, although I never met Felix Klein, who

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I often attended Hilbert's seminar, which generally had a n y w h e r e from forty to seventy participants. Often the speaker was a visitor who had come to talk about his research. It was a daunting experience to speak before such a critical audience, and many who came were justifiably apprehensive. Afterward came the chairman's judicium, and his verdict, usually to the point, could help or harm a young mathematician's standing considerably, at least in the eyes of his colleagues. I once spoke about my work on irrotational waves and was happy that it received a friendly reception. Others were not so fortunate. Young Norbert Wiener, for example, was too insecure and nervous to do justice to his excellent research in harmonic analysis and Brownian motion. Rowe: Are there any particular Hilbert anecdotes that come

to mind? Struik: Oh sure, but a good Hilbert anecdote has to be told with an East Prussian accent, which he never quite lost. Once a young chap, lecturing before Hilbert's seminar, made use of a theorem that drew Hilbert's attention. He sat up and interrupted the speaker to ask: "That is really a beautiful theorem, yes, a beautiful theorem, but who discovered it?--wer hat das erdacht?'" The young man paused for a moment in astonishment and then replied: "Abet, Herr Geheimrat, das haben Sie selbst erdacht!--But, Lord Privy Councilor, you discovered that yourself!" That is a true s t o r y - - I witnessed it myself. Another episode I remember took place in one of Hilbert's lectures on number theory, which I followed during my stay in G6ttingen. The previous day he had written the prime numbers less than 100 on the blackboard, and now he came rushing into class to tell us: "Ach, I made a slip, a bad slip. I forgot the n u m b e r 61. That should not have happ e n e d . These prime n u m b e r s are beautiful; t h e y should be treated w e l l - - m a n muss sie gut behandeln." On another occasion we were waiting for him in the seminar room. He finally came rushing in only to berate us: "Oh, you smug people, here you are sitting around talking about your petty problems. I have just come from the physics seminar where they are playing with ideas that will turn physics upside down!" That was Max Born's seminar, which week after week was attracting a h u n d r e d or more physicists, m o s t l y younger men. Heisenberg and Pauli were then discussing the new matrix theory approach they were developing as an alternative to Schr6dinger's w a v e theory. Rowe: Did you ever get invited to Hilbert' s home? Struik: Yes, he and his wife occasionally invited us to an evening party at their home, usually to meet some visiting celebrity. I have a better recollection of the 20

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989

parties at the Landaus'. He was a stocky fellow and looked more like a butcher than a scientist. Having married the daughter of well-to-do Professor Paul Ehrlich, the famous chemist who found the first effective r e m e d y against syphilis, Landau lived in u p p e r middle-class comfort in a large and splendid home on the outskirts of town. After a sumptuous dinner our host led us to his study, a large room whose walls were covered with books, all of them mathematical. There were complete runs of important journals, collected w o r k s of famous figures, and nearly every imaginable work in number theory and analysis. No frivolous stuff here. There was nothing frivolous about his writing either. He presented his ideas as precisely as possible, in the u n e m o t i o n a l style of Euclid: theorem, lemma, proof, corollary. He lectured the same way: precise, some of us thought pedantically precise. Occasionally he would present a well-known theorem in the usual way, and then while we sat there wondering what it was all about, he pontificated: "But it is false--ist aber falsch"--and, indeed, there would be some kind of flaw in the conventional formulation. Once the guests were assembled with refreshments, Landau started organizing mathematical games. One of them I still like to play once in a while. Suppose you define "A meets B" to mean that at some time A shook hands with B, or at any rate A and B touched each other. N o w construct the shortest line of mathematicians connecting say Euler with Hilbert. Can you shorten it by admitting non-mathematicians in your chain, like royalty or persons who circulated widely and reached old age, like Alexander yon Humboldt? All kinds of variations are possible. Can you forge a link to Benj0min Franklin? To Eleanor of Aquitaine? Rowe: That sounds a little like the present pastime of constructing a mathematician's ancestral tree or determining one's "Erd6s number." Struik: Yes, only the possibilities are much more open-ended. I can't resist telling one more Landau story that my former M.I.T. colleague Jesse Douglass liked to recall. O n e day at G6ttingen Landau was speaking about the so-called Gibbs p h e n o m e n o n in Fourier series, and remarked: "'Dieses Phi~nomen ist von dem englischen Mathematiker Gibbs (pronounced Jibbs) in Yale (pronounced Jail) entdeckt.'" Only my respect for the great mathematician, said Jesse, withheld me from saying: "Herr Professor, what you say is absolutely correct. Only he was not English, but American, he was not a mathematician, but a physicist, he was not Jibbs, but Gibbs, he was not in jail, but at Yale, and finally, he was not the first to discover it." Rowe: Who else did you meet in G6ttingen? Struik: There were many mathematicians from all over

Dirk Struik lecturing on tensor calculus in 1948, the year he published his Concise History of Mathematics and Yankee

Science in the Making. THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989

21

the world: Harald Bohr, Leopold Fej6r, Serge Bernstein, Norbert Wiener, Oystein Ore, and of course, B. L. van der Waerden. I had met him already at the Mathematical Society in Amsterdam. He and Heinrich Grell could often be seen strolling down the Weender Strasse on either side of Emmy Noether. They were sometimes called her Unterdeterminanten (minor determinants). I had some contact with Courant w h e n I first arrived. I had met him in Delft a year earlier, and he and Levi-Civita had both supported my application for a Rockefeller f e l l o w s h i p . C o u r a n t was t h e n working on existence questions connected with the Dirichlet problem as these bore on potential theory and solutions to partial differential equations. These ideas were at that time elaborated in the famous C o u r a n t - H i l b e r t text Methoden der mathematischen Physik. I was very interested in this field, and was already somewhat familiar with it through Ehrenfest's lectures at Leiden. Courant's assistant, Dr. Alvin Walther, took the time to introduce me to the latest developments, which was fortunate considering that Courant was burdened with his many academic obligations. Courant was of course a brilliant man, but to me he seemed then to lack Levi-Civita's talent for organizing his time. R o w e : When did you first begin to take a serious in-

terest in the history of mathematics? Struik: It was on the historic soil of Italy that I met two historians of mathematics, Ettore Bortolotti from Bologna and Giovanni Vacca, and from this point on my interest in the field has grown steadily. I also met Gino Loria, who like Castelnuovo, Enriques, Bianchi, and Severi, was a geometer, though on a more modest scale. We talked about the desirability of having more ancient texts published with commentary. Vacca was a professor in Rome. I remember when we went to visit him and were looking for his apartment along the n a r r o w street that he lived on. Some girls were playing outside, and we asked them where Professor Vacca lived. "'Mamma mia, siamo tutte vacche" ("We are all cows"), they giggled. But we found the house, and talked a m o n g other things about ancient Chinese mathematics, a subject that was then hardly touched. "Learning enough Chinese characters for mathematical purposes is not difficult when you try," he said; but I never tried. Later I met the director of the Dutch archeological institute in Rome, G. J. Hoogewerff, who was then working on Dutch Renaissance painters, the Zwerfvogels (wander-birds). When he heard of my interest in the history of mathematics, he suggested that I take a look at a Dutch Renaissance mathematician who had become an Italian bishop and advisor on calendar reform at the Fifth Lateran Council of 1512-1517. His name was Paul van Middelburg--Paolo di Middel22

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989

borgo. It was more work than I anticipated, as it required reading Latin texts in incunabula and post-incunabula, but it was a nice occasional break from my work on h y d r o d y n a m i c s and function theory. For once I could profit from the Latin I learned in preparation for my entrance at the university. And so I persevered, my research leading me to a number of Rome's antiquarian libraries, like the Alessandrina and the Vatican. To get permission to enter the Vatican archives I had to go through the office of the Netherlands' ambassador, but at least it was no longer necessary to "prostrate oneself before the feet of his Holiness," which, as I was told, had been the case not long before. These libraries are only heated on cold days by a brazier with smoldering charcoal, so that you had to study with your coat on; luckily such ancient palaces had thick walls. In some of them you had to overcome the inertia of custodians who resented the intrusion of readers as an attack on their privacy. I discovered some interesting things about the mathematical bishop who left his native Zeeland because, as he wrote, the people there considered intoxication the summum of virtue. An abstract of my findings was published in the Atti of the Accademia dei Lincei, and the full text appeared later in the Bulletin of the Netherlands Historic Institute. Only a few people have taken the time to glance at it, but let us say that the work was good for my soul. Rowe: When did you begin taking a wider view of the his-

tory of mathematics and science, taking into account the social context that shaped them? Struik: This question interested me from quite early on, and I followed the role played by science, and particularly mathematics, in the wake of the Russian Revolution. In fact, I saw this question of mathematics in society as a testing ground for my newly acquired Marxist views. Did "external" factors actually influence the "internal" structure of science, its growth or stagnation? Until fairly recently, it seems that everyone assumed this was not the case, that mathematics was a purely intellectual undertaking whose development is best understood by analyzing ideas and theories i n d e p e n d e n t of the social system that produced them. But Marxist scholars had already shown that almost equally exalted fields like literature and biology could be successfully tackled using the tools of historical materialism. So what about mathematics? Around the turn of the century mathematics flourished in a state of blissful innocence. One could do geometry, algebra, analysis, and number theory in a delightful social vacuum, undisturbed by any extraneous pressure other than that exerted by one's immediate academic and social milieu. Even as late as 1940 G. H. Hardy could maintain that the "real" mathematics of the great mathematicians had, thank good-

ness, no useful applications. Yet fifty years earlier Steinmetz in the USA and Heaviside in England were already applying advanced mathematical concepts in electrical engineering, and probability and statistics were being utilized in biology, the social sciences, and industry. N o n e of these developments, however, seemed to influence the mathematicians' purist outlook on the field. When I assisted in editing Klein's lectures on nineteenth century mathematics during my stay in G6ttingen, I learned how profoundly the French Revolution had influenced both the form and content of the exact sciences and engineering, as well as the way in which they were taught. This was especially due to the impact of the newly-founded Ecole Polytechnique in Paris, headed by Gaspard Monge. Quite clearly the educational reforms of this period were intended to benefit the middle classes and not the sans culottes. This realization gave me more confidence in the potential efficacy of historical materialism as an approach to the development of mathematics. This confidence was strengthened a few years later when I read Boris Hessen's landmark paper on seventeenth-century English science. Hessen emphasized that even an Olympian figure like Newton was a man of his times who was inspired by problems that were central to the expanding British mercantile economy - - p r o b l e m s posed by mining, hydrostatics, ballistics, and navigation. The British Social Relations in Science M o v e m e n t , w h i c h included such figures as J. D. Bernal, J. B. S. Haldane, J. Needham, L. Hogben, and H y m a n Levy, followed the trail blazed by Hessen, producing a number of germinal ideas for the history of science. These writers were a strong source of inspiration to me in thinking about the historical relationship between mathematics and society, and my views were strengthened by conversations with Levy and J. G. Crowther who were visiting the Boston area from England. Such an attitude also implies concern for the social responsibility of the scientist. In 1936 1 helped to launch the quarterly Science and Society, which for fifty years n o w has been bringing this message of responsibility to the academic world. Some of my contributions to early issues of S&S deal with the sociology of mathematics.

such as politics, he was rather naive. He then seemed to think that the main problem in the world was overpopulation. But at the same time he was fiercely internationalist and detested the way m a n y scientists from the allied c o u n t r i e s still s n u b b e d the G e r m a n s . A n y w a y , we drank beer together and took walks through the woods in the Hainberg overlooking the town. He asked me about my future plans and I admitted that they were rather vague and unpromising. I had spent seven years as an assistant in Delft, which was a very nice job but with no future prospects. Academic openings in those days were few and far between in the Netherlands. Wiener then suggested that I come to the United States. He told me about New England and M.I.T., where he was an assistant professor; they were looking for n e w b l o o d and he thought I might fit in. R o w e : Were you attracted by the prospect of joining the M.I.T. faculty? Struik: Yes, I knew of M.I.T. through the Journal of Mathematics and Physics that it issued, where papers by C. L. E. Moore and H. B. Phillips on projective and differential geometry had appeared. So I knew there were congenial spirits in the mathematics department there. Wiener also made it all sound very attractive by describing the natural beauties of N e w England, his father's farm in the country, and the m o u n t a i n climbing he and his sister Constance had been doing. Of course, I was footloose at the time and this would have been a step up the academic ladder, which was particularly important as it was then, as I said, quite difficult to land a promising job in mathematics. I told him that I might well take him up on this proposition if an offer came my way, and my wife Ruth also liked the idea.

R o w e : I understand that it was through Norbert Wiener

that you first came to the United States. Struik: Yes, Wiener was one of those Americans who had come to G6ttingen in the mid-twenties, and he and I took to each other from the beginning. We talked a good deal of shop, as was wont in G6ttingen and with Wiener. I became acquainted with his work in harmonic analysis and Brownian motion, which made it clear to me that I had met an exceptionally strong mathematician. But in matters of the world,

Dirk and Ruth Struik, 1987. THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989 2 3

Rowe: So you were interested in coming to the United

qualities did you admire most in him?

States, but perhaps open to other offers as well. Struik: Yes, and by the time I heard from M.I.T. I received another tempting offer from the Soviet Union where my brother Anton had been working as an engineer. Otto Schmidt, a mathematician and academician in Moscow, sent me an invitation to give lectures there. My work in differential geometry was not unknown in Russia, as I discovered in 1924 when I was invited to join the committee that was preparing the collected works of Lobatchevsky. Shortly before I left for the United States, Kazan University also bestowed on me its seventh Lobatchevsky prize. I sometimes w o n d e r w h a t might have h a p p e n e d had I accepted Otto Schmidt's offer and gone to work as one of his collaborators. Schmidt was not only a gifted scientist, he was also a first-rate organizer. Not long after I heard from him the conquest of the Arctic became an important part of the Socialist program, leading to the famous airplane expeditions of 1936-37 to the North Pole and the scientific expedition that spent 274 days on an ice floe. Schmidt was one of the leaders of these expeditions and the research that led to settlements in the huge wastelands of Northern Russia and Siberia. Under him I might have turned my attention to soil mechanics, for which my work on hydromechanics could have served as a preparation. Or perhaps I would have gone in for Polar exploration . . . . On the other hand, my natural Dutch obstinacy, also in politics, might have gotten in the way and brought me into conflict with the trend toward conformity typical of the later Stalin years. At any rate, I weighed this decision very carefully, including the factor of Ruth's health, which was not good at the time. We both agreed that life in the United States would be an easier adjustment, both in terms of the economic circumstances and the language and culture. And so I accepted the offer from M.I.T., with the idea that I might consider accepting the offer from the Soviet Union at a later date. Rowe: Was your choice by any chance influenced by an at-

traction to the culture of New England? Struik: Not at all. I really had no idea of New England and Yankees and the whole variety of American cultures at this point. As a matter of fact when I received the i n v i t a t i o n from P r e s i d e n t Samuel Stratton of M.I.T. in September 1926, I had to take out my atlas to see where Massachusetts was located. I was surprised to learn that it was in the northeast and not on the Mississippi--perhaps I confused it with Missouri. Since that time I have always been very tolerant of those Americans who think that Hamburg is in Bavaria, or that Pisa and not Padua is near Venice. Rowe: You were a good friend of Norbert Wiener. What 24

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989

Struik: I would say his courage and his sensitivity. He was a man of enormous scientific vitality which the years did not seen to diminish, but this was complemented by extreme sensitivity; he saw and felt things for which most of us were blind and unfeeling. I think this was partly due to the overly strict upbringing he had as a child prodigy. Wiener was a man of many moods, and these were reflected in his lectures, which ranged from among the worst to the very best I have ever heard. Sometimes he would lull his audience to sleep or get lost in his own computatious--his performance in GOttingen was notoriously bad. But on other occasions I have seen him hold a group of colleagues and executives at breathless attention while he set forth his ideas in truly brilliant fashion. Wiener was among those scientists who recognized the full implications of the scientist's unique role in modern society and his responsibilities to it in the age of electronic computers and nuclear weapons. I well remember h o w u p s e t he was the day after H i r o s h i m a was bombed. When I remarked that because of Hiroshima the war against Japan should now come to a speedy close without much further b l o o d s h e d - - a common sentiment at the time and the official justification still heard t o d a y - - h e replied that the explosion signified the beginning of a new and terrifying period in human history, in which the great powers might prove bound to push nuclear research to a destructive potential never dreamed of before. He also recognized and detested the racism and arrogance displayed in using the bomb against Asians. He just saw further than the rest of us. In Wiener's day robots were largely the stuff of fiction. His favorite parables concerned such robots or similar devices with the capability of turning against those who built them: Rabbi Loew's Golem, for example, or Goethe's Sorcerer's Apprentice, the Genie of the Arabian Nights, and W. W. Jacobs' Monkey's Paw. Today we all know that cybernetics, the science of self-controlling mechanisms, has an increasing impact on industry and emp l o y m e n t , on warfare and the welfare of h u m a n beings. Rowe: You have continued to combine scholarship with political activism since you came to this country. Tell me something about your political activities. Struik: During the Second World War I stayed at M.I.T. and taught mathematics to the "boys in blue" sent to us by the navy. For some time I also spent weekends in Washington working at one of the Netherlands' desks in connection with the war effort, and I participated in the activities of the Queen Wilhelmina Fund, the Russian War Relief Fund, and the Massachusetts Council of American-Soviet Friendship. This

Norbert Wiener (center) and Dirk Struik (right) in the Centennial procession at M.I.T., April 1961.

latter work, which was a logical consequence of the anti-fascist campaign for collective security waged in the late thirties, together with my support for the Indonesians in their fight for independence and for the 1948 campaign of the Progressive Party, attracted the attention of sundry cold and hot warriors of the postwar period. I was called before the witch-hunting committees and an ambitious district attorney had me

and my friend Harry Winner indicted on three counts of 'subversion.' That was in 1951, the beginning of the McCarthy era. There were wild newspaper headlines, M.I.T. suspended me (but luckily not my salary), and I was let out on heavy bail. Bertrand Russell was then lecturing at Harvard. When told that I was accused of attempting to overthrow the governments of Massachusetts and the United States, he murmured gravely, THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. I, 1989 2 5

years of my suspension, I lectured all over the country on the right of free speech, and at home I worked on editing the mathematical works of Simon Stevin. R o w e : You continued to collaborate with Schouten

throughout the 1930s. When did you give up doing differential geometry and concentrate on history? Struik: In the late thirties Schouten and I co-authored

Dirk Struik, 1952.

"Oh, what a powerful man he must be!" R o w e : What became of the charges against you? Struik: The case never came to trial, but it was not until 1955 that the indictment was finally quashed and I regained my position at M.I.T. It might have taken even longer if it had not been for the strong community support that Winner and I received, the dedication of our lawyers, and the Supreme Court ruling in the Pennsylvania case of Steve Nelson, which declared that subversion was a federal offense--Steve and I had been indicted under state law. During the five

26

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989

a two-volume work entitled Einfiihrung in die neueren Methoden der Differentialgeometrie. This gave the first systematic introduction of the kernel-index method and incorporated a number of new techniques--exterior forms, Lie derivatives, etc.--that had since been developed. My last major mathematical publication was Lectures on Classical Differential Geometry, which appeared in 1950. After I became an emeritus in 1960 I gradually gave up following the course of new mathematical developments. I felt a little too old for that. My goal instead has been to learn as much as I can about mathematics up to about 1940. That's a big enough field for one h u m a n being, I think: the history of mathematics from the Stone Age to the outbreak of World War II!

Dirk Jan Struik 52 Glendale Road Belmont, MA 02170 USA

David E. Rowe Department of Mathematics Pace University Pleasantville, NY 10570 USA

The catapult that Archimedes built, the gambling-houses that Descartes frequented in his dissolute youth, the field where Galois fought his duel, the bridge where Hamilton carved quaternions---not all of these monuments to mathematical history survive today, but the mathematician on vacation can still find many reminders of our subject's glorious and inglorious past: statues, plaques, graves, the cafd where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, memorials. Does your hometown have a mathematical tourist attraction? Have you encountered a mathematical sight on your travels? If so, we invite you to submit to this column a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks. Please send all submissions to the European Editor, Ian Stewart.

0

The Kepler Museum and Kepler Monument

o~

Gerhard Betsch The Kepler Museum and Kepler Monument are in Weil der Stadt, 30 km west of Stuttgart (Fed. Rep. Germany). The Kepler M u s e u m is in the house in which the famous mathematicus (astronomer and scientist) Johannes Kepler was born on 27 December 1571. The house, near the marketplace of Weil der Stadt, is owned by the Kepler-Gesellschaft e. V. (Kepler Society), w h o s e p u r p o s e is " t o maintain Kepler's memory and to propagate knowledge of his work." The museum was established in 1940 and has been enlarged considerably in the meantime. On display are pictures and documents concerning Kepler's biography; models to explain his discoveries; the main publications of Kepler (facsimiles as well as first printings); and letters, instruments, etc. Of particular interest is a model of the first known c o m p u t i n g machine, which was designed by Tiibingen professor Wilhelm Schickard (1592-1635) in 1623. Schickard was a close friend and correspondent of Kepler. Opening hours Monday-Friday: 9-12 and 14-16 Saturday: 10-12 and 14-17 Sunday (from October to May only on the first and 3rd Sunday of each month): 11-12 and 14-17. The house at Keplergasse 1 near the museum, also owned by the Kepler Society, contains a special library on Kepler and the Kepler Archives. Here part of the editorial work for the (second) edition of Kepler's collected works is being carried out. 36

The house where Kepler was born, now the Kepler Museum.

The Kepler monument in the marketplace at Weil der Stadt was made by the sculptor August Kreling and erected in 1870. It shows Kepler and smaller statues of Nicholas Copernicus, Tycho Brahe, Michael M/istlin (Kepler's teacher in astronomy in the University of Ti~bingen), and Jost B~rgi. Mathematisches Institut Universitdt Tiibingen D-7400 T~ibingen, Federal Republic of Germany

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO, I 9 1989 Spfinger-Verlag New York

Why the Circle Is Connected: An Introduction to Quantized Topology* E d w a r d G. Effros + Now when Heisenberg noted that, he was really scared. P. A. M. Dirac [4]

s e n b e r g ' s idea of m a t r i x scalars fully into m o d e r n mathematics. O w i n g in large part to the investigations J. von N e u m a n n i n t r o d u c e d the t h e o r y of operator al- of Alain C o n n e s (awarded a Fields Medal in 1982) in gebras over fifty years ago [8]. O p e r a t o r algebras n o w " q u a n t u m g e o m e t r y , " this objective has n o w been replay an i m p o r t a n t role in such diverse fields as geom- alized. Because this a p p r o a c h stresses examples rather etry, algebra, and mathematical physics. It is therefore t h a n general theory, the subject has also b e c o m e cons o m e w h a t surprising that so few mathematicians have siderably m o r e accessible. In fact o n e n e e d s only a become familiar with this important technique. minimal b a c k g r o u n d in abstract analysis to u n d e r P e r h a p s this has b e e n d u e to the technical nature of stand some of the most interesting applications of the the subject, as well as to the perception that the area subject. was just an elegant " n o n - c o m m u t a t i v e " analogue of We will illustrate these d e v e l o p m e n t s b y considfunctional analysis, of interest only to abstract analysts ering a key example, n a m e l y the " q u a n t i z e d torus" and (perhaps) to mathematical physicists. associated with the regular representation of the free Von N e u m a n n a n d his successors had a g r a n d e r g r o u p o n two generators. In the late sixties, R. V. Kap u r p o s e in mind. Their goal was to integrate W. Hei- dison conjectured (see [11]) that this algebra is "conn e c t e d " in the sense that it has no non-trivial projec* This paper is based upon an invited lecture given at the 1987 tions. This was finally p r o v e d fourteen years later in a remarkable p a p e r b y M. Pimsner and D. Voiculescu summer meeting of the Mathematical Association of America. [10], w h o u s e d a geometrical approach. Incorporating f Supported in part by the National Science Foundation. 1. I n t r o d u c t i o n

Edward G. Effros received his undergraduate training at M.I.T. and completed his doctoral studies under the direction of George Mackey at Harvard in 1961. His first position was at Columbia as a Ritt Instructor; in 1964 he became an Assistant Professor at the University of Pennsylvania. In 1979, he accepted a position at UCLA, much to the confusion of the medical faculty (of which his identical twin brother is a member). He first fell in rove with operator algebras as a graduate student after reading some papers by Richard Kadison. He strayed from the area only in the late '60s and early '70s, when Erik Alfsen attracted him to convexity theory and its applications. During the last fifteen years such brilliant younger researchers as Alain Connes and Vaughan Jones have compelled him to try to learn everything else under the sun.

