The Mathematical Intelligencer

Letters to the Editor

The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to either of the editors-in-chief, Chandler Davis or Marjorie Senechal.

More on j

Quine's New Foundations

In connection with the letter of Forest W. Simmons and the Editorial Reply in the Summer 2007 issue of The Mathe­ matical Intelligencer we wish to note the following:

The year 2007 marks the 70th anniver­ sary of Quine's New Foundations (NF), and I urge The Mathematical Intelli­ gencer to take note of it. Although ZF and ZFC still dominate mathematical foundations research, I believe that NF or one of its many variations will in­ creasingly be accepted by mathemati­ cians as a basis for all of the results of practicing mathematicians. The turning point may come with the finding of a generally acceptable alternative to the unrestricted Axiom of Choice (AC). It is true that, although analysis and topology would get along nicely with­ out it, Linear Algebra would have a headache, since the result that every vector space has a basis depends on it. If revising Linear Algebra is unaccept­ able, what can be done about the dif­ ficulties and paradoxes of AC? No mathematician has a comprehensive answer to this question; the beginnings of an answer may be found in the fact that such paradoxes lie in the use of the AC for nondenumerably infinite choice sets. If the unrestricted AC is re­ placed by some restricted form of the AC, then NF could help compen­ sate for this loss of deductive power by enabling, for example, equivalence classes to be used in a more effective way.

1. Karl Menger's Calculus, A Modern Approach has been reissued by Dover

Publications. We have added a Preface and a Guide to Further Reading to this reissue. 2. Several years ago, Springer Verlag published a two-volume selection of Menger's papers, Selecta Mathematica, which we edited jointly with Karl Sig­ mund. Here a number of the papers listed in the previously mentioned Guide are reprinted. The interested reader should also consult our "Com­ mentary of Didactics, Variables, and Fluents," which is in Volume 2 . 3. A s forcefully pointed out b y Menger more than half a century ago, bringing elementary calculus up to date requires more than just the introduction of a symbol for the identity function-for which we obviously prefer his j. For a start, authors should carefully distin­ guish between a functionfand its value j(x) at a point x in its domain. Berthold Schweizer Department of Mathematics University of Massachusetts Amherst, MA 01003 USA e-mail: [email protected] Abe Sklar Department of Mathematics Illinois Institute of Technology Chicago, IL 6061 6 USA

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THE MATHEMATICAL INTELLIGENCER © 2008 Springer Science+ Business Media, Inc.

Robert Jones e-mail: [email protected]

Mathematical

Communities

Happy Birthday! MARJORIE SENECHAL

rom its inception as a pamphlet in 1971 and through 30 years as a magazine The Mathematical Intelligencer has been a chronicle, and chronicler, of the international mathe­ matical community. (Indeed, it has helped to create that communizy.J But The Mathematical Intelligencer doesn't just happen-it's the creation of its lively, imaginative, and hardworking editors and the wonderful�y resourceful and forbearing Springer staff A nd so, for this 30th anniversary issue's "Math­ ematical Communities" column, I've asked mypredecessors and Coeditor-in­ Chiqto rr::flect on theiryears at the helm. -Marjorie Senechal

A Conversation with Klaus and Alice Peters, Founders (with Walter Kaufmann-Buhler) Ibis column is a forum for discussion of mathematical communities throughout the world, and through all time. Our definition of "mathematical community" is the broadest. We include "schools" of mathematics, circles of correspondence, mathematical societies, student organizations, and informal communities of cardinality greater than one. What we say about the communities is just

as

unrestricted.

We welcome contributions from mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others.

Please send all submissions to Marjorie Senechal, Department of Mathematics, Smith College, Northampton, MA 01063 USA e-mai l : [email protected]

6

from the Neue Zurcher Zeitung, a re­ view by C. L. Siegel of Constance Reid's Hilbert, reprinted from the Times Liter­ ary Supplement, excerpted memories of I . Schur, by A. Brauer, and a list of forthcoming Springer books . . . KP: We typed it as we went along. AP: I typed and he talked and that's how it all came about. MS: Where did the name, The Mathe­ matical Intelligencer, come from? KP: Walter, I think. I remember a dis­ cussion about whether people would know what it meant. "Intelligencer" means spy, reporter, and is sometimes used in the names of newspapers; to­ day a lot of people have never heard of it. But Walter had a strong sense of history.

Marjorie Senechal (MS): So this is the original Mathematical Intelligencen

AP: We also asked Peter Hilton if he thought it was appropriate, and he thought it was fine.

Klaus Peters (KP): You've never seen it before?

MS: I think everyone likes it, even if they don't understand it.

MS: Never. Our library has all the is­ sues of the magazine, starting with volume 1 , hut not these pamphlets. They're prehistory.

KP: Walter was the most low-key, unas­ suming person you could ever have known. He would never put himself in the center, he was always very mod­ est, but he knew what he wanted. When we discussed something he was always firm in his opinion, but he would never push himself. One day we published a new edition of a book in the "Yellow Series" by the well-known mathematician Arthur Schoenflies. Wal­ ter had just joined the editorial depart­ ment. He had started at Springer as a marketing person because there wasn't an editorial job, but after a few days, we agreed that we'd both do both, I would do some marketing and Walter some of editorial work, and vice versa. When we decided to do the Schoenflies book he said "Oh, that's really nice, you know, I know him". "How do you know him?" "Oh, he's my grandfather. "

Alice Peters (AP): Unfortunately, it has a lot of holes in it. MS: It's typed on a typewriter! How did you get it into this form? KP: There was a format called an ac­ cordion print or fold; our production department at Springer could do it. AP: We, that is our friend and colleague Walter Kaufmann-Bubier, Klaus, and I, used to do this completely on our own time. Well, being publishers, we don't really think about time as our own time or work time. But we used to do it in the evenings, at our house. We'd open a nice bottle of wine and have a lot of fun and do silly things and come up with crazy ideas of what we might do . . . . It was just sort of, oh let's just sit around and talk and then come up with ideas. MS: The first issue, numbered 0, in­ cludes your editorial statement (Fig. 1), an obituary of Heinz Hopf translated

THE MATHEMATICAL INTELL/GENCER © 2008 Springer Science+ Business Media, Inc.

MS: May I borrow this issue? AP: Guard it with your life! I don't know if even Springer has another one. When Springer moved to its new of­ fices, they might not have kept them.

"to prevent misinformation. " The head of marketing probably came to either Alice or Walter or me, I don't remem­ ber which, and said "what are you say­ ing here, are we misinforming people?"

'R.��--i.

��t IJbt�mcttiMI �utdligtttttt sent at ho�� & abroad to prevent misinformation

AP: I'm certain that someone com­ plained because we were trying to do something a little different and at the same time promote the hooks. But we didn't want to blatantly say, hello, this is promotion.

0

KP : We saw ourselves as part of the mathematical community rather than a publisher out there who makes money off the thoughts of mathematicians. So we decided to go ahead with a vehi­ cle for communication. We wanted feedback from the community on what we did in order to tell them why we did things, etc. It was a communication tool, but of course, in the back of our minds, we also wanted to sell books.

t HAVE BEEN FEELING for some time that we need an informal forum for debating questions of mutual interest to the Ma th�rnatical Community and Springer Ver 1 ' This forum should be frank, amus 'i:bg·; informative, and, of course, re le vant, It is not without hesitati.on that I offer this no. 0 - the product of our spare time - for publi.c criticism.

AP : At the end, after we wrote what­ ever we wrote , we always listed new books that had come out.

Let me say briefly what I·have in mind. Just because of its informal nature,we hope the "Intelligencer" will command interest. by being "historical"· in two senses

(1}

MS: And how was the idea of a joint community of publishing received? KP: I think extremely well. After a few issues, we decided to test whether peo­ ple were really interested; we included a little note in the mailing that said "if you want to continue to get this, you have to send back a postcard. "

backwards - by printing eye-witness accounts of peoP,le and events whi�h have influenced the course of mathe matical research.

(ii)forwards - it could be that some of the things our contributers say ab out current developments ln s.cience in general and mathematics in parti cular will one day acquire histori cal interest.

Figure I.

AP : W e had little postcards printed. I remember those. KP: And we got four thousand back. That reaction was totally unusual. Nor­ mally you would expect to get one per­ cent response on a promotional mail­ ing.

Excerpt from the first issue of The Mathematical ln­

telligencer, 1971.

KP: Unfortunately, this one's not my own, I don't have any of the accordion issues. My thesis adviser, Reinhold Remmert, saved them; he lent them to me . Maybe, in this day and age, one can photocopy the whole thing in this format. MS: How many people did you send the first issue to? KP: Originally, about twelve thousand, I think hut I cannot be sure. MS: That's quite a bit. How did you choose them? KP: Springer had a mailing list and I think we also used the AMS member­ ship list.

AP : Plus Europeans. KP: We saw this initially as a promo­ tion piece, a clever way to have a place to promote our books in a vehicle that people would maybe read. . . . AP: We just played around. And then, of course, these silly headings that we had, like "Sent home and abroad to pre­ vent misinformation. " KP: Not everyone was pleased. AP : So we printed a little note (Fig. 2). KP : This reflects something a little deeper, namely, in any corporation people who deal with the production or marketing are much more serious, and were really offended that we said

AP : We tried to save space, so we just typed it without breaks. KP : And then we got a letter from An­ dre Wei! (Fig. 3). I was really impressed that he took the time to write that. At that time we didn't know him person­ ally; later I knew him very well. I don't know if you knew him, he could be absolutely intimidating. MS: I never knew him. I was totally in­ timidated so I never . . . KP : Many mathematicians were intimi­ dated by him. He was very fierce in his opinions hut he was very nice to me; maybe he thought I wasn't a mathe­ matician anymore, I don't know. Any­ way, we got along really well. But

© 2008 Springer Science+ Business Media, Inc., Volume 30, Number 1, 2008

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People in our promotion department were shocked by the lNTELLIGENCER'S subtitle 'Sent at home and abroad to prevent m isin­ formation 1 , Were all their combined efforts of recent years really a source of misin­ formation and, if so, who were more misin­ formed, people at home or people abroad? The INTELLIGENCER wishes to set on record that, wittingly or unwittingly, the promo­ tion department of Springer-Verlag m isinfor­ med nobody, neither man, beast. nor flower in the luxuriant garden of mathematics.

Figure 2.

Statement for the record.

when he sent this note I had probably not met him yet. MS: And this wonderful map in issue number 4 (Fig. 4)? KP: Walter Kaufmann-BOhler's. It shows the world of Springer publish­ ing then. He was a great guy. Many of the ideas in the Intelligencer in the first years came from him. MS: Did he draw it himself? AP: Yes, we are pretty sure he did.

From Mr. Andre Wei.Z, The Institute for Ad­ vanced Study, School of Mathematics, Princeton, NeiJ) Jersey 08540, USA: ihavereceivedyourmathematicalintellige ncerandfinditinterestingp_articularlyfr omthetypographicalpointofviewwhydoyoub otherstillwithsuchcostlyandsuperfluoU:s innovationsaspunctuationseparationofwo rdsetcwh ichanycompetentgreekepigraphis twilltellyouis reallyquiteunnecessaryan devendisturbingforanyoneusedtothegoodo ofnodehpor tsuobyrtnevethgimuoydohtemdl �reaterconveniencewishingyoueveryluckaw

--==:;;;>o� e> c;;;;:.-Figure 3.

Letter to Andre Wei!.

The Mathematical lntelligencer from

Klaus and Alice Peters in 2007 holding

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THE MATHEMATICAL INTELLIGENCER

KP: But he credits Hilbert. He says Hilbert drew the original version in 1 92 1 , "as an overview of the main en­ deavors of mathematical publishing in the foreseeable future . . . . F. Springer pledged himself and his company to (this one of) Hilbert's programs." MS: Issue Number 4 also has the first "stamp column" and letters from an art historian in Canada, a mathematician in New Mexico, and a mathematician in India. Plus book reviews, in addition to the Springer ads. So the Intelligencer already had the seeds of some of the features it's had as a magazine, and its global reach.

The Intelligencer from 30 years ago.

(Photo

©

2007 Stan Sherer)

F igure 4.

The world of Springer publishing,

1972, by Walter Kaufmann-Bi.ihler (after David Hilbert).

From Mr.John Staples, Department of Mathematics, The AuatroUan Nati011a� University; P. 0. Boa: 4, Canberro, ACT/AustroUa I

am

concerned about the publication and review of

research papers. Such publication usually has three functions 1. A news fi.Uiction: infonning one's contenporaries of one's work 2. An· archival function: preserving and circulating one's work for the infornation of later workers

3. A re'WJ'd funct1on: iroicating that some of one's peers reco@1lize one; s work as a genuine contri­ b ution.

It does not perform a fourth, vital, function 4. A review function: publishing an evaluation of one1 s work by an independent expert . It is absurd that

we

usually tolerate a long delay

in publication so that evaluate the �rk

-

an

independent expert can

and then his evaluation

�

is not published! Instea

it is duplicated after

publication and appears years later - thougn the original evaluation could appear as ·soon as or even before the naoer itself!

Figure 5.

A letter

to

the editor.

KP: The Intelligencer usually avoided political subjects, but in one issue we printed this note: "The following note concerned with the January meeting of the AMS was communicated to the Intelligencer. Should the American Mathematical Society support orga­ nized crime? Presumably not. Orga­ nized prostitution? Again, probably not. But what of the organized ex­ ploitation of man's weakness, cupid­ ity, and stupidity for financial gain? . . . this spectacle is unseemly. Let the in­ dividual be free to choose but let not learned societies lend their re­ spectability to this choice. PJH . " That's Peter Hilton. AP : That had to be 1972, when the meeting was in Las Vegas. We never really anticipated that the community would actually participate in The Math­ ematical Intelligencer. But it kind of evolved that way and it became a lot of fun (Fig. 5). AP: One of the other crazy activities that we did was put together a booklet called "The Underground Guide to Helsinki" for the Helsinki International Congress.

© 2008 Springer Science+Business Media, Inc., Volume 30, Number 1, 2008

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MS: Was that the forerunner of the spe­ cial Intelligencer issues for each ICM? KP: Yes. We were very lucky. We had a secretary in the mathematics depart­ ment who was Finnish and she helped us find all the good, unknown local restaurants that were not on the list of the big restaurants. She also made a language guide to Finnish, how to get a taxi, how to say thank you, and such things, and that went into this booklet. And then we reviewed the best beers in Helsinki. In Finland beers come in, I think, four levels of alcohol content, and we made a review for each level, the best beer in each level, and pub­ lished it. MS: Did you do the grading? KP: No, she did that. And a funny thing happened then! A year before the Con­ gress we had organized a book exhibit. The booksellers in Helsinki did not want us, the publishers, to have an ex­ hibit, for complicated reasons. There's a big mark-up of books in Scandi­ navia-about 20-30o/�r-and the book­ sellers didn't want us to sell at our price. But the head of the Congress said, I'll give you a room if the book­ sellers don't want to do it; Springer can organize it and get all the other pub­ lishers involved. When we told that to the booksellers, they relented and said OK, we'll do it. So anyway, we held this book exhibit, and on the first day a small truck drove up at the back door and started to unload cases of beer and beer glasses and put them in our booth. They said, "This is free for giving our beer the highest ratings; this is the least we can do. " So they provided us with free beer. And then on our first evening in Helsinki we went to one of the restaurants that we had recommended, a small restaurant, and it was packed. We had a hard time finding a table, but finally we sat down and as we were sitting we opened Tbe Underground Guide, and the owner of the restaurant came over and said, "Where did you get this thing?" And we said, "We wrote it." "You wrote it? Everybody comes in here with this guide, and that's why we are so packed. We don't normally have so many people. " So the whole Springer group had a free dinner! AP: Looking through these old issues, I think we began running out of steam after awhile, because number 10 is

1Q

THE MATHEMATICAL INTELLIGENCER

dated 1 975 and number 1 1 is dated 1 976. It began as a quarterly, as it is now. But by 1 975 we were only get­ ting it out once a year, and the issues were getting longer and more compli­ cated. KP: The other thing is that Walter had moved to New York. You know, Alice was the first mathematics editor at Springer in New York. And fortunately, or unfortunately, we decided to get married very soon after I had hired her-this posed a dilemma. I couldn't leave Heidelberg, because I had just been named the scientific director. I ex­ plained this problem to Walter, and he said, "That's not a problem at all. I'll take Alice's job in New York, and she can take my job in Heidelberg. The only problem is, it seems difficult to find a place to live in New York. " Then Alice said "That's not a problem, I have an apartment with low rent." AP: It's really amazing that the man­ agement at Springer allowed us to do that, because the point of hiring me was to have an American mathemati­ cian working as an editor for Springer in the New York office, working on a textbook series and things like that. But they were very accommodating, and Walter and I just switched places. KP: So we decided to turn Tbe Mathe­ matical Intelligencer into a magazine. The first Editors-in-Chief in the new format were Ed Edwards and Bruce Chandler. AP: They did it for a while and then thought they might pass on the scepter. So we asked John Ewing. KP: That was, if I'm not mistaken, in 1 978-we had a meeting about it in Helsinki at the International Congress. MS: After John Ewing, Sheldon Axler was the editor, and then Chandler Davis, is that the right order? KP : Yes. They all carried on and did a really good job. They devoted a lot of time to it. You joined recently? MS: Yes, I edited the "Mathematical Communities" column for many years, and then Chandler invited me to be Coeditor. I was very glad to do it; I en­ joy writing and editing, and also I saw it as a way to prevent Chandler from retiring. After 1 3 years as Editor-in­ Chief-and he'd been the Book Review editor before that-he wanted more

time for his own mathematics and his poetry. So we divided the responsibil­ ities. When you went to the magazine for­ mat, what were your thoughts about it? It had been a newsletter, almost a per­ sonal communication; what did you have in mind for the bigger format? KP: There was no popular magazine for mathematics really. There were the No­ tices and things like that, but nothing like Psychology Today. We thought we could do something like that, and since there was great interest we also thought it could be done a little bit more pro­ fessionally with good editors who re­ ally would devote time, find longer ar­ ticles, things like that. AP : We had always thought, gee it would be nice to have something like that, and we were amazed at how many people encouraged us. Our dream was that it would be sold at newsstands just like Scient({ic American . We looked into what we would have to do for that to be possible. But it was very hard to get into it. KP: Twenty years ago, mathematics was not a very popular item in the lit­ erature. That has changed tremen­ dously. If you look at what is published today, books about mathematicians, a book like Prime Obsession, sells very well. There are lots of popular books in mathematics. MS: Some of them upset mathemati­ cians, but appreciation is a two-way street. If there's an excellent novel or play or nonfiction that can be appreci­ ated by the general public, we should try to appreciate what the author was trying to do. AP: Yes, there was one review of our book, Tbe Honors Class by Ben Yan­ dell, which is otherwise so widely­ acclaimed, that said it didn't present enough of the mathematics. That was, of course, not quite the point of Tbe Hon­ ors Class. So with that kind of a review you say, oh well, they didn't understand. MS: Was that review in the Intelli­

gencer? AP: No, no. MS: We want to broaden the scope of the Intelligencer and broaden the read­ ership but still keep mathematicians writing in and expressing opinions and so forth.