Edward G. Effros

Professor Effros is a past Guggenheim Fellow, and he gave an address at the recent International Congress of Mathematicians at Berkeley.

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1 9 1989 Springer-Verlag N e w York

27

ideas f r o m that p a p e r a n d from [3], C o n n e s t h e n s h o w e d h o w the proof fits nicely into his t h e o r y of "non-commutative differential topology" [2]. Our purpose in this paper is to outline his approach 9 I wish to express m y thanks to Robert Powers, w h o first explained the proof to me, and w h o in particular suggested the apt title. I also apologize to the experts

Von Neumann and his successors had a grander purpose in mind. Their goal was to integrate W. Heisenberg's idea of matrix scalars fully into modern mathematics.

If X is a compact Hausdorff space, we let C(X) C ~ ( X ) be the continuous functions on X. The space C(X) inherits the c o r r e s p o n d i n g *-algebra operations from f'(X). In fact if X is metrizable, then C(X) is isomorphic to a n o r m closed *-subalgebra of foo = f~(N). To see this, we simply let ~ be a discrete probability measure concentrated in a dense countable subset of X, a n d t h e n use the obvious e m b e d d i n g C(X) C, e ~ -~ L~(X,p~)9 A standard result in functional analysis states that conversely, a n y separable norm-closed *-subalgebra of ~ ( X ) m u s t be of the form C(Y) for some compact metric space Y. It was Heisenberg [5] w h o proposed that the natural "scalars" of q u a n t u m physics are infinite matrices

w h o might find the approach oversimplified. Suffice it to say that there are n o w excellent texts for learning the subject the "right w a y " (see, e.g., [6], [9], [13]).

2. S c a l a r s i n m a t h e m a t i c s a n d p h y s i c s

Mathematicians have traditionally used the term scalar for elements of the ground field, which more often than n o t consists of either the reals ~ or the complex n u m b e r s C. We restrict our attention to C, together with the usual operations of addition, multiplication, conjugation, and absolute value: c~ + [3, oL[3, OL* = ~, a n d Io~[for o~, [3 ~ C. Even classically, physicists use the term scalar in a different sense. Classical quantities such as e n e r g y and temperature are generally functions d e p e n d i n g on the time, position, m o m e n t u m , and/or other state parameters. To provide a simple f r a m e w o r k for such scalars a n d their operations, we restrict our attention to the b o u n d e d functions on a state space X. Given any set X, w e let f~(X) d e n o t e t h e b o u n d e d c o m p l e x valued functions f, g . . . . on X with the pointwise *-algebra operations

T =

[ tlltl2" " " ] t21 t22 . . .

with entries tij in C. To be more precise, the appropriate scalars are the linear transformations or operators that these matrices determine. Considering first the finite matrices, we let ~(n) denote the complex n x n matrices with the usual *-algebraic operations of matrix addition, multiplication, and adjoint T* = [~]. The n o r m is best described in terms of the corresponding operators. Using matrix multiplication, one has the usual identification of the n x n matrices with the space of linear transformations ~(n) -~ Lin (C~,C~). Letting C" have the Hilbert n o r m ]J~l] = [~=1 we define

I~k12]89

[f711 = sup {IIT~II:II~II~ 1}. We let ~ ( n ) be the resulting n o r m e d *-algebra.

(f + g)(x) = fix) + g(x), (fg)(x) = f(x)g(x),

f*(x) = fix)a n d the " s u p r e m u m n o r m " I~l~ = sup{~(x)l: x ~ X}

The norm of an operator.

For the infinite dimensional case it is easier to begin with transformations rather than matrices. First we let C ~ = (2 be the vector space of infinite sequences ~ = (~1, ~2. . . . ) of complex numbers that are square summable, i.e., for which ~l~kl 2 < 0% and for these we use the Hilbert n o r m oo 2 89

and dot product The norm of a function.

28 THEMATHEMATICALINTELLIGENCERVOL.11, NO. l, 1989

We t h e n d e f i n e ~o = ~2oo(oo) = ~(C~) to be the bounded linear transformations T: C~176 ---> C~, i.e., those for which

and the dot product

~ " ~ = fx ~(x).~(x)- d~(x). 11731 = s u p {IIT~II:II~II ~ 1} < ~ .

We provide ~oo with this norm and the usual *-algebraic operations, letting T* be the adjoint operator of T. We may associate an infinite matrix [Tq] with T E ~2oo by letting Tij = TSj. 8i, where 81 = (1, 0. . . . ), 82 = (0, 1. . . . ). . . . is the usual orthonormal basis for C= = f2. The natural ordering on ~ is defined by letting T >i 0 (we say T is positive) if ToL. o~I> 0 for all OLE G| We let ~o+ denote the corresponding cone of positive operators. If T 1> 0, then it has a unique positive square root, i.e., an operator S = T~ satisfying S/> 0 and S2 = T (see [12], w for a simple proof). Given an arbitrary operator T, T*T/> 0 because T*ToL9o~ = ToL-ToL /> 0 for any a E C~176 the absolute value of T is defined by ITI =

Given a bounded measurable function h on X, the corresponding diagonal or multiplication matrix M(h) is given by

(M(h)~)(x) = k(x)~(x). In terms of the "continuous basis 8(x) labelled by X" (for non-discrete ~ these are not elements of L2(X,p.)) the "matrix" of M(K) is given by

M(X) =

"~'~(x')

(v'T) ~. Operators in physics frequently behave like functions. Often by choosing an appropriate orthonormal basis they can be diagonalized, i.e., put in the form

T =

I

h l 0~.2 00 ". . .' ' 1

T~(x) = fK(x,y)~(y)d~(y).

0

0

0

~'3" 9 9

where the hj down the diagonal form a bounded sequence, the eigenvalues of the operator. We may regard )'1, h2. . . . as a function ), from ~ into C, and we then have that T is the matrix with respect to the orthonormal basis {~,} of the "multiplication operator" M(K): e2 ~ e 2 given by (M(h)~)(n) = h(n)~(n). It is easy to check that 11/71 = [[h[[oo.More general operators with "nondiscrete spectrum" also play an important part. These may be constructed by replacing C| = ~2 with a Hilbert space of the form L2(X,~) for some positive measure space (X,~x). We think of the functions in the latter space as "X-tupIes'" X

X'

J = (.

~(x)-~(x')

X

~(x) + ~(x)--~(x') + ~(x')

o~ = ( .

3. Group C*-algebras

X'

X

The most striking qualitative difference between classical and q u a n t u m scalars is evident w h e n one considers more than one operator9 Operators S and T need not commute. In particular, if ST # TS, then one cannot simultaneously diagonalize S and T, because diagonal matrices commute 9 Physically, non-commutativity is a reflection of Heisenberg's Uncertainty Principle (see [7], pp. 77-79). It is useful to regard Zoo as a "quantized analogue of ~%" By a separable C*-algebra, we shall mean a unital (containing/) separable closed *-subalgebra of ~% Because we saw that the algebras C(X), where X is a compact metric space, arise as the separable closed *-subalgebras of e ~, it is only natural to think of a separable C*-algebra as a "quantized analogue of C(X)." The *-algebra C(X) itself may be regarded as a C*-algebra because the map f ~-> Mf provides an isometric *-isomorphism of e ~ into ~ .

.)

with the corresponding algebraic operations

~+n=(.

It is worth noting that one can associate more general operators T with functions or generalized functions K(x,y) by letting

Xr

o~(x)--o,~(x') ~ )

.)

All infinite dimensional separable Hilbert spaces are isometric. Thus for any such Hilbert space H, we have that ~(H) ~ ~(C ~176= ~o~. We have for example that C~ =

~2 ~

~2(~) ~

L2(~1)

where 1F1 denotes the circle with the Lebesgue meaTHE MATHEMATICALINTELLIGENCERVOL. 11, NO. 1, 1989

29

sure d~(~) determined by ff(Od~(O

= (2"rr)-~ f~-~f(e i~ dO

for f ~ C(~). A specific isometry between ~2 and e2(Z) m a y be constructed by rearranging the basis vectors. To see directly that/?2(77) a n d L2(]J-1) are isometric, one recalls that the functions ~ ~ ~ form an orthonormal basis for L2(-~-~). Thus we have an isometry ~:L2(~ 1) --> f2(Z) determined by ~0) = (~k), where

sentation p of G on e2(G) determined by p(s)8 t = 8t~_,. Clearly p(s) commutes with K(s), and thus with K(CG) and MC*~d(G)). Assuming that d(s) = ;~(a)8~ 98s = 0 for all s, it follows that

K(a)8~ 9 8t = h(a)p(s-1)81 9 8t = P(S-1))~(a)8l " 8t(3.1 ) = )~(a)~l"~t~-, = d(ts - I ) = 0, i.e., all the matrix entries of Ma) are zero, and thus a = 0. We will write

~k = f " ~k = ff(O~-kd~(~).

Given a countable group G, the group algebra CG is defined to be the vector space of finite linear combinations of group elements ~,o~(Sk) (Sk ~ G, a k ~ C) with the *-algebraic operations (x~<s~>)(~(t~>) (~;(s))*

= Y,~f~ds~t~> = ~<s~-l>.

(3.2)

a ~ ~d(s)(s>.

Turning to some examples, consider first the C*-algebra C'red(27). Arrange the elements k of ~, or the corresponding basis vectors 8k of e2(Z), in a graph ...

-------I~)~-.-----.1~o------.-~o-------~ -2 -1 0 1 2

...

~(1)

The graph of 77.

Since the group elements form an algebraic basis for this vector space, CG has for its dimension the cardinality IGI of G. It is a standard elementary fact from algebra that if G is finite, then CG is isomorphic to a s u m of matrix algebras:

The arrows correspond to the addition by the generator 1, or in the vector space e2(Z), the application of the translation operator ~(1). We m a y use the isometry ~:L20 -1) ~ t72(77) described above to prove that

CG -~ ~(nl) @ . . ~ ~(np),

a n d in particular, we obtain the well-known formula {G] = ~ nk2 (where G has p distinct irreducible repres e n t a t i o n s , the c o r r e s p o n d i n g d i m e n s i o n s b e i n g nl~

. . . 9 rip).

To place a n o r m on CG we m a y regard the elements of G as "'translation" operators. We let ~2(G) be the Hilbert space with orthonormal basis {Ss} indexed by elem e n t s s ( G. For each t ( G we have a corresponding unitary translation operator K(t):~2(G) ---+ ~2(G) determined by h(t)Ss = 8ts. The m a p K:G ~ ~(f2(G)) extends linearly to CG:

a n d it is easy to check that this is a *-homomorphism of CG into ~(f2(G)). h is injective because if K(Xaj(sj)) = 0, t h e n Xo~jSs~ = ME~j(sj)) 81 = 0, and thus ~xj = 0 for all j. Pulling back the operator n o r m to CG, we let C*~ed(G) be the corresponding metric space completion of CG. Since ~(f2(G)) is complete, ~ extends to an isometry h:C*,ed(G ) --~ ~(f2(G)), with image the n o r m closure of K(CG). The latter is obviously a separable C*-algebra, a n d we let C*~ed(G) have the corresponding C*algebraic structure. This is called the reduced C*-algebra of G. The elements of this completion m a y be regarded as infinite sums of group elements. Given a ~ C*~d(G), we define its Fourier coefficient at s to be d(s) = K(a)8~ 9 8s. If a = E%(s> ~ CG, t h e n it is immediate that d(s) = %. For general a ( C*~ed(G) we have that if ~(s) = 0 for all s, then a = 0. To see this we employ the right repre30 THE MATHEMATICAL INTELUGENCER VOL. 11, NO. 1, 1989

C*roa(Z) = C('r In fact we have that ~-1~(1)~ = M(O, because applying the left side to a basis vector ~k E L2(T1) we get ~-~,(1)~

k = ~-~(1)Sk

= ~-18~+,

= ~k+,.

It follows that a = ~,e~k(k> ~-~ X~k~ k

determines an isometric *-isomorphism of C77 onto the *-algebra ~ generated by the function f(~) = ~ in C(T~). Because C*~d(Z ) is the completion of C77, the StoneWeierstrass Theorem shows that C'red(Z) ~ C(TI). The group element (k> m a p s to the function ~k SO the association (3.2) corresponds to the usual assignment of a Fourier series to a continuous function f: f ~ Xf(k)~ k.

As is well-known 9 if f is continuously differentiable, the Fourier coefficients will converge fairly rapidly to zero, a n d in addition, the Fourier series converges uniformly to f. It w o u l d be natural to turn next to Z 2, the free abelian group on two generators. The above arguments m a y be a d a p t e d to s h o w that C ' r e d ( 7 / 2 ) ~ C ( - ~ - 2 ) , We are, h o w e v e r , more interested in n o n - c o m m u t a t i v e C*-algebras, so we consider instead B:29the free group on two generators. The resulting C*-algebra C'red0:2) m a y be regarded as a "non-commutative 2-torus." De-

note the generators of ~2 by u and v. We can again "picture" the elements of ~2 (or the corresponding basis elements of 22(0:2)) and the action of the operators k(u) and k(v) by a graph.

v~

u

F2 - {1} u-l'~ v 2

~

Iu

v-~u-1

4- 4 -p2

x(~)l

k(u)

v u

I ko(U)

The graph and punctured graph of D:2. Note that the graph is homogeneous: it "looks the same from any vertex." However, it has a more surprising property that will play a critical role in what is to follow. Let us delete the vertex I from the graph, as well as the edges incident to that vertex, and then reconnect u-1 directly to u and v -1 directly to v. In this manner we obtain a two-component graph, each component of which is isomorphic to the original graph. Taking the linear spans of these components we see that

22(0:2) = H u 9 H v (~ C~ 1,

tion if e = e* = e2. We denote the set of projections in by proj ~. If ~ = C(X) and e ( proj ~ , then e is a continuous real-valued function that satisfies e(x)2 = e(x), and thuse(x) = 0 or l for a l l x ~ X. T h u s d t i s evident that the projections are in one-to-one correspondence with the open-closed subsets of X. At the other (i.e., non-commutative) extreme, we have that the projections e E ~ are in one-to-one correspondence with the closed subspaces of C% With the commutative case in mind, we define a C*-algebra to be connected if it has no non-trivial projections (i.e., other than 0 and 1). We can now state Kadison's Conjecture: THEOREM [10]: C*red(~n) is connected. For the case n = 1, this is just the statement that the circle is connected, and thus our title. For simplicity we only consider the proof for n -- 2. H o w can we use functional analytic methods to test for connectivity in C*-algebras? In the commutative case, if a compact metric space X comes equipped with a positive measure ~ whose support is all of X, it suffices to show that an open and closed set G satisfies either ~(G) = 0 or ~(X/G) = 0. We will take this approach in the non-commutative theory, and it is therefore necessary to introduce the elements of non-commutative integration theory. The two most elementary measures in analysis are the counting measure and Lebesgue measure. Each of these has a simple non-commutative analogue. Consider first the classical counting measure, which assigns to each finite subset S of ~ the number 7(S) = card S. We let coo _C 2| be the *-algebra of functions vanishing off finite sets. Identifying the finite subsets of r~ with projections in coo C f ' , the corresponding map 7: proj Coo--+ N U {0}

where H, (resp., H~) is spanned by words ending in a non-zero power of u (resp., v). If we modify h(u) and h(v) to send 8 - 1 to ~ and ~v-1 to 8v and let both be zero on ~1, then we obtain operators ko(U) and ko(V) which restrict to unitary operators on H~ 9 H v. These determine a representation of I:2 on H u G H~, and thus ho(U) and ko(V) determine a degenerate (0 on the onedimensional subspace C~1) representation h 0 of B:2 on 22(Y2). It is apparent from the graph that in fact ho(u ) and ko(V) restricted to H, (resp., H~) are simulta- is additive, i.e., if ej are orthogonal projections (ei 9 ej neously unitarily equivalent to k(u) and k(v) on all of = 0 for i ~ j), then ~l(Xej) = Yq(ej). Letting 21 be the e2(g:2). It follows that we have a unitary equivalence functions f E 2" for which Xj[((/')[ < 0% we obtain an extension of 7 to a map E:21 --+ C by letting Y,~ = k 0 = k ~ k @ 0 c s ,. E.~(j). Y~is linear, and is positive in the sense that f I> 0 It follows that the linear extension of X0 to C0:2 is implies that E(f)/> 0. bounded, and thus extends to C*,,d(0:2). The non-commutative analogue of the counting What can we say about the C*-algebra C*~a(~:2) gen- measure is the dimension function, which assigns to erated by k(u) and X(v)? R. Powers proved that it is each subspace V of C" the integer dim V. We let Coo C algebraically simple [11], and in that sense has no ~" be the *-algebra of matrices of finite rank (or equiv" p o i n t set topology" (there is only one "point"!). alently the matrices with finite dimensional range). Nonetheless, simple C*-algebras can have interesting Identifying the finite dimensional subspaces of C" topological properties, if the latter notions are suitably with proj Coo, we obtain a corresponding additive map formulated. The simplest is "connectivity," which we dim: proj Coo--~ t~ U {0}. will illustrate below by showing that C*~(0:2) has that property. Because isomorphic subspaces have the same dimension, we have that for any invertible U ~ ~| dim 4. C o n n e c t i v i t y a n d traces U-1EU = dim E. The classical notion of trace (i.e., the An element of e of a *-algebra ~ is said to be a projec- sum of the diagonal elements) makes sense on Coo and

THEMATHEMATICAL INTELLIGENCER VOL.11,NO. 1, 1989 31

restricts to the d i m e n s i o n f u n c t i o n on proj Coo. The s i m p l e s t result in n o n - c o m m u t a t i v e integration t h e o r y is that in fact d i m has a natural extension to a function called trace: ~| --* [0, ~] t h a t is linear for p o s i t i v e scalars a n d has the additional p r o p e r t y trace U S U -~ = trace S

(4.1)

for a n y invertible U. Letting ~ C ~= be the integrable or trace class o p e r a t o r s T, i.e., those for w h i c h [[TIh = trace ITI < % one t h e n p r o v e s that dim: proj Coo--* ~I U {0} a n d trace: Coo ---* C h a v e a c o m m o n extension to a m a p trace: ~ ~ G. In fact w e h a v e trace T = ET~k" ~k for a n y o r t h o n o r m a l basis ~k, i.e., trace T is again the s u m of the diagonal elements. If U ~ ~ " is invertible a n d T ~ ~1, t h e n U T U -1 ~ ~2I, a n d (4.1) is still valid. G i v e n projections E,F ~ ~ ( n ) (n < oo), w e h a v e trace (E - F) = d i m E - dim F

(4.2)

is a n integer. Additional care is required in the inifinite d i m e n s i o n a l case b e c a u s e if E - F is not integrable, t h e n trace (E - F) is u n d e f i n e d , a n d one can h a v e the m e a n i n g l e s s e x p r e s s i o n ~ - ~ on the right side of (4.2). T h e formula is valid if E - F is integrable. The r e a d e r m a y prefer to skip the proof, w h i c h u s e s the s p e c t r a l t h e o r e m for c o m p a c t self-adjoint o p e r a t o r s (see [12], w 4 . 1 : Suppose that E and F are projections in and that E - F is a trace class operator. Then trace

Lemma

~,

(E - F) ( Z.

P r o o f : W e leave it to the r e a d e r to check the identities

E(E - F) 2 = (E - F)2E, F(E - F) 2 = (E - F)2F. Because (E - F) 2 is a positive c o m p a c t operator, w e m a y write (E - F) 2 = EKkP k, w h e r e P1, P2, 9 9 9 are finite d i m e n s i o n a l projections, a n d t h e s e q u e n c e of e i g e n v a l u e s k 1 > k 2 > . . . is either finite or c o n v e r g e s to 0. Because each Pk h a s the f o r m fk((E -- F) 2) for a c o n t i n u o u s function fk, E comm u t e s w i t h each Pk, a n d thus letting Q = I - EP k, E = Y,kEPk + EQ. The s a m e applies to F, a n d because (E - F)Q = 0 (a simple exercise) w e h a v e that E-

F = El(E--

F)Pk

p r o v i d e s a d e c o m p o s i t i o n of E - F into finite d i m e n sional projections. 32

THE MATHEMATICAL 1NTELLIGENCER VOL. 11, NO. 1, 1989

Letting ~k be a basis that s t e p - b y - s t e p s p a n s PIH, (P~ + P2)H . . . . . it is e v i d e n t that trace E - F = E k trace (E - F)P k. But trace (E - F)P k = trace EP k - trace FPk = d i m EP k - d i m FP k is a difference of n o n - n e g a t i v e integers, h e n c e the partial s u m s for trace E - F contain infinitely m a n y integers. We c o n c l u d e their limit m u s t be an integer, c o m p l e t i n g the proof. R e t u r n i n g to the classical theory, w e recall that if X is a c o m p a c t metric space, probability m e a s u r e s on X m a y be d e f i n e d to be positive linear functionals ~ on C(X) satisfying p,(1) = 1. For the n o n - c o m m u t a t i v e analogue, s u p p o s e that ~ is a separable unital C*-subalgebra of ~=. Letting ~l + = ~l f3 ~ + , a normalized trace on si is a linear m a p ~: ~ --~ C w h i c h is positive (i.e., a I> 0 ~ ~(a) /> 0), unital (i.e., -r(1) = 1), a n d satisfies T(u-lau) = ~-(a) for a n y invertible u E ~ . The g r o u p C*-algebras always c o m e e q u i p p e d with a canonical n o r m a l i z e d trace. In fact w e m a y simply let 9 (a)

=

d(1)

=

k(a)81

9 81 ,

w h e r e 1 is the identity for G. -r is positive because T(a*a) = k(a)~ 1 9 k(a)81 i> 0. For a n y s,t E G w e h a v e that ~((s}(t)) = ~-((t)(s)), because "r((st)) equals 0 unless s a n d t are inverses, in w h i c h case it is equal to 1. By linearity a n d continuity, it follows that "r(ab) = ~(ba) for a,b ~ CG, a n d thus for a,b E C*red(G ). F r o m this it is i m m e d i a t e l y s e e n that ~"is a n o r m a l i z e d trace o n C*r~d(G). For G = Z the identity is 0 a n d the f o r m u l a for a = ~,OLk(k) E C77 is given b y 9 (a) = k(a)8o 9 ao = M(E~k~k) 1" 1 = fEo~k~kdp,(O. T h u s u n d e r the identification of C'red(77) w i t h C(T1), "r is identified w i t h the usual L e b e s g u e m e a s u r e . We say that a n o r m a l i z e d trace ,r is faithful if ~(a*a) = 0 implies t h a t a = 0. T h e g r o u p C*-algebra trace is faithful because if "r(a*a) = 0, t h e n IIh(a)8lll2 = k(a)81" k(a)81 = k(a*a) = "ffa*a) = 0 a n d so k(a)81 = 0. T h u s X(a)as = p ( s - 1 ) X ( a ) a l

= 0,

i . e . , k(a) = 0, a n d a = 0.

O u r interest in the n o r m a l i z e d trace s t e m s f r o m the

following lemma. L e m m a 4.2: Suppose that ~ is a unital C*-algebra with a faithful normalized trace T and that ~o is a *-subalgebra of ~ . If T(proj ~o) C ~_, then ~o does not have any non-trivial projections. Proof: If e ~ -~/0 is a projection, t h e n so is 1 - e, and

thus we have 0 ~ e ~ 1. From the positivity of T it follows that 0 ~ T(e) ~ 1. If T(e) = 0, then since e = e'e, we have e = 0. On the other hand, if T(e) = 1, then ~'(1 - e) = 0 and e = 1, completing the proof. We will s h o w that C*red(~2) has a subalgebra ~0 such that "r(proj ~/0) C Z, a n d that if C*red(~2) contains a non-trivial projection, then the same is true for s/0, contradicting Lemma 4.2. 5. R e l a t i n g

the traces

The strategy will be to relate our two examples of traces, trace and T. Given a 6 C*red(~2) , it is easy to

but this will be valid only if we k n o w that X(a) - X0(a) is integrable. In a n y event, the formula is correct for a in the set s~0 of a 6 C~'red(~2) for which X(a) - It0(a) We have that Xo(u) and Mu) agree at all but two basis elements: ~1 and ~u_~, and thus they agree on the ort h o g o n a l c o m p l e m e n t of the t w o - d i m e n s o n a l subspace s p a n n e d by these two vectors. A simple induction shows that for a n y group element s 6 ~2, ko(s) and Ms) also coincide on the c o m p l e m e n t of a finite dimensional subspace, and thus the same is true for Ma) and ~o(a) for all a ~ C~:2. Because finite rank maps are integrable, we conclude that C~:2 C ~o. Our little tour is almost at an end. The set ~0 is a *-subalgebra of C*red(~2) because ~1 is an ideal in ~ (just as fl is an ideal in e~). Furthermore sg0 is dense in C'red0:2) because it contains C~:2. But sg0 has an enorm o u s advantage over CB:2: because ~1 is complete (just as e 1 is complete), sg0 is closed u n d e r certain analytic manipulations. If C*r~d(B:2) has a non-trivial projection, t h e n a simple spectral approximation a r g u m e n t shows that there is a self-adjoint element a ( CB:2 with dis-

F

sp a

The construction of a projection.

describe the matrix of X(a) relative to the canonical basis. In particular, from the calculation in (3.1), the diagonal terms are given by ~,(a)%. ~s = ~,(a)~l " al = T(a).