AP: In the early years, the mathemati­ cal community was much more closely knit. There was a real community sense. When we were going to math­ ematics meetings there was a different spirit than I now find. That's one of the difficulties, I guess. Naturally, you can't maintain that and have growth. KP: But there are things that one can do. For instance, it just occurs to me that you might publish important ex­ pository lectures. And announce forth­ coming lectures in the Intelligencer. There's a lot that can be done to in­ crease the Intelligencers circulation, but with the current culture of more in­ terest in mathematics in the general au­ dience, one would have to make a lit­ tle switch in the Intelligencer. You might want it expanded to include more literary things. But we've strayed from the original subject. MS: No, actually we haven't. When did you leave Springer? KP: In 1 979. We came to the United States and started Birkhauser Boston. Then after many years Birkhauser was sold to Springer, and we left again and went to Academic Press. Then when Academic Press was sold to General Cinema we left again. MS: General Cinema? KP: Yes. and then we spent a couple of years with Jones and Bartlett, but it turned out that they were more inter­ ested in textbooks than in research mathematics. Don Jones, Sr. said, "you know, if you feel comfortable, why don't you start your own company' You can take all the books that you de­ veloped here and purchase them from us, including the open contracts. " So we decided to form A. K. Peters, Ltd. We'll celebrate our fifteenth anniver­ sary at the same time as the hztelli­ gencer celebrates its thirtieth.

Bruce Chandler and Harold Edwards, Coeditors, 1978 The second volume of The Mathemat­ ical Intelligencer lists the two of us as "Founding Editors" and lists no other editors except a "Research News Edi­ tor" (Fritz Hirzebruch) and four "Con­ sulting Editors. " Readers no doubt took this to mean that we were the editors overseeing the publication of these four issues, and that the magazine had been

of us, Bruce and Ed, who were both at­ tending an international congress on the history of science in Edinburgh in the summer of 1 977, took a day trip by plane from Edinburgh to London's Heathrow airport where we met Klaus, who made a stopover of a couple of hours on his way from Springer head­ quarters in Heidelberg to the US. We discussed ideas for the proposed magazine and found ourselves in suffi­ cient agreement that, at the end of the meeting, it was decided that we would go to work on the project with the in­ tention of producing the promised first issue for January 1 978 to coincide with the Joint Meeting of the AMS, MAA, and SIAM in Atlanta. Springer chose as man­ aging editor Irene Heller, a promising graduate student, and the three of us went to work in earnest in the Fall of 1 977. We soon discovered that it was not easy to find the kind of material we hoped to publish, especially not for a magazine that did not yet exist. There were some tense times that Fall as we wondered whether we could find enough material of the quality we wanted to fill the issue, but looking at the finished product thirty years later we feel we did do well. One help was the death of three outstanding figures in mathematics-Paul Bernays, ]. E . Mathematical Intelligencer. Littlewood, and Marston Morse-for Volume 0 already offered subscrip­ whom we found excellent mathemati­ tions-$9.50 for four issues of the pro­ cians to write brief obituaries. jected first volume, shipping and han­ Another help was our willingness to dling included-so Springer was deal with controversial topics. Perhaps committed to going ahead. Because the we could even be accused of deliber­ magazine was to be mostly if not ex­ ate provocation. The excerpt from the clusively in English, and because the book "Why the Professor Can't Teach," main office for Springer books in En­ by Morris Kline, was sure to provoke a glish was in the Flatiron Building in response-which can indeed be read in New York, Klaus and company decided the following issues. And John Guck­ to look for an editor for the projected enheimer's article on the controversy magazine who lived in or near New surrounding "catastrophic theory" ad­ York. Walter was their man in New dressed what was at that time a hotly York, and he proposed Bruce Chandler debated topic. Even Erwin Neuen­ schwander's historical article about Rie­ for the job. Bruce was not interested in taking mann's example of a continuous, non­ on such a big job by himself. He al­ differentiable function (solicited as an ready had a full-time faculty position. accompaniment to Riemann's picture But he said he might reconsider if on the cover) turned out to provoke Harold (Ed) Edwards joined in as coed­ some intellectually vigorous responses. itor and if Springer would furnish a We made no secret of our intention. managing editor to do the heavy lifting As we wrote in our first editorial: "Our associated with the production of the primary goal in terms of the style of the magazine. Springer accepted his terms. magazine is readahili�y. If it comes to In this way it came to pass that the two a choice-and it probably will-be-

our idea beginning with the first issue. Both of these impressions are wrong. The original Mathematical Intelli­ gencer was the conception of Klaus and Alice Peters and Walter Kaufmann­ Bi.i hler, with the later participation of Roberto Minio. They wrote and/or as­ sembled twelve. pamphlet-sized publi­ cations, numbered 0 through 1 1 , that they called The Mathematical Intelli­ gencer. These issues, appearing spo­ radically in the early 1970s, succeeded in attracting the attention of mathe­ maticians-they were amusing, infor­ mative, and unpredictable. They drew attention to Springer publications, and they seemed to provide the small band of editors a great deal of pleasure. The Mathematical Intelligencer had yet another incarnation before we took over as "Founding Editors. " Volume 0 of a NEW Mathematical Intelligencer appeared in the summer of 1 977. It con­ tained an editorial labeled "Please Com­ ment" that referred to the now-forgot­ ten 1975 publication by the Conference Board of the Mathematical Sciences of a "mock issue" of a proposed publica­ tion called Mathematical World, and the editorial says that Volume 0 was created when "the financial committee of MAA decided by one vote not to continue the efforts toward publication of Mathe­ matical World." One vote launched The

© 2008 Spnnger Science+ Business Media, Inc., Volume 30, Number 1 , 2008

11

tween articles which 50% of the read­ ers will throw down in disgust because they don't understand them and articles which 50% will throw down because they disagree with them, we will always choose the latter. Indeed, with an arti­ cle of the latter type, 1% of the readers will probably take pen in hand to pro­ vide what they feel are necessary cor­ rections or refutations, and this sort of intellectual give-and-take, provided it is conducted with the appropriate degree of tolerance and civilized respect for dif­ fering points of view, is the lifeblood of a scholarly community." This year is the 300th birthday of Euler, as well as the 30th of The Math­ ematical Intelligencer, so it seems fit­ ting to quote the next paragraph of our editorial as well: "The patron saint of The Mathematical Intelligencer is Leon­ hard Euler, not so much because of his enormous contribution to mathematics as because of his open, give-and-take style. Euler published theorems without proofs but with intriguing plausibility arguments, published critical examina­ tions of various aspects of controversial subjects such as divergent series or log­ arithms of imaginary numbers, and even published things that were striking but just plain wrong, along with his many lasting contributions. Euler hardly ever published the last word on anything. At the end of one of his articles one feels that he has simply chosen a convenient place to stop and that soon either he or someone else will have something further to say and Euler, confident of his standing and eager to know the truth, does not much care whether it is he or someone else who does take the next step. " Our third issue coincided with the International Congress of Mathemati­ cians in Helsinki in the Summer of 1 978. Much of the material for the fourth and final issue of 1978 had already been lined up, but it was agreed between us and Klaus Peters in Helsinki that, al­ though we would remain as the editors of that fourth issue, we would end our active involvement, and Roberto Minio would take our place. The reasons for this unexpected turn of events were complex and not altogether clear to us, either then or now, but we look back on it without regret. A change in command at Springer re­ sulted in interruption at the Intelli-

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THE MATHEMATICAL INTELLIGENCER

gencer-the following three years saw only two years' worth of issues-and it was not until John Ewing took over as Editor in the Fall of 1 982 that the Intel­ ligencer resumed regular publication. We continued to be listed as "Founding Editors" through the end of Volume 4. Our thanks to Chandler Davis and Marjorie Senechal for remembering our role at the Intelligencer and for inviting us to provide this look back on the be­ ginnings 30 years ago. And our best wishes to them for the fourth decade of the magazine!

John Ewing, Editor, 1979-1986 Writing a book is a creative act, like painting or sculpting; it may take years to finish, but the work of art comes to life all at once, to be measured and ad­ mired (or critiqued) by the world soon after its completion. Editing a journal or a magazine is more like raising a child. A journal is conceived; it develops slowly over time, passing through phases, and it matures into something that often cap­ tures the spirit of the original concept, but with its own personality. Books are created; journals grow up. The Intelligencer was conceived by Klaus and Alice Peters and Walter Kauf­ mann-Bubier, and the first eleven num­ bered pamphlets (not easily found these days) represented its infancy. Chandler and Edwards set the rules for the In­ telligencer's early childhood when they packed into the first volume surveys, history, opinion, and whimsy, all mixed together in a slightly disjointed format that reminded the reader of the maga­ zine's roots as a typewritten pamphlet. Although they parented for only a sin­ gle volume, their influence was felt far beyond. And then came adolescence. I began as an editor of The Mathematical Intel­ ligencer in 1 980, just as it headed into its teenage years-lazy in some ways, rebellious in others, and often unap­ pealing. It was lazy because it was un­ structured; the commitment to the con­ cept of the Intelligencer had not been matched by a commitment to editorial structure or production support. It was rebellious because it lacked a clear un­ derstanding of who had authority; Springer's in-house editor seemed to make all final decisions, when they were made at all. And it was unap­ pealing to many potential authors be-

cause it offered little assurance of cer­ tain readership and absolutely no as­ surance of prestige. I was new to editing and made many mistakes. When material was unsuit­ able, I wrote long letters explaining the reasons, to which authors replied with even longer letters explaining why I was wrong. I soon learned to be polite and concise ("Thanks for thinking of the Intelligencer, but your article isn't suit­ able."). I frequently confused the job of editing (making decisions) with the job of copyediting (rewriting), and spent vast amounts of time working on each and every article. Some authors were appreciative; others were offended; and some huffed away to find a journal that didn't confuse the tasks of editing and authoring. And I made some bad deci­ sions . . . as well as enemies. To a large extent, both the bad de­ cisions and the enemies arose from des­ peration. The Intelligencer had been darling as an infant, but it quickly lost its charm. The nuggets of ready-made material, new and old, were gone. The amateurish format, which was quaint in the first few issues, soon became off­ putting to readers. The lack of editorial and production support caused many ty­ pographical errors. Few manuscripts flowed in for consideration (in fact, none did!). As a consequence, I sent out hundreds of letters to mathematicians around the world: Such and such would make a terrific Jntelligencer article; peo­ ple have expressed a real interest in this topic; wouldn't you like to express your opinion about this? Most produced noth­ ing; some eventually brought a response and occasionally an article as well. Alas, solicited articles can be both wonderful and dreadful. When they are great, they provide not only good ma­ terial for the journal, but satisfaction for the editor who initiated the process. When they are not so great, however, they provide a dilemma that often ends badly. Can they be fixed? Sometimes, but the author is frequently annoyed at the extra work an editor demands. And once the fixing process is started, it's hard to reject the article, once the au­ thor has done not one but several "fa­ vors" for the editor. But immediately re­ jecting a solicited article is hard as well, and there often are no good options for dealing with a poorly written solicited article. In those days, virtually every ar-

tide was solicited. Some were brilliant when they arrived; some required lots of extra work; some remained dreadful and were published nonetheless; and some were rejected-not an easy task for any editor. Articles were only part of the Intel­ ligencers personality, however, and in many ways they were less important than all the other material-opinion, photos, quotes, cartoons, contests, news items, book reviews, and odd bits of historic documents. To a great ex­ tent, the impetus for shaping the Intel­ ligencers personality in this way-for giving it a quirky personality that would persist for the rest of its life-came from Walter Kaufmann-Bi.ihler. By the Intelligencers teenage years, the Peterses had left Springer-Verlag, and only Walter remained. He had del­ egated control of the Intelligencer to others, but he watched over it like an indulgent parent, forgiving the fauxpas and providing a steady supply of en­ couragement and advice. It was Walter who brought the In­ telligencer to adulthood. He wrote to me regularly with ideas: Attached is a short summary paper by P . D.T.A. Elliott, which might, with some hut not extraordinarily much effort, he expanded into a pa­ per for the Intelligencer. I just saw the attached diagram in the proofs of N. Koblitz's forthcom­ ing new book. This might make a nice page filler for the MI. MacLane might be somebody to ask for a paper on the question of whether there are good and had ar­ eas (deserving and undeserving) of mathematics. He has been interested in these things: an article by him could he quite entertaining and sharp. There were dozens of such letters every year, filled with ideas for articles and fillers. But Walter's letters were also filled with wit and a wry sense of hu­ mor that made our correspondence a pleasure: Your Intelligencer bill (office and in­ cidental expenses) is reasonable, even though any non-negative num­ ber is too large. Many thanks for your letter of June 15 and Truesdell's review [from the

Monthly, for which I was then Book Review Editor] . We do get copies of the Monthzy, actually more than we would like, but they come so often that it is important that we throw them away quickly to make sure that we won't be buried. There is trouble ahead. I have heard from a third party ( strictly speaking, Serge Lang) that is prepar­ ing an article which he wants to submit to the MI, pointing out how ' s article was bad are enemies: Springer and 's side. is on Walter had a fine intellect and an en­ cyclopedic knowledge of mathematics and mathematicians. He always protested that he was not a mathe­ matician, but I've known few people over the decades who were more mathematician than he. Walter died at age 42 in 1986 from an asthma attack. l3y the middle of 1 982, it became clear that the informal editorial and pro­ duction support was not working. Vol­ ume 4 was falling further and further behind, and subscribers had not re­ ceived issues for which they had paid more than two years before. After some tough discussion, we decided to make changes: I became the Editor-in-Chief and a Springer staff person was put in charge of production. The first issue of Volume 5 began with a brief piece by me with the title "Not-an-Editorial . " Beginning with Issue 5.1 the Intel­ ligencer changes (once again) both in format and in editorial organiza­ tion. At one time, I thought of writ­ ing a lengthy editorial detailing these changes and outlining future plans. I will spare you; such editorials are interesting often to editors, some­ times to publishers, and seldom to readers. ___

Good or bad, the changes will speak for themselves. Making promises for the future will not convince you that the Intelligencer is better now; we hope you agree in the future that it is a better, more reliable journal. The purpose of the Intelligencer re­ mains the same: to inform, to en­ tertain, and to provoke. It is our deep conviction that mathematicians are intdlectually curious about mathematics as a whole, and that

satisfying this curiosity is a worth­ while endeavor. The new Intelligencer had more struc­ ture: 50 and 100 Years Ago (edited by Jeremy Gray), regular Editorials (by Ian Stewart or me), Book Reviews (Gary Cornell and Ian Stewart), the Problem Corner (Murray Klamkin), the Stamp Corner (Robin Wilson), the Evidence (Stan Wagon), and the quirky "old In­ telligencer," which often contained strange old material , and sometimes contained even stranger new things (see "Odd to Obscurity" by M. Gemignani in 5.2). The magazine ap­ peared regularly, four times a year, al­ most on schedule . Each issue had the same format, which now looked as if someone had designed it (somebody did), and each issue had a cover that tied to something inside. The Intelli­ gencer had grown up. The covers of Volume 5 were atro­ cious in one respect. In an effort to make the new style different from the old, the shade of yellow was changed­ to a bilious mustard color that every­ one hated from the first issue. In one last act of teenage rebellion, the lntel­ ligencer adopted a different color for each subsequent issue, carefully pre­ viewed in advance so as not to repeat the mistake of Volume 5. I stayed on through Volume 8 in 1986. My final editorial (with the title "A Final Editorial") ret1ected on my ex­ perience as Editor over the previous seven years. It began hy recalling Hardy's introduction to A Mathemati­ cian's Apology, in which he wrote that "exposition, criticism, and appreciation" was work for "second-rate minds. " I replied: Hardy was wrong. Was Poincare a second-rate mind? Weyl? Artin? They all devoted time to writing about mathematics, to explaining mathe­ matics both to other mathematicians and to the outside world. Many oth­ ers have done the same. Should all these people have spent their time proving theorems instead? We might as well suggest that musicians should spend all their time writing music rather than performing it. Is there something suspicious about mathematicians who want to know more about their mathematical cul­ ture and heritage? If so, then we ought to be equally suspicious of

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musicians who want to listen to mu­ sic and study it. A painter may de­ spise art-critics, as Hardy says, but a painting that is never viewed is also never admired. Over the ensuing years, 1be Mathe­ matical Intelligencer moved on and in­ deed was admired. It has matured into a magazine that is known to mathe­ maticians around the world for its sparkling, informative, and sometimes quirky articles and commentary, and it is a magazine that sits on coffee tables in mathematics departments every­ where. Living with the Intelligencer while it was growing up was a pleasure, and it shaped my professional career from that point forward. Hardy also wrote in A Mathematician 's Apology that he thought writing about mathematics rather than doing it was a melancholy experience. I suppose he would judge editing such work to be even sadder. But I never did, and I still don't.

Sheldon Axler, Editor, 1987-1991 Changes

1be Mathematical Jntelligencer was al­ ready a terrific publication when I be­ came Editor-in-Chief, inheriting that po­ sition from John Ewing for issues starting in 1 987 and continuing for five years. Although I had loved reading 1be Mathematical Intelligencer, I could not resist tinkering. Five noticeable changes occurred during my first year: The Opinion column replaced the Editorial column. The Editorial col­ umn had been written by either the Editor-in-Chief or one of the other editors, while the Opinion column was potentially open to anyone who wanted to present a strong view on a topic of interest to mathematicians. Disagreements and controversy were welcomed. • Mathematical Entertainments re­ placed the Problem Corner, with Steve Weintraub as the new column editor. The name change here sug­ gested that this column would con­ tain more than just problems. For example, one issue included a math­ ematical acrostic. As another exam­ ple, this column ran a contest to name the five most influential math­ ematicians of the period 1800-1914, with the winner (who received a free •

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THE MATHEMATICAL INTELLIGENCER

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Springer book) decided by the entry that most agreed with the total vote from all entries. According to the votes received, the five most influ­ ential mathematicians from 1800 to 1 9 1 4 were Cauchy, Gauss, Hilbert, Poincare, and Riemann. 50 and 100 Years Ago was renamed Years Ago, with Allen Shields as the new column editor. The name change allowed for more flexibility in focus­ ing on important developments in mathematics from the past, without a restriction to two particular years. The Book Reviews section was re­ named Reviews, with Chandler Davis as the new column editor. This name change was intended to en­ courage reviews of more than just books (software, movies, plays, etc.). The Mathematical Tourist was a new column, edited by Ian Stewart, high­ lighting sights for traveling mathe­ maticians. Here is the description of the kind of material that this column sought: The catapult that Archimedes built, 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 re­ minders of our subject's glorious and inglorious past: statues, plaques, graves, the cafe where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, memorials.

In 1 988 Springer hired Madeline Kraner to improve the design and pro­ duction of their magazines. With Made­ line's help, The Mathematical Intelli­ gencer became even more visually appealing. The journal soon began winning recognition for best all around scholarly publication, production qual­ ity, and design, including a highest achievement award from the American Association of Publishers and best-in­ category awards for the covers.

joyed publishing during my time as Ed­ itor-in-Chief: • The Personal Column was aimed at lovelorn mathematicians, with en­ tries such as the following: 30, 6'2", Lebesgue look-alike seeks attractive, affectionate, 24-34, non­ smoker analyst for causal integra­ tion. Measurements not important.