On the other hand, consider the " p e r t u r b e d " representation ~0- The isometry of e2(~:2) onto H, carrying K(a) to the restriction of Xo(a) sends the canonical basis of f2(G) onto an orthonormal basis for H u. It follows that if a ~ C*red(~:2), and 8s E H u, there exists a t ( D:2 with k0(a)Ss 9 8~ = X ( a ) ~ t 9 ~t = T(a).

The same applies to H v, and we conclude that X0(a)Ss'~s = Iv(a) 0 ss #= 11 In particular, X(a) - k0(a) has only one non-zero diagonal element, T(a). It would be tempting to argue that trace IMa) - k0(a)] = T(a),

(5.1)

connected spectrum containing 0 and I in distinct components. But using a line integral we m a y " d e f o r m " a into a nontrivial projection e that is still in sg0. In fact, let F be a curve disjoint from the spectrum of a which winds once about I a n d not at all about 0. We define e = (2Tri)-1 fr (~1 -- a)-ld~, where the line integral is defined as a uniform limit of Riemann sums in the usual way. The whole point is to check that the Riemann sums give terms converging in the ~ - 1 n o r m to k(e) - X0(e), thus proving that X(e) X0(e) is integrable, a n d e ( sg0. Once this not very difficult computation is done, we have from Lemma 4.1 that (because X(e) a n d X0(e) are projections), T(e) = trace X(e) - X0(e) is an integer, contradicting Lemma 4.2. Note that the whole a r g u m e n t is reminiscent of differential topology. Just as it often possible to prove p u r e l y topological results by a p p r o x i m a t i n g w i t h s m o o t h m a p p i n g s a n d a p p l y i n g differential techniques, we m a y regard sg0 as "the s m o o t h " elements in C'red0:2). This is not just a coincidence. Non-cornTHE MATHEMATICALINTELLIGENCERVOL. 11, NO. 1, 1989 33

m u t a t i v e differential t o p o l o g y is a thriving subject, which must be discussed elsewhere.

6. Why the circle is connected The above a r g u m e n t may be applied to ~:,,, the free group on n generators. Deleting the origin from the usual graph of this group, one obtains n components, each a copy of the original graph, and the a r g u m e n t a p p l i e s as before. In particular, this implies that C*r~d(7/) = C(T) is connected, i.e., T is connected. For a general c o m m u t a t i v e discrete g r o u p G, we have that C*red(G) = C(G), where G is the dual group of G, and it is well k n o w n that G is connected if and only if G is torsion free. This leads us to a problem that has yet to be solved: if G is a non-commutative torsion free group, does it always follow that C*r~d(G) has no projections? For a general C*-algebra ,~ it is also of great interest to look for projections in d | ~d(n). If ,~/ = C(X), X compact, these correspond to the vector bundles over X, and in the general case give meaning to " q u a n t i z e d vector bundles." This belongs to C*-algebraic K-theory, another area of great interest (see [1]).

References [1] B. Blackadar, K-theory for Operator Algebras, Mathematical Sciences Research Institute Publications, Vol. 5, Springer-Verlag, New York, 1986. [2] A. Connes, Non-commutative differential topology, Publ. Math. IHES 62 (1985), 257-360. [3] J. Cuntz, K-theoretic amenability for discrete groups, J. Reine Ang. Math. 344, 180-195. [4] P. A. M. Dirac, The Derivation of Quantum Theory, Gordon and Breach, London, 1971. [5] W. Heisenberg, Uber quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen, Zeit. fiir Physik 34 (1925), p. 879. [6] R. Kadison and J. Ringrose, Fundanwntals of the Theory of Operator Algebras, Academic Press, 1986. [7] G. Mackey, Mathematical Foundations of Quantum Mechanics, W. A. Benjamin, New York, 1963. [8] J. von Neumann, Zur Algebra der Funktionaloperationen und Theorie der normalen Operatoren, Math. Ann. 102 (1929), 370-427. [9] G. Pedersen, C*-algebras and their Automorphism Groups, Academic Press, 1979. [10] M. Pimsner and D. Voiculescu, K-groups of reduced crossed products by free groups, J. Op. Theortl 8 (1982), 131-156. [11] R. Powers, Simplicity of the C*-algebra associated with the free group on two generators, Duke Math. J. 49 (1975), 151-156. [12] F. Riesz and B. Sz.-Nagy, Functional Analysis, Frederick Ungar Pub., 1955. [13] M. Takesaki, Theory of Operator Algebras, I, SpringerVerlag, 1979.

Mathematics Dept. UCLA Los Angeles, CA 90024 USA

34

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. l, 1989

this year's Crafoord prize, together with a significant sum of money, jointly to Pierre Deligne (who was my student) and myself. Nevertheless, I regret to inform you that I do not wish to accept this (or any other) prize for the following reasons. 1) My salary as professor, even my pension starting next October, is more than sufficient for my own material needs as well as those of my dependents; hence I have no need for money. As for the distinction given to some of my work on foundations, I am convinced that time is the only decisive test for the fertility of n e w ideas or views. Fertility is measured by offspring, not by honors. 2) I note moreover that all researchers of high level, to which a prestigious award such as the Crafoord prize addresses itself, have a social standing that provides them with more than enough material wealth and scientific prestige, with all the power and privileges these entail. But is it not clear that superabundance for some is only possible at the cost of the needs of others? 3) The work that brought me to the kind attention of the Academy was done twenty-five years ago at a time when I was part of the scientific community and essentially shared its spirit and its values. I left that environment in 1970, and, while keeping my passion for scientific research, inwardly I have retreated more and more from the scientific "milieu." Meanwhile, the ethics of the scientific community (at least among mathematicians) have declined to the point that outright theft among colleagues (especially at the expense of those who are in no position to defend themselves) has nearly become the general rule, and is in any case tolerated by all, even in the most obvious and iniquitous cases. Under these conditions, agreeing to participate in the game of "prizes" and "rewards" would also mean giving my approval to a spirit and trend in the scientific world that I view as being fundamentally unhealthy, and moreover condemned to disappear soon, so suicidal are this spirit and trend, spiritually and even intellectually and materially. This third reason is to me by far the most imperative one. Stating it is in no way meant as a criticism of the Royal Academy's aims in the administration of its funds. I do not doubt that before the end of the century, totally unforeseen events will completely change our notions about "science" and its goals and the spirit in which scientific work is done. No doubt the Royal Academy will then be among the institutions and the people who will have an important role to

play in this unprecedented renovation, after an equally unprecedented civilization collapse. I regret the inconvenience that my refusal to accept the Crafoord prize may have caused you and the Royal Academy, especially because a certain amount of publicity was already given to the award prior to the acceptance by the chosen laureates. Yet, I have never failed to make my views about the scientific community and the "official science" of today known to this same community and especially to my old friends and young students in the mathematical world. They can be found in a long reflexion R~coltes et Semailles (Reaping and Sowing) on my life as a mathematician, on creativity in general, and on scientific creativity in particular; this essay unexpectedly became a portrait of the morals of the mathematical world from 1950 up to today. While awaiting its publication in book form, a provisional edition of 200 preprints has been sent to mathematical colleagues, especially algebraic geometers (who now do me the honor of remembering me). Under separate cover, I send you the two introductory parts for your personal information. Again I thank you and the Royal Academy of Sciences of Sweden and apologize for the unwanted inconvenience. Please accept my best regards. A. Grothendieck Department of Mathematics Univ. Montpellier 2 PI. Eugene Bataillon 34060 Montpellier Cedex, France

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989

35

Polynomial Identities and the Cayley-Hamilton Theorem Edward Formanek

My aim in this article is to publicize a theorem of C. Procesi [P] and Y. P. Razmyslov [R] that describes all the polynomial identities (or identical relations) satisfied by the algebra of n x n matrices over a field K. I hope to make clear h o w their theorem was derived from the Cayley-Hamilton Theorem and the invariant theory of n x n matrices. I. Kaplansky defined algebras satisfying a polynomial i d e n t i t y in his seminal article [K]. An area of mathematics does not usually have its origin unequivocally in a single source, but this is true of polynomial identity algebras. M. D e h n [D] in 1922 and his student

W. Wagner [W] in 1937 did work that belongs to the area; however, their contributions were largely overlooked. Definition. Let A be an algebra over a field K. We say that A satisfies a polynomial identity if there is a nonzero polynomial (with coefficients in K) f ( x 1. . . . . xn) in n o n c o m m u t i n g variables xl . . . . . xn such t h a t f(al . . . . . a,) = 0 for all a 1. . . . . a n in A. Here are s o m e examples of algebras satisfying a polynomial identity, or PI-algebras. (For convenience, we sometimes use variables x , y , z instead of x l , x 2. . . . . ) 1. A n y c o m m u t a t i v e algebra satisfies Ix,y] = 0, where [x,y] = x y - y x . 2. The algebra of upper triangular r x r matrices over K satisfies [x,y] r = O. 3. Let V be a K-vector space with basis { v l , v 2 . . . . }, and let A(V) be the exterior algebra on V. (A(V) m a y be characterized as the free algebra on {vi} modulo the relations Vi 2 = 0 , ViVj = - - V j V i ) . A ( V ) satisfies the polynomial [[x,y],z] = ( x y - y x ) z - z ( x y - y x ) = O. 4. M2(K), the algebra of 2 x 2 matrices over K, satisfies [[xy - yx]2,z] = ( x y - y x ) 2 z - z ( x y - y x ) 2 = O. 5. A n y subalgebra or h o m o m o r p h i c image of a PIalgebra is a H-algebra. The remaining examples require a definition. The standard polynomial of degree r is ~r(Xl .....

Xr) = ~

sign(~)x,~(~)x~(2) 99. X.rr(r),

~'r r

where S r is the symmetric group of permutations of {1. . . . . r}. 6. If A is finite dimensional over K of dimension r, t h e n A satisfies ~r+l(xl . . . . . Xr+l) = 0. 7. If A is algebraic over K of b o u n d e d degree r (each THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1 9 1989 Springer-Verlag New York

37

element of A satisfies a polynomial of degree r over K), then A satisfies 5Dr+x(X, xy, x y 2. . . . . x y r - l , x y r) = O. 8. (Amitsur-Levitzki Theorem [A-L]). M,(K) is annihilated by the standard polynomial of degree 2n and no polynomial of lower degree. In all the examples, except the final one, it takes at most a few lines to verify that the given identity is satisfied. The Amitsur-Levitzki Theorem now has four essentially different proofs, all requiring some work and cleverness. Algebras with polynomial identity are analogous to groups satisfying an identical relation. This area of group theory, called varieties of groups, has developed quite independently of PI-theory, with very little interaction between the two areas. Even basic terminology is sometimes different. For example, the subgroup (of a free group on countably many generators) of identical relations satisfied by a given group is called a f u l l y invariant subgroup, while the ideal (of the free algebra on countably many generators) of identities satisfied by a given algebra is called a T-ideal. Aside from the preferences of researchers in the two areas, an intrinsic reason w h y the two areas have developed differently is that the concept of a prime ideal is fundamental in ring theory, but has no analogue in group theory. It turns out that if K is infinite and A is a prime K-algebra (i.e., the zero ideal is a prime ideal of A), then for some positive integer n, A satisfies exactly the same identities as M,(K), the algebra of n x n matrices over K. Moreover, the T-ideal of identities of M , ( K ) is a prime ideal dR,, the ideals {At,} form a strictly descending chain ~1 D d~2 D ~t 3 D . . . . and they are the only nonzero prime T-ideals. The ideals {~,} have remained a basic object of study. For example, the Amitsur-Levitzki Theorem says that the standard polynomial of degree 2n is the polynomial of least degree in d~,. Although m a n y p r o b l e m s concerning ~ , r e m a i n open, there is a theorem, discovered independently by Procesi [P] and Razmyslov [R], which completely describes ~t,. The theorem says, in a sense that will not be made completely precise here, that all polynomial identities satisfied by n x n matrices are consequences of the Cayley-Hamilton Theorem. A good way to understand their theorem is to look at the case of 2 x 2 matrices. (From now on we will assume that K has characteristic zero. In fact there is no explicit description of ~ , in characteristic p > 0.) Let U be a 2 x 2 matrix over K. Then the Cayley-Hamilton Theorem says that U2 - T(U) + det(U) = 0,

functions of oq and o~2 in terms of the power symmetric functions of a s and o~2 gives det(U)

= c q e 2 = 1/2[(~1 + ~2) 2 -

= 1/2IT(U)2 -

U2 -

T(LOU + V2[T(U)2 - T(U2)] = 0.

(3)

Now we multilinearize (polarize, in the language of the last century) by substituting U = U 1 + U2 in (2) and deleting all but the multilinear terms. The result is equal to (b(U1 + U2) - qb(U1) - (b(U2), where qb(U)

The concept of a prime ideal is fundamental in ring theory, but has no analogue in group theory. denotes the characteristic polynomial of U. UIU2 + U2U1 - T(U1)U2 - T(U2)U~ + T(U1)T(U2) - T(U~U2) = O.

(4)

This formula (4) is the multilinear form of the CayleyHamilton Theorem for 2 x 2 matrices. Note that the denominator in (3) has v a n i s h e d - - a l l of the coefficients in (4) are integers. In fact the coefficients are all _ 1, and this also is true for the multilinear form of the Cayley-Hamilton Theorem for n • n matrices. Multiply (4) on the right by another 2 x 2 matrix U3 and take traces. T(UIU2U3) + T(U2U1U3) - T(U1)T(U2U3) - T(U2)T(U1U3) + T(U1)T(U2)T(U3) - T(UIU2)T(U3) = O.

(5)

Equation (5) is valid for any 2 x 2 matrices U1,U2,U3, and thus gives an identical relation among traces of 2 x 2 matrices. We will now rewrite (5) in a way which makes its form transparent. (123)

(213)

T(U1U2U3) +

T(U2UIU3)

(2)(13)

(1)(23) -

T(U1)T(U2U3)

(1)(2)(3)

-

T(U2)T(UIU 3) + T(UI)T(U2)T(U3)

-

T(U~U2)T(U3)

(6)

(12)(3)

(1)

where T(U) denotes the trace of a ~natrix U. If c,1 and ot2 are the characteristic values of U, then applying N e w t o n ' s formulas for the elementary symmetric

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. I, 1989

(2)

Then (1) becomes

=

38

(~12 + ~22)]

T(U2)].

~,

sign(~r)T~,(Ul,Uz, U3) = O.

~rES3

Definition: Let "rr e St, where S, is the symmetric group of p e r m u t a t i o n s of 1. . . . . r. Write ~r as a

product of disjoint cycles (including 1-cycles, so that each of the digits 1. . . . . r occurs exactly once): "rr = (a 1 . . . ak, ) (bl . . . bk2) ( q . . . Ck3) . . . .

The associated trace f u n c t i o n of ~r is T,,(U~ . . . . .

U,) = T(U~ . . . LI~,)T(Ub, . . . U G )

T(G,

U~ ....

Note that T~, is a function Mn(K) ~ ~ K, where Mn(K) r is the set of r-tuples of elements of M , ( K ) . Let ~(n,r) denote the K-vector space of all such functions. Then

All polynomial identities satisfied by n x n matrices are consequences of the CayleyHamilton Theorem.

for all invertible n x n matrices P over K. Here the first fundamental theorem is that t h e r i n g of invariants is generated by the traces of monomials in U 1. . . . . U r. A second f u n d a m e n t a l theorem t h e n gives the relations a m o n g the generators. The stated theorem apparently only gives the multilinear relations a m o n g the traces, but it is a general principle that in characteristic zero the multilinear relations generate all relations. For matrix invariants over a field of characteristic p > 0, t h e r e is n o t y e t a first f u n d a m e n t a l theorem, so of course there is also not a second fundamental theorem. Finally, we can give the promised description of ~t n, the T-ideal of polynomial identities satisfied by M,(K), where K has characteristic zero. This requires only a simple but very important observation of B. Kostant [Ko], w h o was clearly aware of the second fundamental theorem although he did not explicitly state it in the above form. Suppose that fix1 . . . . . x~) is a multilinear polynomial with coefficients in K. Then

(*) we can define a K-linear function = qr(n,r) : K[Sr] ~

T~(n,r)

by ~(Xa,,'rr) = EG, T~. Multilinearizing the Cayley-Hamilton Theorem for 2 x 2 matrices showed that G{sign(w)'a-: ~r 9 $3} lies in the kernel of ~(2,3)--i.e., it defines a function which vanishes on triples of 2 x 2 matrices. More generally, multilinearizing the Cayley-Hamilton Theorem for n x n matrices shows that X{sign(~)-~: ~ 9 S,+I} lies in the kernel of ~ ( n , n + 1). We n o w have e n o u g h notation to state the main result of Procesi [P] and Razmyslov [R]. T H E O R E M (Second F u n d a m e n t a l Theorem of Matrix Invariants. [P], Theorem 4.3; [R], Proposition 1). Let K be a field of characteristic zero. Then the kernel of ~(n,r) : K[Sr] ~ @(n,r) is zero if r _-< n, and is the twosided ideal of K[Sr] generated by Z{sign(-~)'m ~ 9 Sn+l} ifr~n

+ 1.

The reason it is called a second f u n d a m e n t a l theorem is the following. In invariant theory, a first f u n d a m e n t a l theorem gives a g e n e r a t i n g set for some ring of invariants. In our set-up the ring of invariants is the set of p o l y n o m i a l functions in @(n,r) that are invariant u n d e r s i m u l t a n e o u s c o n j u g a t i o n - - i . e . , polynomial functions f : M , ( K ) r ~ K that satisfy f(pu1p-1 .....

PUr P - l )

Xr) is a polynomial identity for M , ( K ) if and only if T[f(U1 . . . . . U~) 9 Ur+l] = 0 for all U1, .... UF+ 1 ~- M n ( K ).

f ( x 1. . . . .

= f(U1, . . . , U~)

This is n o t h i n g more than the n o n d e g e n e r a c y of the trace as a bilinear form on M , ( K ) : If U 9 M , ( K ) , then U = 0 if and only if T ( U V ) = 0 for all V e M,(K). Combining (*) with the second fundamental theorem gives a description of all the multilinear polynomial identities satisfied by Mn(K).

References [A-L] S. Amitsur and J. Levitzki, Minimal identities for algebras, Proc. Amer. Math. Soc. 1 (1950), 449-463. [D] M. Dehn, Ober die Grundlagen der projectiven Geometrie und allgemeine Zahlsysteme, Math. Ann. 85 (1922), 184-193. [K] I. Kaplansky, Rings with a polynomial identity, Bull. Amer. Math. Soc. 54 (1948), 575-580. [Ko] B. Kostant, A theorem of Frobenius, a theorem of Amitsur-Levitski and cohomology theory, J. of Math. and Mech. 7 (1958), 237-264. [P] C. Procesi, The invariant theory of n x n matrices, Adv. in Math. 19 (1976), 306-381. [R] Y.P. Razmyslov, Trace identities of full matrix algebras over a field of characteristic zero, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 723-756 (Russian). English Translation: Math. USSR Izv. 8 (1974), 727-760. [W] W. Wagner, Uber die Grundlagen der projectiven Geometrie und allgemeine Zahlsysteme, Math. Z. 113 (1937), 528-567. Department of Mathematics The Pennsylvania State University University Park, P A 16802 USA

THE MATHEMATICAL INTELLIGENCER VOL. I L NO. 1, 1989

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An Algebraist Commuting in Berkeley* Craig Huneke

1. Introduction This article was occasioned by a three-week program in commutative algebra held at the Mathematical Sciences Research Institute (MSRI) in Berkeley from June 15-July 2, 1987. Those who have not been to the Institute have been deprived of a spectacular view of the entire bay area from the Golden Gate bridge to Oakland. The price for this view is that the institute is somewhat inaccessible by foot, but a regularly run bus provides good service. The program was attended by 140 mathematicians from around the world. Fifty-six main talks and thirtytwo seminar talks were given. For those w h o are counting this comes to approximately 6.286 talks per day, mainly on commutative algebra, but also including many talks on related fields such as algebraic K-theory, representation theory, combinatorics, and, especially, algebraic geometry. Did this heavy infusion of talks make the conference successful or not? Surely neither necessarily--but it did give us a unique and probably unprecedented chance to gain a vision of most of the subject as it is developing today. As a coorganizer (the others were Mel Hochster, chair of the organizing committee, and Judith Sally), I naturally hope that the conference was a success in terms of the mathematics that will come out of the propinquity of so many mathematicians over three weeks. In any case it was a gastronomic success for me. In particular, the Thai restaurants in Berkeley were great. We also had the strongest gathering of table tennis players I've ever seen among mathematicians and so, of course, a tournament, won by Gerd Almkvist of Sweden, was a sine qua non. But what about the mathematics?

This article is intended to describe some of the new work in commutative algebra that was presented at the conference. Obviously, I must choose some work over others and so I will focus on two n e w techniques: one introduced within the last year, the other in the last four years. I am concentrating on methods rather than results in the belief that the techniques I will describe will yield more than they have already, which is impressive as is. The techniques deal with diametrically opposite cases arising in commutative algebra. One is a method to study rings containing fields; the other has been chiefly used in commutative algebra to prove theorems concerning rings that do not contain fields. From this description alone one might expect the first method to be more geometric and the latter

* I'd like to blame my wife, Edith Clowes, for suggesting this title. 40

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1 9 1989 Springer-Verlag New York

Mathematical Sciences Research Institute, Berkeley.

Judith Sally

method to be more arithmetical, but the exact opposite holds. Of course both of these methods are limited and by no means apply to many subjects that were discussed at the conference. Many talks were devoted to the study of maximal Cohen-Macaulay modules and there has been progress in the study of how many equations are needed to define algebraic sets. By algebraic set I mean the set of zeroes of a set of polynomial equations, with coefficients in an algebraically closed field. In fact constant progress has been made in this area ever since the proofs of Daniel Quillen and A. A. Suslin that projective modules over polynomial rings are free. Recently nice w o r k h a s b e e n d o n e c o n c e r n i n g the Betti numbers of modules and noncommutative algebraic geometry. Other time-honored questions were spoken on and neither method described in this article is likely to be applicable to some of them. For instance, perhaps the most famous unsolved problem is the Jacobian conjecture which states that, if fx. . . . . fn are polynomials in n variables x 1. . . . . xn over the complex numbers such that the determinant of the Jacobian matrix, (Ofi/Oxj), is a nonzero constant, then the subring generated by the fi is equal to the entire polynomial ring. In other words, the variables x i are polynomials in the fi. Rather than giving an immediate description of these techniques I'll provide a brief historical survey of the field, suitably slanted to make what comes later appear more natural. In fact I have to look no further than a speech given at the conference by Irving Kaplansky, the director of MSRI and an important figure in the development of commutative algebra in this

country. A survey article by Kaplansky [K] is also a good source. He indicated in his speech a path of major theorems in commutative algebra, each at approximately fifteen year intervals, from the time of David Hilbert to the present. These theorems provide exactly the history I need and an opportunity to make suitable definitions needed in the rest of this paper. Before the historical digression I'd like to say a word concerning what commutative algebra is about. A fatuous answer which, although true, provides no information, is that commutative algebra studies rings that are commutative. A better answer will be provided in the historical sketch below--commutative algebra focuses on certain kinds of commutative rings (e.g., polynomial rings and power series rings), for many different reasons arising from the study of invariant theory, n u m b e r theory, and algebraic g e o m e t r y , which are some of the fields that have influenced its development. Ultimately, of course, commutative algebra is what its practioners do--especially its best practioners. Therefore it is easiest to quote Mel Hochster. He wrote in [H, pg. 2] "Finally, I want to remark t h a t . . , algebra, after all, has to do with solving equations. Abstract algebra is the daughter of the theory of equations (in the broadest sense) and perhaps its best theorems still deal with that subject."

2. A Little Light History Several of the most fundamental theorems of commutative algebra were proved by Hilbert around 1890, most of them inspired by his research into invariant THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989

41

theory. Probably the most basic theorem is the Hilbert basis theorem. In modern language it says that if R is a Noetherian ring then so is the ring R[X1, . . . , Xn] consisting of polynomials with coefficients in R. A s u i t a b l y recast v e r s i o n of the proof s h o w s t h a t R[[XI . . . . . Xn]], the ring of formal power series with coefficients in R, is also Noetherian. Already this theorem provides a good class of rings to study; later theorems will show just how good this original class is. Of course one has to start with a Noetherian ring somewhere to use this theorem to build new ones, and natural candidates are any field, rings of algebraic integers, or even rank one discrete valuation rings. A second important theorem of Hilbert is his Nullstellensatz. In pure field theory Oscar Zariski restated it as follows: THEOREM 2.1. (Hilbert Nullstellensatz) Suppose that K is an algebraically closed field and L is a field extension of K that is finitely generated as an algebra over K. Then L = K. One can convert this theorem into a statement about polynomial rings and solutions of equations in the following manner. The assumption that L is finitely generated over K means that L is isomorphic to a polynomial ring K[X1 . . . . . Xn] modulo a maximal ideal M. The conclusion that L = K is equivalent to the existence of elements Yl. . . . . yn in K such that M is contained in (and therefore equal to, since it is maximal) the ideal generated by {X i - Yi}. By the Hilbert basis theorem M is finitely generated, say by polynomials F1. . . . . Fm, and what Nullstellensatz shows is the existence of y = yl . . . . . y, in K such that y is a solution of the equations F i = 0,1 ~ i ~ n. A more revealing way of stating the theorem is first to remark that a field K is algebraically closed if and only if every polynomial with coefficients in K in one variable has a solution in K. The Nullstellensatz then states that any finite set of polynomials {Fi} with coefficients in K and in several variables has a solution in K, provided it is possible to have a solution in any field extension of K. The only case where it is clearly impossible to have a common solution is if there are auxiliary equations {Gi} such that FIG 1 + . . . + FroGm = 1, e.g., the equations F1 = X = 0, F2=XY-I=O

,

clearly cannot have a common solution. Obviously YF 1 F2 = 1. Hilbert's theorem says that this restriction is the only obstruction to the existence of a common zero of the polynomials F1. . . . . F m. Two more theorems proved by Hilbert have been of great importance. The first of these, the Hilbert syzygy theorem, foreshadowed the characterization due to Jean-Pierre Serre of regular local rings as those having finite global dimension. A local ring R with maximal -

42

THE MATHEMATICAL INTELLIGENCER VOL, 11, NO. 1, 1989

ideal m has finite global dimension if for every finitely generated module M, there exists an exact sequence: 0--~ Fn---* Fn_ 1 ~

. . . ---* Fo---~ M---~ 0

where the F i are finitely generated free R-modules. This property is equivalent to saying Tor ~ (R/m, M ) = 0 for all but finitely many i. The existence of such a resolution of M in some sense linearizes problems concerning M: the properties of M are intertwined with the behavior of the maps ~i, which after choosing bases of F i and F i_ 1 can be thought of as matrices with coefficients in R. The second theorem is Hilbert's proof that rings of invariants of the general linear group acting linearly on a polynomial ring are Noetherian. This solved the first main problem of invariant t h e o r y - - t h e finite generation of rings of invariants. Masayoshi Nagata gave a counterexample to show that without restrictions on the group, the invariants need not be finitely generated. Hilbert's proof is so simple and beautiful that I will present a sketch here based on David Mumford's account of Hilbert's fourteenth problem [M]. Let R be a polynomial ring over a field of characteristic zero and let G be the general linear group of some size acting linearly on R. Put S = R G, the ring of invariant polynomials. All that is needed from representation theory is that every representation of G is completely reducible; the proof works equally well for any such group (i.e., for a linearly reductive group). This property allows us to decompose R = S G C as an S-module where C is a G-complement of S in R. In particular, the injection i of S into R splits: there is an S-linear map ~ from R to S, namely the projection onto S, such that ~i = identity map on S. Now we can abstract to a completely general situation and finish the proof by proving: THEOREM 2.2. Let R be any Noetherian ring and let S be any subring of R such that the injection of S into R splits via an S-hornomorphism f~. Then S is Noetherian. PROOF: The simple proof is based on the observation that ideals in S are contracted from their extensions in R. Let I be an ideal of S and suppose that s is an element of IRNS. Write s = Efig i where )~ belong to I (and hence are in S) and the gi belong to R. Apply 6; s = f3(s) = EBffigi) = EfiB(gi) and this element is in I. Hence I = I R N S . Now consider an infinite ascending chain {Ii} of ideals in S. The chain of ideals {IiR } must stop since R is Noetherian and contracting back gives that the original chain of ideals also stops. Hence S is Noetherian. In fact, in the case of invariants of groups acting linearly on polynomial rings, the invariants will be a graded algebra over the base field K and in this case Noetherianness implies that the algebra is even finitely generated over K.

We shall later return to properties of direct summands of rings, especially those which are direct summands of regular rings. Clearly Hilbert was a dominant force in the later formation of commutative algebra. The next fifteen-year period included the work of Emanuel Lasker, who was then world chess champion and was to remain so until 1921 when Capablanca beat him. As a boy I read a book on Lasker's life and games entitled Emanuel Lasker and subtitled The life of a chess master by J. Hannak. It had a foreword by Albert Einstein and an appendix that described in layman's terms his contribution to mathematics, Lasker's proof that ideals in polynomial rings over fields decompose as intersections of primary ideals. (An ideal q is primary if whenever ab E q either a ~ q or bn E q for some n I> 1.) This generalized Kummer's ideal numbers and provided a fundamental tool for commutative algebraists, especially after Emmy Noether gave an insightfully simple proof of primary decomposition valid for any Noetherian ring. During the early part of the twentieth century an English mathematician named F. S. Macaulay was

E v e n s o m e a l g e b r a i c g e o m e t e r s t e n d t o have g l a z e d eyes when a c o m m u t a t i v e algebraist starts explaining C-M rings.

hard at work studying the primary decompositions of ideals in polynomial rings. In particular he isolated an extremely important property of polynomial rings that n o w goes by the name Cohen-Macaulay (C-M for short). Macaulay's work is more relevant today than in his own time. Cohen-Macaulayness or the lack of it lies at the center of commutative algebra, but unfortunately it has always been a hard concept to understand. Even some algebraic geometers tend to have glazed eyes when a commutative algebraist starts explaining C-M rings. This is partly because there are at least six equivalent characterizations of the property, none of which is obviously equivalent to the others, and to understand fully how this property is used one must understand these equivalences. Of course it is exactly the fact that Cohen-Macaulayness appears in many different guises that makes it so useful. I will begin with a definition that is not as narrow as it first appears. Suppose that R is a local ring (has a unique maximal ideal) and contains a formal power series ring A = k[[xI . . . . . xe]] over a field k in such a way that R is finite as an A-module. Then R is C-M if and only if R is free as an A-module. The same definition applies if A is a polynomial ring over a field K and R is a K-algebra containing A and finite as an Amodule. To give the general definition of C-M we

must first define a regular sequence on a module. Let R be a commutative ring (always with identity) and let M be an R - m o d u l e (always unital). A s e q u e n c e x 1. . . . . x n of elements of R is called a regular sequence on M (or an M-sequence) if: (1) (x 1. . . . . Xn)M ~ M and (2) for each i, 1 ~ i <~ n, x i is not a zero divisor on M/(xl . . . . . xi_l)M. The dimension of the ring R, denoted by dim(R), is the supremum of lengths of chains of prime ideals in R. If (R, m) is local, then dim(R) is the least integer n such that there exists n elements x I. . . . . x, in m and a positive integer N with mN contained in the ideal generated by the x;. The set of elements x I. . . . , x~ is called a system of parameters (s.o.p.). A local ring R is C-M if and only if some (equivalently every) s.o.p, is a regular sequence on R. As the reader can see, the first definition, although in a restricted situation, is much easier to digest in one sitting. Introducing dimension and C-M rings is actually taking a giant forward leap chronologically. In the 1930s, soon after Noether's wonderful simplifications, Wolfgang Krull introduced dimension and localization and the theory of local rings. In many ways he is the founder of commutative algebra. Krull observed the process by which, given a prime ideal p in a ring R, we can form the rings of quotients Rp by using denominators not in p to obtain a local ring whose prime ideals are in 1-1 correspondence with those of R that are contained in p. He also introduced the completion of a local ring. Completion and localization are two of the most important techniques of the field. Let R be a local ring with maximal ideal m. Introduce a pseudometric on the ring by letting d(x,y) = 1/2", where n = m a x { i : x - y is in mi}. If R is Noetherian then d is a metric and the completion of R gives another local ring /~. Krull proved that this pseudo-metric is a metric in the Noetherian case. Because we will refer to this theorem later, we isolate it here. THEOREM 2.3. (Krull's intersection theorem) If R is a Noetherian local ring with maximal ideal m, then n~l mn = O.

The premier example of completion is obtained by letting R be k[X 1. . . . , Xn] localized at the maximal ideal (X1. . . . . Xn); in this case/~ is nothing but the formal power series ring k[[X 1. . . . . Xn]]. Krull made several conjectures concerning complete local rings that were proven in the late thirties and early forties by a brilliant student of Oscar Zariski's named I. S. Cohen. These conjectures showed the full importance of completion and also first truly showed the distinction between rings containing fields and those not containing fields, at least in so far as the general theory was concerned. (A great difference was already evident for affine rings from the work of Zariski. THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989

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By "affine" we mean any ring finitely generated as an a = (al . . . . . an) is a zero of each f,., and let m = (xi - ai) be algebra over a field. Zariski proved many theorems the maximal ideal of R corresponding to a. Then R m is regular detailing the algebraic structure of such rings. An if and only if rank (J(a_)) = n - dim(R), where ](a) is the aside: Zariski is well-known for introducing algebra Jacobian matrix (3f/Oxj) evaluated at the point a. into algebraic geometry, partly to make complete and A problem that was fairly easy to solve if R contains precise m u c h of w h a t the Italian g e o m e t e r s had "proved." An intriguing quote of Isaac Newton in his a field was to show that every regular local ring is a Arithmetica Universalis gives an opposite viewpoint: "Equations are expressions belonging to arithmetical computation, and in geometry properly have no place. 9 . . Therefore these two sciences ought not be con- W h e n I w a s a g r a d u a t e s t u d e n t I f e l t t h a t unfounded.") Cohen's structure theorem for complete less o n e p r o v e d a r e s u l t t h a t u s e d in an e s s e n local rings gave birth to a standard technique in com- t i a l w a y t h e p r o p e r t i e s o f r e g u l a r rings, e s p e mutative algebra: localize, complete, and then use the c i a l l y t h e f a c t t h a t m o d u l e s o v e r t h e m h a v e Cohen structure theorem to study a problem. Cohen f i n i t e p r o j e c t i v e d i m e n s i o n , t h e n one h a d n o t proved the following statements for complete local really done anything substantial. rings (R,m) that contain a field. 2.4 There is a field L contained in R such that the composition of the injection of L into R followed by the surjection of R onto Rim is an isomorphism. Such an L is called a coefficient field. 2.5. R is a homomorphic image of a formal power series ring with coefficients in L. 2.6. If x I . . . . . x n is any s.o.p, of R, then the complete subring A = L[[x 1. . . . . xn]] of R is isomorphic to a formal power series ring in n variables over L and furthermore R is a finite A-module. This last condition is the analogue in the complete case of the Noether normalization theorem for affine varieties. Cohen also gave a theorem for complete local rings that do not contain fields. Essentially the same statements hold but one must replace L by a complete rank one discrete valuation ring. This seemingly minor change, however, has caused no end of headaches for researchers trying to prove results in this case. Two concepts studied extensively in the fifties and sixties were the notions of regularity and multiplicity. In the last section of this article we will consider multiplicity, so here we'll treat regularity. A local ring R is said to be regular if its maximal ideal can be generated by a s.o.p. If R contains a field this condition is equivalent to saying that the completion of R is isomorphic to a formal power series ring over a field. An arbitrary ring is said to be regular if each of its localizations at its maximal ideals is regular. The corresponding classical notion is that of a simple or non-singular point. An easy criterion for affine rings is the famous Jacobian criterion.

Xn]/(fl. . . . . fro) be an affine ring over a field K of characteristic O. Suppose that

J a c o b i a n C r i t e r i o n : Let R = K/XI . . . . .

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THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989

unique factorization domain. It was not until Maurice Auslander and David Buchsbaum in 1958 used homological algebra that it was shown to be true in general. The work of Auslander, Buchsbaum, and Serre in the late fifties ushered in an era of extensive use of homological algebra. It has recently been again the case that to solve several homological problems for rings that do not contain fields n e w powerful techniques needed to be introduced. When I was a graduate student I felt that unless one proved a result that used in an essential w a y the properties of regular rings, especially the fact that modules over them have finite projective dimension, then one had not really done anything substantial. I have since revised this point of view but it has a germ of t r u t h - - r e g u l a r rings are the fundamental building blocks of commutative algebra. The sixties b r o u g h t a m o n u m e n t a l w o r k that in many ways was the best commutative algebra ever done: Heisuke Hironaka's proof of resolution of singularities in characteristic zero. Unfortunately this work is little understood today, especially by most commutative algebraists, including myself. Resolving singularities and understanding the process of this resolution is to me one of the fundamental goals of commutative algebra--that of solving equations in a broad sense. Parts of the proof have been singled out for special study, but other parts have by and large not been fully treated, e.g., there have only been a few papers on the behavior of Hilbert functions under blowing up. There have been several proofs of resolution in low dimensions (especially dimension 2). Nonetheless I find it a bad sign that the proof as a whole has never been significantly simplified. It is against all my experience that the first proof of a result could be the best, nor that much cannot be learned by improving upon the first proof. The early seventies gave birth to several striking results, for instance the work of Buchsbaum and Ei-

senbud on the structure of free resolutions, but I will concentrate on two of them that ushered in the technique that will occupy the next section. The first of these papers was the paper of Christian Peskine and Lucien Szpiro in 1973, Dimension projective finie et cohomologie locale [P-S]. This paper introduced reduction to characteristic p. The second paper was the 1975 CBMS notes of Hochster, Topics in the homological theory of modules over commutative rings, which refined this technique. Both works relied heavily on a theorem Michael Artin proved in 1969 known as the Artin Approximation Theorem. Another work of Hochster and Joel Roberts used reduction to characteristic p in a different way, to prove a beautiful theorem concerning rings of invariants of groups, complementing the early work of Hilbert. The theorem states: THEOREM 2.7. [H-R]. Let G be a linearly reductive group acting linearly on a polynomial ring R over a field K. Let S = R G be the ring of invariants. Then S is C-M. As discussed above, S being C-M is connected with S being free over a polynomial subring. This theorem can be stated without using the word "Cohen-Macaulay" at all; the theorem becomes (see [Ke] who put it into this form): THEOREM 2.8. Let R and S be as in the above theorem. Then there exist fundamental homogeneous invariants G 1. . . . . Ga and auxiliary homogeneous invariants F 1. . . . . Fm such that any invariant H in S can be written uniquely in the form EJi(G 1. . . . . Ga)Fi, where Ji are polynomials over K in d variables. A simple example of this theorem is the following: Example 2.9. Let R = K[x,y,u,v] and let G = K*, the nonzero elements of K, act on R as follows: if a E K*, then x---~ ax, y--~ ay, u --~ a-lu, and v---~ a-iv. The ring S of invariants is K[xu,xv,yu,yv]. The fundamental invariants of the theorem can be taken to be xu, yv, and xv + yu, and the auxiliary invariants can be taken to be 1 a n d xv. For instance (xv) 2 can be written as (xv + yu)(xv) - (xu)O/v)l. As in the proof of Hilbert's theorem, what is important in the assumption is that S is a direct summand of R as an S-module. Based on the paper of Hochster and Roberts the following conjecture was made: CONJECTURE 2.10. Let R be a regular ring and let S be a ring contained in R such that S is a direct summand of R as an S-module. Then S is C-M. Partial results were obtained by George Kempf [Ke] w h e n e v e r R and S are affine algebras and by JeanFrancois Boutot [B], who proved the stronger result that S will have rational singularities in the case that both rings are affine over a field of characteristic zero.

Only within the last year has the conjecture been shown in [H-H] for rings that contain fields by using the technique of reduction to characteristic p. It is still open if the rings do not contain fields. The reader might wonder how I could have almost finished this short history without mentioning Alexander Grothendieck. Now I have! His influence is so general that it is often overlooked, much like the name of a country on a map that is difficult to perceive because it stretches across the entire map. Specific contributions he made, such as the development of local cohomology and the theory of excellent rings, play an important role in commutative algebra. The two techniques described in the next sections can seem formidable to a beginning student. Students are faced with the prospect of learning a machine in order to use it or else accepting on face value the requisite theorems and applying them. Many are not willing to accept theorems they do not understand; my gut feeling is that this is laudable and should be encouraged. Moreover, machines often gobble up those who try to understand their engines, and people spend their lives becoming mechanics for the machine. It is no longer reasonable to understand everything one might use. The most obvious example is the classification of finite simple groups, which I would imagine few people have ever read. An example closer to home for me is the proof of resolution of singularities. This result is used all the time by algebraic geometers (less often by commutative algebraists), but almost no one has read the proof. I find this trend both inevitable and disturbing, especially as it is likely to become much worse. The problem is particularly severe for graduate students who simply do not have the luxury of m a n y years. For any mathematician there is a play-off between the effort required to learn a new method and what its practical use will be. For the most part this inertia is a good form of self-criticism for mathematics. Results that are not extremely useful tend to become forgotten. Unfortunately it is possible for a result that is not obviously useful when it is born to become useful but unused at a later date, until someone dusts off the old volumes to find it. In commutative algebra I feel that good examples should be a motivating factor to decide what one should learn or not learn. W. Heinzer reminded me of a visitor we had once at Purdue who asked a lecturer for an example of what he was talking about; he added to his question, "or is that not done?" The attitude that a good example is not worth as much as a bad theorem can be found throughout mathematics, the one exception being a counter-example to a famous problem. Personally I prefer the good example. Is it any clearer what commutative algebra is about? It would be nice to be able to partition commutative algebra into a small number of boxes. Such a compartTHE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989 4 5

mentalization is probably impossible but let me try; I'll choose four "boxes" off the top of my head: Homological Questions, Theory of Primary Decomposition, Birational Study of Rings, and Invariant Theory. Probably if I picked up the latest issue of Mathematical Reviews and tried to fit every article in section 13 into one of these categories I would fail miserably. However, I am taking a broad view of each category. For instance, under the theory of primary decomposition I include work on Cohen-Macaulay rings, formal fibers, linkage theory, the study of unique factorization domains, and the theory of multiplicities.

3. Reduction to Characteristic p Before beginning a description of reduction to characteristic p, we will give a proof of a theorem of Jo61 Brian~on and Henri Skoda to illustrate why it is often useful to work in characteristic p. The proof we will give in characteristic p was found this year [H-H] and is extremely simple. By reduction to characteristic p we can immediately claim the validity of the theorem for any ring containing a field. All previous proofs were, however, difficult and the first proof (over C) rested on a deep transcendental result of Skoda. The theorem originally arose in the following context. Let R be a formal (or convergent) power series ring in variables z1. . . . . za over C and let f be an element of R. Setj(f) = (Of/Oz1. . . . . 3f/3za)R, the Jacobian ideal of f. It was known that some power fk of f lies in j(f). Mather raised the problem of finding the least such k such that for a n y n o n - u n i t f, fk is in j(f). Brian~on and Skoda proved that d, the number of variables, bounds k for any non-unit f. In fact they showed considerably more; to explain what more they showed we need the concept of the integral closure of an ideal. As it turns out f is integral over j(f).

convergent power series over C, then f is integral over I = (gl . . . . . gn) if and only if there exist a constant K and an open set around 0 in C a for which

Klxl. The theorem Brian~on and Skoda proved was: THEOREM 3.4. [B-S] Let R be either the formal or convergent power series ring in d variables and let I be an ideal of R. Then (Ia) * C I. If f is integral over I, then fa is integral over Ia and hence by this theorem lies in I. This solves the original problem of Mather. The above theorem appeared in 1974. It became a challenge to algebraists to find an algebraic proof of this purely algebraic statement. This was finally accomplished in 1981 in a series of two papers, by Joseph Lipman and Bernard Teissier [L-T] and Lipman and Avinash Sathaye [L-S] who generalized this theorem to: THEOREM 3.5. Let R be a d-dimensional regular local ring and let I be an ideal of R. Then (Id) * C I. To give a proof of this theorem in characteristic p we must use one crucial fact about the Frobenius map in regular rings. If R is a ring of characteristic p > 0, let f : R ~ R be the map which sends r to rP, the Frobenius map. If R is regular, an extremely important theorem, due to Ernst Kunz, is that the Frobenius map is flat. (A homomorphism of rings R ~ S is fiat if for any injective map q~ : M --* N of R-modules, the induced map | 1 : M | S --~ N @R S is injective). One consequence of this that we shall need is the following. If I is an ideal of R and x is in R, then I : x denotes {r ~ R : xr E I}, the annihilator of x into I. If I is an ideal generated by Yl. . . . . Yn and q = p~ is a power of the characteristic p of R, then I[q] denotes the ideal generated by (yl) q (yn) q or equivalently, I [qI is the ideal generated by all the qth powers of elements of L We need: .

DEFINITION 3.1. Let R be a commutative ring and let I be an ideal of R. An element x in R is said to be integral over I if it satisfies (3.2)

x k + alx k-1 + " ' ' + a k = O,

where a i is in I i for 1 ~ i ~ k. The set of all elements integral over I forms an ideal denoted by/*. Notice that if x is i n / * then certainly xk is in I for some k. In fact an easy induction using equation (3.2) shows that if x is integral over I, then there exists a k (the same k as in (3.2)) such that for all nonnegative integers n, (3.3)

x k+n E P.

Less obviously, (3.3) is equivalent to x being integral over I. We shall in practice use (3.3). If R is the ring of 46 THE MATHEMATICAL 1NTELLIGENCER VOL. 11, NO. 1, 1989

.

.

.

.

THEOREM 3.6. If R is a regular local ring of characteristic p > O, I is an ideal of R, and x is an element of R, then ltqI : xq = (I : x)tq I. Theorem 3.6 looks innocuous but is the basis of most of the applications of the reduction to characteristic p. The converse is also true, as was shown first by Kunz. A n o t h e r reduction one can make is due to D. G. Northcott and David Rees IN-R]. As long as the residue field of R is infinite (which is a harmless ass u m p t i o n in the context of the Brian~on-Skoda theorem), any ideal of R is integral over an ideal generated by d elements. One now easily sees that the following theorem is sufficient to prove (3.5). THEOREM 3.7. Let R be a regular local ring of character-

Normally one considers systems of equations with height constraints that should not h a v e any solution. Some examples of these p h e n o m e n a are: E x a m p l e 3.9: The most famous type of a system with height constraint comes from Krull's principal ideal theorem. Consider the equation, d-1

-

Y,X, =

o.

i=1

A solution x 1. . . . .

xe,y i . . . . . Ye-1 of this equation with height condition w o u l d be a ring A in which the height of (x 1. . . . . xa)A = d but in which x a is nilpotent over the first d - 1 of the x i. This is impossible by the t h e o r e m of Krull.

D a v i d Rees

istic p > 0 and let I be an ideal generated by n elements, Yl. . . . . y,. Then (P)* C I. P r o o f : ([H-HI) Let x be a n o n - z e r o e l e m e n t of (P)* with x ~ I. T h e n b y (3.3) there is an integer k such that for all m >1 O, X m + k ~ (In)m = Inm. This latter ideal is generated by all monomials II (yi)bi of degree n m = Ebi in Yi. . . . . y,. Because there are exactly n of the Yi, at least one of the bi must b e / > m w h i c h shows that I nm ((yi)m. . . . . (y,)m)R for every m and in particular for m = pe = q for every e. Thus, (xt)(xq) E /lql for every q = /Y, and so x t E/Iql : xq = (I : x)Iql C_ mq for every q = pe, and hence xk is in the intersection of all the powers of the maximal ideal of R. By Krull's Intersection Theorem 2.3, x t (and thus x) is 0.