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Shy combinatorialist, 37, just coming out GWM, seeks same. I'm tired of going through life not knowing whether I'm included-excluded. Let's get together for coffee and see if we're a complete match. Naturally, I'm discrete. The Cartoon Contest sought and published original cartoons related to mathematics or mathematicians. The Poetry Contest sought and pub­ lished original poetry related to mathematics or mathematicians. The readers' vote on which theorem is most beautiful out of 24 theorems on the ballot led to the following re­ sults: 1 . ei7r - 1 2 . Euler's theorem for a polyhedron: V+F = E+2 3. There are exactly five regular polyhedra. 4. l�=I ( 1/n2) = Ji2!6 Several pieces of mathematical fic­ tion showed that mathematicians can write more than theorems and proofs. The Summer 1 989 issue of The Math­ ematical lntelligencer contained an article by Carolyn Gordon entitled "When You Can't Hear the Shape of a Manifold." So that readers could hear the results, Dennis DeTurck had produced music that depended upon the shape of a manifold. Each copy of that issue contained a plastic record (the kind that one plays on a phonograph) with the manifold mu­ sic. Today one would put these sounds on a web site and provide a link in the article, but at the time this was a unique addendum to a math­ ematics article. =

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Having fun

The spirit of The Mathematical Intelli­ gencer is to have fun, to be irreverent, and to publish items of interest to math­ ematicians that would not appear in a traditional research journal. Here are a few of the nonstandard items that I en-

Covers

The cover of each issue, like the cover of a book, should not matter much, but it seemed to make a huge psychologi­ cal difference in the way people per­ ceived each issue. Thus I paid attention

to what went on the cover. Before each issue, I would send to the Springer of­ fice in New York one or two pieces of artwork that might make good covers. These choices depended upon finding visually interesting artwork or graphics that accompanied an important article. The talented Springer staff would then usually send me back two potential lay­ outs for the cover, and I would choose one of them. Usually the covers were printed in shades of two colors. However, when I had a compelling graphic that needed full color on the cover, Springer was willing to spend the extra money. Two articles while I was Editor-in-Chief ab­ solutely needed full color inside, which at the time was considerably more ex­ pensive than full color on the cover, and both times Springer generously agreed. One of those two articles was David Hoffman's "The Computer-Aided Discovery of New Embedded Minimal Surfaces," which later won the Chau­ venet Prize of the Mathematical Associ­ ation of America. Only once did I reject the suggested Springer cover layouts. For the Summer 1 990 issue, we were publishing a fasci­ nating interview with Hoang Tvy, Di­ rector of the Hanoi Mathematical Insti­ tute. Professor Tvy is the author of what is probably the first mathematics book published by a guerrilla movement. This book, a geometry textbook, was pub­ lished by the Viet Minh resistance press in 1949 during the Vietnamese struggle against French occupation. I had a copy of one page from that hook, and I thought that it would make a splendid cover. But the Springer staff in New York said that the copy was not of suf­ ficiently high quality to reproduce well, and they sent me two other potential cover designs using other artwork that accompanied the interview. However, I badly wanted to put on the cover that page from the first math­ ematics book published by a guerrilla movement, because I thought that it was a dramatic part of the story. So I asked the Springer staff to try again. They came up with an outstanding de­ sign, making one of the best covers dur­ ing my time as Editor-in-Chief. The main part of the cover shows three Viet­ namese schoolgirls, smiling in front of a computer that they are using. The page from Professor T�ry's 1 949 geom-

etry textbook appears in the lower right corner at about 20% of its actual size, but quite legible and making a beauti­ ful juxtaposition with the photo. Controversies

Controversies can make for interesting reading, especially in mathematics where we rarely argue about the sci­ entific validity of a result. I was happy to air controversy within the pages of The Mathematical Intelligencer: excit­ ing controversies helped keep the pub­ lication edgy. Because The Mathemati­ cal Intelligencer is published only every three months, I often had time to send a controversial article to someone with an opposing viewpoint and publish a response in the same issue. Sometimes I was able to bounce things back and forth several times. The record for this within one issue was the Summer 1987 issue, which included a back-and-forth on constructive mathematics. The last item in the string of rejoinders was sub­ titled "Ian Stewart rebuts Fred Rich­ man's reply to Ian Stewart's response to Fred Richman's reply to Ian Stewart's review. " Three of the controversies, dis­ cussed below, were not so whimsical. Controversy One

In 1 986 and 1987 Serge Lang had con­ ducted successful and highly publicized campaigns within the National Acad­ emy of Sciences to reject the member­ ship nomination of Samuel Huntington, a social scientist. Lang felt that Hunt­ ington had misused mathematics in his scholarly work, presenting pseudo­ mathematics more for mystification than for explanation. I thought that this dis­ pute would make an interesting article for The Mathematical Intelligencer, so I asked Lang to write something about the Huntington affair. Lang told me that he had already written everything he wanted to say on the subject, but he suggested that I ask Neal Koblitz to write an article. Koblitz's earlier article, "Mathematics as Propa­ ganda'' (published in Mathematics To­ morrow) had in fact first alerted Lang to Huntington's use or misuse of math­ ematics. Thus I asked Koblitz, who produced a fascinating article titled "A Tale of Three Equations; or The Emperors Have No Clothes. " Naturally I sent a copy ( prepublication) of Koblitz's arti-

de to Huntington and told him that I would be happy to publish a response from him. Huntington replied that he would not write a response but that Herbert Simon, a Nobel Prize winner in Economics, would be willing to re­ spond to Koblitz. So I wrote to Simon, who indeed wrote a defense of Hunt­ ington in an article titled " Unclad Em­ perors: A Case of Mistaken Identity. " Koblitz's article and Simon's response, along with a brief reply from Koblitz to Simon's response, were all published in the Winter 1988 Mathematical Intelli­

gencer. The next issue of The Mathematical Intelligencer included further back-and­ forth between Simon and Koblitz, this time starting and ending with Simon. All of this generated a lot of mail-in the next three issues I published a total of twelve letters to the editor on this nasty dispute. I saw no need at the time to weigh in with an editorial comment of my own, but I can say now that it was absolutely clear to me that Koblitz and Lang were completely correct in their analysis of Huntington's work. This controversy had a sad aftermath. After the articles and responses had been published, Lang changed his mind and told me that he wanted to submit an article about the Huntington affair. I had already offered him the opportu­ nity to do so, before I had approached Koblitz, but Lang had declined then. I told Lang that to be accepted, his arti­ cle would need to contain new mater­ ial not contained in Koblitz's excellent account. When Lang did submit an article, it contained nothing new that was rele­ vant. Unfortunately I could not use the limited space in The Mathematical In­ telligencer for repetition. I tried gently telling Lang that we could not publish his article, but he became furious with me. This was painful because I had known and liked Lang since my senior year as an undergraduate, when Lang spent some time at Princeton. I had wanted to study for my senior com­ prehensive exam from Lang's Algebra, but I could not find a copy. Lang gra­ ciously gave me a copy of Algebra, with the provision that I would then go to a bookstore and buy a copy of his Real Analysis book (which I did, thus get­ ting two good books for the price of one).

© 2008 Springer Science+Business Media, Inc., Volume 30, Number 1 , 2008

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Lang refused to speak with me for a few years after I rejected his article, but gradually his anger dissipated and we were again able to have pleasant con­ versations. Controversy Two

In 1 988 Steven Krantz submitted a re­ view of The Science of Fractal Images (edited by Heinz-Otto Peitgen and Dietmar Saupe) and The Beauzy ofFrac­ tals (by Heinz-Otto Peitgen and Peter Richter) to the Bulletin of the A merican Mathematical Society. Krantz's review was accepted for publication in the Bul­ letin, and he circulated preprints of it. Benoit Mandelbrot took exception to Krantz's review in preprint form and wrote a rebuttal. Krantz was willing to have Mandelbrot's rebuttal published in the Bulletin along with his review, but the editorial policy of the Bulletin does not allow responses to reviews. The Bulletin then took the unusual step of retracting its acceptance of Krantz's re­ view. The Mathematical Intelligencer, which welcomes controversy and encourages rebuttals, was happy to publish both Krantz's review and Mandelbrot's re­ sponse in the Fall 1 989 issue. As I had expected, this controversy generated a fair amount of mail. I published letters to the editor about the Krantz/Mandel­ brot dispute in the next four issues. Controversy Three

The Spring 1 989 issue of The Mathe­ matical Intelligencer contained an in­ terview with the Soviet mathematician Igor Shafarevich, conducted by Smilka Zdravkovska, who was an Associate Ed­ itor at Mathematical Reviews and who had been an undergraduate at Moscow State University. Shafarevich had been elected as a Foreign Associate of the U.S. National Academy of Sciences for his outstanding work in number theory, algebra, and algebraic geometry. As part of this interview, Zdravkovska asked Shafarevich about his long essay Russophohia, adding as part of her question about this essay that "some consider it unfair, and even accuse you of anti-Semitism. " I believe that this question and its response by Shafarevich was the first time that Rus­ sophohia was brought to the attention of mathematicians in the English lan­ guage.

16

THE MATHEMATICAL INTELLIGENCER

Soon after the interview with Sha­ farevich was published, Lawrence Shepp and Eugene Veklerov submitted an ar­ ticle to The Mathematical Intelligencer, detailing what they considered to be er­ rors and anti-Semitism in Russophohia. I read Russophohia and found it to be badly done history containing a huge dose of anti-Semitism. However, unlike Huntington's writings that had led to controversy a year earlier in the pages of The Mathematical Intelligencer, Rus­ sophohia contained no pseudomathe­ matics. There was nothing remotely mathematical in the pages of Russo­ phobia; the author did not use mathe­ matics to buttress his arguments. In fact, there is no way to tell from reading Rus­ sophohia that the author knows any mathematics. In other words, Russophohia is a po­ litical document discussing history, with no connection to mathematics except that the author is a mathematician (which was not stated anywhere on my copy). Russophohia comes to what seemed to be absurd conclusions, but I did not think that an article dissecting a purely nonmathematical political/his­ torical document was appropriate for publication in The Mathematical Intel­ ligencer-too many good articles with some connection to mathematics had to be turned down because of lack of space. Thus I rejected the article by Shepp and Veklerov, although I agreed to pub­ lish a critical letter to the editor from them about Shafarevich and Russopho­ hia. That letter appeared in the Sum­ mer 1 990 issue. Then Shafarevich submitted to The Mathematical Intelligenceran expository article titled "Abelian and Nonabelian Mathematics. " This was the kind of arti­ cle appropriate for The Mathematical In­ telligencer, and experts told me that the content was very good. The content was purely mathematical, with no political as­ pects and no political/historical com­ ments. One prominent mathematician advised me not to publish the article be­ cause of Shafarevich's anti-Semitism, warning that there would be repercus­ sions if Shafarevich's article appeared in

The Mathematical Intelligencer. I did not like Shafarevich's political views, and I found Russophohia to be highly offensive. But no one had ever asked me about my political views

when I submitted a mathematics paper for publication, and I was not about to start subjecting authors of papers sub­ mitted to The Mathematical Intelli­ gencer to political screening. Thus I accepted Shafarevich's article for publi­ cation. It appeared in the Winter 1 99 1 issue, and I got ready to hear the criti­ cism. I was pleasantly surprised when no complaints arrived. Tragedies

Two tragic deaths marred my term as Editor-in-Chief: • Walter Kaufmann-Bubier was Math­ ematics Editor at Springer New York. He was one of the founders of The Mathematical Intelligencer, which could not have survived and flour­ ished without Walter's support. Wal­ ter had appointed me as Editor-in­ Chief. Shortly before my first issue came out, Walter died suddenly of heart failure caused by a severe asthma attack. Mathematics had lost a good friend who cared far more about scholarly quality than the bot­ tom line. The Fall 1987 issue of The Mathematical Intelligencer was ded­ icated to Walter, with several articles of reminiscence from Walter's col­ leagues and friends. • Allen Shields, who wrote the Years Ago column during my first three years as Editor-in-Chief, died of cancer in September 1 9R9. Allen's columns sparkled with insight and demonstrated his unusual knowledge of history as well as mathematics. He and I were good friends and mathe­ matical collaborators-we wrote six research papers together. The Spring 1 990 issue of The Mathematical In­ telligencer was dedicated to Allen, with several articles of reminiscence from Allen's colleagues and friends. Soliciting and refereeing articles

During my time as Editor-in-Chief, about half the articles published in The Mathematical Intelligencer originated because I asked the author to write an article, with the other half arriving un­ solicited. The rejection rate among un­ solicited articles was high because of space limitations and the large number of articles submitted. I quickly rejected over half the unsolicited articles with­ out sending them to referees; these ar­ ticles were simply not appropriate for

The Mathematical Intelligencer, and I saw no reason to waste referees' time. My most unpleasant duty as Editor-in­ Chief was having to send rejection let­ ters to many people who wanted to contribute. The rejection rate among so­ licited articles was low, partly because I was careful whom I asked to write the articles and partly because it's tricky to twist someone's arm to write an article and then reject it. In the rare case where a solicited article turned out poorly, I usually asked for extensive revisions, and then more extensive revisions if the second version still was not good , and so on, until either a good article was produced or the author gave up in frus­ tration at all the requested revisions. My agreement with Springer was that I and the column editors that I appointed would have complete control of the con­ tents of The Mathematical Intelligencer, except for advertisements. Springer scrupulously adhered to this agreement, never pressuring me to stitle a contro­ versy or suppress a review that might adversely affect Springer's interests as a commercial publisher of mathematics. I'm truly grateful to Springer for the cre­ ative opportunity it gave me as Editor­ in-Chief. Editing The Mathematical In­ telligencer was lots of fun!

Chander Davis, Editor 1991-2004, Coeditor 200&­ When the possibility was floated that I might become Editor-in-Chief of The Mathematical Intelligencer, I was al­ ready in pretty deep. In addition to the material I was generating as column editor and occasional contributor, I was reading every issue in its entirety and hashing it over-delightedly but frankly-with the Editor, my old friend Sheldon Axler. I hope I wasn't such a burden as to speed Sheldon's decision to leave his position! But leave it he did, and he recommended me to succeed him. So I leapt at the opportunity, right? Well, yes, I spoke up for it eagerly­ hut after considerable hesitation. Other editing jobs I had taken on with no fuss-even, years ago, the enormously challenging one at Mathematical Re­ uiews. The difference I felt about The Intelligencer was that the duties are largely the editor's to invent. In an or­ dinary mathematical editing position, one knows what the community ex-

pects. The task is still many-faceted, and I wouldn't slight the creativity required. The difference at The Intelligencer is that one doesn't only follow the flight plan: one writes it. The creativity needed was what made the enterprise exciting, but it was a little scary, and it still is. Most of my jobs, teaching as well as editing, have been living up to the expectations of those in austere, ivy-covered institu­ tions. Here we are with no ivy-covered walls. We pull up our soap-box and say our piece. Does anyone other than me still re­ member Henry Morgan's radio show in the USA? He sassed establishments with such glee, in the spirit of the soap-box orator-yet there he was, on national radio! And here we are, in glossy mag­ azine format' So I began my editorship with a con­ crete and lofty image of the position, hut with no great certainty that I could live up to it. I had a head start, with a tradition already there and with a con­ tinuing editorial team: Jan Stewart, David Gale, and the rest-and Bob Burckel, ever-vigilant, checking every manuscript. Best of all, the tradition, the team, and the readers were international and diverse. Naturally enough. I also inherited an unresolved debate or two. The Sha­ farevich controversy was still bubbling when I took over. Smilka Zdravkovska had interviewed I. R. Shafarevich just as he was turning toward activity in a po­ litical movement that was disturbing to many of his colleagues. She learned of his privately circulated Russophobia and with great diplomacy secured his per­ mission to insert a question or two about it into the published interview. I agreed with Sheldon that it was not in­ cumbent on The Intelligencer to demo­ nize Professor Shafarevich, or to sani­ tize him; we cast about for ways to display the contradictions. Readers came to our rescue with passionate, di­ verse opinions. I published letters pro and con from different lands; but I had to call "time's up'' : I couldn't let Russ­ ian politics crowd mathematics out of our pages. I'm still proud of the clincher I found : a sorrowful tribute by Boris Moishezon to his revered teacher Sha­ farevich, in an obscure Russian emigre paper, from which I translated a long excerpt (see our vol. 1 4 . no. 1 , 61-62).

The tribute recognizes the evil, and it remains fair to everyone involved. The tragic complexity of life is in view. I recall another controversy that elicited more earnest letters than I could justifiably print. I had accepted an arti­ cle from a Texas numerical analyst ex­ pounding the "Intelligent Design" (ID ) position: that the organisms we observe could not have arisen by random mod­ ifications with natural selection, but must be the intentional product of a guiding intelligence. A hornet's nest of outraged Intelligencer readers swarmed to rebut. I busily edited the letters to reduce duplication and to keep the to­ tal length of the debate within reason; this involved many friendly e-mail and telephone exchanges with readers. Af­ ter a couple of issues, the proponent of ID was given the last word. Some of my correspondents thought I ought to have rejected his article in the first place (as I might have done-not for being too outrageous but for having too little relationship with mathematics). One friend and mentor, a leading applied mathematician, put it this way privately: "I'm sorry you were taken in. " I replied, "I wasn't taken in. I thought we'd have a good debate, and we did . " The argu­ ments were incisive, and some not at all familiar. I especially relished Alexan­ der Sherr's (see vol. 23, no. 4, 3). But in hindsight, some things were disap­ pointing. It would have been much bet­ ter drama if at least one of the responses I received had supported ID. None did. And some of my own views on evolu­ tion happened not to be expressed by any of the letter-writers, nor did I pre­ sume to interject them-leaving me feeling somehow let down. More important, The Intelligencer has been a forum for exploring the un­ certainties we feel about the nature of mathematics, and society's input to mathematics. How can it be that our subject, a plainly social enterprise, con­ sists so largely of apparently certain statements, seemingly invariant under any change in society? Some mathe­ maticians see no puzzle here-but we need to draw them into the dialogue, too, because some of them are con­ spicuously offended by it, and we wish to understand and learn from the sources of their hostility. Remember the strong feelings let loose by the "Sakal hoax," or the contempt some mathe-

© 2008 Springer Science+Business Media, Inc., Volume 30, Number 1 , 2008

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maticians display for the history of mathematics-especially social history. This is just the sort of disputation The Intelligencer likes to host. We've had such debate, yet I feel we might have had more! Look around. The bases of mathematics are in turmoil: the challenge of intuitionism has not really been met; the challenge of experimen­ tal mathematics is crowding on its heels; and most mathematicians are now will­ ing in principle to incorporate physical truth within mathematical truth, but don't see how. We would like the dif­ ferent views to confront each other in The Intelligencers pages, and now and then we achieve this confrontation, as with Doran Zeilberger's provocative piece (vol. 16, no. 4, 1 1-14) and Martin Gardner's (vol. 23, no. 1 , 7-8). Both drew sharp ripostes, as did the sympo­ sium set up by Marjorie Senechal in the Communities column, vols. 22-23, on social construction of mathematics. Surely there is much more you have to say, and we look forward to hearing it. When you feel your colleague's viewpoint is preposterous, whether or not you are the type to say so in con­ versation, you may say, "You must be kidding," in this magazine; please do­ that very phrase was used in at least one of the debates I've mentioned. A major purpose of The Intelligencer from the outset has been to talk math­ ematics to each other, without contro­ versy, in a discourse uniting all of us across national borders and transcend­ ing divisions into fields. Here's how I put it years ago in an editor's note "Our Own Babel" (vol. 19, no. 2, 4): There is a famous joke about a boy in a cultivated Central European

family. His mother spoke to him in French, his father in German, and his nursemaid in Hungarian. The child understood them all, but didn't say a word himself until he was four: he thought he was supposed to have his own language. Alas, the joke is true of us. Each of us is entitled to make up a new private language and start speaking it. Is there salvation for us? Well, maybe there is. Let's see. We should really try, here and there, to create an is­ land of comprehension in the mid­ dle of the din-a privileged space where mathematicians speak to each other and are understood. It is the highest aim of my editorship that one such island shall be-shall con­ tinue to be- The Mathematical In­

telligencer. Or, as I often exhort authors, here we must try to do what a good collo­ quium talk is supposed to do: make sense to everyone in the audience. Vi­ suals help. Informality, descending at times to silliness, helps. Expressing everything in English doesn't help, re­ ally; sorry about that. I'm consoled by seeing other magazines occasionally translate our articles into other lan­ guages (in the last few years, often in­ cluding Chinese), but I doubt that the silliness translates well. I have a private game of sometimes sneaking an on­ translated bit of another language into our pages: retaining English as the lin­ gua franca yet acknowledging that other linguae stand on their own on a par with English. So I took a crack at this daunting ed­ itorship in 1 99 1 , and, as my predeces-

ft foll

18

THE MATHEMATICAL INTELLIGENCER

sors had found, it was wonderful fun. If I have kept careening along this ill­ marked highway so unduly long, it's not just because there was fun. It's certainly not because I think I've lived up to the aspirations I saw and undertook. Rather the reverse: my awareness of how much we ought to have achieved makes me thirst to have another go. Sharing re­ sponsibility with Coeditor Marjorie Senechal-we've been in this together since 2005-adds to the fun, enlarges the vision, and increases my optimism that the vision can be achieved. Wait till next year!

Chandler Davis and Marjorie Senechal Thank you, Alice and Klaus, Bruce and Ed, John, and Sheldon! We're glad you look back with such pleasure on your years of editing this remarkable maga­ zine, despite all the hassles, controver­ sies, scrambles, and scrapes. It's still a pleasure in 2007. In 1 97 1 , when the first accordion is­ sue unfolded from Alice Peters's type­ writer, the mathematical community was small enough to send postcards to, yet large enough to need an Intelli­ gencer. In the years since then, the mathematical community has bur­ geoned and diversified, like mathemat­ ics itself. What is the role of The Math­ ematical Intelligencer in an e-mail age, in an ever-growing community, in an ever-growing mathematical landscape? Should the Intelligencer go online, or remain in the reading room? Can we, should we, reach a wider public? These are questions for you, its read­ ers. We welcome your responses, now more than ever.

nt

Refuge from M isery and Suffering REUBEN H ERSH AND VERA JOHN-STEINER

\ \ \ \

jl

l

hile working on our new book, Loving and Hat­

ing Mathematics: The Emotional Side of Mathematical Life, we were surprised at how many well­

known mathematicians have created mathematics while in prison. We found five prisoners of war from three different wars, plus two political prisoners, and one prisoner con­ victed of evading military service. Alongside these, there was one mathematician who used mathematics to escape an ex­ cruciating toothache, one who was revived to life by a math­ ematical problem while bedridden and almost 90 years old, a novelist who was distracted by mathematics from his decades-long writer's block, and an idealistic youngster who was helped to endure the agony of participating in a sense­ less, brutal bombing war. Of all escapes from reality, mathematics is the most suc­ cessful ever. It is a fantasy that becomes all the more addictive because it works back to improve the same re­ ality we are trying to evade. All other escapes-sex, drugs, hobbies, whatever-are ephemeral by compari­ son. . . . The mathematician becomes totally committed, a monster, like Nabokov's chess player who eventually sees all life as subordinate to the game of chess [ 1 3] .

Absorption Mathematics is sometimes a safe hiding place from the mis­ eries of the world. In its mild form, escape is absorption. You know you're finally, really getting into your problem when you dream about it every night. (No guarantee your dream will give you the solution!) They say Newton some­ times forgot both to eat and to sleep. To outsiders, this is called "absent-mindedness. " They say that Norbert Wiener, when walking down a corridor at M.I.T. with a mathemat­ ical paper in his right hand, would come to an open class­ room door, walk through the doorway and around the four walls of the classroom, and then out again, guiding him­ self with his left hand against the wall, while still reading.

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THE MATHEMATICAL INTELLIGENCER © 2008 Springer Science+ Business Media. Inc.

Blaise Pascal had renounced mathematics and science in favor of ascetic devotion to the Blessed Virgin, but he was still able to turn to mathematics in an emergency. Among other miseries which afflicted the wretched Pas­ cal were persistent insomnia and bad teeth . . . . Lying awake one night [in 1658] in the tortures of a toothache, Pascal began to think furiously about the cycloid to take his mind off the excruciating pain. To his surprise he noticed presently that the pain had stopped. . . . Inter­ preting this as a signal from Heaven that he was not sin­ ning in thinking about the cycloid rather than his soul, Pascal let himself go. For eight days he gave himself up to the geometry of the cycloid and succeeded in solv­ ing many of the main problems in connection with it [3]. John Littlewood at the age of 89 got a new lease on life from a mathematical challenge: In 1 972 he had two bad falls and he fell again in Janu­ ary 1 975. He was taken to the Evelyn Nursing Home in Cambridge but he had very little interest in life. In my desperation I suggested the problem of determining the best constant in Burkholder's weak L2 inequality (an ex­ tension of an inequality he had worked on). To my im­ mense relief (and amazement) he became interested in the problem. He had never heard of martingales but was keen to learn about them. And all this at the age of 89, and in bad health! It seemed that mathematics did help to revive his spirits and he could leave the nursing home a few weeks later. From then on he kept up his inter­ est in the weak inequality and worked hard to find suit­ able constructions to complement an improved upper bound [4]. The American novelist Henry Roth, author of Call It Sleep, lived for many years in a remote village in Maine, in the far northern United States. He was suffering from writer's block, and he was attempting to help support his family by raising and slaughtering ducks and geese. To mentally survive the

Maine winters, he did calculus problems. In fact, he did all the problems in Thomas's influential calculus text, and he later visited Professor Thomas at M.I.T. to tell about this feat. Freeman Dyson was able to find some solace from the hardship of war through absorption in mathematical activities: In 1 943 I had left Cambridge and was working for the Royal Air Force as a statistician . . . . Hardy knew that I was interested in the Rogers-Ramanujan identities. So he sent me a paper to referee. The paper was by W. N . Bailey and contained a new method o f deriving identi­ ties of the Rogers-Ramanujan type . . . . I never met Bai­ ley. During those months, he was at Manchester and I was at the Royal Air Force Bomber Command head­ quarters in the middle of a forest in Buckinghamshire. It was a long, hard, grim winter. I was working a sixty­ hour week at Bomber Command. The bomber tosses which I was supposed to analyze were growing steadily higher. The end of the war was not in sight. In the evenings of that winter I kept myself sane by wander­ ing in Ramanujan's garden, reading the letters I was re­ ceiving from Bailey, working through Bailey's ideas, and discovering new Rogers-Ramanujan identities of my own. I found a lot of identities of the sort that Ramanujan would have enjoyed. My favorite was this one: "" + x4) . . . (1 + x n + x2 n) � x n2+ n ( 1 + X + x2)( 1 + x2� �� ( 1 - x)( l - x2) . . . ( 1 - x 2 r + l) n=O ---------

-----

=

IT

n= I

----

( 1 - x2 n) . ( 1 - XII)

In the cold dark evening, while I was scribbling these beautiful identities amid the death and destruction of 1 944, I felt close to Ramanujan. He had been scribbling even more beautiful identities amid the death and de­ struction of 1 9 1 7 [6]. John Todd and Olga Taussky, while in London during the " Blitz," took advantage of the German air raids to get some work done. "During the war, Olga and I wrote sev­ eral papers in bomb shelters. Our bomb shelter was the

ground floor of our apartment building. During raids we wrote papers--about six in all--while the other twenty to thirty people chatted, slept, or read" [ 1 ] .

Prison Stories Quite a few well-known mathematicians have served time as prisoners of war, from the Napoleonic War to World War II. At least two have been political prisoners--in the U . S . , and i n Uruguay. A n impressive amount o f beautiful math­ ematics has in fact been created in prison, where it served to help the imprisoned mathematician to survive his ordeal. A major part of projective geometry was created in prison. In November of 1 8 1 2 Jean-Victor Poncelet, a young officer in the exhausted remnant of Napoleon's army retreating from Moscow under Marshal Ney, was left for dead on the frozen battlefield of Krasnoi. A Russian search party found him still breathing. In March of 1813, after a five-month march across the frozen plains, he entered prison at Saratov on the banks of the Volga. When "the splendid April sun restored his vi­ tality," he commenced to reproduce as much as he could of the mathematics he had learned at the Ecole Polytechnique, where he had been inspired by the new descriptive geom­ etry of Monge and the elder Carnot. In September of 1814, Poncelet returned to France, "carrying with him the material of seven manuscript notebooks written at Saratov in the pris­ cms of Russia together with diverse other writings, old and new." Bell writes that this work "started a tremendous surge forward in projective geometry, modern synthetic geometry, geometry generally, and the geometric interpretation of imag­ inary numbers that present themselves in geometric manip­ ulations, as ideal elements of space" [3]. Leopold Vietoris, the Austrian topologist who died in 2002 at the age of 1 1 1 , was serving as a mountain guide for the Austro-Hungarian army in World War I while working on his thesis "to create a geometrical notion of manifold with topological means. " Just before the Armistice, on November 4, 1918, he was captured by the Italians. He completed his thesis while a prisoner of war [ 1 1].

REUBEN HERSH AND VERA JOHN-STEINER are both emeritus pro­

fessors at the University of New Mexico, he in Mathematics and she in Language, Literacy, and Socio-cu�ural Studies. Hersh is the author of The Mathematical Experience and Descartes' Dream (both with Philip J. Davi s) ,

of 1 8 Unconventional Essays on the Nature of Mathematics (editor) , and of What Is Mathematics, Really? John-Steiner is the author

of Creative

Collaboration and Notebooks of the Mind, for which she received the

William James Award ofthe American Psychological Association in 1 990. The two have just finished a coauthored book Loving and Hating Math­ ematics: The Emotional Side of Mathematical Life. One chapter from this

book provided the basis for the present arti c le . Reuben Hersh Vera John-Steiner I 000 Camino Rancheros

Santa Fe, NM 87505 USA e-mail (Hersh): [email protected] e-mail Qohn-Steiner): [email protected]

© 2008 Springer Science+ Business Media, Inc., Volume 30, Number 1, 2008

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Two other mathematicians in the Austro-Hungarian army in the First World War were taken prisoner, not by the Ital­ ians but by the Russians. Eduard Helly of Vienna and Ti­ bor Rad6 of Budapest met in a prison camp near Tobolsk in 1 9 1 8 . Rad6 had just begun university when he enlisted as a lieutenant and was sent to the Russian front. Helly was then already a research mathematician; he had proved the so-called Hahn-Banach theorem in 1 9 1 2 , before either Hahn or Banach. In the Russian prisoner-of-war camp, Helly be­ came Rad6's teacher. Escaping from the prison camp near Tobolsk, Rad6 made his way north to the Arctic regions of Russia. There Es­ kimos befriended him and offered him hospitality as he slowly made his way westwards. After a trek of many thousands of miles Rad6 reached Hungary in 1 920. It was five years since he had been studying as a student in Budapest, but now he returned to his studies, this time at the University of Szeged. Helly had shown him the fascination of research-level mathematics, so now it was mathematics rather than civil engineering that he studied [7]. Rad6 assisted Frigyes Riesz in his great book on func­ tional analysis, and in 1 929 he emigrated to the United States. He founded the graduate program in mathematics at Ohio State University, and he became a leading author­ ity on the theory of surface measure. Being a mathematician in prison could have serious, detrimental consequences. The French analyst and applied mathematician Jean Leray was a German prisoner of war for 5 years in World War II. "He feared that if his compe­ tence in fluid dynamics and mechanics were known to the Germans, he might be required to work for them, so he turned his minor interest in topology into a major one . . . during those five years he carried out research only in topol­ ogy" [8]. In fact, Leray created sheaf theory, which soon be­ came one of the principal tools in algebraic topology. Nev­ ertheless, after he was free, Leray returned to analysis, leaving topology to others. The French number-theorist Andre Wei!, like his compa­ triots Poncelet and Leray, had a spectacularly productive time in prison. In the summer of 1 939, war with Germany was im­ minent, and Wei! was under orders for military service. "This was a fate that I thought it my duty, or rather my dharma, to avoid, " he wrote in his autobiography. He departed for Finland. By bad luck, the Russians invaded Finland a few months later. "My myopic squint and my obviously foreign clothing called attention to me. The police conducted a search of my apartment. They found several rolls of stenotypewrit­ ten paper at the bottom of a closet. . . . There was also a let­ ter in Russian, from Pontryagin" [ 1 5]. After three days in prison, Wei! was unexpectedly re­ leased at the Swedish border. He only learned 20 years later how his life was spared. The Finnish function-theorist Rolf Nevanlinna told Wei! he was present at a state dinner also attended by the chief of police. When coffee was served the latter came to Nevanlinna saying: 'Tomorrow we are executing a spy who claims to know you. Ordinarily I wouldn't have troubled you with such trivia, but since we're both here anyway I'm glad to have the opportunity to consult you.' 'What is his name?'

24

THE MATHEMATICAL INTELLIGENCER

'Andre Weil . ' Upon hearing this, Nevanlinna told me, he was shocked. 'I know him,' he told the police chief. 'Is it really necessary to execute him?' 'Well, what do you want us to do with him?' 'Couldn't you just escort him to the border and deport him?' 'Well, there's an idea; I had­ n't thought of it.' Thus was my fate decided. Shipped back to France by way of Sweden and Scot­ land, Wei! spent three months in jail in Rouen. His friend Henri Cartan wrote to him, "We're not all lucky enough to sit and work undisturbed like you . " On April 7, 1 940, he wrote to his wife Eveline, "My mathematics work is pro­ ceeding beyond my wildest hopes, and I am even a bit worried-if it's only in prison that I work so well, will I have to arrange to spend two or three months locked up every year?" On April 22, he wrote her, "My mathematical fevers have abated . . . before I can go any further it is in­ cumbent upon me to work out the details of my proofs . . . " On May 3, 1940, he was sentenced to five years in prison, which was immediately commuted if he agreed to serve in combat. On June 1 7, 1 940, "the command came to aban­ don our machine guns and join our regiment on the beach. We were boarded on a small steamship . . . the next morn­ ing we were in Plymouth. " Wei! eventually reached the United States to continue his illustrious career [ 1 5] .

Mathematics and Politics Chandler Davis, an editor of this magazine, was my (R. H . ) schoolmate, a math graduate student a t Harvard when I was an undergraduate English major. During the Mc­ Carthyite red scare in the 1 950s, Chan's career was inter­ rupted when he refused to answer questions asked by the U.S. House of Representatives Committee on Do-American Activities. He proudly referred to his revolutionary ances­ try-in the American Revolution-and refused to cooper­ ate in proceedings that violated the first amendment to the U.S. Constitution that guaranteed freedom of speech. Chan­ dler was fired by the University of Michigan from his job as Instructor of Mathematics. He was convicted of "con­ tempt of Congress," and, after exhausting appeals, he was confined for 6 months in the Federal Prison in Danbury, Connecticut. Then, when he was released, he was totally blacklisted by universities in the U.S. The great Canadian geometer Donald Coxeter invited him to apply to the Uni­ versity of Toronto. "Initially the government refused Davis's entry but ultimately, after a letter-writing campaign, they re­ lented" [12]. He moved to Canada to teach at the Univer­ sity of Toronto. A 1 994 special issue of Linear Algebra and Its Applications, celebrating his contributions to matrix the­ ory, describes his time in prison. "Throughout this ordeal, Chandler maintained his research interest in mathematics. He also maintained his sense of humor. A footnote in his paper on an extremal problem, conceived while he was in prison but published afterward, reads: 'Research supported in part by the Federal Prison System. Opinions expressed in this paper are the author's and are not necessarily those of the Bureau of Prisons' " [5] . [Editor's Note. It is true that I had the whimsical idea to give prison as the institutional affiliation for that paper. However, the elegant phrasing was suggested to me by Peter Lax-thank you, Peter !-Chan­ dler Davis.]

A Singular Anomaly

Let it be understood that not every imprisoned mathe­ matician fared as well as Andre Wei! or Chandler Davis. The Uruguayan analyst, Jose Luis Massera, writes that he was drawn into political activity under the influence of the refugees from Fascism, Luis Santal6 from Spain and Beppo Levi from Italy. "In that epoch, around the great movement of solidar­ ity with the Spanish people, I began the political activity which I shared with mathematics for the rest of my life" [10]. In 1943, Massera became a Communist. After completing his degrees in Montevideo, Umguay, Massera won a Rockefeller grant to go to Stanford, where he worked with P6lya and Szego. But then he became more in­ terested in differential equations, so he transferred to the East , where he commuted between New York and Princeton, work­ ing simultaneously with Richard Courant on minimal surt�tces and with Solomon Lefschetz on topological methods for or­ dinary differential equations. In 1966, after returning to Uruguay, he published his well-known hook Linear nifer­ ential Equations and Function Spaces with his student ]. J Schaffer [9]. He was also elected as a Communist Deputy to the Parliament of Uruguay. When the milit
walked conversing with another prisoner. . . . As must be clear, human relationships were almost exclusively limited to the cell . . . . In it one could read books of the good li­ brary that had been formed with donations that families of prisoners had made when it was permitted. Of course, po­ litical books were excluded . . . as well as mathematics books. Who knows what mysterious messages could be conveyed by those odd and incomprehensible symbols? My cellmates were various including Communists and Tupamaros, with whom I conversed freely on the most varied topics [ 10] . "Tupamaros" was a movement of "urban guerillas," a tac­ tic rejected by the Communists. A paper-factmy worker, a Communist, was with me for years and we became great friends; he was very intelli­ gent and restless, we talked on the most diverse themes. I could give him little courses on physics, chemistry, etc . , which h e absorbed with passion. In other cells there were prisoners who were young mathematicians like Markar­ ian and Accinelli, whom I saw only during the recreations and collective tasks; nmning some risks we produced some small mathematical works like one entitled 'Is it true that two plus two is always four?' , which might interest and intrigue the non-mathematician prisoners. During all this, Marta, my wife, had also been im­ prisoned, she was tortured and interned in a women's jail, in what was formerly a monastery. She was there for 3 years until some time during the year 1979; she was able to recover our apartment, which had been oc­ cupied and sacked by the military [ 1 0] . We can a d d to Massera's own memoir a note called "Re­ cuerdos" ( Memories . . . ), part of an article written by Elvio Accinelli and Roberto Makarian, mathematicians who shared with Massera more than 3 years of imprisonment [2] . Written in secret, with tiny handwriting, manuscripts were carried from cell to cell by the prison inmate who deliv­ ered bread or tools, 1 who risked with this audacity being punished and sent to the 'Isla'2 [isolation] . Those little pa­ pers circulated in open defiance and Massera wrote about dialectic, logic and mathematics, making real our affir­ mation that science and culture cannot be destroyed. At that time and place, to think was entirely prohibited. To demonstrate by some means what was thought, was an act of defiance and bravery, beyond the intrinsic value that the written or demonstrated material might have. And in those conditions we conquered a space to think and discuss . . . . It was as if, despite everything, in the interstices of repression one lived freely. One clay , a book on Hilbert's Problems, which had been sent and dedicated to Massera by Lipman Bers (then President of the American Mathematical Society) came to be seen by some prison inmates. Who knows how much pressure on the part of many international organizations allowed Massera to receive such a mag-

------· - ------- -

1 Each

day, for some hours, each prisoner could have tools in his cell, that were delivered by an inmate, with the objective of making handicrafts. The sale of these,

during the vexing and frustrating work toward release, helped with family sustenance. 2The "Isla" (island) was a place of punishment where the prisoner stayed 1n a state of complete isolation; the only thing within the walls was his body. At night they de­ livered a blanket and quilt. And three times a day a soldier, with whom he was prohibited communication, delivered a meager food ration. It was the coldest, least hos­ pitable place in the prison.