The sixties brought a monumental work that in many w a y s was the best commutative algebra ever done: Heisuke Hironaka's proof of resolution of singularities in characteristic zero.

The t e c h n i q u e of r e d u c t i o n to characteristic p realizes a m e t a t h e o r e m , given in [H]. To describe the t h e o r e m w e n e e d s o m e definitions. By a s y s t e m of p o l y n o m i a l e q u a t i o n s w i t h h e i g h t condition over a ring A w e m e a n a system of polynomial equations, Fx(X, Y) = 0

E x a m p l e 3.10: This example reflects w h a t is k n o w n as the analytic i n d e p e n d e n c e of a s.o.p. First some notation is n e e d e d . If c = (q . . . . . ca) is a d-tuple of nonnegative integers, let Xc denote II(Xi)Ci and set [c[ = Ec i. Consider the equations, XM=.YCXr = 0, III~I=.(Y~Z~ - 1) = O. If these e q u a t i o n s h a d a solution in a local ring in which the x i were a s.o.p., t h e n the equations say that some m o n o m i a l of degree n in the x i is in the ideal g e n e r a t e d by all of the other monomials of degree n in the x i. This is impossible because of the so-called analytic independence of s.o.p.'s. F:xample 3.11:

Consider the equation d

d

I-[ (x,)t i=1

yi(xi),§

= o.

i=1

If A is a local ring containing a field, t h e n no s.o.p. x 1. . . . . xa can satisfy this equation. This has b e e n p r o v e d b y M. Hochster, w h o also s h o w e d that m a n y homological theorems follow from the veracity of this simple statement. It is still open w h e t h e r or not in a general Noetherian commutative ring this equation with height condition can have a solution. The conjecture that no such solution is possible is called the Monomial Conjecture. The m e t a t h e o r e m given in [HI about equations with height condition can n o w be stated.

(3.8)

~:,(X, Y) = 0 Metatheorem: (see [HI) Let P be a theorem about Noetherian

w h e r e X = X 1. . . . . Xd and Y = YI . . . . . Ym and the F i h a v e coefficients in a ring A. If R is an A-algebra w e say that x = x 1. . . . . xa and y = Yl. . . . . Ym is a solution of (3.8) in R if ht(x I . . . . . xe)R = d (but (xR) ~ R) a n d Fi(x,y ) = 0. H e r e if I is an ideal in R, ht(I) = min{dim(Rv) : I C P, P prime}.

rings that is true for finitely generated algebras over finite fields. Suppose that P is equivalent to the statement that for a certain family of systems of equations with height condition over Z no system in the family has a solution. Then P is true for all Noetherian rings R containing a field, regardless of characteristic. THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989

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has a solution in R. (2) If a system of polynomial equations over R has a solution (Sl. . . . . sn) with si in R, then for every integer t > 0 there is a solution (rl . . . . . rn) with the ri in R such that r i =-- s i m o d mta, 1 <~ i <~ n.

M e l v i n Hochster

A full list of the constraints satisfied by a system of parameters in a local ring is far from complete. M a n y constraints are known; almost always the constraints represent important properties with far-reaching consequences. A simple example of a statement that I can s h o w over the complex numbers by using reduction to char-

Machines often gobble up those who try to understand their engines, and people spend their lives becoming mechanics for the machine. It is no longer reasonable to understand everything one might use.

acteristic p, but for which I do not k n o w of a simple direct solution, is the following. If C is an n by n matrix with coefficients in some ring R, define the ith characteristic function of C as follows: write det(zI - C) = E ( - 1)ichi(C)z n-i, where I is the n x n identity matrix. Then chi(C ) are in R and of course ch,(C) = det(C) and Chl(C ) = tr(C). N o w suppose that R = C[[x,y]] a n d A and B are two commuting matrices such that chi(Ax -tBy) is in (x,y) 2i for all i. Then necessarily chi(A ) and ch.{B) are in (x,y)i for all i. This statement is not an isolated fact but is really a translation, suggested by J. Lipman, of a statement concerning the integral closures of ideals. It surely m u s t follow from some basic polynomial identity on the characteristic polynomials, but I d o n ' t k n o w w h a t that identity might be. O n l y one ingredient is missing before we can describe the main procedure, n a m e l y the Artin approximation theorem. We first need some definitions. Say a local ring (R,m) is an approximation ring if it satisfies the following two equivalent conditions: (1) W h e n e v e r a s y s t e m of p o l y n o m i a l e q u a t i o n s over R has a solution in R, the completion of R, then it 48

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989

The conditions are equivalent because the extra congruences can be expressed by polynomial equations, but the second condition shows w h y R is called an approximation ring. Let R = K[X 1. . . . . X,] be a polynomial ring over a field K of characteristic 0 and let R = K[[X1. . . . . X,]] be the completion of R with respect to the maximal ideal m of R generated by X 1. . . . . X,. Let S be the integral closure of R in R (i.e., S consists of all formal p o w e r series in the X i which satisfy monic polynomials over R), let n = m/~ N S, and let R h = S n. Then R h is a local ring called the Henselization of R. The ring R h is m u c h bigger t h a n R; for instance (1 + x) 1/" can be expanded as a formal power series and is clearly integral over R and hence belongs to R h. THEOREM 3.12. (Artin[A]) R h is an approximation ring. Recently Christel Rotthaus has s h o w n a n y excellent Henselian local ring containing the rationals is an approximation ring. We can n o w outline the process of reduction to characteristic p > 0. Suppose we are trying to prove a theorem, e.g., the Briangon-Skoda theorem, for a ring containing Q. A s s u m e the theorem is false and we are given a counterexample that we can express in terms of equations being zero or not zero, possibly with height conditions. First localize at a prime that captures all the information needed and then complete so that there is a contradiction in a complete local ring R. By Cohen's structure theorem R is m o d u l e finite over a power series ring B = K[[x 1. . . . . Xd]], a n d if necessary we m a y even in general assume that K is algebraically closed. Let S = K[x 1. . . . . Xd] a n d A = SM, w h e r e M = (xI . . . . . Xd)S. Note that A = B. We make a reduction of the problem by obtaining a counterexaml~le in a ring that is a finite m o d u l e over A h, instead of A. To do this we apply the Artin approximation theorem not only to our original set of equations, giving the existence of a counterexample, but also to the entire ring structure of R over B. We approximate the multiplication table of the (finite) set of generators of R over B. Next we use the fact that A h consists of algebraic elements over A, so that we m a y write A h as a direct limit of localizations of affine K-algebras at maximal ideals to show there is a counterexample in a finite stage and hence in the localization of an affine algebra at a maximal ideal. Then we remove our localization to obtain a counterexample in an affine K-algebra. Next choose a finitely generated field extension L of Q that contains all relevant coefficients a n d so obtain a

counterexample in an affine algebra over L. Then find a finitely generated 77-subalgebra C of L with L as its quotient field so that there is a counterexample in a finitely generated C-algebra. Finally work modulo the maximal ideals of C; if m is a maximal ideal of C then necessarily C/m is a finite field and we will be able to preserve the counterexample modulo almost all maximal ideals of C, so that we have finally reduced to characteristic p for a contradiction! A quick set of steps for this process, which should be a litany for many commutative algebraists is: localize, complete, approximate, reduce to affine over Q, then affine over 77, and kill p. I thought that such a useful process deserves a poem, however bad (see

box).

from the definition, the fundamental fact 3.6, and the Krull Intersection Theorem 2.3. For if x is in the tight closure of I but not in I, then there is a nonzero c in ~/Iql : xq = ~ (I : x) [qI = 0. Contradiction. Thug in characteristic p the tight closure of an ideal I in a ring R is contained in the intersection of the contractions to all regular rings containing R of the extension of I. Using tight closure in characteristic p, together with reduction to characteristic p, many theorems were reproven with simpler proofs and vastly generalized in [H-H]. Apparently much more study is needed of this concept and related ones. Perhaps the most remarkable use of the tight closure is the following theorem, proved in [H-H] by reduction to characteristic p and then using the tight closure. It heuristically says that, homologically speaking, there is no overlap between finite extensions of regular local rings and extensions that are themselves regular local rings. Easy corollaries of this result include the general theorem that direct summands of regular rings that contain fields are C-M and the Monomial conjecture (3.11). THEOREM 3.14. (Vanishing Theorem). Let A ~ R ~ S be injective maps of Noetherian rings which contain fields such that A and S are regular domains and R is a module finite extension of A. Let M be any A-module of finite type. Then for all i >1 1, the map TorA(M,R)---~ Tor iA(M,S) is the z e r o map.

This powerful method has one big drawback. Often a theorem does not easily lend itself to direct translation into equations. A good example of this is the t h e o r e m and ensuing conjecture of Hochster and Roberts d i s c u s s e d above. As we remarked, this theorem has been recently proved for rings containing a field by using reduction to characterize p [H-H]. The trick was proving a much stronger theorem which allowed translation into equational constraints. Both this proof and the proof of the Brian~on-Skoda theorem above came through the study of a new concept called the tight closure of an ideal, introduced this year in [H-H]. The tight closure lies between an ideal and its integral closure but in general is much tighter (i.e., closer to the original ideal) than the integral dosure. In characteristic p it is easy to define although somewhat technical. DEFINITION 3.13. Let R be a domain in characteristic p and let I be an ideal of R. We say an element x in R is in the tight closure of I if there is a nonzero element c such that cxq E I Iql for all sufficiently large q = pe. Notice that if R is a regular local ring, then the tight closure of any ideal is just itself. This follows at once

The overwhelming drawback of these methods is the failure of them to work when the ring does not contain a field. Luckily there has been recent progress by Paul Roberts in proving several old conjectures for rings that do not contain fields. This is the subject of the last section.

4. Multiplicities and Rings Not Containing Fields Ever since the difficulties in proving that regular local rings that do not contain a field have unique factorization, rings of this class have been a bugbear for many c o m m u t a t i v e a l g e b r a i s t s . There is a long (and growing) list of theorems that are true for rings containing a field but are open otherwise. The worst type of regular local ring in this regard is the so-called ramified one. These are regular local rings (R,m) of characteristic 0 but whose residue field Rim has characteristic p > 0, so that p must be in m. The word "ramified" refers to those such that p is in m2. This tends to make life difficult for several reasons. First of all, there is no chance to use reduction to characteristic p - - p a r t of the success of that method was that, in a sense, we could choose the characteristic we wanted: here there is never a choice. The only prime we can kill is p and w h e n we do we no longer have a regular ring: we THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989 4 9

have introduced singularities, a n d have, in general, decreased the dimension of our ring. One of two theorems that Paul Roberts has recently proved involves intersection multiplicities; both of his p r o o f s rely on the w o r k of F u l t o n on intersection theory. The other theorem Roberts was able to prove, [R2], w h i c h we shall not discuss here, was the socalled " n e w intersection t h e o r e m . " The night before the first of his three talks at the Berkeley meeting several of the participants including Roberts and myself were eating dinner at a Taiwanese restaurant. Paul received a very apropos fortune cookie that night. It said something close to, "You will be noticed by those w h o count." Multiplicities, which have played a f u n d a m e n t a l role in commutative algebra since the papers of Chevalley [C] a n d Samuel [S], were brought into the limelight in the sixties from different directions by Serre [Se] a n d the work of D. Rees. A good introduction to i n t e r s e c t i o n t h e o r y a n d m u l t i p l i c i t i e s is g i v e n in Fulton's CBMS notes IF]. We first give a local definition of multiplicity of an ideal in a local ring.

as a polynomial in n of degree d whose highest term is na/d!. In fact e(R) = 1 for all regular local rings, and the converse is true for complete domains. Suppose n o w that S is any complete local ring that contains a field, let xl . . . . . x a be a s.o.p, for S generating an ideal I, and choose a coefficient field K for S as in the Cohen structure theorem. Then S will be a finite module over the power series subring R = K[[xI . . . . . xd] ]. The following theorem gives an imp o r t a n t c o n n e c t i o n b e t w e e n S being C-M a n d the value of e(/): THEOREM 4.4. S is C-M if and only if e(I) = ranka(S).

(By definition, rankR(S) = vector space dimension of S | over L, where L is the quotient field of R.)

L

PROOF: We will prove one direction. If S is C-M then S is free of rank r = ranka(S ) as an S-module. Since S/I n = S | Rim n, the dimension of S/I" over K is simply r times the dimension of Rim n over K. By our calculation above this shows that the multiplicity is exactly r. The other direction is harder and more useful. In general rankR(S ) t> e(I).

D E F I N I T I O N 4.1. Let I be an m-primary ideal in a local H o w does one compute intersection multiplicities of ring (R,m). Then R/I has a finite filtration whose successive quotients are copies of the residue field, R/m; the number of two curves in the plane? Let R = K[x,y] be the polynothese copies is unique and called the length of R/I, denoted by mial ring in two variables over a field K and let f and g r The function f(R/I n) becomes a polynomial function be two elements of R, so that the solutions of f = 0 of degree equal to the dimension d of R for large n; by defini- and g = 0 define curves. Given a point (a,b) in K2 we tion e(I) is d! times the leading coefficient of this polynomial wish to compute the intersection multiplicity of f and g at P = (a,b). Classically N e w t o n [N] used the resultant and is called the multiplicity of I. Alternatively e(I) = lim n----~oo f(R/In)d!/n a. If I = m, the maximal ideal of R, then e(m) is also denoted by e(R) and is called the multiplicity of R. This integer measures the singularity of R. Below we shall show that if R is regular then e(R) = 1 and the converse is true for complete domains.

Example 4.2:

If f is a p o l y n o m i a l in one variable x over a field K and a is in K, then the ideal generated by x - a is a maximal ideal in R = K[x]. Let S(a) be the local ring given by localization at this ideal. Then the multiplicity of the ideal f S(a) in S(a) is exactly the usual multiplicity of the element a w i t h respect to f. The multiplicity is e if (x - a) e divides f but (x - a) e+l does not divide f.

Example 4.3:

Let R = K[[xl . . . . . xa]] be a formal power series ring over a field. Let m be the (unique) maximal ideal of R. What is e(R)? W h e n the ring contains a coefficient field, length is simply the dimension over the coefficient field, so we w a n t to k n o w w h a t dimK(R/mn) is. This module has a K-basis the monomials in the x i of degree less than n and there are exactly (n+da-1) of these. It follows that the multiplicity is exactly 1, as we m a y expand the binomial coefficient 50

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989

to compute the points of intersection and their multiplicities. A m o d e r n w a y is as follows. Let m(P) be the maximal ideal of R generated by x - a and y - b. Let R(P) be the local ring obtained by localizing R at m(P). Then i(P;f,g), the intersection multiplicity of f and g at P, is dimK(R(P)/(f,g)R(P)). Unfortunately, a similar definition no longer works for higher dimensional intersection, for instance for the intersection of two surfaces in 4-space. One might hope for the following. Let R be a polynomial ring in d-variables, and let p and q be distinct prime ideals of R whose zero sets define algebraic subvariefies of d-space. Let P = (a 1. . . . . aa) be an isolated point of the intersection of these varieties. Algebraically the statement that P is isolated means that if we let R(P) be the local ring obtained by localizing R at the maximal ideal generated by x i - a i, t h e n dimk(R(P)/(p + q)R(P)) = dimK(R(P)/pR(P ) | R(P)/qR(P)) is finite. One could hope that this dimension should be the intersection multiplicity of the two varieties. Unfortunately, this is not the case; in fact u s u a l l y this d i m e n s i o n is too big. To rectify this problem Serre proposed an algebraic definition that corrected the excess f o u n d in higher dimensions. His definition works equally well for modules. A geometric definition correcting the excess h a d been given earlier.

DEFINITION 4.5. Let R be a regular local ring and let M present an outline of Roberts's proof here, mainly beand N be two finitely generated R-modules such that cause I know it better than the proof of Gillet and f ( M | N) is finite. We set x(M,N), the intersection multi- Soul6. Both proofs use a type of Riemann-Roch plicity of M and N, equal to Y,(- 1)ie,(Tori(M,N)). This sum theorem. Szpiro had prophesied t h a t a n appropriate is finite because regular local rings have finite global dimen- Riemann-Roch theorem could yield a proof of the vansion. ishing conjecture. The nonvanishing conjecture is still If M = R/p and N = R/q then this number gives an open if R is a ramified regular local ring. accurate count of the intersection multiplicity of the Roberts's proof relied heavily on the intersection variety determined by the zero set of p and the zero set t h e o r y d e v e l o p e d by Fulton, M a c h p h e r s o n , and of q at the point corresponding to the maximal ideal of Baum. A nice presentation of this theory is given in the local ring. Tor0(M,N) is isomorphic to M | N so Fulton's book IF2]. The proof relies on the theory of the first term is exactly the length of M @ N. If R is a local Chern characters. polynomial ring in two variables and M = R/fR, N = The C h e r n characters take values in the C h o w R/gR as above then all the Tor's except Toro vanish, group of a ring (or scheme) X. The Chow group is an which explains the success of the classical definition of abelian group denoted by A(X) = (~n=0 i At(X), where intersection of curves in the plane. n = dim X. The actual definition of this group is not One of the properties a theory of intersection multi- important for our discussion here. We need only that plicities should have is the principle of continuity--if Ai(X) = 0 for i > dim X and that AnX is isomorphic to we move one of the varieties in a continuous (i.e., flat) 2~, generated by a class denoted by [X]. In general family then the intersection multiplicity should not Ai(X ) is the free abelian group on the classes of vary (in an appropriate sense). One consequence of /-dimensional closed irreducible subvarieties of X mothis is that if the dimensions of what we are inter- dulo "rational equivalence." secting add up to less than the dimension of the amRoberts's proof of the vanishing conjecture comes bient space, then we should get 0 for the intersection almost immediately from four properties and the exismultiplicity, because we should be able to move one of tence of a Chern character that satisfies them. Let X be the varieties away from the other. Algebraically this an algebraic scheme and suppose that G is a bounded means that if dim(M) + dim(N) < dim(R), then x(M,N) complex of locally free coherent sheaves on X. (If X = should be zero. Here the dimension of a module M is by Spec(R) we just mean a bounded complex of finitely definition the dimension of its support, or equiva- generated projective R-modules.) The support of G, lently is dim(R/ann(M)). On the other hand if dim(M) supp(G), = {p : (G)p is not exact}, with the corre+ dim(N) = dim(R), then they should have to inter- sponding definition if X is not the spectrum of a ring. sect: hence we should expect that x(M,N) > 0. Serre Now suppose we have a function, called the Chern [Se] proved both of these properties for regular local character, satisfying the following properties: rings that contain a field using reduction to the diagonal. However, his proof broke down in the ramified case of 3.9. Given any G as above, there is an associated map chi(G) : regular local rings that do not contain a field. Over the A,X--~ A , _ i(Y), where Y = supp(G). intervening years two conjectures were formulated: 3.10. The Chern character is multiplicative. If E and F are 3.6 Vanishing Conjecture. If R is a local ring and M and N bounded complexes of locally free modules (or sheaves), then are two finitely generated R-modules such that f(M | N) is chp(E | F) = Ei+j=pchi(E)chj(F). finite and dim(M) + dim(N) = dim(R), then x(M,N) > 0 provided R is a regular local ring (or even if M and N have 3.11. The Chern character is commutative. If E and F are as finite projective dimension). in (3.10), then chi(E)chj(F) = chj(F)chi(E). 3.7. Nonvanishing Conjecture. If R is a local ring and M and N are two finitely generated R-modules with f ( M | N) finite and dim(M) + dim(N) < dim(R), then x(M,N) = 0 provided R is a regular local ring (or even if M and N have finite projective dimension). Recently Paul Roberts [R1] was able to prove the vanishing conjecture in the case that R is a regular local ring or even in the more general cases that R has an isolated singularity or is a complete intersection. Independently Gillet and Soul6 proved the vanishing conjecture when R is a complete intersection. Their proof uses the tools of algebraic K-theory. I will

3.12. (Local Riemann-Roch) Suppose that X = Spec(R), where (R,m) is a local ring of dimension n, and assume that supp(E) = {m}. Then ch~(E) : AnX--~ Ao{m } = 27, and we require that ch,(E)([X]) = • = E(-1)i(Hi(E)). We will prove the vanishing theorem under the assumption that such a map with properties (3.9)-(3.12) exists. In fact no such map exists in this generality. However, the essence of the proof is the same and in fact (3.12) is correct (at least with Q-coefficients) when R is a regular local ring. Fulton does give a Chern character satisfying (3.9) and (3.10) that almost satisfies (3.12). One must correct the formula given in (3.12) by THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989

51

a local T o d d class. The resulting formula is called the local Reimann-Roch theorem. Roberts showed that the C h e r n character given by Fulton satisfies (3.11). In practice one also needs several other properties of the Chern character such as functoriality and additivity, but (3.9)-(3.12) suffice for our outline. We will confine ourselves to the case in which R is regular so we don't have to worry about the local Todd class. Let R be a regular local ring and suppose that M and N are t w o finitely g e n e r a t e d R - m o d u l e s such t h a t f(M | N) is finite. Let E (respectively F) be a minimal projective resolution of M (respectively N). It is well k n o w n that Tori(M,N) is equal to the i th homology of the total complex E | F. If dim(M) + dim(N) < dim(R) = n we wish to show that the Euler characteristic of this complex is 0, i.e., (3.13)

~,(- 1)iF,(Hi(E | F)) = O.

Acknowledgment I'd like to t h a n k Bill Heinzer a n d Mel Hochster for reading a preliminary version of this manuscript and m a k i n g valuable s u g g e s t i o n s . The p h o t o g r a p h s of Melvin Hochster, David Rees, and Judith Sally were taken by J~irgen Herzog.

Bibliography [A]

M. Artin, Algebraic approximation of structures over complete local rings, I.H.E.S. Publ. Math. 36 (1969), 23-58. [B] J.F. Boutot, Singularit6s rationnelles et quotients par les groupes r6ductifs, Inventiones Math. 88 (1987), 65-68. [B-S] J. Brian~on and H. Skoda, Sur la cl6ture int~grale d'un id6al de germes de functions holomorphic en un point de C n, C.R. Acad. Sci. Paris S~r. A 278 (1974), 949-951. [C] C. Chevalley, On the theory of local rings, Annals of Math. 44 (1943), 690-708. [F] Fulton, Introduction to intersection theory in algebraic geometry, CBMS 54 (1984), American Math. Society. [F2] W. Fulton, Intersection Theory, Ergeb. Math Grenzgb., 3. Fo!ge, Springer-Verlag. [H] M. Hochster, Topics in the homological theory of modules over commutative rings, CBMS 24 (1975), American Math. Society. [H-H] M. Hochster and C. Huneke, Tight closure, invariant theory, and the Briangon-Skoda theorem, in preparation. [H-R] M. Hochster and J. L. Roberts, Rings of invariants of reductive groups acting on regular rings are CohenMacaulay, Advances in Math. 13 (1974), 115-175. [K] I. Kaplansky, Commutative rings, LNM 311 (1973), Springer-Verlag, Berlin,153-164. [Ke] G. Kempf, The Hochster-Roberts theorem of invariant theory, Michigan Math. ]. 26 (1979), 19-31. [L-S] J. Lipman and A. Sathaye, Jacobian ideals and a theorem of Brian~on-Skoda, Michigan Math. J. 28 (1981), 199-222. [L-T] J. Lipman and B. Teissier, Pseudo-rational local rings and a theorem of Brian~on-Skoda about integral closures of ideals, Michigan Math. J. 28 (1981), 97-116. [M] D. Mumford, Hilbert's fourteenth problem--the finite generation of subrings such as rings of invariants, Proceedings of Symposia in Pure Math. 28 (1976), 431-444, American Math. Society. [N] I. Newton, Geometrica analytica, 1680. [P-S] C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale, I.H.E.S. Publ. Math. 42 (1973), 323-395. [R1] P. Roberts, The vanishing of intersection multiplicities of perfect complexes, Bulletin A.M.S. 13(2) (1985), 127-130. [R2] - - - , Le Th6oreme d'intersection, preprint (1987). [S] P. Samuel, La notion de multiplicit6 en Alg6bre et un G6om6trie alg6brique, Journ. de Math. 30 (1951), 21-205. [Se] J.P. Serre, Alg~bre locale multiplicit~s, LNM 11 (1965), Springer-Verlag, Berlin.