© 2008 Springer Science + Business Media, Inc., Volume 30, Number 1, 2008

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nificent present in his celJ.3 And who knows how much compassion there was in the prison guard that permit­ ted that such a book be found in the prison wing. To be a Communist was dangerous. And in that place it was also dangerous to be a mathematician. . . . Life and mathematics kept those of us who were the pro­ tagonists of this story united. Massera was for us a teacher, beyond the strictly scientific arena. And a friend with whom we shared with pride many joyous moments as well as other kinds. Today we scarcely write or speak of these things. Nevertheless we remember those years, that some con­ sider 'empty,' often with a smile on our faces. But we forget neither the pain nor the learning of life that we experienced. It is strange to say, but those years of im­ prisonment and isolation had positive aspects for us. We wouldn't be who we are, including professionally, with­ out the 'stain' of that period. An international campaign of protest on Massera's behalf was carried on for years, led by Laurent Schwartz in France, by Lee Lorch and Israel Halperin in Canada, and by Lipman Bers and Chandler Davis in the United States. On March 3, 1 984, Massera was set free. "For months, my house was in­ vaded by hundreds of friends who came to greet me. "

My Thoughts Are Free Let's end with an uplifting story. In his youth Paul Turan was imprisoned in a Fascist labor camp. The Hungarian la­ bor forces into which Turan was drafted were formed to support the Army's operations. These forces consisted of people considered untrustworthy to be given arms in the regular army-Communists, Gypsies, and Jews. The young men who made up these forces were unarmed, and they served under regular army officers. They were ordered to clear railway lines and to build staging areas close to com­ bat zones. When war came, without guns they were help­ less when attacked. They perished in great numbers. For Turan, it was a time of great pain, but he continued to work, using mathematical problem-solving as his refuge. In September 1940 I was called for the first time to serve in a labor camp. We were taken to Transylvania to work on building railways. Our main work was carrying railroad ties. It was not very difficult work, but any spectator would have recognized that most of us did it rather awkwardly. I was no exception. Once one of my more expert com­ rades said so explicitly, even mentioning my name. An of­ ficer was standing nearby, watching us work. When he heard my name, he asked the comrade whether I was a mathematician. It turned out that the officer, Joseph Win­ kler, was an engineer. In his youth he had placed in a mathematical competition; in civilian life he was a proof­ reader at the print shop where the periodical of the Third

Class ofthe Academy (Mathematical and Natural Sciences) was printed. There he had seen some of my manuscripts. All he could do for me was to assign me to a wood­ yard where big logs for railroad building were stored and sorted by thickness. My task was to show incoming groups where to find logs of a desired size. This was not so bad.

I was walking outside all day long, in the nice scenery and the unpolluted air. The [mathematical] problems I had worked on in August came back to my mind, but I could not use paper to check my ideas. Then the formal ex­ tremal problem occurred to me, and I immediately felt that this was the problem appropriate to my circum­ stances. I cannot properly describe my feelings during the next few days. The pleasure of dealing with a quite un­ usual type of problem, the beauty of it, the gradual ap­ proach of the solution, and finally the complete solution made these days really ecstatic. The feeling of intellectual freedom and of being, to a certain extent, spiritually free of oppression only added to this ecstasy [ 14] . After Turan's cry of ecstasy, any more words may seem anticlimactic. But we will conclude by pointing out the ad­ vantage imprisoned mathematicians have over other scien­ tists. Poncelet needed only pencil, paper, and his memories of the Polytechnique to lose himself in projective geometry. Turan , without even pencil or paper, was able to recreate his world of combinatorial identities and estimates. Accord­ ing to Vladimir Arnold, mathematics is just "the part of physics where experiments are cheap. " No need for a lab or even for a library, just your mind and its contents! REFERENCES

1 . Albers, Don (2007). "John Todd - Numerical Mathematics Pioneer," The College Mathematics Journal 38: 1 1 .

2. Accinelli, Elvio and Roberto Makarian (1 996). Recuerdos (Memo­ ries). In lntegrando (Integrating), Centro de Estudiantes de lnge­ nieria (Center of Engineering Students). 3. Bell, E. T. (1 937). Men of Mathematics, New York: Simon & Schuster. 4. Bollobas, Bela, {ed.), Littlewood's Miscellany, Cambridge University Press, 1 986.

5. Choi, Man-Duen and Peter Rosenthal (1 994). "A Survey of Chan­ dler Davis," Linear Algebra and Its Applications, 208/209:3-1 8. 6. Dyson, F (1 988). Pp. 7-28 of Ramanujan Revisited, George E. An­ drews, et at. , editors, Boston: Academic Press. 7. Helly, Eduard and Tibor Rad6. MacTutor On-line Mathematics Bi­ ographies. 8. Jean Leray. MacTutor On-line Mathematics Biographies. 9. Massera, J. L. and J. J. Schaffer. (1 966). Linear Differential Equa­ tions and Function Spaces, New York: Academic Press.

1 0. Massera, J. L. (1 998). "Recuerdos de mi vida academica y polftica (Memories of my Academic and Political Life)." Lecture delivered at the National Anthropology Museum of Mexico City, March 6, 1 998, and published in "Jose Luis Massera. The scientist and the man. Mexican Prize of Science and Technology" Ed: Faculty of Engi­

neering, Montevideo, Uruguay. 1 1 . Reitberger, H. (2002). "Leopold Vietoris (1 89 1 -2002), " Notices of the AMS, 49/1 0 (November): 1 232.

1 2. Roberts, Siobhan, The King of Infinite Space: Donald Coexter, the man who saved geometry, New York, Walker, 2006.

1 3. Rota, G. C. (1 990). "The Lost Cafe," in Indiscrete Thoughts, Boston: Birkhauser. 1 4. Turan, P. (1 997). "Note of Welcome," Journal of Graph Theory 1 /1 .

1 5. Weil, A. (1 992). The Apprenticeship of a Mathematician, Boston: Birkhauser.

3The entry of books written in languages besides Spanish was always prohibited. And in the epoch of which we write, every mathematics book was prohibited from entering: they might have codes that the censor couldn't interpret.

26

THE MATHEMATICAL INTELLIGENCER

@fi?W·J.[.M

D av i d E . R ow e , E d i t o r

Re membering an Era: Roger P e nrose's Paper on "Gravitational C o l lapse : The Ro l e of G eneral Re lativity"

l

EDITOR'S NOTE: Back in the 1 960s, Einstein's theory of general relativity re-emerged as a field of important research activity. Much of the impetus behind this resurgence came from powerful new mathematical ideas that Roger Penrose and Stephen Hawking applied to prove general sin­ gularity theorems for global space-time structures. Their results stirred the imag­ inations of astrophysicists and gave rel­ ativistic cosmology an entirely new re­ search agenda. A decade later, black holes and the big-bang model were on the tongues of nearly everyone who fol­ lowed recent trends in science. As pop­ ular expositions dealing with quasars, pulsars, and the geometry of black holes began to appear in magazines and text­ books, Stephen Hawking reached a wide audience in 1988 with a lucid little book called A Brief History C!f Time. Twenty years later, it has emerged as one of the greatest scientific best-sellers of all time with some 10 million copies in print. Roger Penrose's talents as an ex­ positor became widely known with the appearance of The Emperor�\· New Mind in 1 9H9 . More recently, he displayed the vast breadth of his interests in mathe­ matics and physics in The Road to Re­ alizv ( 2004), a tour de force that bears the distinctive style of writing that Pen­ rose has made all his own. Having long recognized that our contemporary sci­ entific culture is not necessarily con­ ducive to the pursuit of creative ideas when these ideas run counter to main­ stream trends, Penrose makes a per­ suasive case for openness in scientific discourse. His friend Michael Atiyah has

given an apt description of how Pen­ rose's intense pursuit of truth has made him a leading defender of unorthodox research agendas: These days most physicists follow the latest band-wagon, usually within microseconds. Roger steers his own path and eschews band-wagons. He may not always be right, but it is im­ portant that we have individuals who stick to their guns. Future progress with ideas, as in evolu­ tionary genetics, depends on a suf­ ficient stock so that some good ones will survive and prosper. Roger is one of those who are helping to di­ versify our "gene pool" of ideas. While Penrose continues to forge ahead, we take this opportunity to re­ call where he was in 1 969, the year that saw the publication of his expository paper on "Gravitational Collapse: The Role of General Relativity" in a special issue of Revista de Nuovo Cimento. Those familiar with his more recent popular writings will surely recognize in the excerpt reprinted here that fa­ miliar Penrose style that captures the excitement surrounding the new sin­ gularity theorems in the 1 960s. In fact, this paper has since come to be re­ garded as a classic account, having been reprinted as a "Golden Oldie" in General Relativi�V and Gravitation 34, 2002. The present excerpt, republished with the kind permission of the author, omits a few of the more technical pas­ sages in the original article. But the di­ agrams, which always enliven Pen­ rose's writings, have been reprinted in their entirety. D.E.R.

G ravitational Collapse: The Role of General Relativity by Roger Penrose I shall begin with what I think we may now call the "classical" collapse picture as presented by general relativSend submissions to David E. Rowe, Fachbereich 17-Mathematik,

ity. Objections and modifications to this picture will be considered afterward. The main discussion is based on

Roger Penrose is Emeritus Rouse Ball Professor of Mathematics at Oxford University. He has received many

Johannes Gutenberg University,

prizes and awards, including the 1 988 Wolf Prize in physics, which he shared with Stephen Hawking for their

055099 Mainz, Germany.

joint contribution to our understanding of the universe.

© 2008 Springer Sc1ence+Bus1ness Media, Inc., Volume 30, Number 1 , 2008

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Schwarzschild's solution of the Einstein vacuum equations. This solution repre­ sents the gravitational field exterior to a spherically symmetrical body. In the original Schwarzschild coordinates, the metric takes the familiar form

t

time

ds2 = ( 1 - 2 m/r)dt2 - ( 1 - 2 mlr) - 1 dr2 - r2 (d82 + sin2 8dcp2). ( 1 ) Here e and cp are the usual spherical polar angular coordinates. The radial coordinate r has been chosen so that each sphere r = const, t = const has in­ trinsic surface area 47Tr2 . The choice of time coordinate t is such that the met­ ric form is invariant under t � t+const and also under t � - t. The static na­ ture of the space-time is thus made manifest in the formal expression for the metric. The quantity m is the mass of the body, where "general-relativistic units" are chosen, so that

..... "' c 0 u

II

1..

vacuum

c= G= 1 that is to say, we translate our units ac­ cording to

1 s = 3 · 1 0 1 0 em = 4 lO.'lH g. ·

When r = 2 m, the metric form ( 1 ) breaks down. The radius r = 2 m is re­ ferred to as the Schwarzschild radius of the body. Let us imagine a situation in which the collapse of a spherically symmetri­ cal (nonrotating) star takes place and continues until the surface of the star approaches the Schwarzschild radius. So long as the star remains spherically symmetrical, its external field remains that given by the Schwarzschild metric ( 1 ) . The situation is depicted in Figure 1 . Now the particles at the surface of the star must describe timelike lines. Thus , from the way that the "angle" of the light cones appears to be narrow­ ing down near r = 2 m, it would seem that the surface of the star can never cross to within the r = 2m region. However, this is misleading. For sup­ pose an observer were to follow the surface of the star in a rocket ship, down to r = 2 m. He would find (as­ suming that the collapse does not dif­ fer significantly from free fall) that the total proper time that he would expe­ rience as elapsing, as he finds his way down to r = 2 m, is in fact finite. This is despite the fact that the world line he follows has the appearance of an "infinite" line in Figure 1 . But what does

28

THE MATHEMATICAL INTELLIGENCER

Figure I. Spherically symmetrical collapse in the usual Schwarzschild co-ordinates.

the observer experience after this finite proper time has elapsed? Two possi­ bilities which suggest themselves are: (i) the observer encounters some form of space-time singularity-such as in­ finite tidal forces-which inevitably de­ stroy him as he approaches r 2 m; (ii) the observer enters some region of space-time not covered by the (t, r, O, cp) coordinate system used in ( 1 ) . (It would be unreasonable to suppose that the observer's experiences could simply cease after some finite time, without his encountering some form of violent agency.) In the present situation, in fact, it is possibility ( ii) which occurs. The easi­ est way to see this is to replace the coordinate t by an advanced time pa­ rameter v, given by =

v = t + r + 2m log( r - 2 m) , whereby the metric ( 1 ) i s transformed to the form (Eddington [1], Finkelstein

[2]) ds2

=

(1 - 2 m/ r)dv2 - 2drdv - r2 (dOl+sin28dcp2).

(2)

This form of metric has the advantage that it does not become inapplicable at r = 2 m. The whole range 0 < r < oo is encompassed in a nonsingular fashion by (2). The part r > 2 m agrees with the part r > 2 m of the original expression ( 1 ). But now, the region has been ex­ tended inwards in a perfectly regular way across r = 2 m and right down to­ wards r = 0. The situation is as depicted in Fig­ ure 2. The light cones tip over more

and more as we approach the centre. In a sense, we can say that the gravi­ tational field has become so strong, within r = 2 m, that even light cannot escape, and is dragged inwards towards the centre. The observer on the rocket ship, whom we considered above, crosses freely from the r > 2m region into the 0 < r < 2 m region. He en­ counters r = 2m at a perfectly finite time, according to his own local dGck, and he experiences nothing special at that point. The space-time there is lo­ cally Minkowskian, just as it is every­ where else ( r > 0). Let us consider another observer, however, who is situated far from the star. As we trace the light rays from his eye, back into the past towards the star, we find that they cannot cross into the r < 2m region after the star has col­ lapsed through. They can only intersect the star at a time before the star's sur­ face crosses r 2 m. No matter how long the external observer waits, he can always (in principle) still see the sur­ face of the star as it was just before it plunged through the Schwarzschild ra­ dius. In practice, however, he would soon see nothing of the star's surface­ only a "black hole"-since the ob­ served intensity would die off expo­ nentially owing to an infinite red shift. But what will he the fate of our orig­ inal observer on the rocket ship? After crossing the Schwarzschild radius, he finds that he is compelled to enter re­ gions of smaller and smaller r. This is clear from the way the light cones tip over towards r = 0 in Figure 2, since the observer's world line must always remain a timelike line . As r decreases, the space-time curvature mounts ( in proportion to r - 5), becoming theoreti­ cally infinite at r = 0. The physical ef­ fect of space-time curvature is experi­ enced as a tidalforce: objects become squashed in one direction and stretched in another. As this tidal effect mounts to infinity, our observer must eventually1 he torn to pieces-indeed, the very atoms of which he is com­ posed must ultimately individually share this same fate!

t

time

J

'

� I

=

Figure 2.

lapse

Spherically symmetrical col­ co-ordi­

in Edd ington-Finkelstein

nates.

Thus, the true space-time singular­ i(y, resulting from a spherically sym­ metrical collapse, is located not at r = 2 m, hut at r = 0. Although the hyper­ surface r 2m has, in the past, itself =

been frequently referred to as the "Schwarzschild singularity", this is re­ ally a misleading terminology since r = 2 m is a singularity merely of the t co­ ordinate used in ( 1 ) and not of the space-time geometry. More appropri­ ate is the term "event horizon", since r = 2m represents the absolute bound­ ary of the set of all events which can he observed in principle by an exter­ nal inertial observer. The term "event horizon" is used also in cosmology for

essentially the same concept (cf. Rindler [3]) . In the present case, the horizon is less observer-dependent than in the cosmological situations, so I shall tend to refer to the hypersurface r = 2 m as the absolute event horizon2 of the space-time ( 2) . This, then, i s the standard spheri­ cally symmetrical collapse picture pre­ sented by general relativity. But do we have good reason to trust this picture? Need we believe that it necessarily ac­ cords, even in its essentials, with phys­ ical reality? Let me consider a number of possible objections : (a) densities in excess of nuclear den­ sities inside, (b) exact vacuum assumed outside, (c ) zero net charge and zero magnetic field assumed, (d) rotation excluded, (e) asymmetries excluded, (f) possible A-term not allowed for, ( g ) quantum effects not considered, (h) general relativity a largely untested theory, (i) no apparent tie-up with observa­ tions. As regards (a), it is true that for a body whose mass is of the order of Mo, its surface would cross r = 2 m only af­ ter nuclear densities had been some­ what exceeded. It may be argued, then, that too little is understood about the nature of matter at such densities for us to be at all sure how the star would behave while still outside r = 2m. But this is not really a significant consider­ ation for our general discussion. It could be of relevance only for the least massive collapsing bodies, if at all. For, the larger the mass involved, the smaller would be the density at which it would be expected to cross r = 2 m. It could be that very large masses in­ deed may become involved in gravita­ tional collapse. For m > 1 0 1 1 M0 (e.g., a good-sized galaxy), the averaged density at which r = 2 m is crossed would he less than that of air! The objections (b) , ( c) , (d), ( e), and, to some extent, ( f ) can all he partially handled if we extract, from Figure 2, only that essential qualitative piece of

1 1n fact, if m is of the order of a few solar masses, the tidal forces would already be easily large enough to kill a man in free fall, even at r = 2m. But for m > 1 08 Mo the tidal effect at r = 2m would be no greater than the tidal effect on a freely falling body near the Earth's surface. 2 1n a general space-time with a well-defined external future infinity, the absolute event horizon would be defined as the boundary of the union of all timelike curves which escape to this external future infinity.