By (3.12) the value of the left-hand side of (3.13) is exactly chn(E | F)([X]), where X = Spec(R). By (3.10) this is the s u m Xi+j=nchi(E)chj(F)([X]). The support of F is equal to the support of N a n d has dimension equal to the d i m e n s i o n of N. Hence if j < codim(N) = n dim(supp(N)), then by (3.9) chj(F)([X]) is an element in A~_j(supp(N)) = 0, because n - j > dim(supp(N)) by a s s u m p t i o n on j. On the other h a n d suppose that i < codim(M). Then chi(E)chj(F)([X]) = chj(F)chi(E)([X]) by (3.11), chi(E)([X]) is in A,_;(supp(M)), a n d n - i > dim(supp(M)) by our assumption, so that this element is 0. Thus if either j < n - dim(N) or if i < n - dim(M), the corresponding term chi(E)chj(F)([X]) is zero. The entire s u m will be zero if one of these inequalities m u s t hold. Suppose that j ~ n - dim(N) and i ~ n dim(M). Then n = i + j ~ 2n - dim(M) - dim(N) > n by our a s s u m p t i o n on the s u m of the dimensions of the two modules. This contradiction proves the vanishing theorem. The proof m a y seem strange in the following sense: one shows a s u m of integers is 0 by showing each one of t h e m is zero! It would seem more natural that individual terms could be nonzero but the s u m was nonetheless 0. H o w e v e r , in the t h e o r y of multiplicities there are several instances of the same p h e n o m e n o n , where some s u m is proved to be zero by showing each term is 0; conversely it often h a p p e n s that if one assumes the s u m is 0 then all the terms are forced to be 0. Thus it seems to be a theme in nature. Although the proof as I presented it is formal a n d simple, the steps (3.9)-(3.13) are not at all simple a n d one m u s t develop a great deal more just to obtain these statements. Both proofs of the vanishing theorem use heavy machines: Roberts uses the machine of Fulton, and the proof of Gillet a n d Soul6 uses the machinery of algebraic Kt h e o r y a n d , in particular, uses a n o t h e r " R i e m a n n - Department of Mathematics Roch" theorem. It would be nice to find a simple proof Purdue University of such a basic fact. West Lafayette, IN 47907 USA

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THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989

Steven H. Weintraub* For the general philosophy of this section, see Volume 9, No. 1 (1987). A bullet (o) placed beside a problem indicates a submission without solution; a dagger (~) indicates that it is not new. Contributors to this column who wish an acknowledgment of their contribution should enclose a self-addressed postcard. Contest entries and problem solutions should be received by I May 1989. The prize (a free subscription to the Intelligencer) for the best contribution to this column in 1988 is awarded to Krzysztof Ciesielski for Problem 88-3, which appeared in Volume 10, No. 1. Such a prize will again be awarded for the best contribution in 1989. The winner of Contest 89-1 below will be awarded a free Springer book of his~her choice, up to a value of $60.

Problems Mathematicians' birth years: Contest 89-1 by the Column Editor Give the year of birth of each of the following mathematicians: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

value given in an entry, then the score of the entry will be f ( x l , y l ) + 9 9 9 + f(x16,Y16). Of course, low score wins. (Ties will be broken by lot. Joint entries are welcome, but readers are on their honor not to consult reference materials.)

Repeated subtraction: Problem 89-2 b y Henry Lulli (Gardena High School, USA)

L. Euler J.-L. Lagrange C.-F. Gauss A. L. Cauchy P. L. Chebyshev B. Riemann S. Lie H. Poincar4 D. Hilbert G. H. Hardy S. Lefschetz J. E. Littlewood S. A. Ramanujan S. Banach J. von N e u m a n n K. GOdel

For a 4-tuple of non-negative integers A = (al,a2,a3,a4), let D(A) = (la2 - a,l,la3 -

a2l,la4

-

a3Hal -

a4l).

Given such a 4-tuple A, not identically zero, define a sequence {A~ by A0 = A, A i+l = D(Ai) for i I> 0. Let h = h(A) be the unique non-negative integer such that A h = (s,s,s,s) for some positive integer s = s(A). Determine all possible values for the pair (h,s).

Contest entries will be scored as follows: Let fix,y) = - 5 if x = y; otherwise fix,y) = min (Ix - yl,30). If x i is the true year of birth of mathematician i, and Yi the

* C o l u m n editor's address: D e p a r t m e n t of Mathematics, Louisiana State University, Baton Rouge LA 70803-4918 USA THE MATHEMATICALINTELLIGENCERVOL. 11, NO. I 9 1989Springer-VerlagNew York 53

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THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989

HUYGENS' PRINCIPLE AND HYPERBOLIC WAVE EQUATIONS Paul G/inther

A N A L Y T I C PROPERTIES OF A U T O M O R P H I C L-FUNCTIONS

BEILINSON'S C O N J E C T U R E S ON SPECIAL V A L U E S OF L-FUNCTIONS

S t e p h e n Gelbart and F r e y d o o n Shahidi

edited by M. R a p o p o r t , N. S c h a p p a c h e r , and P. S c h n e i d e r

F o r t h e first t i m e in b o o k form, the author presents the results obtained by several mathematic i a n s t o w a r d s t h e s o l u t i o n of Hadamard's problem concerning t h e v a l i d i t y of H u y g e n ' s p r i n c i p l e for h y p e r b o l i c differential equations. 1988, 847 pages, $69.00 ISBN: 0-12-307330-8

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THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 1, 1989 55

Existence Theorems in Mathematics A. M. W. Glass*

The lesson for today is taken from the book M o r e Beasts f o r W o r s e C h i l d r e n by Hilaire Belloc reading from "The Microbe": All these have never yet been seen-But Scientists who ought to know, Assure us that they must be s o . . . Oh! let us never, never doubt What nobody is sure about! All of us who have confronted existence theorems in mathematics will recognize the relevance of the words of today's lesson. Yet, in our daily lives as mathematicians, we all too easily forget them and settle for weaker results than necessary. To illustrate this point, let me subdivide existence theorems into four kinds: (1) (2) (3) (4)

Mere existence. Effective existence. Constructive existence. Complete solution.

Often passage from one of these to the next is extremely difficult. Yet moving to the next level offers far greater insight and is well worth pursuing instead of

merely settling for a weaker result. I will illustrate each of the above categories with examples from algebra or number theory. As this is a tea-time chat, I have selected examples for their familiarity or ease of presentation and hope they prove to be of interest. I have also, for the same reason as well as my own incompetence, confined myself to this crude subdivision rather than far more subtle ones provided by proof theory and computational complexity theory.

(1) Mere Existence Even mere existence can be extremely difficult to establish. For example, just showing that for each natural number n ~ 3, x n + yn = z" has only a finite number of primitive solutions in the positive integers warranted a Fields Medal for G. Faltings [Faltings]. If we knew an explicit bound on z in terms of n (rather than merely the existence of such a bound), w e ' d maybe have more of a chance of establishing Fermat's Last Theorem (see (4) below). One small consolation in this direction has been obtained in [Heath-Brown]. For each natural number n /> 3, let X ( n ) = {m ~ n: x m + ym = z m has a solution in the positive integers}.

* This paper is an amplified version of colloquia talks (tea-time chatter) that I gave at Oklahoma State University, the University of Waterloo, and Michigan State University. I wish to thank these universities for the w a r m hospitality they afforded me. I am most grateful to A. Seidenberg for many helpful remarks, corrections, and considerable improvements to an earlier draft and to V. Frederick Rickey for supplying several historical facts and reading the manuscript. 56 THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 1 9 1989Springer-VerlagNew York

THEOREM 1: IX(n)l/n --~ 0 as n --~ oo. Proof (outline): For each odd prime p, xP + yP = zP has only a finite n u m b e r of primitive solutions in the set of positive integers by [Faltings]. So there is B(p), a positive integer, such that if x~ + y6 = z~ and (Xo,Yo, Zo) = 1, then xo,y o, z o <~ B(p). Hence if x~ + y~ = z~ with (xl,Yl,Zl) = 1, then "~,k , k .,k _< B(p). Thus 2 k -< z k ~ B(p) ~ l , Y l , "~ 1 ~ so k ~ Iog2B(p ). Consequently Fermat's Theorem holds from m = kp if k > Iog2B(p ). Elementary number theory n o w implies the result (see [Heath-Brown] for the details).// Of course, Theorem 1 does not imply Fermat's Last Theorem; for all we k n o w to the contrary, IX(n)l > log n for n sufficiently large. A k n o w l e d g e of B(p) would, however, lead to a far sharper result; a more constructive proof of Mordell's conjecture is sorely needed.

(2) Effective Existence Sometimes one can do s o m e w h a t better than mere existence. In this section I will be interested in examples for which it is possible to establish a procedure that, in theory, can be carried out on an idealized computer to o b t a i n s o m e or all s o l u t i o n s of the e x i s t e n c e problem; i.e., effective existence. Such an idealized computer is called a Turing Machine (not a Star Wars Device); it is p r e s u m e d to have at its disposal an infinite a m o u n t of tape, m e m o r y , etc. and have no a priori b o u n d on h o w long it can r u n to do a calculation, t h o u g h for any computation it is only to use a finite a m o u n t of time, tape, m e m o r y , etc. In more technical language, effective existence is a recursive procedure. If we can p u t a b o u n d a priori on the r u n n i n g time, a m o u n t of tape and m e m o r y used, etc. (i.e., an a priori b o u n d on the searching) we will regard such an existence proof as a constructive one in that it can be carried out in some technologically highly advanced universe with e n o u g h molecules etc. to build and run such a h i g h - p o w e r e d computing machine. For example, suppose I wish to test membership in some set X. W h e n I feed in a potential element of X, m y Turing machine's cogs, circuits, or whatever bec o m e a c t i v e a n d e v e n t u a l l y will p l a y O r l a n d o Gibbons's Jubilate Deo if the element actually belongs to X or Thomas Tallis's Lamentationes Jeremiah if it does not. (It is a s s u m e d that the listener can distinguish bet w e e n t h e s e t w o s u p e r b pieces of E n g l i s h T u d o r church music!) This is an example of an effective process. If I k n e w in advance that the machine would, for a n y i n p u t n, p l a y Jubilate Deo w i t h i n say, 10 l~176 years if it played it at all, I'd just have to live a little longer t h a n that to k n o w w h e t h e r or not an element n belongs to X. This would then be an example of a constructive process. I will n o w give some specific examples.

(i) Commutative Rings In this article all rings will have an identity, 1. A finitely presented commutative ring is the quotient of-a finitely generated commutative ring by a finitely generated ideal. Hence it has the form z[xl .....

xm]/(fl(x) . . . . .

f.(x))

where Z[X1 . . . . . Xm] = Z[X] is the ring of polynomials in commuting indeterminates X 1. . . . . Xm over the ring of integers Z, h(X) . . . . . f,(X) E Z[X], and (]:I(X). . . . . fn(X)) is the ideal of Z[X] that they generate. I wish to solve the word problem uniformly for finitely presented commutative rings; that is, I wish to give an effective procedure which, w h e n given m and h(X) . . . . . fro(X) in Z[X] = Z[X 1. . . . . Xm] and g(X) E Z[X], determines w h e t h e r or not g(X) + I = I in Z[X]/ /, where ! = (fl(X) . . . . . fn(X)). So I w a n t an effective m e t h o d for determining whether or not g(X) E I. T H E O R E M 2 [Simmons]: The word problem for finitely presented commutative rings is uniformly soluble. In her m o n u m e n t a l 1926 paper on effective solutions to problems about ideals of polynomials over fields, Greta H e r m a n n , a student of E m m y Noether, gave an elementary constructive proof of the following by induction on the n u m b e r of variables X 1. . . . . X m (see [Seidenberg 1]): There are explicit (recursive) functions b1 a n d b2 such that for any field K (a) there is a basis for the solution set of E~=1 f/(X)hj(X) = 0 all of whose elements have degree not exceeding b1 (re,d) where d = max{deg(fj): 1 j ~ n}; (b) if g(X) E (fl(X), . . . , fn(X)), there are polynom i a l s gj(X) E K[X] of d e g r e e n o t e x c e e d i n g b2(m,d, deg(g)) such that g(X) = E~= 1 f/(X)gj(X). The bother is that ~ is not a field a n d H e r m a n n ' s results do not apply directly. However, by combining (a) and (b) it is easy to construct such bI and b2 for the ring Z/(r) for any integer r/> 2. (*) The other main ingredient of the proof is that ;[X] satisfies the ascending chain condition on ideals. This is established by "mere existence" (see, e.g., N. Jacobson, Basic Algebra 1). Now for any integer r ~> 2, h(r) = {f ~ 27[X]: rf @ I} is an ideal of 77[X] a n d I C_ I:(r) C h(r 2) C . . . . Hence for some s / 1, I:(rs) = I:(rs+l) a n d for such an s it is immediate that I = (I + (r~)) A (h(rS)). Consequently, if g(X) E h(r) but g(X)~ I, then g(X) ~ I + (r ~ ) f o r s o m e k ~ s . THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989 5 7

Proof of Theorem 2: In order to test whether or not g(X) ~ (FI(X) . . . . . f,(X)) = I for fl(X) . . . . . f,(X) 77[X], we first note that we can list all n-tuples of polynomials in Z[X]. STEP 0: By (b), test if g(X) E I (~ Q, the ideal in Q[X] generated by fl(X) . . . . . fdX). If it is, use (b) to find an integer r / 1 so that rg(X) E I (so g(X) E h(r)); if it isn't, g(X) ~ I so play Lamentationes Jeremiah and stop. STEP k:

Determine if g(X) = Y~=I fj(X)hkj(X) where hk,(X)) is the kth n-tuple in the list. If so, play Jubilate Deo and stop. If not, use (*) to determine if ~(X) b e l o n g s t o the ideal of (27/(rk))[X] g e n e r a t e d by h(X) . . . . . f,(X), where h(X) --~ h(X) is the natural map of Z[X] onto (Z/(rk))[X]. If it doesn't, g(X) ~ I + (rk), SO g(X) ~ I. Consequently, play Lamentationes Jeremiah a n d stop. If it does, go to the next step. As I noted above, this procedure must stop and the theorem is proved.// (hkl(X) .....

Note that if we could b o u n d s a priori, w e ' d have a constructive algorithm instead of merely an effective one. This requires considerably more work. I will outline a proof from [A. Seidenberg 2] in the next section. It is possible to effectively list all finitely presented commutative rings by appropriate coding.

T H E O R E M 3 [Blass and Glass]: There is an effective procedure to determine for an arbitrary finitely presented commutative ring in the list whether or not it is a domain. The proof, like that of Theorem 2, is elementary and d e p e n d s on three extra facts besides Theorem 2: (a) [Seidenberg 2] Given h(X) . . . . . f,(X) E 77[X] = 77[X1. . . . . Xm], one can explicitly (constructively) find a prime P0 > 0 in 77 such that I:(p) = I for all primes p E Z such that p > P0, where I = (fl(X) . . . . . f,(X)). (b) [Seidenberg 2] With the notation of (a), one can explicitly (constructively) find fpl(X) . . . . . fp,p(X) 77[X] such that I:(p) = ~pl(X) . . . . . fp,p(X)) for each prime p (~P0). (c) [van den Driess] There is an effective procedure ( i n d e p e n d e n t of the choice of a prime field K) that determines (from the prime field K) whether or not an arbitrary finitely generated ideal (fl(X) . . . . . f,(X)) of K[X] = K[X1 . . . . . Xm] is prime. Indeed, by (a) and Theorem 3, we have: C O R O L L A R Y 4 [Blass a n d Glass]: There are effective procedures to determine for an arbitrary finitely presented commutative ring in the list (i) whether or not it is a domain of characteristic O, (ii) whether or not it is a domain of fixed prime characteristic p ~ O. 58

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989

(ii) A b e l i a n Lattice-ordered G r o u p s Let ~ be a lattice order on an Abelian group G such that f ~ g implies f + h ~ g + h for all f,g,h E G. Then G is said to be an Abelian lattice-ordered group. If we use ~ / a n d / X for the least upper b o u n d and greatest lower b o u n d operations, we can omit ~ a n d obtain a finitely axiomatized algebraic theory. By universal algebraic considerations [Cohn, Corollary IV.3.3], free Abelian latice-ordered groups exist in all ranks. However, as noted in [K. Baker], the free Abelian lattice-ordered g r o u p F m on m g e n e r a t o r s can be characterized as follows: Let ~ be the real line and ~1 . . . . . grm. ~m ~ ~ be the m s t a n d a r d projections. Let H m be the additive Abelian group of functions from ~m into R generated by ~t1. . . . . ~m" Then Fm is the additive group of continuous finite piecewise elements of Hm; i.e., functions from ~m into ~ of the form ~yY,~'= laijk~k, where aijk E Z (i E I, j E J, 1 ~< k ~< m), I and J are finite sets and G ( y ) denotes the greatest lower b o u n d (least upper bound) u n d e r the order f ~< g if and only if fiX) ~< g(X) in R for all X E ~m. Note that if ~yE~'= lai/k~rk = w E Fm, then the zero set Z(w ~ / 0 ) of w X/0 is given by Z(w ~/O) = {X ~ Rm: (w ~/0)(X) = 0} = y ~ Flip where IIij = {X E Rm: Y,km=laqkXk ~ 0}, a half space of R m through 0. Thus Z ( w ~/0) is a closed polyhedral cone of ~m with vertex 0. Conversely, given a n y closed polyhedral cone of ~m with vertex 0 associated with rational coefficients, I can obviously write d o w n an explicit element of F m w h o s e zero set is precisely the closed polyhedral cone. Moreover, if f,g >i 0, then Z(g) C Z(f)if and only i f f ~ rg for some positive integer r. If Ihl = h ~/ - h , then IhI I- 0; and IhI = 0 if a n d only if h = 0. Thus for w I . . . . . w , ~ F m, IWlI v . 9 9 v IWnI = 0 if and only if (w I = 0 & . . . & w, = 0). Kernels of h o m o m o r p h i s m s b e t w e e n Abelian lattice-ordered groups are called ideals and comprise those subgroups that are closed u n d e r the lattice operations and are convex (if fl ~ g ~
due to [K. Baker] and [Beynon]. For an altogether different proof, see [Khisamiev]. In contrast, if I explicitly (constructively) list all tengenerator one-relator Abelian lattice-ordered groups F10/(w0), Flo/(Wl) . . . . . we have:

tive pairs of triangulations of 4 dimensional polyhedra. In this light, the result seems less surprising. Because any 4 dimensional polyhedron can be triangulated, the isomorphism problem for Abelian latticeordered groups (like that for groups) falls into the "mere existence" and not "effective existence" category.

THEOREM 6 [Glass and Madden]: Although the word problem for the finitely axiomatized variety of Abelian lattice-ordered groups is uniformly soluble, there does not exist (iii) Nilpotent Groups an effective procedure which when presented with two arbitrary ten-generator one-relator Abelian lattice-ordered The main tool for handling algorithmic problems of nilgroups will determine whether or not they are isomorphic. potent groups is a result of [Philip Hall]:

(Informally this says that although there is a single machine that will tell us all we need to know about any individual finitely presented Abelian lattice-ordered group, there cannot exist a Turing machine which will infallibly distinguish between pairs of particularly simple ones!)

Proof: It is easy to see that Flo/(Wi) ~ F1o/(Wj) if and only if there is a piecewise linear homeomorphism with integer coefficients fixing 0 and mapping Z(Iwil ) onto Z(Iwjl ). Any closed polyhedron of dimension 4 can be constructively e m b e d d e d in ~2• and so gives rise to a closed polyhedral cone in ~10 with vertex 0 in a natural way. Thus if there were an effective solution for the isomorphism problem for ten-generator one-relator Abelian lattice-ordered groups, there would be an effective procedure to determine if two arbitrary (combinatorial) closed polyhedra of dimension 4 were piecewise linear homeomorphic. But [Markov] coded the insolubility of the isomorphism problem for finitely presented groups into the piecewise linear homeomorphism problem for d o s e d polyhedra of dimension 4, proving that the latter is not effectively soluble. I can now deduce that the isomorphism problem for ten-generator one-relator Abelian lattice-ordered groups cannot have an effective solution.//

If Markov's result seems surprising, first notice that by vertical projection three triangles with one common vertex and one side in common to any pair

If F is a finitely generated free nilpotent group of any class, f, gl . . . . . gn E F, and f does not belong to the normal subgroup of F generated by g l , - 9 9 , gn, then there is a finite homomorphic image F of F in which the images g~. . . . . g--~..... are all 1 but ~ the image of f, is not 1. Any finite (nilpotent) group is finitely p r e s e n t e d - for the relations, simply write down the entire multiplication t a b l e - - a n d I can list them all in a constructive way by any reasonable coding. Given, f, gl . . . . . g, in a free nilpotent group F of some class, I constructively list the members of the normal subgroup N of F generated by gl . . . . . g~. If f E N, it will appear in this list. On the other hand, if f ~ N, there will be a finite nilpotent group in which gl . . . . . g~ are all 1 but f is not. So I also run through all finite nilpotent groups and check if gl = 9 9 9 = g, = 1 # f occurs in one of them. Running the two processes alternately one step at a time provides an effective solution to the word problem for nilpotent groups (of any class). Philip Hall's result can be generalized [Blackburn]: If G is a finitely presented nilpotent group and fl,f2 E G, then fl and f2 are not conjugate in G if and only if they are not conjugate in some finite homomorphic image. The proof just given s h o w s that the conjugacy problem for finitely presented nilpotent groups is effectively soluble as observed in [Blackburn]. Far deeper work [Grunewald & Segal] gives an effective solution of the isomorphism problem for finitely presented nilpotent groups. The a b o v e p r o o f of the solubility of the w o r d problem for nilpotent groups is, of course, equally valid for any finitely axiomatized class of algebras that is residually finite in the strong sense that given a finitely generated normal subgroup (ideal, etc.) N in a finitely generated free algebra F and f E F \ N , there is a finite homomorphic image of F whose kernel includes N but omits f.

(3) Constructive Existence is piecewise linear homeomorphic to a triangle. Thus Markov's Theorem boils down to a question of effec-

In this section I wish to examine two instances of constructive processes in mathematics; that is, processes THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989 5 9

that can specifically be carried out provided we had sufficient time, space, and inclination. Such processes are firmly entrenched even in Gauss's work: he k n e w the procedure for constructing n-gons for certain large n b u t d i d n o t u n d e r t a k e the actual c o n s t r u c t i o n [Gauss].

(i) Polynomial Ideals over Fields In the last section, I promised to outline Seidenberg's proof [A. Seidenberg 3] that for a n y finitely generated ideal I of 7/[X] one can constructively find a natural

Often passage from one of these to the next is extremely difficult. Yet moving to the next level offers far greater insight and is well worth pursuing instead of merely settling for a weaker result. n u m b e r s such that I:(r ~) = I:(r s + 1). Indeed, Seidenberg gives a constructive proof of the ascending chain condition on Z[X].

Proof: The first thing to notice is that given ideals A 1 C A 2 C . . . in 7/(or 7//(m) for some integer m), one can explicitly find j such that Aj = Aj+ 1 (because (nl) C (n2) if a n d only if n 2 is a divisor of nl). Moreover, any finitely generated ideal of 7/is the homomorphic image of Z by a finitely generated ideal that can be explicitly given. This latter condition easily extends to: any finitely generated submodule of a free 7/module N of a finite rank is isomorphic to the quotient of a finite rank free Z-module A of finite rank by a finitely generated ideal L where A, I, and the isomorphism can be given explicitly. It n o w follows by induction on the rank of N that one can constructively obtain for each chain {Mj: j = 1,2,3 . . . . . } an integer i such that M i = Mi+ v From this it follows (see [A. Seidenberg 3, page 58]) that a n y finitely generated ideal in Z[X] is isomorphic to the quotient of a finite rank free 7/[X]-module A of finite rank by a finitely generated ideal I, where again A, I, a n d the isomorphism can be explicitly given. I n o w s h o w how, given A 1 C 32 C . . . finitely generated ideals in 7/[X], to construct k so that A k = Ak+ 1. For ease of exposition, I assume that X is a single variable (otherwise use induction on the number of variables). For each i, let L(Ai) be the set of leading coefficients of elements of A i. Then L(Ai) and a basis for A i from which it is obtainable can be constructed. Let n i be the m a x i m u m of the degrees appearing in such a basis. I m a y assume that n 1 < n 2 < . . . and, by the above, explicitly obtain iI such that All and All+ 1 contain the same elements of degree at most n 1, so Lm(Ai) 60

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989

= Lm(ai2 ) if m ~ n 1. Replace n I by nil + 1 a n d repeat the construction a n d obtain a j such t h a t L n. +1(Ai.+1) = L,,ik+,(Ai k+ 1) for k = j + 1. Hence L,,i;+ l'IAik+ 1) = L(Aik+l ), so Lm(Aik ) = Lm(Aik+l ) for al~ m. Consequently, Aik = Aik + 1 as desired.// Theorem 2 can n o w be strengthened to give a constructive uniform procedure to determine whether or not g(X) E I = (~I(X). . . . . f,(X)).

Proof: By H e r m a n n ' s result, I can determine if g(X) E I | Q. If it isn't, g(X) ~ I. If it is, I can find an integer r such that rg(X) E I. By the above, I can construct an integer s such that I:(r ~) = I:(rs+l) and also give explicitly a finite basis for I + (rS). Now I can test to see if g(X) E I + (rs) or not; i.e., if g(X) ~ I or g(X) ~ I. This proves the claim.// Note that this m e t h o d makes possible a "2 step" proof rather than a groping technique. Also, Theorem 3 and Corollary 4 can be obtained constructively. Indeed, [A. Seidenberg 2] shows h o w to construct, from a given finitely generated ideal of Z[X], the associated primes (i.e., h o w to explicitly obtain finite bases for them) (Theorem 5, op. cit.) and h o w to explicitly get a normal decomposition for any given ideal (Theorem 6, op. cit.). Then I C Z[X] is prime if and only if it is primary and equal to its associated prime. This yields an alternative proof of Theorem 3.

(ii) Linear Forms in Logarithms Hilbert's 7th P r o b l e m is: If c~ a n d [3 are algebraic numbers with ~ # 0, 1 and [3 irrational, is oL~transcendental? This question was answered in the affirmative by [Gelfond] and, independently, [Schneider]. More r e c e n t l y , in his g r e a t w o r k of 1966, [A. Baker 1] proved: If otI. . . . . Otn, [31. . . . . [3, are algebraic numbers with c~1. . . . . oLn (~ {0,1} and {1,B1. . . . . B,} linearly i n d e p e n d e n t over the rationals, t h e n oL~l . . . c~, is transcendental. The result follows quite easily [A. Baker, page 11] from: (*) If o~1. . . . . oLn are non-zero algebraic n u m b e r s such that log c~1. . . . . log c~, are linearly i n d e p e n d e n t over the rationals, t h e n 1,log oh . . . . . log c~, are linearly i n d e p e n d e n t over the field of algebraic numbers. Another easy consequence of (*) is [A. Baker, page 11]. A n y n o n - v a n i s h i n g linear c o m b i n a t i o n of logarithms of algebraic n u m b e r s with algebraic coefficients is transcendental; equivalently, if B0. . . . . ~Bn,CXl. . . . . cr are non-zero algebraic numbers with oL1. . . . . c~, # 1 and E~'=I ~i log o~/~ 0, then [3o + E~'=I ~j log oq # 0. This work leads to lower b o u n d s for linear forms in logarithms. Specifically, let oq . . . . . o~, be non-zero

algebraic n u m b e r s with degrees at most d and heights at most A ; let 13o. . . . . 13n be algebraic numbers with degrees at most d and heights at most B (/>2). Let/~ = 130 + E~=I 13jlog Otj. T h e n / ~ = 0 or IA] > exp( - C log B) for some constructively computable C which d e p e n d s only on n, d, and A, where log denotes the principal d e t e r m i n a t i o n of the logarithm function [A. Baker, Theorem 3.1]. A closer analysis leads to further i m p r o v e m e n t s . Specifically, if/~ # 0, then

I/N[i>exp(-Cl(n,D)(log B

+ C2(n,D)),

non-negative integers, where dl t> 2 and dj[dj+ 1 (1 ~< j ~< k - 1) if k > 0. In the case of a finitely presented Abelian group, this canonical form can be explicitly derived by matrix algebra, as I nowfllustrate. First notice that the subgroup generated by h~ and h2 is the same as that generated by h 1 and h2 + nh 1 for a n y integer n. Clearly it is the same as that generated by h2 a n d h 1. These suggest matrix reduction. Let G be the quotient of the free Abelian group F on {Xl,X2,X3} by the subgroup H of F generated by 2x 1 + x2, 6x~ + x2 + 12x3, and 6Xl - x2 + 24x3. We write this in the form

x112

6

6j

2x 1 + x 2 6xl q- x2 q- 12x3 6x1 - x2 + 24x 3 w h e r e C I ( n , D ) <~ (27n2D log A)" 221nD2 log log A, C2(n,D) is small and D = [Q(e~1. . . . . an,130. . . . . 13n): x2 1 1 - 1 Q] [Blass et al]. x3 0 12 24 A l t h o u g h all these results give explicit construcfible constants, the techniques used are extremely subtle. It w o u l d be inappropriate for me to produce an outline here. H o w e v e r , as I n o w d e m o n s t r a t e , t h e y can be where the ij entry is the coefficient of x i in the jth genu s e d to obtain complete solutions to certain classes of erator of H. I perform column operations to reduce the generators of H to i n d e p e n d e n t ones of nicer form. equations.

(4) C o m p l e t e S o l u t i o n s Consider the equations Y~7=OajxJyn-j= ----- 1, where a n = 1 a n d a 0. . . . . an-1 are integers. As a consequence of Alan Baker's result on lower b o u n d s for linear forms in logarithms, the number of integer solutions is finite and one can give an explicit upper b o u n d on the absolute value of these solutions [A. Baker, Chapter 4, Sections 2 and 5]. The improvements of [Waldschmidt] or [Blass et all can be used to lower this upper b o u n d considerably. This b o u n d can t h e n be further lowered using the basis reduction algorithm of [Lenstra, Lenstra, & Lov~isz] so that only r e l a t i v e l y s m a l l s i z e d s o l u t i o n s (~<103~ n e e d be s e a r c h e d for. This is easily a c h i e v a b l e on a n y of today's average mini-computers using convergents. Specifically:

(i) [Blass et al 2] The only integer solutions of x 3 63xy2 + 180y3 = 1 are (1,0) and ( - 1 1 , - 2 ) . (ii) [Blass et al 1] The only integer solutions of x4 + x3y _ 3x2y2 _ xy3 + y4 = _+1are (__1,0), (0,-+1), (1,_1), (-1,_+1), ( - 2 , 1 ) , ( 2 , - 1 ) , (1,2)and ( - 1 , - 2 ) . The equation in (ii) is the Thue equation associated with the real quartic field of least discriminant over the rationals. The r u n n i n g times for these programs on a VAX-785 were approximately 5 minutes and 1 hour, respectively.

1 6 12

Ii

2 nd C o l u m n = - I st Column + 2na C o l u m n

-- ~1 24

2 1 0

4 0 12

6 -1 24

L J E24 1 1 0

0 12

24

3rdColumn = I st Column + 3rd C o l u m n

3 ra C o l u m n = 3rd C o l u m n - 2 x 2 nd C o l u m n

Ii4i] 0 12

I can also do the same to the generators of F to obtain a nicer form for the generators of H; i.e., I perform row reductions as well. Thus I get

Istrow

li 12 4010 00 3~arow= I! 4iJo 0

=

1st row - 2 x 2 nd row

3rd row - 3 x I st row

E1001

(ii) A b e l i a n Groups

Interchange I st a n d 2 na rows

0 0

4 0

0 0

Every finitely generated Abelian group can be written uniquely in the form 77/(dl) ( ~ . . . ( ~ 7//(dk) (~ 7/(~... 7 / w i t h r (the n u m b e r of copies of 2/), k,d 1. . . . . dk

Thus I obtain G ~ Z/(4) Q2e via the equations Yl = 0, 4y2 = 0, w h e r e y l = 2Xl + x2, Y2 = Xl + 3x3, a n d y 3 = x3. THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989

61

Note that two arbitrary finitely presented Abelian groups will be isomorphic if a n d only if their canonical forms are the same. Thus for Abelian groups, a n d quite unlike Abelian lattice-ordered groups, the w o r d a n d isomorphism problems are quite soluble a n d can easily be accomplished on a n y m o d e r n computing machine.// The solution just given is taken from [Magnus, Karrass, a n d Solitar, Section 3.3].

Concluding Remarks To s u m up, some proofs (often indirect) give the existence of solutions but no clue h o w to obtain them. These fall into the scope of Belloc's poem, our Lesson for Today. The theoretical existence of an effective algorithm is slightly better, t h o u g h even that m a y be impossible in certain cases (e.g., the Word Problem for certain finitely p r e s e n t e d g r o u p s [Boone] or [Novikov], a n d the Isomorphism Problem for Abelian lattice-ordered g r o u p s - - s e e above). Much better is an explicit algorithm with an a priori b o u n d on the runn i n g t i m e a n d size of s o l u t i o n s , w h e r e the o n l y problem m a y be the impracticality of completing such a search. The algorithm I gave for membership in a finitely generated ideal of Q[X] is such an example in Section (3); they are in sharp contrast to the algorithm I gave for the problem in 7/IX] in Section (2) where the algorithm was merely effective (no a priori b o u n d was given for h o w long to search; all I k n e w was that one of the two procedures would eventually answer each question). Finally, I gave two examples w h e r e improvements in the theory lead to practical solutions of important problems in mathematics. Such examples are the only ones whose solutions we can be "sure a b o u f ' in a real sense; shorter and quicker computer programs then become important. I h o p e that by the use of familiar examples I have illustrated the importance of finding optimal information about solutions. Thank y o u for your patience.

References

[N. Blackburn] Conjugacy in nilpotent groups, Proc. Amer. Math. Soc. 16 (1965), 143-148. [J. Blass & A. M. W. Glass] Work in progress, unpublished manuscript. [J. Blass, A. M. W. Glass, D. Manski, D. B. Meronk & R. P. Steiner] Constants for lower bounds for linear forms in the logarithms of algebraic numbers I & II, Acta Arith. (to appear). [J. Blass, et al 1] Practical solutions of Thue equations over the rational integers (unpublished manuscript). [J. Blass et al 2] On Mordell's Equation y2 + k = x3, (unpublished manuscript). [W. W. Boone] The word problem, Annals of Math. 70 (1959), 207-265. [P. M. Cohn] Universal Algebra, Harper & Row, London, 1965. [L. P. D. van den Driess] The model theory of fields: decidability and bounds for polynomial ideals, Ph.D. Thesis, University of Utrecht, 1978. [G. Faltings] Endlichkeitss/itze ffir abelsche Variethten fiber Zahlk6rpern, Inven. Math. 73 (1983), 349-366. [C. F. Gauss] Disquistiones Arithmeticae, 1801. [A. O. Gelfond] On Hilbert's 7th Problem, Doklady Akad. Nauk SSSR 2 (1934), 1-6. [A. M. W. Glass & J. J. Madden] The word problem versus the isomorphism problem, J. London Math. Soc. 30 (1984), 53-61. [F. J. Grunewald & D. Segal] Some general algorithms II: Nilpotent Groups, Annals of Math. 112 (1980), 585-617. [P. Hill] Nilpotent Groups, Queen Mary College Maths Notes, 1969. [D. R. Heath-Brown] Fermat's Last Theorem for "almost all' exponents, Bull. London Math. Soc. 17 (1985), 15-16. [G. Hermann] Die Frage der endlich vielen Schritte in der Theorie der Polynomideale, Math. Ann. 95 (1926), 736- 788. [N. G. Khisamiev] Universal theory of lattice-ordered abelian groups, Algebra i Logika 5 (1966), 71-76 (in Russian). [A. K. Lenstra, H. N. Lenstra, Jr., & L. Lovasz] Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), 515-534. [w. Magnus, A. Karrass & D. Solitar] Combinatorial Group Theory, Wiley, New York, 1966. [A. A. Markov] Insolubility of the problem of homeomorphy, Proc. International Congress of Math, 1958 (ed. J. A. Todd, FRS), University Press, Cambridge, 1960, 300-306. [P. S. Novikov] On the algorithmic unsolvability of the word problem in groups, Trudy Mat. Inst. Steklov 44 (143) 1955 (in Russian). [T. Schneider] Transzendenzuntersuchungen periodischer Funktionen, J. reine Angew. Math. 172 (1934), 65-69. [A. Seidenberg 1] Constructions in algebra, Trans. Amer. Math. Soc. 197 (1974), 273-313. [A. Seidenberg 2] Constructions in a polynomial ring over the ring of integers, American J. Math. 100 (1975), 685- 703. [A. Seidenberg 3] What is Noetherian?, Rend. Seminario Mat. Fis. Milano 44 (1974), 55-61. [H. Simmons] The solution of a decision problem for several classes of rings, PacificJ. Math. 34 (1970), 547-557. [M. Waldschmidt] A lower bound for linear forms in logarithms, Acta Arith. 37 (1980), 257-283.

[A. Baker 1] Contributions to the theory of Diophantine equations: I On the representations of integers by binary forms. II The Diophantine equation y2 = x3 + k, Phil. Trans. Royal Society London A263 (1967/8), 173-191 and 193-208. [A. Baker] Transcendental Number Theory, University Press, Cambridge (2nd edition) 1979. [A. Baker & H. Davenport] The equations 3x2 - 2 = y2 and 8x2 - 7 = z2, Quart. J. Math. Oxford, 20 (1969), 129-137. [K. Baker] Free vector lattices, Canad. J. Math. 20 (1968), 58-66. [H. Belloc] More Beasts for Worse Children, Duckworth & Co., London, 1898. [W. M. Beynon] Applications of duality in the theory of fi- Department of Mathematics and Statistics nitely generated lattice-ordered abelian groups, Canad. J. Bowling Green State University Bowling Green, Ohio 43403-0221 USA Math. 29 (1977), 243-254. 62

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO, 1, 1989

Chandler Davis*

Littlewood's Miscellany edited by B~la Bollobas Cambridge: Cambridge University Press, 1986

Reviewed by Lawrence Zalcman For a third of a century or more, G. H. Hardy and J. E. Littlewood dominated British mathematics. Each of these remarkable men also wrote a nontechnical book about mathematics. Hardy was a gifted writer, "the better journalist" of the two (in Landau's phrase); and his meditation on mathematics and mathematicians, A Mathematician's Apology, has attained something of the status of a classic. Excerpted in the extremely popular anthology, The World of Mathematics, it was reprinted some time ago with a memoir of Hardy by no less a figure than C. P. Snow. Littlewood's A Mathematician's Miscellany was not so fortunate. Lacking both the polish and the continuity of the Apology, it was actually rejected by Cambridge University Press (p. 134). W h e n it was finally published (by the commercial house of Methuen), it enjoyed a modest success and then, after a decent interval, was allowed to go out of print. Yet there have always been those of us w h o have found the robust intelligence evident on every page of the Miscellany infinitely more attractive--and stimulating--than the melancholy sense of loss and m o u r n i n g for w a n i n g p o w e r s that s u f f u s e s the Apology. For us, the reprinting of the Miscellany in a greatly amplified version, edited and with a foreword by B61a Bollob~s, is truly cause for rejoicing. The original Miscellany was a diverse collection of observations, anecdotes, jokes, mathematical and nonmathematical one-liners, essays, reviews, and

* C o l u m n editor's address: Mathematics Department, University of Toronto, Toronto, O n t a r i o M5S 1A1 C a n a d a

J. E. Littlewood other occasional pieces, aimed at the "amateur," selected according to the dual criteria of relative unfamiliarity and lightness and presented with an admirable brevity and dryness of wit. "I must leave this to the judgment of my readers," writes Littlewood in his introduction, "but I shall have failed where they find anything cheap or trivial." And then he adds a line which has become famous: "A good mathematical joke is better, and better mathematics, than a dozen mediocre papers." Littlewood did not fail, nor were his readers disappointed. It is a pleasure, therefore, to report that the new edition of Littlewood's Miscellany retains all the material of the old Miscellany (updated, here and there, but very sparingly), with the exception of a single brief piece (consisting of three short reviews) that will not be missed. The new edition runs 200 pages, 115 of which correspond to the original text. The remainder of the volume comprises Bollob~s's foreword, actually an informal memoir of Litt l e w o o d (22 pages); "The m a t h e m a t i c i a n ' s art of work," a charming essay by Littlewood reprinted from

THE MATHEMATICALINTELLIGENCERVOL. 11, NO. I 9 1989Springer-VerlagNew York 63

Rockefeller University's magazine (12 pages); and previously unpublished material, arranged under the rubrics " P e o p l e , " " A c a d e m i c Life," and " O d d s and Ends" (51 pages). Inevitably, it will be to these last-mentioned sections that aficionados of the original Miscellany will first turn. Here they will find a rich lode of anecdotal material about such well-known Cambridge characters as Hardy, Russell, Besicovitch, and Housman, as well as a host of others. Littlewood's account of Russell's anguished reaction on hearing the theory of relativity explained to him ("To think I have spent my life on absolute muck") is alone worth the price of the book, as is the anecdte (p. 130) of Russell's encounter with Einstein. Augmenting the ample material on Hardy and Russell and the frequent mention of Besicovitch are vignettes of (among others) G. D. Birkhoff, Harald Bohr, M. L. Cartwright, A. A. Markov, Ostrowski, Ramanujan, D. C. Spencer, van der Corput, Weyl, W. H. Young, and Zygmund, and a marvellous extended sketch of Landau (who comes off extremely well). On the other hand, Paley (who was Littlewood's student) and Wiener are conspicuous by their absence; what is one to make of these strange omissions? There is also a fair amount of academic gossip, including behind-the-scenes accounts of the (Trinity) fellowship elections of Russell, Ramanujan, Besicovitch, and Wittgenstein (amazingly, none of these was considered an open-and-shut case; in fact, in each instance, considerable political maneuvering was required to insure the desired outcome) and an amusing rehearsal of how J. B. S. Haldane managed to hang on to his Trinity readership through a divorce scandal despite unanimous condemnation by the standing board of i n q u i r y - - t h e n styled, appropriately enough, the Sex Viri (later changed, "pusillanimously" as Littlewood puts it, to the Septem Viri). A short section entitled "Queer ways that theorems get proved" contains some fascinating comments on Bloch's theorem: "One of the queerest things in mathematics, and one might judge that only a m a d m a n I could do i t . . . the proof itself is crazy." The new material ranges far beyond mathematics and mathematicians, including tales of dons, junior bursars, and manciples long forgotten, animal stories, toasts, sermons ("Shall we wake with the wise virgins or sleep with the foolish virgins?"), various mots ("Perfect greed casteth out fear," "He that bloweth not his own trumpet, his trumpet shall not be blown," "I cannot see that a purely intellectual study of Ethics has any bearing on one's conduct") and much, much more. Readers who, like Hardy (p. 118), are unfamiliar with the anecdote of the Curate's Egg as well as those who think that plus-fours are a kind of golfer's handicap would be well-advised to keep a copy of Brewer's Dictionary of Phrase and Fable within arm's reach. Here 64 THEMATHEMATICALINTELLIGENCERVOL. 11, NO. 1, 1989

and there, lack of familiarity with local custom makes for rough going even with Brewer; but, on the whole, one's efforts are repaid. This said, it must be admitted that the new material is rather u n e v e n and would have benefited from a heavier editorial hand. In a couple of instances (pp. 128,141), stories that must have been hilarious when recounted orally fall flat on the printed page, where they appear m e r e l y vulgar or embarrassing. And, while editorial reluctance to tamper with the ipsissima verba of the master (extending, apparently, even to notations on old photographs, cf. p. 155, I. 6) is perhaps understandable, failure to correct errors of spelling, agreement2, and fact is not. Thus, "Ostrowski'" is misspelled ("Ostrovsky") five times on p. 127--"Besicovitch" also manages to get mutilated in the same s t o r y - - a n d "terrific" comes out "teriffic" on p. 130. Forgivable, s u r e l y - - e x c e p t for

" A g o o d m a t h e m a t i c a l j o k e is better, and

better mathematics, than a dozen mediocre papers. ""

Littlewood's own strictures on Hardy's spelling (p. 118) and his claim that "we could all spell all but perfectly from the age of 12." And then what is one to make of a sentence (p. 153) like "But the lives of 'failed F.R.S.'s' is poisoned"? Most embarrassing of all is the characterization (p. 157) of Bieberbach as "a Jew, more Nazi than the Nazis," a description that would presumably have been almost as offensive to the (nonJewish) Nazi Bieberbach as it remains to Jews. There are also several curious errors in transcription. For instance, an exchange between Moore and Russell (p. 127) is dated 1876. Russell was four years old in 1876; surely 1916 is intended. The fourth word of the second line of the second stanza of the poem (p. 168) Littlewood remembered "effortlessly for 55 years" should be "last," not "first." For "this" on the first line of p. 149, read "thesis," and correct Bromwich's third initial (p. 148) from "L." to ' T A . " The introduction also has its share of spelling errors ("concensus" p. 5, 1. 17; "desparation," p. 17, I. 7) and at least one mistran-

1 See Douglas M. Campbell, "Beauty and the Beast: the Strange Case of Andr6 Bloch," The Mathematical tntelligencer 7, no. 4 (1985), 36-38; Henri Cartan and Jacqueline Ferrand, "The Case of Andr6 Bloch", ibid. 10, no. 1 (1988), 23-26. 2 In this connection, it may be noted that the answer to Littlewood's query (p. 152) as to "whether the Son of God could make a mistake in grammar" is affirmative. The only case which requires discussion is that in which omnipotence is assumed. But an omnipotent being can do anything.

scription ("visit" for "'wait" on p. 11, 1.24). After such a rich complex of horrors, a simple typographical error ("mathemtical," p. 150, 1. -3) comes as something of a relief. Unfortunately, a number of these have been introduced into the reset text of the original Miscellany. Thus "precision" on p. 56, 1. -8 should be "precisian"; one should have Sn + e instead of Sn+~ on p. 82, I.-15; and the exponent in the displayed formula on p. 88 should be l-e, not 1-r. Surely Littlewood, and the reader, deserve better from Cambridge University Press. The n e w edition of the Miscellany is decorated with eight photographs of Littlewood, including his Senior Wrangler portrait, the famous photo taken in N e w Court, Trinity College, of H a r d y and Littlewood looking for all the world like Laurel and Hardy, and a photo of Littlewood dining in Davos with his daughter Ann. Samples of the handwriting of Hardy, Littlewood, and Ramanujan are also reproduced. The frontispiece is a p h o t o g r a p h of a b u s t of Littlewood sculpted by Mrs. Bollob~is. The cover photo (in the paperback edition) shows the elderly Littlewood, very dapper in dark suit cum umbrella, looking severe yet avuncular. One imagines that he has just finished admonishing a junior colleague, "Well, all I can say is that a y o u n g m a t h e m a t i c i a n w h o can't live for pleasure as much as he wants, and do a proper job of work at the same time, shows a gross incompetence in the art of life.'" Buy this book (but don't forget to write to CUP to complain about the misprints). Department of Mathematics Bar-Ilan University 52 100 Ramat Gan, Israel

Chaos: Making a N e w Science by James Gleick Viking Penguin Inc., N e w York, 1987, xi + 352 pp., US $19.95