© 2008 Springer Sc1ence+ Business Media, Inc., Volume 30, Number i, 2008

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information which characterises the so­ lution (2) as describing a collapse which has passed a "point of no re­ turn" . I shall consider this in more de­ tail shortly. The upshot will be that if a collapse situation develops in which deviations from (2) near r = 2m at one time are not too great, then two con­ sequences are to be inferred as to the subsequent behaviour. In the first in­ stance, an absolute event horizon will arise. Anything which finds itself inside this event horizon will not be able to send signals to the outside worlds. Thus, in this respect at least, the qual­ itative nature of the " r = 2 m" hyper­ surface in (2) will remain. Similarly, an analogue of the physical singularity at r = 0 in (2) will still develop in these more general situations. That is to say, we know from rigorous theorems in general-relativity theory that there must be some space-time singularity result­ ing inside the collapse region. How­ ever, we do not know anything about the detailed nature of this singularity. There is no reason to believe that it re­ sembles the r = 0 singularity of the Schwarzschild solution very closely. In regard to (c), (d), and (f) we can actually go further in that exact solu­ tions are known which generalize the metric (2) to include angular momen­ tum (Kerr [5]) and, in addition, charge and magnetic moment (Newman, et. a!. [6]), where a cosmological constant may also be incorporated (Carter [7]). These solutions appear to be somewhat special in that, for example, the gravi­ tational quadrupole moment is fixed in terms of the angular momentum and the mass, while the magnetic-dipole moment is fixed in terms of the angu­ lar momentum, charge, and mass. However, there are some reasons for believing that these solutions may ac­ tually represent the general exterior asymptotic limit resulting from the type of collapse we are considering. Any ex­ tra gravitational multipole moments of quadrupole type, or higher, can be ra­ diated away by gravitational radiation; similarly, extra electromagnetic multi­ pole moments of dipole type can be radiated away by electromagnetic radi­ ation. (I shall discuss this a little more later.) If this supposition is correct, then (e) will to some extent also be covered by an analysis of these exact solutions. Furthermore, (b) would, in effect, be

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THE MATHEMATICAL INTELLIGENCER

covered as well, provided we assume that all matter (with the exception of electromagnetic field-if we count that as "matter") in the neighbourhood of the "black hole", eventually falls into the hole. These exact solutions (for small enough angular momentum, charge, and cosmological constant) have absolute event horizons similar to the r = 2 m horizon in (2). They also possess space-time curvature singular­ ities, although of a rather different structure from r = 0 in (2). However, we would not expect the detailed struc­ ture of these singularities to have rele­ vance for a generically perturbed solu­ tion in any case. It should be emphasized that the above discussion is concerned only with collapse situations which do not differ too much initially from the spherically symmetrical case we originally consid­ ered. It is not known whether a gravi­ tational collapse of a qualitatively di:f ferent character might not be possible according to general relativity. Also, even if an absolute event horizon does arise, there is the question of the "sta­ bility" of the horizon. An "unstable" horizon might be envisaged, which it­ self might develop into a curvature sin­ gularity. These, again, are questions I shall have to return to later. As for the possible relevance of grav­ itational quantum effects, as suggested in (g), this depends, as far as I can see, on the existence of regions of space-time where there are extraordi­ nary local conditions. If we assume the existence of an absolute event horizon along which curvatures and densities remain small, then it is very hard to be­ lieve that a classical discussion of the situation is not amply adequate. It may well be that quantum phenomena have a dominating influence on the physics of the deep interior regions. But what­ ever effects this might have, they would surely not be observable from the out­ side. We see from Figure 2 that such ef­ fects would have to propagate outwards in space/ike directions over "classical" re­ gions of space-time. However, we must again bear in mind that these remarks might not apply in some qualitatively different type of collapse situation. We now come to (h), namely the question of the validity of general rel­ ativity in general, and its application to this type of problem in particular. The

inadequacy of the observational data has long been a frustration to theorists, but it may be that the situation will change somewhat in the future. There are several very relevant experiments now being performed, or about to be performed. In addition, since it has be­ come increasingly apparent that "strong" gravitational fields probably play an important role in some astro­ physical phenomena, there appears to be a whole new potential testing­ ground for the theory. Among the recently performed ex­ periments, designed to test general rel­ ativity, one of the most noteworthy has been that of Dicke and Goldenberg [8], concerning the solar oblateness. Al­ though the results have seemed to tell against the pure Einstein theory, the in­ terpretations are not really clear-cut and the matter is still somewhat controver­ sial. I do not wish to take sides on this issue. Probably one must wait for fur­ ther observations before the matter can be settled. However, whatever the final outcome, the oblateness experiment had, for me, the importance of forcing me to examine once more the founda­ tions of Einstein's theory, and to ask what parts of the theory are likely to be ''here to stay" and what parts are most susceptible to possible modification. Since I feel that the "here to stay" parts include those which were most revolu­ tionary when the theory was first put forward, I feel that it may be worthwhile, in a moment, just to run over the rea­ soning as I see it. The parts of the the­ ory I am referring to are, in fact, the geometrical interpretation of gravity, the curvature of space-time geometry, and general-relativistic causality. These, rather than any particular field equa­ tions, are the aspects of the theory which give rise to what perhaps appears most immediately strange in the collapse phe­ nomenon. They also provide the phys­ ical basis for the major part of the sub­ sequent mathematical discussion . . . . So I want to admit the possibility that Einstein's field equations may be wrong, but not (that is, in the macro­ scopic realm, and where curvatures or densities are not fantastically large) that the general pseudo-Riemannian geo­ metric framework may be wrong. Then the mathematical discussion of the col­ lapse phenomenon can at least be ap­ plied. It is interesting that the general

mathematical discussion of collapse ac­ be worthwhile to present the "generic" This convergence is taken in the sense tually uses very little of the details of general-relativistic collapse picture as I that the local surface area of cross-sec­ Einstein's equations. All that is needed see it, not only as regards the known tion decreases, in the neighbourhood of is a certain inequality related to posi­ theorems, but also in relation to some each point T, as we proceed into the tive-definiteness of energy. In fact, the of the more speculative and conjectural future. (These null geodesics generate, near T, the boundary of the set of adoption of the Brans-Dicke theory in aspects of the situation. To begin with, let us consider what points lying causally to the future of place of Einstein's would make virtu­ ally no qualitative difference to the col­ the general theorems do tell us. In or­ the set T ) Such a T is called a trapped der to characterize the situation of col­ suiface. lapse discussion. We may ask whether any connec­ The final listed objection to the col­ lapse "past a point of no return'', I shall lapse picture is (h) , namely the appar­ first need the concept of a trapped sur­ tion is to be expected between the ex­ ent lack of any tie-up with observed face. Let us return to Figure 2 . We ask istence of a trapped surface and the astronomical phenomena. Of course it what qualitative peculiarity of the re­ presence of a physical space-time sin­ could be argued that the prediction of gion r < 2 m ( after the star has col­ gularity such as that occurring at r 0 the "black hole'' picture is simply that lapsed through ) is present. Can such in ( 2 ) . The answer supplied by some we will not see anything-and this is peculiarities be related to the fact that general theorems (Penrose [4], [9], precisely consistent with observations, everything appears to he forced in­ Hawking and Penrose [10]) is, in effect, since no "black holes'' have been ob­ wards in the direction of the centre? It that the presence of a trapped surface served! But the real argument is really should be stressed again that apart always does imply the presence of the other way around. Quasars are ob­ from r = 0, the space-time at any in­ some form of space-time singularity. There are similar theorems that can served . And they apparently have such dividual point inside r = 2 m is perfectly large masses and such small sizes that regular, being as "locally Minkowskian" also be applied in cosmological situa­ it would seem that gravitational col­ as any other point ( outside r = 0 ) . So tions . . . . The main significance of theorems lapse ought to have taken over. But the peculiarities of the 0 < r < 2m re­ quasars are also long-lived objects. The gion must he of a partially "global" na­ such as the above is that they show light they emit does not remotely re­ ture. Now consider any point T in the that the presence of space-time singu­ semble the exponential cut-off in in­ ( u, r)-plane of Figure 2 ( r < 2 m ) . Such larities in exact models is not just a fea­ tensity, with approach to infinite red a point actually represents a spherical ture of their high symmetry, hut can he shift, that might be inferred from the 2-sw:face in space-time, this being expected also in generically perturbed spherically symmetrical discussion. traced out as the e, 'P coordinates vary. models. This is not to say that all gen­ eral-relativistic curved space-times are This has led a number of astrophysi­ The surface area of this sphere is 47Tr2 cists to question the validity of Ein­ We imagine a flash of light emitted si­ singular-far from it. There are many stein's theory, at least in its applicabil­ multaneously over this spherical sur­ exact models known which are com­ face T For an ordinary spacelike 2- plete and free from singularity. But ity to these situations. My personal view is that while it is sphere in flat space-time, this would those which resemble the standard certainly possible ( as I have mentioned result in an ingoing nash imploding to­ Friedmann models or the Schwarz­ model sufficiently earlier) that Einstein's equations may he wards the centre ( surface area de­ schild-collapse wrong, I feel it would he very prema­ creasing) together with an outgoing closely must be expected to be singu­ ture indeed to dismiss these equations f1ash exploding outwards ( surface area lar (A :S 0). The hope had often been just on the basis of the quasar obser­ increasing) . However, with the surface expressed (cf. Lindquist and Wheeler vations. For, the theoretical analysis of 1; while we still have an ingoing f1ash [ 1 2] , Lifshitz and Khalatnikov [13]) that collapse, according to Einstein's theory, with decreasing surface area as before, the actual space-time singularity oc­ is still more or less in its infancy. We the "outgoing'' t1ash, on the other hand, curring in a collapsing space-time just do not know with much certainty is in effect also falling inwards (though model might have been a consequence what the consequences of the theory not as rapidly), and its surface area also more of the fact that the matter was all are. It would be a mistake to fasten at­ decreases. The surface T ( u = const, hurtling simultaneously towards one tention just on those aspects of general­ r = const < 2 m) of metric ( 2 ) ) serves central point, than of some intrinsic fea­ relativistic collapse which are known as the prototype of a trapped surface. ture of general-relativistic space-time and to assume that this gives us es­ I f we perturb the metric ( 2) slightly, in models. When perturbations are intro­ sentially the complete picture. ( It is per­ the neighbourhood of an initial duced into the collapse, so the argu­ haps noteworthy that many general-rel­ hypresurface, then we would still ex­ ment could go, the particles coming ativity theorists have a tendency, pect to get a surface T with the fol­ from different directions might "miss" each other, so that an effective themselves, to be a hit on the scepti­ lowing property: T is a spacelike closed" 2-surface "bounce" might ensue. Thus, for ex­ cal side regarding the ·'classical'' col­ such that the null geodesics which ample, one might envisage an "oscil­ lapse picture ! ) Since it seems to me that meet it orthogonally all converge ini­ lating" universe which on a large scale there are a number of intriguing, largely tially at T resembles the cydoidal singular beunexplored possibilities, T feel it may =

------- ------- ----- -------

------ ---- -------

---- ---

-----

3By a "closed" surface, hypersurtace, or curve, I mean one that is "compact without boundary".

© 2008 Springer Science+ Bus1ness Media, Inc., Volume 30, Number 1 , 2008

31

haviour of an "oscillating" spatially closed Friedmann model; but the de­ tailed behaviour, although perhaps in­ volving enormous densities while at maximal contraction, might, by virtue of complicated asymmetries, contrive to avoid actual space-time singularities. However, the theorems seem to have ruled out a singularity-free "bounce" of this kind. But the theorems do not say that the singularities need resemble those of the Friedmann or Schwarzschild so­ lutions at all closely. There is some evi­ dence (cf. Misner [14], for example) that the "generic" singularities may be very elaborate and possess a qualitative struc­ ture very different from that of their smoothed-out counterparts. Very little is known about this, however. . . . Most of the theorems (but not all, cf. Hawking [1 1 ]) require, as an addi­ tional assumption, the nonexistence of closed, timelike curves. This is a very reasonable requirement, since a space­ time which possesses closed timelike curves would allow an observer to travel into his own past. This would lead to very serious interpretative diffi­ culties! Even if it could be argued, say, that the accelerations involved might be such as to make the trip impossible in "practice" (cf. Gdel [ 1 5]), equally seri­ ous difficulties would arise for the ob­ server if he merely reflected some light signals into his own past! In addition, closed timelike curves can lead to un­ reasonable consistency conditions on the solutions of hyperbolic differential equations. In any case, it seems un­ likely that closed timelike curves can substitute for a space-time singularity, except in special unstable models . . . . Finally, it should be remarked that none of the theorems directly estab­ lishes the existence of regions of ap­ proaching infinite curvature. Instead, all one obtains is that the space-time is not geodesically complete (in timelike or null directions) and furthermore, cannot be extended to a geodesically complete space-time. ("Geodesically complete" means that geodesics can be extended indefinitely to arbitrarily large values of their length or affine para­ meter-so that inertially moving parti­ cles or photons do not just "fall off the edge" of the space-time.) The most "reasonable" explanation for why the space-time is not extendible to a com­ plete space-time seems to be (and I

32

THE MATHEMATICAL INTELLIGENCER

would myself believe this to be the most likely, in general) that the space-time is confronted with, in some sense, infinite curvature at its bound­ ary. But the theorems do not quite say this. Other types of space-time singu­ larity are possible, and theorems of a somewhat different nature would be re­ quired to decide which is the most likely type of singularity to occur. We must now ask the question whether the theorems are actually likely to be relevant in the case of a collapsing star or superstar. Do we, in fact, have any reason to believe that trapped surfaces can ever arise in grav­ itational collapse? I think a very strong case can be made that at least some­ times a trapped surface must arise. I would not expect trapped surfaces nec­ essarily always to arise in a collapse. It might depend on the details of the sit­ uation. But if we can establish that there can be nothing in principle against a trapped surface arising-even if in some very contrived and out­ landish situation-then we must surely accept that trapped surfaces must at least occasionally arise in real collapse. Rather than use the trapped-surface condition, however, it will actually be somewhat easier to use the alternative condition of the existence of a point

whose light cone starts "converging again. " From the point of view of the general theorems, it really makes no es­ sential difference which of the two con­ ditions is used. Space-time singularities are to be expected in either case. Since we are here interested in a collapse sit­ uation rather than in the "big bang," we shall be concerned with the future light cone C of some point p. What we have to show is that it is possible in princi­ ple for enough matter to cross to within C, so that the divergence of the null geodesics which generate C changes sign somewhere to the future of p. Once these null geodesics start to con­ verge, then the "weak energy condi­ tion" will take over, with the implica­ tion that an absolute event horizon must develop (outside C) . As a conse­ quence of the stronger "energy condi­ tion" it will also follow that space-time singularities will occur. Since we ask only that it be possi­ ble in principle to reconverge the null rays generating C, we can resort to an (admittedly far-fetched) "Gedanken ex­ periment". Consider an elliptical galaxy containing, say, 1011 stars. Suppose, then, that we contrive to alter the mo­ tion of the stars slightly by eliminating the transverse component of their ve­ locities. The stars will then fall inwards

+

time

\

F igure 3. The future light cone of p is caused to recon­ verge by the falling stars.

towards the centre. We may arrange to steer them, if we like, so as to ensure that they all reach the vicinity of the centre at about the same time without colliding with other stars. We only need to get them into a volume of diameter about fifty times that of the solar sys­ tem, which gives us plenty of room for all the stars. The point p is now taken near the centre at about the time the stars enter this volume (Figure 3). It is easily seen from the orders of magni­ tude involved that the relativistic light deflection ( an observed effect of gen­ eral relativity) will he sufficient to cause the null rays in C to reconverge, thus achieving our purpose. Let us take it, then, that absolute event horizons can sometimes occur in a gravitational collapse. Can we say anything more detailed about the na­ ture of the resulting situation? Hopeless as this problem may appear at first sight, I think there is actually a rea­ sonable chance that it may find a large measure of solution in the not-too-dis­ tant future. This would depend on the validity of a certain result, which has been independently conjectured by a number of people. I shall refer to this as the generalized 4 Israel conjecture (abbreviated GIC). Essentially GIC would state: If an absolute event hori­ zon develops in an asymptotically flat space-time, then the solution exterior to this horizon approaches a Kerr­ Newman solution asymptotically with time . . . . The following picture then suggests itself. A body, or collection of bodies, collapses down to a size comparable to its Schwarzschild radius, after which a trapped surface can he found in the re­ gion surrounding the matter. Some way outside the trapped surface region is a surface which will ultimately he the ab­ solute event horizon. But at present, this surface is still expanding some­ what. Its exact location is a complicated affair, and it depends on how much more matter (or radiation) ultimately falls in. We assume only a finite amount falls in and that GIC is true. Then the expansion of the absolute event hori­ zon gradually slows down to stationar-

0 G

0

Figure 4.

Spatial view of spherical "black hole" (Schwarz­ schild solution) .

ity. Ultimately the field settles down to becoming a Kerr solution (in the vac­ uum case) or a Kerr-Newman solution (if a nonzero net charge is trapped in the "black hole") . . . . But suppose GIC is not true, what then? Of course, it may be that there are just a lot more possible limiting so­ lutions than that of Kerr-Newman. This would mean that much more work would have to be done to obtain the detailed picture, but it would not im­ ply any qualitative change in the set­ up. On the other hand there is the more alarming possibility that the absolute event horizon may be unstable! By this I mean that instead of settling down to become a nice smooth solution, the space-time might gradually develop larger and larger curvatures in the neighbourhood of the absolute event horizon, ultimately to become effec­ tively singular there. My personal opin­ ion is that GIC is more likely than this, but various authors have expressed the contrary view." If such instabilities are present, then this would certainly have astrophysical implications. But even if GIC is true,

the resulting "black hole" may by no means be so "dead" as has often been suggested. Let us examine the Kerr-Newman solutions, in the case m2 > a 2 + e2, in a little more detail. But before doing so, let us refer back to the Schwarzschild solution (2) . In Figure 4, I have drawn, what is in ef­ fect, a cross-section of the space-time, given by v r const. The circles rep­ resent the location of a flash of light that had been emitted at the nearby point a moment earlier. Thus, they in­ dicate the orientation of the light cones in the space-time. We note that for large r, the point lies inside the circle, which is consistent with the static na­ ture of the space-time (i.e . , one can "stay in the same place" while retain­ ing a timelike world line). On the other hand, for r < 2 m, the point lies outside the circle, indicating that all matter must be dragged inwards if it is to remain moving in a timelike direction (so, to "stay in the same place" one would have to exceed the local speed of light). Let us now consider the corresponding picture for the Kerr-Newman solutions with m2 > a2 + e2 (Figure 5). I shall -

=

41srael conjectured this result only in the stationary case, hence the qualification "generalized". In fact, Israel has expressed sentiments opposed to GIC. However, Is­ rael's theorem [1 6] , [1 7] represents an important step towards establishing of GIC, if the conjecture turns out to be true.

5Sorne recent work of Newrnan [1 8] on the charged Robinson-Trautman solutions suggests that new features indicating instabilities may arise when an electromag­ netic field is present.

© 2008 Spnnger Science +Bus1ness Media, Inc., Volume 30, Number 1 , 2008

33

·.

Figure S. Rotating "black hole" (Kerr-Newman solution with m2 > a2 + e2 ) . The inhabitants of the structures 5 and 5* are extracting rotational energy from the "black hole".

not be concerned here with the curi­ ous nature of the solution inside the absolute event horizon H, since this may not be relevant to GIC. The hori­ zon H itself is represented as a surface which is tangential to the light cones at each of its points. Some distance out­ side H is the "stationary limit" L, at which one must travel with the local light velocity in order to "stay in the same place". I want to consider the question of whether it is possible to extract energy out of a "black hole". One might imag­ ine that, since the matter that has fallen through has been lost forever, so also is its energy content irrevocably trapped. However, it is not totally clear to me that this need be the case. There are at least two methods (neither of which is very practical) which might be constructed as mechanisms for extract­ ing energy from a "black hole" . The first is due to Misner [19]. This requires, in fact, a whole galaxy of 2 N "black holes", each of mass m. We first bring

them together in pairs and allow them to spiral around one another, ultimately to swallow each other up. During the spiraling, a certain fraction K of their mass-energy content is radiated away as gravitational energy, so the mass of the resulting "black hole" is 2 m( l K). The energy of the gravitational waves is collected and the process is repeated. Owing to the scale invariance of the gravitational vacuum equations, the same fraction of the mass-energy is col­ lected in the form of gravitational waves at each stage. Finally we end up with a single "black hole" of mass 2Nm(l K)N. Now, the point is that however small K may in fact be, we can always choose N large enough so that (1 K)N is as small as we please. Thus, in principle, we can extract an arbi­ trarily large fraction of the mass-energy content of Misner's galaxy. But anyone at all familiar with the problems of detecting gravitational ra­ diation will be aware of certain diffi­ culties! Let me suggest another method, -

-

-

which actually tries to do something a little different, namely extract the "ro­ tational energy" of a "rotating black hole" (Kerr solution). Consider Figure 5 again. We imagine a civilization which has built some form of stabilized structure 5 surrounding the "black hole" . If they lower a mass slowly on a (light, inextensible, unbreakable) rope until it reaches L, they will be able to recover, at 5, the entire energy con­ tent of the mass. If the mass is released as it reaches L, then they will simply have bartered the mass for its energy content. (This is the highest-grade en­ ergy, however, namely wound-up springs!) But they can do better than this! They also build another structure 5*, which rotates, to some extent, with the "black hole". The lowering process is continued, using 5*, to beyond L. Fi­ nally the mass is dropped through H, but in such a way that its energy con­ tent, as measured from S, is negative! Thus, the inhabitants of S are able, in effect, to lower masses into the "black hole" in such a way that they obtain more than the energy content of the mass. Thus they extract some of the en­ ergy content of the "black hole" itself in the process. If we examine this in detail, however, we find that the an­ gular momentum of the "black hole" is also reduced. Thus, in a sense, we have found a way of extracting rotational energy from the "black hole". Of course, this is hardly a practical method! Certain im­ provements may be possible, e.g., us­ ing a ballistic method. 6 But the real sig­ nificance is to find out what can and what cannot be done in principle, since this may have some indirect relevance to astrophysical situations. Let me conclude by making a few highly speculative remarks. In the first place, suppose we take what might be referred to, now, as the most "conser­ vative" point of view available to us, namely that GIC is not only true, but it also represents the only type of situ­ ation that can result from a gravitational collapse. Does it follow, then, that nothing of very great astrophysical in­ terest is likely to arise out of collapse? Do we merely deduce the existence of a few additional dark "objects", which

6Calculations show that this can indeed be done. A particle p0 is thrown from S into the region between L and H, at which point the particle splits into two particles

p 1 and p2 . The particle p2 crosses H, but p1 escapes back to S possessing more mass-energy content than p0!

34

THE MATHEMATICAL INTELLIGENCER

do little else but contribute , slightly, to the overall mass density of the uni­ verse? Or might it be that such "ob­ jects", while themselves hidden from direct observation, could play some sort of catalytic role in producing ob­ servable effects on a much larger scale. The "seeding" of galaxies is one possi­ bility which springs to mind. And if "black holes" are born of violent events, might they not occasionally be ejected with high velocities when such events occur? (The one thing we can be sure about is that they would hold together!) I do not really want to make any very specific suggestions here. I only wish to make a plea for "black holes" to he taken seriously and their consequences to be explored in full detail. For who is to say, without careful study, that they cannot play some important part in the shaping of observed phenom­ ena? But need we be so cautious as this? Even if GIC, or something like it, is true, have we any right to suggest that the onlv type of collapse which can occur is one in which the space-time singu-

larities lie hidden, deep inside the pro­ tective shielding of an absolute event horizon? In this connection, it is worth examining the Kerr-Newman solutions for which m2 < a2 + e2. The situation is depicted in Figure 6. The absolute event-horizon has now completely dis­ appeared! A region of space-time sin­ gularity still exists in the vicinity of the centre, but now it is possible for in­ formation to escape from the singular­ ity to the outside world, provided it spi­ rals around sufficiently. In short, the singularity is visible, in all its naked­ ness, to the outside world! However, there is an essential dif­ ference between the logical status of the singularity marked at the centre of Figure 6 and that marked at the cen­ tres of Figures 4 and 5. In the cases of Figures 4 and 'i, there are trapped sur­ faces present, so we have a theorem which tells us that even with generic perturbation, a singularity will still ex­ ist. In the situation of Figure 6, how­ ever, we have no trapped surfaces, no known theorem guaranteeing singular­ ities, and certainly no analogue of GIC.

0 0

stationary

()

Figure 6. A "naked with m2 < a2 + e 2 ) .

singularity" ( Kerr-Newman solution

0

So it is really an open question whether a situation remotely resembling Figure 6 is ever likely to arise. We are thus presented with what is perhaps the most fundamental unan­ swered question of general-relativistic collapse theory, namely: does there ex­ ist a "cosmic censor" who forbids the appearance of naked singularities, clothing each one in an absolute event horizon? In one sense, a "cosmic cen­ sor" can be shown not to exist. For it follows from a theorem of Hawking [ 1 1 ] that the "big bang" singularity is, in principle, observable. But it is not known whether singularities observ­ able from outside will ever arise in a generic collapse which starts off from a perfectly reasonable nonsingular initial state. If in fact naked singularities do arise, then there is a whole new realm opened up for wild speculations! Let me just make a few remarks. If we en­ visage an isolated naked singularity as a source of new matter in the universe, then we do not quite have unlimited freedom in this! For although in the neighbourhood of the singularity we have no equations, we still have nor­ mal physics holding in the space-time surrounding the singularity. From the mass-energy t1ux theorem of Bondi, et. al. [20] and Sachs [21], it follows that it is not possible for more mass to be ejected from a singularity than the orig­ inal total mass of the system, unless we are allowed to be left with a singular­ ity of negative total mass. (Such a sin­ gularity would repel all other bodies, but would still be attracted by them!) While in the realm of speculation concerning matter production at singu­ larities, perhaps one further speculative remark would not be entirely out of place. This is with respect to the man­ ifest large-scale time asymmetry be­ tween matter and antimatter. It is often argued that small observed violations of T- (and C-) invariance in funda­ mental interactions can have no bear­ ing on the cosmological asymmetry problem. But it is not all clear to me that this is necessarily so. It is a space-time singularity (i.e . , presumably the "big bang") which appears to gov­ ern the production of matter in the uni­ verse. When curvatures are fantastically large-as they surely are at a singular­ ity-the local physics will be drastically

© 2008 Springer Sc1ence+ Business Media, Inc., Volume 30, Number 1, 2008

35

physics which must take place at a space-time singularity.

altered. Can one be sure that the asym­ metries of local interactions will not have the effect of being as drastically magnified? When so little is known about the geometrical nature of space-time sin­ gularities and even less about the na­ ture of physics which takes place there, it is perhaps futile to speculate in this way about them. However, ultimately a theory will have to be found to cope with the situation. The question of the quantisation of general relativity is of­ ten brought up in this connection. My own feeling is that the purpose of cor­ rectly combining quantum theory with general relativity is really somewhat different. It is simply a step in the di­ rection of discovering how nature fits together as a whole. When eventually we have a better theory of nature, then perhaps we can try our hands, again, at understanding the extraordinary

[1 1 ] S. W. Hawking, Proc. Roy. Soc., A300, 1 87 (1 967). [ 1 2] R. W. Lindquist and J. A. Wheeler, Rev. Mod. Phys., 29, 432 (1 957).

REFERENCES

[ 1 ] A. S. Eddington, Nature, 1 13, 1 92 (1 924). [2] D.

Finkelstein, Phys. Rev.,

[ 1 3] E. M. Lifshitz and I. M. Khalatnikov, Ad­ vances in Phys., 1 2, 1 85 (1 963).

1 1 0, 965

[1 4] C. W. Misner, Phys. Rev. Lett. , 22, 1 07 1

(1 958).

(1 969).

[3] W. Rindler, Roy. Astr. Soc. Month. Not., 1 1 6, 6 (1 956).

[1 5] K. Gdel, in: Albert Einstein Philosopher Scientist, edited by P. A. Schilpp (New

[4] R. Penrose, in Battelle Rencontres (Eds.

York, 1 959), p. 557 .

C. M. De Witt and J. A. Wheeler) New

[1 6] W. Israel, Phys. Rev., 1 64, 1 776 (1 967).

York, 1 968. [5] R. P. Kerr, Phys. Rev. Lett., 1 1 , 237 (1 963).

[1 7] W. Israel, Commun. Math. Phys., 8, 245 (1 968).

[6] E. T. Newman, E. Couch, K. Chinnapared,

A. Exton, A. Prakash, and R. Torrence,

[1 8] E. T. Newman, personal communication

Journ. Math. Phys., 6, 9 1 8 (1 965). [7] B. Carter, Phys. Rev. , 1 14, 1 559 (1 968).

(1 969). [1 9] C. W. Misner, personal communication (1 968).

[8] R. H. Dicke and H. M. Goldenberg, Phys.

[20] H. Bondi, M. G. J. van der Burg, A. W. K.

Rev. Lett., 1 8, 3 1 3 (1 967). [9] R. Penrose, Phys. Rev. Lett., 14, 57 (1 965).

Metzner, Proc. Roy. Soc., A269, 21 (1 962).

[1 0] S. W. Hawking and R. Penrose, Proc. Roy.

[21 ] R. K. Sachs, Proc. Roy. Soc., A270, 1 03 (1 962).

Soc., A (in press) (1 969).

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36

THE MATHEMATICAL INTELLIGENCER

Sem idynam ical Systems and H i l bert1S Fifth P roblem

J . W. NEUBERGER

�

� emidynamical system, semiflow, one-parameter semi­ group-all the same thing. In this note "semigroup" -"'-is used in place of all three names. Denote by X a Polish space ( separable, metric, com­ plete). A semigroup T on X is a function with domain [O,oo) so that for each t 2: 0, T(t) : X ---> X, and • T(O) is the identity transformation on X, • T(t) T( s) T(t + s) , t, s 2: 0 ( indicated product of T( t) and T(s) is composition). Such a semigroup T is called strongly continuous provided that =

if x E X and Rx (t)

=

T( t)x, t 2: 0, then g,. is continuous.

(1)

T is called joint�y continuous provided that if g : [O,oo)

X X---> X and g ( t,x) T( tlx, t 2: 0, x E X, then g is continuous . =

For many semigroups, strong continuity implies joint con­ tinuity. A function gx as in Equation ( 1 ) is called a trajec­ tory of T If T has domain all of R instead of just [O,oc), we refer to T as a "group."

Hilbert's Challenge In his Fifth Problem, Hilbert asked whether assumptions of differentiability that Sophus Lie made were actually a con­ sequence of Lie's algebraic and continuity assumptions. It is common to say that the celebrated results of Montgomery, Zippin, Gleason (see, for example, [9)) entirely settle the Fifth Problem. But Hilbert was asking more. Quoting from the translation in [21], he asked "how far Lie's concept of continuous groups of transformations is approachable in our investigations without the assumption of differentiabil­ ity of the functions . " Then he stated the famous part of the problem; he then proceeded, "Moreover we are thus led to the wide and interesting field of functional equations which have been heretofore investigated usually only under the assumption of differentiability."

Hilbert was asking whether assumptions on structure au­ tomatically implied differentiability, but he was asking this question not only for group manifolds, but also for more general "functional equations, " such as those under dis­ cussion here. The open-ended part of Hilbert's challenge should give inspiration for a long time. Before going further, a reader might keep in mind the following problem associated with Abel's equation: Find functions f : [O,oo) ---> R so that .f(O)

=

1 , .f(x)((y)

=

f(x + y) , x,y 2: 0.

(2)

If no continuity is specified, there is a tremendous collec­ tion of highly discontinuous solutions (think about Hamel bases), hut if f is assumed to be continuous and satisfy Equation (2), then f must be differentiable, and, in fact, there must be a number a so that

f(x)

=

exp( ax) , x 2: 0.

Here, then, the structure is not such as to render the dif­ ferentiability hypothesis redundant. Developments since the mid-1 930s for semigroups pro­ vide a case study for the general version of Hilbert's Fifth Problem. Let us start with Marshall Stone's characterization [18] of strongly continuous groups T of unitary transforma­ tions on a complex Hilbert space in terms of generators ob­ tained from T by means of differentiation at zero. The gen­ erator is always iA, where A is a closed, densely defined, self-adjoint operator, and each such transformation "gener­ ates" a unitary semigroup in the manner of Theorem 1 that follows. Stone's work was surely motivated by the Schriidinger equation of quantum mechanics. Stone's theo­ rem has only group and continuity hypotheses. He proves the needed differentiability. In [19], von Neumann studies nonlinear Hamiltonian sys­ tems on a Euclidean region !1 that give rise to representa­ tions by means of groups of linear transformations defined on Liil) . Von Neumann's paper was an inspiration for the

© 2008 Spnnger Science +Business Media, Inc., Volume 30, Number 1, 2008

37

main part of the present note, in that his Hamiltonian sys­ tem yields its own group-a nonlinear one. The Hille­ Yosida-Phillips Theorem, a version of which is Theorem 1 that follows, carries over Stone's ideas to semigroups, mo­ tivation for which may be taken from the case of the heat equation and other processes, which are essentially not re­ versible in time. Many major problems in PDE depending on time are essentially irreversible, so it is imperative to ex­ tend the analysis beyond the study of groups. This note seeks to give some perspective on such semi­ groups and also to indicate some developments. In partic­ ular, it will indicate that there is sufficient differentiability available to permit an analysis (without any additional as­ sumptions) of a jointly continuous semigroup T as in the opening paragraph of this note. Let me lead off with a slightly specialized version of the Hille-Yosida-Phillips Theorem (sometimes called the Hille-Yosida Theorem) (see, for example, [8], [1 5]). The re­ sult actually implies the general version. This version is a theorem about linear semigroups ( T( t) is a linear transfor­ mation for t 2: 0) on a Banach space X. Denote by Q(X) the collection of all linear, nonexpan­ sive, strongly continuous semigroups T on X (nonexpan­ sive here means that T(t) has norm not exceeding one for t 2: 0). Denote by G(X) the collection of all closed, densely defined linear transformations B on X to X (B closed means that ((x, Ex) : x E D(B) } is closed in X X X), such that if A 2: 0, then the resolvent (I - AB)- 1 exists, has domain all of X, and is nonexpansive. THEOREM 1 If TE Q(X) and

B

1

=

{(x,y) E X X X : y = lim - ( T(t) - J )x}, t--->0 + t

(3)

then B E G(X). Moreover, if B E G(X), then there is T E Q(X) such that t T(t)x = n---> lim00 (1 - - B) - n x, t 2: 0, x E X, n and (3) holds. Note that if all trajectories of T in the theorem are differ­ entiable, then B is everywhere defined, and is consequently bounded, thanks to the closed graph theorem. In applica-

tions to PDEs, B is generally a differential operator on an L2 space and is not defined everywhere. This theorem illustrates how algebraic and continuity conditions imply differentiability in the sense of Equation (3) . It is what I call a complete theory, in that it gives a col­ lection of semigroups Q(X) and a collection of vector fields G(X) in one-to-one correspondence via constructive rules. For a given TE Q(X) , B is defined by differentiating T at 0. Recovery of a semigroup T from a member B of Q(X) is constructively given by an exponential formula (which to a numerical analyst is inverse Euler's method). How does differentiability on a dense set of trajectories of T arise? Suppose that TE Q(X), x E X, t > 0, and y

lim

t---> 0 +

e-mail: jwn@untedu

38

THE MATHEMATICAL INTELLIGENCER

t

lt T(')x = x, x E X, 0

(4)

A Linear Semigroup May Serve as Stand-in for a Nonlinear One For orientation, let me provide a simple example of linear representation.

merical analysis, functional analysis, and superconductivrty. His teaching

Denton, TX 76203- 1 430

1 -

B.

Trajectories of T which are not differentiable are limits of differentiable trajectories and are sometimes called weak solutions of Equation (4). A principal motivation for the present strain of semi­ group theory is the identification of classes of autonomous evolution equations (such as systems of PDE describing temporal processes for which the law of evolution does not change form with time) that may be solved. For many in­ tended applications of semigroups, X is a collection of func­ tions, and B is a differential operator (maybe nonlinear) on a subset of X (often a dense subset).

worl< on PDEs relates to his worl< on semigroups, quasi-analyticrty, nu­

USA

T(·)x.

0

u'(t) = B ( u( t)), t 2: 0.

ing a central point of view for partial differential equations (see his /n­ te//igencer article, vol. 27 (2005), no. 3, 47-55)-an uphill struggle. His

Department of Mathematics

t

D(B) is dense in X. This leads to a connection with dif­ ferential equations: given any y E D(B) , the domain of B, the trajectory u starting at y for the corresponding semi­ group T satisfies the differential equation

Texas under the supervision of H. S. Wall. He has been patiently seek­

Universrty of North Texas

l

Some calculations yield that y E D(B)-the domain of Because

JOHN W. NEUBERGER did his doctoral worl< at the Universrty of

is never by the lecture method.

1

= t

Consider T to be the nonlinear semigroup on [O,::JO) with T( t)x

=

t + x, t,x ::::::: 0,

and denote by C([O,oo)) the Banach space of bounded real­ valued continuous functions on [O,oo). A corresponding lin­ ear representation S of T is given by

(S(t)f)(x)

=

f( T(t)x)

=

fU + x) , t,x ::::::: 0 , /E C([O,oo)),

a translation semigroup on C([O,oo)). This semigroup S does not fit the HYP theorem, not being strongly continuous in the specified Banach space, but it does fit Theorem 2 that follows, because S is strongly continuous in another topol­ ogy. This is an example of how choice of topology can have substantial consequences concerning density of do­ main of generators. Having this example in mind will help guide the reader through the history described in the next paragraphs. In the late 1950s, when I was a graduate student, we were studying nonautonomous versions of linear evolution equa­ tions such as Equation (4). I asked myself, "Why do things have to he linear?", and in my thesis [10] I took up nonlin­ ear Stieltjes-Volterra integral equations. It was not long un­ til I turned to nonlinear semigroups, publishing my first re­ sults with a nonlinear resolvent in [ 1 1 ] with " hypotheses tres restrictives' according to [1], p. 168. (I assumed differentia­ bility that I would have liked very much to have proved.) The next decade saw an explosion of work on nonlinear semigroups, hut complete theories were restricted to strongly continuous, nonexpansive semigroups on a convex subset of Hilbert space; see [1], [3], and [17] for summaries. (For nonlinear transformations, "nonexpansive" means Lipschitz norm not exceeding 1 . ) In these papers, generators were maximal monotone operators, nonlinear analogues to linear generators such as B in the HYP theorem. All this work can he characterized as an attempt to cre­ ate a theory of nonlinear semigroups in analogy to the es­ tablished linear theory. However, Webb in [20] produced an example of a simple-looking nonlinear semigroup on a non-Hilbert Banach space, for which the corresponding "generator" B had sparse domain, seemingly inadequate for use in recovering the semigroup. This tended to discour­ age further efforts to develop nonlinear analogues to the linear theory. In [2], a wide class of nonlinear monotone operators on certain subsets of Banach spaces are shown to give rise to nonexpansive semigroups, hut [2] fails to as­ sign a generator to each such semigroup and hence does not circumvent difficulties pointed out by Webb's example. The present development of generator-resolvent theory for nonlinear semigroups is essentially an application of lin­ ear theory, in contrast with earlier efforts to develop such a theory in analogy with linear theory. Around 1970, while reading Sophus Lie's work on con­ struction of integrating factors for ODEs, I was surprised to find that although Lie had (local) nonlinear groups in abun­ dance, his "generators" were linear. It became clear that his "generators" were those of an associated linear representa­ tion. I was led to attempt the application of Lie's eighty­ year-old ideas to nonlinear semigroups, with the aim of cir­ cumnavigating obstacles indicated by Webb's example. Having seen Lie's structure, I worked toward a complete

theory of strongly continuous nonlinear semigroups on X in terms of generators of linear representations (acting on a space of functions on X) . I worked out half of the the­ ory I wanted in [ 1 2] . I could start with a nonlinear semi­ group T on X, derive a generator for a linear representa­ tion S of T, and even recover T itself from this linear generator. However, I wasn't even close to characterizing these generators to develop a complete theory. Nothing much happened in this direction until 1992 at a semigroup conference in Curac;:ao. During a slow "problems" session at the conference, I presented my half-solution. Bob Dor­ roh started asking penetrating questions. In a year or so of collaboration, Dorroh and I arrived at the results in [4] , which I will now describe: a complete theory of jointly con­ tinuous semigroups on a Polish space.

Semigroups on a Polish Space For a Polish space X, denote by SG(X) the collection of all jointly continuous semigroups on X. Denote by CB(X) the Banach space (under sup norm) of all bounded continu­ ous real-valued functions on X. A sequence lfk)k= o in CB(X) {3-converges to f E CB(X) if the corresponding sequence of sup norms of members of the sequence is bounded and the sequence itself converges to f uniformly on compact subsets of X. Denote by LG(X) the collection of all linear transformations A on CB(X) to CB(X), enjoying the fol­ lowing four properties: • A is a derivation in the sense that if f, g E L(A) , then fg E D(A) and A( fg) f(Ag) + (Af)g. • D(A) is {3-dense in CB(X). • If A ::::::: 0 then (1 - AA)- 1 exists, is nonexpansive, and has as domain all of CB()(). A n 1 ,2 , . . . } is • If y > 0 then {(/ - -A) - : 0 $ A � ')', n n uniformly {3-equicontinuous.