Reviewed by John Franks This is a book about new ways in which mathematics is used to model phenomena in the real world. It is intended for a general audience. The author is James Gleick, formerly a science reporter for the New York Times. He does a good job explaining what constitutes a mathematical model (by which he means a differential equation or a difference equation) and what it does. The theme of the book is that even rather simple non-linear models can, and typically do, exhibit extremely sensitive dependence on initial conditions. What this means is that two solutions of a non-linear ordinary differential equation can start with very close initial conditions but then diverge rapidly from one another while remaining in a bounded region. The ef-

fect is that after a moderate amount of time the position of a solution can appear to be a random function of the initial condition. This seems quite paradoxical because the differential equation certainly generates a deterministic system. Only recently have scientists begun to realize the extent to which simple dynamical systems produce this random-seeming, complex, "chaotic" behavior. The story that this book recounts is the sometimes painful process of acquiring this understanding and the histories of the scientists involved. The result, in Gleick's view, has been a revolution in the way science views n a t u r e - - a "paradigm shift.'" The book is subtitled "Making a new science." We are told, "Where chaos begins classical science s t o p s " and "Chaos poses problems that defy accepted ways of working in science.'" This revolution, Gleick says, was carried out in the face of n u m e r o u s obstacles by "a few freethinkers, working alone, unable to explain where they are heading." These stories of individual achievement, in diverse disciplines but with a common theme, are the heart of his book. We are presented with an exciting and romantic story, but I suspect that Gleick has greatly overestimated the achievements of chaos theorists. Only time will tell. Also I think he missed the obvious explanation of the sudden interest in chaos in many branches of science. Thirty years ago there was very little a scientist could do with a typical nonlinear model. It is generally impossible to solve non-linear differential equations. And it is not even clear what a scientist could do with a complicated, explicit, closed-form solution if one could be found. It would probably be no more enlightening than the original differential equation. As a result researchers have emphasized linear systems, which are much more tractable but greatly limited in the range of phenomena they can model. This may even have reached the point, as Gleick suggests, of scientists training themselves not to see non-. linearity in nature. The necessity of this oversimplification changed with the advent of inexpensive, easy-to-use digital computers. N o w one can numerically approximate the solutions of non-linear differential or difference equations and, equally important, graphically display the results. The computer is a viewing instrument for mathematical models that will, in the long run, be more significant than the microscope to a biologist or the telescope to an astronomer. Nearly all of the scientists whose work is discussed in this book made heavy use of computers and were among the first to do so in their discipline. It is no more surprising that numerous types of complex dynamical phenomena have been discovered in the last twenty years than would be the discovery of n u m e r o u s kinds of bacteria if thousands of biologists were, for the first time in history, given microscopes. THE MATHEMATICAL INTELLIGENCER VOL. l I , NO. I, 1989 6 ~

Most researchers now understand that seemingly random behavior is an inherent element of non-linear dynamical systems and not just the result of experimental error or "noise." This is an important insight. I w o u l d suggest that its widespread acceptance has more to do with readily accessible computers than the brilliance or courage of Gleick's protagonists. What is surprising and fascinating is the resistance to accepting it that is documented in this book. Indeed, it was well known to Henri Poincar6 and George David Birkhoff, but only when computers made it impossible to ignore did it gain acceptance. We are probably nowhere near the end of the exciting n e w insights into nature that can be gained by computer viewing of mathematical models. All of the models discussed in Chaos:Making a new science are relatively simple ordinary differential equations or difference equations. As computers become more powerful and supercomputers become widely accessible and easy to use, we can expect similar insights into nonlinear partial differential equations. If non-linear ordinary differential equations are intractable w i t h o u t computers the typical non-linear partial differential equation is hopeless. The breakthroughs came first in simple ordinary differential equations because a personal computer (in some cases even a hand calculator, as we learn in this book) is adequate to experiment with them. Much greater computational power is necessary for partial differential equations, because their numerical solution requires many more arithmetical steps. Despite these advances, I would speculate that the real evolution in scientific modeling and the greatest impact of computers on science is probably yet to come. Computers have made it easy to view the solutions of non-linear differential equations. But now that we have computers it is no longer clear that differential equations are the best models to use, especially in areas like biology or the social sciences. One can envisage a wholly new mathematical construct, better suited to digital computers, which could provide a new type of model. The emergence of such a construct capable of accurately m o d e l i n g a variety of phen o m e n a w o u l d i n d e e d be a true paradigm shift. Nothing this dramatic appears to have happened yet. There is indeed a revolution in progress, but it is not, as this book suggests, the "Chaos revolution." Instead it is the computer revolution, and the current popularity of chaos is only a corollary. This revolution may still be in its infancy, but computers have already taught us one remarkable fact. There is mounting evidence that sensitive dependence on initial conditions may well be as close to the rule as to the exception for non-linear dynamical systems. This is a surprising fact, well documented in this book. It surprises us because it was invisible before the computer, but with computers it is easy to see, even hard to avoid. 66

THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989

I would have liked to see more information on the mathematics behind chaos. Much of this mathematics can be readily described at the same technical level as the rest of the book. For example, a discrete-time dynamical system discovered by Stephen Smale and commonly called the "Smale horseshoe" is described, but its significance is never discussed. It was one of the first examples of sensitive dependence on initial conditions to be completely and rigorously understood. But more important, a theorem of Smale shows that the horseshoe or its analogue for continuous time systems occurs as a subset of almost any system that displays chaotic behavior. An even more serious omission is the absence of any discussion of the close connection of the Smale horseshoe with the doubling map on the interval. The doubling m a p is the function on the unit interval whose value at x is defined to be the fractional part of 2x. An understanding of this simple dynamical system goes a long way toward demystifying the paradox of deterministic randomness. If we think of x as written as a binary number with a leading "decimal point" then this function simply shifts the decimal point to the right one place and deletes the first digit. Clearly, if an initial x is chosen whose value is significant to 16

In reading this book I w a s repeatedly struck by the parallels between chaos and catastrophe theory.

binary places, then after 16 iterates its value will appear to be completely random and independent of the starting value. This is essentially the same mechanism that underlies the complex behavior of most chaotic dynamical systems. The opportunity Gleick missed here was to give the non-specialist reader a real insight into the nature of chaos. The doubling map is readily understandable and closely related to the dynamics of the Smale horseshoe. The Smale horseshoe, in turn, is an essential ingredient of almost all chaotic dynamics! I was also surprised to note that, despite a lengthy discussion of the Lorenz attractor, there was no mention of the work of John Guckenheimer and Robert Williams concerning this remarkable example. It is a great achievement to be the first to observe an important phenomenon in nature, but it is equally important to be the first to understand and explain it. Edward Lorenz wrote a system of three first-order ordinary differential equations as a simplified model of atmospheric convection. When he studied the system with a computer he observed all the hallmarks of chaos: very complicated trajectories of solutions, sensitive dependence on initial conditions, seemingly random behavior in a deterministic system, etc. His paper [3]

on this example was published in the Journal of Atmospheric Sciencein 1963. Gleick tells of the extraordinary influence that the work of Lorenz, and this example in particular, has had in the development of non-linear dynamics. However, there is no mention of the work of m a t h e m a t i c i a n s w h o explained the remarkable things Lorenz had discovered and catalogued. This work by Guckenheimer and Williams [2] represents one of the outstanding achievements of mathematics in the understanding of so-called strange attractors. It reduces the dynamics of a system of three non-linear differential equations to the study of a class of func-

One of Gleick's main p o i n t s is that nature seems to have a penchant for one-dimensional dynamics, but the one-dimensionality is often well disguised.

tions from the interval to itself, in fact, one not unlike the doubling map described above. This omission is especially surprising because one of Gleick's main points is that nature seems to have penchant for onedimensional dymanics, but the one-dimensionality is often well disguised. Guckenheimer and Williams certainly showed this to be the case for the Lorenz attractor. The sections on fractal geometry and its inventor, Benoit Mandelbrot, are the only parts of the book that have little to do with chaotic dynamics. Fractal geometry is a static theory only marginally related to any kind of dynamics. Mathematicians may find it disconcerting to learn that the term fractal geometry does not refer to a body of mathematics, as, for example, the terms projective geometry or algebraic geometry do. In fact, there appears to be no formal mathematical definition of the word fractal. This book mentions no theorems in fractal geometry. A highly regarded mathe m a t i c a l text [1] w i t h the title The Geometry of Fractal Sets deals primarily with mathematics developed prior to 1950, which is certainly not what Gleick is describing. There may be recent mathematical resuits in fractal geometry, but, if so, they clearly play a secondary role. Instead, fractal geometry is viewed by its proponents as a new framework and set of tools for describing nature. Gleick tells us, "In the end, the word fractal came to stand for a way of describing, calculating, and thinking about shapes that are irregular and fragmented, jagged and broken-up." The principal tenet of this descriptive framework is that nature is extemely irregular and the degree of irregularity remains constant when viewed on different scales. That is, natural phenomena exhibit a self-similarity of complexity across scales. "Above all fractal

meant self-similar," according to Gleick. The most important tool for fractal geometry is Hausdorff dimension. This is a numerical invariant of metric spaces that is defined as a limit of numbers obtained from covers of the space by smaller and smaller bails. For manifolds it equals the topological dimension, but in general it is not an integer. The connection of Hausdorff dimension with the ordinary concept of dimension is somewhat tenuous, but to the uninitiated the idea that an object can have a dimension of 1.2618 has a kind of science-fiction-like appeal. It would have been better if the word "dimension" had never been attached to this number. Often in mathematics or physics a common word will assume a special or technical meaning sometimes quite different from its usual sense. The use of the word dimension here is one example, and I am told that "universal" is used by physicists in a different sense than its common use. Any exposition of mathematics or physics for non-specialists should take pains to explain which words are being used in an unusual way. Unfortunately, this has not been done by Gleick in this book nor in general by the adherents of fractal geometry. On the contrary, I feel that the fact that Hausdorff dimension assumes fractional values may have been emphasized to glamorize the concept. This sort of mystification should be avoided in science, rather than catered to. Incidentally, Gleick suggests that mathematicians prefer the term Hausdorff dimension to fractal dimension or fractional dimension because they spitefully want to deny credit to Mandelbrot. This is both false and insulting. One of the principles of fractal geometry holds that Hausdorff dimension is an important measure for physical objects. As Mandelbrot himself acknowledged, his program describedbetter than it explained. He could list elements of nature along with their fractal dimensions--seacoasts, river networks, tree bark, galaxies--and scientists could use those numbers to make predictions. The mathematical definition of Hausdorff dimension, involving limits as size goes to zero, may not make sense for a physical object, but one can make similar philosophical objections to measuring length. In any case, Gleick tells us, Mandelbrot specified ways of calculating the fractional dimension of real objects, given some technique of constructing a shape or given some data, and he allowed his geometry to make a claim about the irregular patterns he had studied in nature. The claim was that the degree of irregularity remains constant over different scales. Surprisingly often, the claim turns out to be true. Over and over again, the world displays a regular irregularity. Like fractal geometry this book describes better than it explains. The reader may be entertained, but is unlikely to gain new insights into nature. My greatest disappointment was the way in which mathematics is THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989

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portrayed. One could read this book and come away with the view that mathematical proofs are an obstacle to the pursuit of t r u t h - - a sort of self-imposed mental straitjacket worn by stodgy old pedants. This probably overstates Gleick's view somewhat, but he definitely feels that mathematics has greatly suffered from an interest in rigor. A reader for w h o m the concepts of proof and rigor are vague at best could easily conclude they are better avoided. I had hoped for a more sympathetic view of the discipline. Mathematics has a methodology unique among all the sciences. It is the only discipline in which deductive logic is the sole arbiter of truth. As a result mathematical truths established at the time of Euclid are still held valid today and are still taught. No other science can make this claim. The phrase "mathematical certainty'" is commonly used in general discourse to repr e s e n t the highest s t a n d a r d of truth. A t h e o r e m proven today may be forgotten in the future because it is uninteresting but it will never cease to be true (barring drastic changes in our standards of rigor). I would contend that an important criterion for judging a scientific discipline is the half-life of its truths. Mathematics does extremely well by this measure and mathematicians are justifiably proud that their standards of truth are higher than those of other sciences. The use of deductive logic rather than empiricism is taken for granted by mathematicians to such an extent that they rarely contemplate alternatives. Scientists in other disciplines seldom appreciate this methodology and in some cases even disdain it. With few exceptions, Gleick's treatment of mathematics echoes this attitude. Rigor is blamed for increased narrow specialization in mathematics and for decreased contact bet w e e n mathematicians and other disciplines. This seems to me to be implausible; in particular I note no lack of narrow specialization in empirical disciplines and no greater tendency toward interdisciplinary research.

There is mounting evidence that sensitive dependence on initial conditions may well be as close to the rule as to the exception for nonlinear dynamical systems.

In fairness, these views may only be reflections of the attitudes of some of the scientists interviewed by Gleick. We are told, for example, that Mandelbrot felt it necessary to emigrate from France because of the stifling influence of Bourbaki. Curiously, this book contains a fair amount of Bourbaki b a s h i n g - - n o t because of Bourbaki's pedagogical style, where, in my view, it might be deserved, but for their rigor. 68

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The methodology of mathematics does raise valid philosophical questions that need more attention. There is an apocryphal story of a meeting between a youthful Albert Einstein and an aging Henri Poincar6 at which Einstein said, "I considered taking up mathematics, but I decided against it because there is often no connection with the real world in mathematics and it is impossible to tell what is important." To which Poincar4 replied, "Well, in my youth I considered becoming a physicist, but I decided against it because in physics it is impossible to tell what is true." Clearly there is a tradeoff between high standards of truth and applicability. Sadly, this issue attracts little attention from either side. This book contains considerable discussion of the dynamics of complex functions. I believe that, as far as we know today, this is pure mathematics that models nothing in nature, yet has great appeal for even the

Gleick suggests that mathematicians prefer the term Hausdorff dimension to fractal dimension or fractional dimension because they spitefully want to deny credit to Mandelbrot. This is both false and insulting. most down-to-earth experimentalist. It is difficult to view pictures of Julia sets or Mandelbrot sets without becoming something of a Platonist. These sets seem to have a reality of their own even if they don't reflect some part of what we glibly refer to as the real world. What constitutes a good mathematical model? This question needs to be paid much more attention by all scientists. H o w do we judge if a given piece of mathematics accurately reflects some aspect of nature? This is not a mathematical question; at least, it is not a question whose answers are subject to proof. Historically the ability to make non-trivial quantitative predictions has been an important test for models. My own view is that a good model must explain, at least to some extent. For example, one could videotape some phenomenon, digitize the result, and use that to produce a "mathematical model" capable of reliably describing the process. But such a model would add nothing new to our understanding. Similarly, it is not enough to find a mathematical system that exhibits similar behavior to a physical experiment, but has no apparent connection with it. I fear that many of the models of chaos may be faulted for this deficiency. In reading this book I was repeatedly struck by the parallels between chaos and catastrophe theory. Gleick quotes extravagant claims of chaos p r o p o n e n t s , such as "'twentieth-century science will be remembered for just three things: relativity, quantum mechanics, and chaos." Fifteen years ago similar claims were made for catastrophe theory. I

must admit that I was one who at least entertained the possibility that those claims were correct and that catastrophe theory represented a new "'paradigm shift.'" N o w such a view seems extremely improbable. Gleick himself yields to the temptation to engage in hyperbole on behalf of chaos. After a brief account of

It is difficult for professional scientists, much less the general public, to distinguish excessive hype from solid scientific advances. Chaos has given us both.

the inadequacy of pre-chaos science in dealing with complex natural phenomena, he says, Now all that has changed. In the intervening twenty years, physicists, mathematicians, biologists, and astronomers have created an alternative set of ideas. Simple systems give rise to complex behavior. Complex systems give rise to simple behavior. And most important, the laws of complexity hold universally, caring not at all for the details of a system's constitutent atoms. Despite having carefully read this book, I do not know to what he refers when he speaks of universal laws of complexity. Like chaos, catastrophe theory was highly interdisciplinary, claimed breakthroughs in numerous areas, was warmly received by researchers in several disciplines, and received a great deal of media attention. Both chaos and catastrophe theory are based on some interesting and beautiful mathematics, dynamical systems and singularity theory, respectively. But both

G l e i c k h a s g r e a t l y o v e r e s t i m a t e d the a c h i e v e m e n t s of chaos t h e o r i s t s .

have been rather phenomenological in their modeling, finding a similarity in a mathematical model and a physical p h e n o m e n o n to be an adequate basis for adopting that model. In discussions with colleagues I have encountered considerable objection to the comparison of chaos to catastrophe theory. Some feel it is unfair to chaos while others think it is unfair to catastrophe theory. Probably both groups are right. Comparing catastrophe theory to fractal geometry may be unfair to catastrophe theory since it has a great deal more substantial mathematics behind it than fractal geometry. On the other hand a comparison between catastrophe theory and non-linear dynamical systems may under-

rate the impact of non-linear dynamics. The insight that seemingly random, complex behavior is inherent in non-linear dynamics and not a result of error or noise is an important one. For me it is difficult to think of it as a recent breakthrough or a revolutionary idea since it was taught to me in a routine fashion over twenty years ago when I was a graduate student. But the full impact of this idea on the conduct of scientific research is only n o w being felt. Catastrophe theory has not had a comparable impact. R e s e a r c h e r s in n u m e r o u s disciplines are n o w obliged to study non-linear dynamics if they hope to provide good mathematical models. They have to know about disorder if they are going to deal with it. This new attitude is a very good thing--I believe that non-linear dynamics is the place most researchers should be looking for models. Unfortunately, our knowledge of chaotic systems, beyond the fact that they exist in profusion, is still extremely limited. This is true in both a theoretical and a practical sense. I am concerned that extravagant claims like those quoted above raise unrealistic expectations that have no chance of being met in the near future. Despite its portrayal in this book, chaos is not a new tool that can solve the problems of every discipline. It is difficult for professional scientists, much less the general public, to distinguish excessive hype from solid scientific advances. Chaos has given us both. Whether in the long run chaos enjoys a greater success than catastrophe theory in providing useful mathematical models remains to be seen. This judgment will not be made by mathematicians or by popular science writers. It will be made by the practitioners of the various disciplines to which the techniques of chaos are being applied. And the criterion by which chaos will be judged is the quality of the models it provides for physical phenomena.

References 1. K. J. Falconer, The Geometry of FractaI Sets, Cambridge Tracts in Math. #85, Cambridge University Press (1985). 2. J. Guckenheimer and R. F. Williams, Structural Stability of Lorenz Attractors, Publ. Math. IHES, 50 (1979) 59-72. 3. E. N. Lorenz, Deterministic non-periodic flow, Jour. Atmos. Sci. 20 (1963) 130-141. 4. S. Smale, Diffeomorphisms with many periodic points, Differential and Combinatorial Topology (S. S. Cairns, ed.), Princeton: Princeton University Press (1963), 63-80.

Department of Mathematics Northwestern University Evanston, IL 60201 USA

James Gleick Replies. Is Professor Franks serious? In my book Chaos, he read the story of Edward Lorenz, who, running a primitive THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1, 1989

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weather model on his n e w Royal McBee computer, made his surprising discoveries about the misbehavior of simple nonlinear systems. Then he read h o w Michel H6non used an IBM 7040 to create a new strange a t t r a c t o r - - h o w , for H6non, experimentation meant " f r e e d o m to play with the problem on a primitive computer." He read about the analog computer in a Santa Cruz basement that led to the work of the notorious Dynamical Systems Collective. He read about the use of more and more powerful computers to peer into the chaos of the logistic m a p - - h o w the limitations of hand-cranked mechanical calculators gave way, for example, to Frank Hoppensteadt's movie, made on a Control Data 600. He read about Philip Marcus's eye-opening model, on a Cray supercomputer, of the Great Red Spot of Jupiter. He read about the HP-65 calculator that made possible Mitchell Feigenbaum's initial insights into the universality of the transition to chaos, and the heavy computation that then helped Feigenbaum develop his intuition about chaotic systems: "His accelerator and his cloud chamber were the computer." He read about the efforts of Feigenbaum and others to develop a n e w style of using computers, a style that favored the interactive use of graphics over more traditional terminals.

This is one of those little peccadilloes of the reviewing business: take an author's theme, claim it as your own, and contend t h a t the author "'missed" it. He read how Benoit Mandelbrot used computers at Harvard and IBM to reveal shapes that earlier mathematicians had to treat as aberrations. And likewise for John Hubbard, able, with computation, to analyze the behavior of Newton's method, where Arthur Cayley had earlier been stymied: "Hubbard, a century later, had a tool at hand that Cayley lacked." Professor Franks read all that and m o r e - - a central theme of my b o o k - - a n d then he wrote: "I think [Gleick] missed the obvious explanation of the sudden interest in chaos . . ." And what is this obvious explanation? "The advent of inexpensive, easy to use, digital computers." This is one of those little peccadilloes of the reviewing business: take an author's theme, claim it as your own, and contend that the author "missed" it. I wrote about new techniques of exploring systems with "the adjustable lens of a computer" and about "the computer replacing laboratories full of test tubes and microscope"; Professor Franks w r i t e s - - a n d contends he is revealing a truth overlooked in my b o o k - that "the computer is a viewing instrument for mathe70

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matical models which will, in the long run, be more significant than the microscope to a biologist . . . . " I think that Professor Franks's review misleads readers about some others matters as well. I hope that most people w h o read my portraits of Steve Smale, David Ruelle, John Hubbard, or Benoit Mandelbrot will not feel that m y view of mathematicians is unsympathetic. My view of the importance of rigor is more complicated and many-sided than Professor Franks suggests. He believes I have a disdain for rigor. This is not so. His own defense of rigor might be summed up as follows: "Every serious mathematician understands that rigor is the defining strength of the discipline, the steel skeleton without which all would collapse. Rigor is what allows mathematicians to pick up a line of thought that extends over centuries and continue it, with a firm guarantee." That's h o w I put it in Chaos, at any rate. I w o n ' t quibble a b o u t Professor Franks's other quibbles, except one: it is not "false," as he says, but rather quite true that some people consciously displayed their resentment of Mandelbrot by eschewing his fractal terminology in favor of such terms as Hausdorff dimension. If Professor Franks really doubts this, he needs to have some intimate, late-night conversation with a few of the leading players in the strange, still-unfolding history of chaos.

11 GardenPlace Brooklyn NY 11201 USA

John Franks Responds to James GleickMr. Gleick and I agree that most of the scientists discussed in his book made innovative use of computers in their research. He documents this well. We would agree as well that computers are having a profound effect on the w a y science is done. However, I do not believe it is accurate to call this a central theme of his book. If we are to judge by the book's subtitle, his central theme is that we are witnessing the making of a new science and that "a true paradigm shift, a transformation in a w a y of thinking" is taking place. In my review I suggested that something less grandiose is occurring. We are not witnessing a Chaos revolution, but only one aspect of the c o m p u t e r revolution. Rather than a new science we are seeing the use of an important n e w tool in science. The observation that computers are having a profound effect on science is hardly deep or original, and contrary to Mr. Gleick's assertion, I certainly do not claim it for my own. I did not say that Mr. Gleick was unsympathetic to mathematicians. I did suggest that he might be unsympathetic to mathematics. I was disappointed that there is almost no description or discussion of any mathematical theorem in his book. The sole exception I found is a brief aside about the fact that Adrien

Douady and John Hubbard were able to prove that the Mandelbrot set is connected when the issue of its connectedness could not be resolved by computer. I believe that mathematical theorems and their proofs have played a central role in the understanding of non-linear dynamics. I would have liked to see this role sympathetically discussed. I do not believe that the m e t h o d o l o g y of m a t h e m a t i c s - - t h e o r e m s and p r o o f s - - g o t its due credit. The success of Douady and H u b b a r d mentioned above is only one of many instances where the methods of mathematics were successful when other techniques were not. Mathematicians are described sympathetically by Mr. Gleick, but their theorems are not.

Editor's comment: The Letters-to-the-Editor column of the Mathematical Intelligencer welcomes comments about the issues raised by John Franks and James Gleick.

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by Robin Wilson*

Hipparchus (c. 180-125 BC) was probably the greatest astronomical observer of antiquity. He calculated the length of a year to within 61/2 minutes, constructed the first known star catalogue, and discovered the precession of the equinoxes. Because he had a method for solving spherical triangles, he has also been called the "father of trigonometry." His work led to the construction of a "table of chords" giving, essentially, the sine of each angle from 0~ to 90~ in steps of 1/4~ This Greek stamp, showing Hipparchus and an astrolabe, was issued in 1965 to c o m m e m o r a t e the opening of the Evghenides Planetarium in Athens.

Euclid (flourished c. 300 BC), one of the most influential mathematicians of all time, taught at (and may have founded) the celebrated school of mathematics in Alexandria, during the reign of Ptolemy I. He wrote a number of treatises, including Optics, Porisms, On Divisions, and Conics. His most important work is, of course, the Elements, which contains 13 books devoted to arithmetic, geometry (plane and solid), elementary number theory, and the theory of proportion; it has been translated into many languages and used as a text in mathematics for over two thousand years. Euclid has given his name to several mathematical concepts, including Euclidean geometry, Euclidean space, and the Euclidean algorithm. This stamp, issued by Sierra Leone in 1983, shows Euclid and his pupils, a detail from R a p h a e l ' s w e l l - k n o w n painting "The School of Athens.'"

* C o l u m n editor's address: Faculty of M a t h e m a t i c s , The O p e n University, Milton K e y n e s M K 7 6AA E n g l a n d

72 THE MATHEMATICALINTELLIGENCERVOL. 11, NO. 1 9 1989Springer-VerlagNew York