=

=

THEOREM 2 Given TE SG(X), if we set

A

=

{( j,g) E CB(X)

X

CB(X) : 1

g(x) = lim - ( f( T( t)x) - f(x)) , x E X), t-->0+ t

(5)

then A E LG(X) . Conversely, for any A E LG()() there is a unique TE SG(X), such that t lim ((! - - A)- n f )(x), f( T( t)x) = n�:xJ n x E X, t :2:: 0, f E CB(X),

(6)

and (5) holds. In this situation the transformation A is called the Lie gen­ erator of T A proof is in [4] . The first part of the argument is rela­ tively routine for those familiar with [6], [7], [8] , [14] , and [1 5], but I must say a few words about the converse, in which one starts with A E LG()() and produces TE SG()() . First show that A generates a linear semigroup S on the norm closure of D(A), then use the fourth condition on A to extend S to all of CB(X). Observe that if x E X, A E [O,oo), then the transformation 71 defined by 71 ( }' ) = ( S(A)()(x) , f E CB(X)

© 2008 Springer Science + Business Media, Inc., Volume 30, Number 1 , 2008

39

is a ,8-continuous multiplicative linear transformation from CB(X) to R (its null space is thus a maximal ideal in CB(X)) , and that, happily, TJ is given by point evaluation, that is, there is Yx,A E X so that

Here is a sketch of some other possible applications. For X, a subset of a Banach space, and TE SG(X), suppose that T has a densely defined generator B in the conven­ tional sense, that is,

TJ ( /) = fCYx,A), /E C B(X).

Ex = lim

t-->O+

Then take

and check that T is indeed a jointly continuous semigroup on X whose Lie generator is A. Separability of X is needed in concluding that real-valued ,8-continuous multiplicative linear functions on X must be given by point evaluations. See [4] and [16] for details.

Given any TE SG\X), there is a second linear semigroup even more closely related to T Denote by M(X) the set of compact regular Borel measures on the Borel sets B(X) of X, and define the linear semigroup U on M(X) by

J.L( T( t) - 1 ,0} , t ?. 0, J.L E M(X) , fl E B(X).

Following [ 1 3], one can interpret U as the semigroup ad­ joint to S (see [5], for example), where

(S(t)j)(x) = j( T(t)(x) ) , x E X, t ?. 0, fE. CB(X) is again the linear representation; that is,

U(t)

=

S(t)*.

(7)

Care is needed in the choice of topology here: S is a semi­ group on CB(X) with the ,8-topology; members of the dual of CB(X), under the ,8-topology, may be represented by in­ tegration with respect to compact regular measures; using this dual space, S(t)*, t ?. 0, may be defined, and it turns out that Equation (7) holds (see [16]). The semigroup U has the property that if x E X, and 8x is the Dirac measure associated with x, then

U(t)8x

=

8T( t)X•

t ?. 0.

This property allows us to interpret U as a linear extension of the nonlinear semigroup T Namely, if one first identi­ fies X with Dirac measures on X, then U(t) agrees with T(t) acting on those points. To summarize what has been done: We start with T, we take its linear representation S, we take the adjoint semigroup U of S, and we find that U is a lin­ ear extension of T An informal recapitulation: The points x E X are first identified with corresponding Dirac measures concentrated at x. Then more measures, compact regular measures, are introduced to keep company with the Dirac measures. The resulting measure space has enough ele­ ments to provide a generator, defined by differentiation, in terms of which T can be recovered, thus circumventing the difficulties associated with Webb's example.

Possible Applications Close relationships of semigroups U and S in the previous section with Kolomogorov's forward and backward equa­ tions of stochastic differential equations can be seen (see [5], for example). This structure helped me to write a code for a fully nonlinear filter-but that is another story.

40

THE MATHEMATICAL INTELLIGENCER

(�f)(x) = j'(x)Bx, x E D(B).

(9)

Sophus Lie would recognize (9) immediately. However, members of D(A) need not be Frechet differentiable. Know­ ing that /E D(A) tells one only that 1

Related Semigroups on Spaces of Measures

=

(8)

Then the corresponding Lie generator A of T can be given, provided /E CB(X) is a C 1 function in D(A), by

T(A)x = Yx,A

( U( t)J.L)([!)

1

- ( T(t)x - x), x E. D(B). t

(Aj)(x) = lim - (j( T(t)x) - .f(x)), x E t-->0+ t

X.

Even if x E D(B), it can be determined only that (Aj)(x) is a directional derivative of f in the direction Bx- with Frechet differentiability probably not holding. So even though Equation (9) suggests a form for a Lie generator of T in this case, it is still not clear how to articulate precisely the nature of D(A). Nevertheless, I offer the following con­ jecture (in which I don't have particular confidence, but which may suggest other more reasonable speculation). CONJECTURE. Suppose X is an open subset of a Ba­ nach space Y, and B is a closed, densely defined trans­ formation on X to Y Denote by A the collection of all (j,g) E CB(X)2 so that 1

g (x) = lim+ - ( j(x + tBx) - 'f(x)), x E D(B). t-->O t If A satisfies the four conditions preceding Theorem 2, then there is a unique semigroup T on X with conventional gen­ erator B as in (8) and Lie generator A. I also do not expect Theorem 2 to be particularly easy to apply: It is going to be fairly difficult to show how given concrete examples fit the hypothesis. I am reminded of Ralph Phillips's comment to me: "It is not easy to apply semigroup theory to differential equations. " It has always been a noteworthy occasion when semigroup theory is ap­ plied to PDE in a substantial way. However, there have been a number of such occasions.

Final Comment In the summer of 2005, I greatly profited from a stay at the Max Planck Institute in Leipzig. I was asked to speak in Sophus Lie Seminarraum. I still regret that I didn't seize the chance to talk about the present topic there. REFERENCES

[1] H. Brezis, Operateurs maximaux monotones, North Holland, Am­ sterdam, New York (1 973). [2] M. Crandall and T. Liggett, "Generation of semi-groups of nonlin­ ear transformations on general Banach spaces, " Amer. J. Math. 93 (1 97 1 ), 265-298.

[3] G. da Prato, Applications croissantes et equations d'evolutions dans /es espaces de Banach, Academic Press, London, New York

1 976.

[4] J . R. Dorroh and J. W. Neuberger, "A theory of strongly continu­

[1 3] J . W. Neuberger, "A complete theory for jointly continuous non­

ous semigroups in terms of Lie generators," J. Funct. Anal. 1 36

linear semigroups on a complete separable metric space, " J. Ap­

(1 996), 1 1 4-1 26.

plicable Analy. 78 (200 1 ) , 223-231 .

[5] E. B. Dynkin, Markov Proceses-1, Springer, Grund. Math. Wiss.

[1 4] A. Pazy, "Semigroups of linear operators and applications to partial differential equations," Springer, Appl. Math. Sci. 44 New York (1 983).

1 2 1 , Berlin, Heidleberg, Gbttigen (1 965). [6] K.-J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, Springer, New York, Berlin, Heidleberg (1 999).

[7] J. Goldstein, Semigroups of linear operators and applications, Ox­ ford University Press, New York (1 994).

[1 5] F. Riesz and Sz.-Nagy, Functional analysis, Ungar (1 965), Dover, New York (1 990). [1 6] D. Sentilles, "Bounded continuous functions on a completely reg­ ular space, " Trans. Amer. Math. Soc. 1 68 (1 972), 31 1 -336.

[8] E. Hille and R. Phillips, Functional analysis and semigroups, Amer­ ican Mathematical Society, Providence (1 957). [9] D. Montgomery and L. Zippin, Topological transformation groups, lnterscience, New York, London (1 955).

[1 7] R. E. Showalter, Monotone operators in Banach spaces and non­ linear partial differential equations , Amer. Math. Soc. Math. Sur.

Monogr. 49 Providence (1 996). [1 8] M. H. Stone, "On one-parameter unitary groups in Hilbert Space,"

[1 0] J. W. Neuberger, "Continuous Products and Nonlinear Integral Equations, " Pac. J. Math. 8 (1 958), 529-549. [1 1 ] J. W. Neuberger, "An exponential formula for one-parameter semi­ groups of nonlinear transformations, " J. Math. Soc. Japan 1 8 (1 996), 1 54-1 57.

Annals Math. 33 (1 932), 643-648. [1 9] J. von Neumann, "Dynamical systems of continuous spectra," Proc. Nat. Acad. Sci. 18 (1 932), 278-286.

[20] G. F. Webb, "Representation of semigroups of nonlinear nonex­ pansive transformations in Banach spaces," J. Math. Mech. 1 9

[1 2] J . W. Neuberger, "Lie generators for one parameter semigroups of transformations, " J. Reine Ang. Math. 258 (1 973), 1 33-1 36.

(1 969/1 970), 1 59-1 70. [2 1 ] B. H. Yandell, The Honors Class, A. K. Peters, Nantick (2002).

ScientificWorkPiace d Proces�

•

�

Typesetting Computer Algebra •

The Gold Standard for Mathematical Publishing

© 2008 Springer Science+ Business Media, Inc., Volume 30, Number 1 , 2008

41

Mathemati c a l ly Bent

Co l i n Adams , Ed itor

]

WILLIAMS COLLEGE Student Course Survey Form: Instructor Modified Version COLIN c. ADAMS The proof i s i n the pudding. To the Instructor Opening a copy of The Mathematical lntelligencer you may ask yourself

uneasily, "What is this anyway-a mathematical journal, or what?" Or you may ask, " Where am !?" Or even

To aid in the evaluation of teaching, all instructors are required to provide stu­ dent course survey forms to all students who have been enrolled in a class with them in a given semester. This year, the College is experimenting with Instructor Modified forms, which allow the instructor the latitude to ask questions that may be particularly relevant to the type of course that she/he is teaching. Please make certain that students know to return all forms to the Course Survey office before the last day of final exams.

"Who am !?" This sense of disorienta­ tion is at its most acute when you open to Colin Adams's column. Relax. Breathe regularly. It's mathematical, it's a humor column, and it may even be harmless.

Williams Marking Instructions • Use a Number 2 . 63 pencil

only.

• Make a dark mark resembling a mole. • CHOOSE ONE ANSWER ONLY TO EACH QUESTION. • MAKE NO STRAY MARKS ON THIS FORM. YOU DO NOT WANT TO KNOW WHAT HAS HAPPENED TO STU DENTS WHO DID . . .

To the Student These results are used to determine the fate of each instructor. Someone who does well on this Survey is showered with Applebees gift certificates and shaving samples that come in small mailed boxes. Someone who does poorly is taken behind the Administration Build­ ing and paddywacked. As you answer these questions, please keep in mind that courses vary widely in their aims and methods. Some courses attempt to teach you information and thought processes that will help you for the rest of your life. Others are excuses to keep you off the street and to allow time for faculty to engage in research they would rather be doing. But both serve a valuable role within the College. Please give careful consideration to each question.

Column editor's address: Colin Adams, Department of Mathematics, Bronfman Science Center, Williams College, Williamstown , MA 01267 USA e-m a i l : Colin.C [email protected]

42

THE MATHEMATICAL INTELLIGENCER © 2008 Splinger Science+ Business Media. Inc.

1. My gender is: 1 . Female. 2. Male. 2. The grade I expect to receive in this course is: 1 . A + (Fat chance). 2. A. 3. B. 4. Any passing grade, just make it a pass. 5. I don't care about your stinking grades. I am here to further my knowledge, not to pad my transcript so I can get some job that makes me miserable for the rest of my life. 3. Before the course began, my interest in the course material was:

1 . Nonexistent. 2. How would I know? I didn't yet know the course ma­ terial. 3. Nothing like what it became the minute the professor started teaching. 4. Compared to other courses I have taken at Williams, the amount of work expected of me was:

1 . Just right. 2. Absolutely perfect. 3. Adams, er, I mean the instructor made us work a tremendous amount, and I have to tell you, I am so glad he did. I learned a dumpster's worth of material. It was truly amazing, and I don't regret one nanosec­ ond of it. 5. The effort that I put into this course was: 1 . Comparable to the sum total of all the effort put into all other courses by all of the other students on campus. 2. There wasn't any effort. I loved this class so much that I found myself working on it day and night just for the pure pleasure of it. 6. Compared to Colin C. Adams, the instructor in this course was:

1 . Comparable. 2. Much better. 3. Not nearly as good. 7. In accessibility, I would rate the instructor:

1 . Highly accessible. Almost always in his office during office hours. 2. Very highly accessible. Available in his office during office hours and at Starbucks much of the rest of the time. 3. Extremely very highly accessible. I 'm the student who came by his house at 2:00 in the morning, and, yes, he was there. 8. If we change the color of the blue comment sheets,

which of the following colors would you choose?

1 . Fermented cider 2. Sand trap

3. Alpine avalanche 4. Burnt cinder 5. Lichen 9. Was the instructor rigorous? 1. Oh, yes. He had us doing jumping jacks and pushups. I was exhausted.

2. Yes, every statement was proved in meticulous detail using Boolean logic. 3. No, I had Adams. 10. If you were having an imaginary conversation with Karl Friedrich Gauss, and he asked you about the mathematical merits of this course, you would say:

1 . Karl, my man, what Adams knows about math couldn't fit in your little pinky. 2. I cannot explain to you, even though you are a great accountant or something, the depth of Adams's math­ ematical knowledge. 3. Oh my God. I am talking to a dead man. Help me, somebody, help me! 11. Rate the instructor's board technique: 1 . That dude shreds the slopes. 2. The way he boosts air out of a halfpipe is truly sick. 3. I have seen some faculty who could ride a stick-take the whole English department, for instance-but Adams is in a class by himself. 12. My favorite part about this course was: 1 . The group hugs. 2. The field trip to Atlantic City. 3. The way Professor Adams gave us advice about how to understand the world, and turned what could be deadly dull material into life lessons. I had no idea that an integral is a metaphor for our existence on earth. Professor Adams explained that an integral is a limit of a sum. And a life is just the sum of its parts, which can't really be assessed until the life is over. So you must take a limit of the sum that is your life, as you approach death. And that is why, every time you integrate a function, you should consider what your life will be worth after you are gone. Well, perhaps, I am going on too long here for this multiple choice problem. Suffice to say that I am a changed human being. My perspective on life, the world, and the fu­ ture is absolutely and irrevocably changed for the bet­ ter, and all thanks to Professor Adams. 13. My overall rating of the quality of instruction of this course is:

1. Phenomenal, better than any other instructor I have ever had. 2. Superior to phenomenal. Better than the best instruc­ tor anyone has ever had at Williams. 3. Super-superior. Better than any instructor that has ever lived on the face of the earth from the beginning of time. 4. Kick butt incredible super-superior. Better than any in­ structor that has ever lived on this planet or any other planet, from the beginning of time through the pres­ ent and into the future until the universe ceases to exist. 5. Exceptional. Better than any instructor conceivable in the mind of an omnipotent God. 14. Would you recommend this course to a friend?: 1 . What course? 2. If I had any . . . . 3. Not only that, but I would shout it from the rooftops. "Take Colin C. Adams! Take Colin C. Adams! You will be so very glad you did!"

© 2008 Springer Science+Business Media, Inc., Volume 30, Number 1, 2008

43

[email protected]§ 6'h¥11i.illl?il ..

An Excurs i o n Arou nd the Nati onal Mal l i n Washi ngton D C, U SA JOE HAMMER

Does your hometown have any mathematical tourist attractions such as

statues, plaques, graves, the cafe

where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels? If so, we invite you to submit an essay to this column. Be sure to include 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 Mathematical Tourist Editor, Dirk Huylebrouck, Aartshertogstraat 42, 8400 Oostende, Belgium e-mail: dirk.huylebrouck@pi ng.be

44

D i rk H uylebro u c k , Editor

he Mall is the cultural center of Washington, DC, USA. Sixteen museums, art galleries, libraries, and several monuments surround this grassy parkland, which is about one mile long and 400 feet wide (Fig. 1). Here you will also find the headquar­ ters of the prestigious Smithsonian In­ stitution, with nine of its 19 museums on the Mall itself, including the Air and Space Museum. At the time of this writ­ ing, the Air and Space Museum is the most visited museum in the world. At the east end of the Mall is the National Capitol, affectionately called "the wed­ ding cake," dominating Washington on "The Hill," and the Mall in particular. Behind the Capitol building is the vast Library of Congress with the world's most comprehensive collection of printed material and other media. For mathematicians, there is a special must­ see item on permanent display­ George Washington's high school math workbook. The Mall has another distinguishing feature. It provides ample space for pic­ nicking, jogging, and games. There is even a carousel in summer and an ice skating rink in winter. Anyone who watches TV would recognize the Mall as the backdrop of pageants and demonstrations held there for a wide spectrum of local and international causes. It is fitting to note that the MALL (in Washington) is an acronym for Mu­ seum, Art gallery, Library, Leisure. The 200-year history of the Mall's tur­ bulent and controversial development is a remarkable story [1]. Each of its sur­ rounding buildings has some architec­ tural significance. In this article I ex­ plore two of the newer buildings: the East Building (EB) of the National Gallery of Art (NGA) and the Museum of the American Indian (NMAI) (Fig. 1). The architecture of each is of geomet­ ric interest.

The East Building of the National Gallery of Art The NGA comprises two buildings and a sculpture garden. The older West Building (WB) was designed in 1937 by

THE MATHEMATICAL INTELLIGENCER © 2008 Springer Science +Business Media, Inc.

I

the architect John Russell Pope, a lead­ ing architect of neoclassical buildings who was best known for the Jefferson Memorial along the Tidal Basin in DC. Like many other buildings of the period in Washington, WB is neoclassical in style, and is a rectangular complex with dimensions 780 feet X 303 feet. Its main feature is a domed, 103-foot diameter rotunda, 135 feet from ground to top, supported by 24 marble Ionic columns. The whole building is symmetric around the rotunda, in sharp contrast to the East Building. Symmetric fa�ade, asymmetric interior

Across a plaza, 1 00 yards from the WB, is the EB. This complex was designed by the modernist architect, I. M. Pei, the recipient of the Pritzker Architecture Prize. This prize is equivalent in archi­ tecture to the Nobel Prize. The EB was built about 30 years after the WB, when the WB could no longer provide ade­ quate storage or exhibition space for the new acquisitions, bequests, and dona­ tions in a burgeoning, prosperous, post­ war nation. The NGA now houses about 1 10,000 objects. Pei's brief was to design a complex for two functions: one for exhibition galleries and one as a research center for visual arts, including a library. He had to address two principal problems. First, the shape of the site is an asym­ metric trapezoid, a serious challenge for any architect. The second problem was to align the fac;:ade of the new building with that of the old. This was impera­ tive, since 100 yards is close proximity for buildings of these sizes. The main difficulty in alignment was that the ma­ jor part of the EB site lies to the south of the east-west central axis of the WB . Pei solved both problems in one mas­ terstroke. For the development plan (Fig. 2), he inscribed a new trapezoid within the site, aligning its western side with the eastern fac;:ade of WB. He then divided the new trapezoid diagonally from its northeast to its southwest cor­ ner into two triangles. He designated the larger northwest wing-an isosce-

Figure I.

les triangle-for galleries, and the southeastern wing-a right-angled sca­ lene triangle-for the research center. Next, on the base of the isosceles tri­ angle (the western side of the trapezoid) Pei designed an H-shaped symmetric fa�:ade, such that the (conceptual) ex­ tended plane of the central east-west axis of the WB passes through the sym­ metric line of the H-fa�:ade. In addition, he situated the main entrance of the new building exactly opposite the en­ trance of the WB. To provide an even

Map of the Mall in Washington, DC.

closer relationship between the two buildings, the walls of the EB, both in­ side and out, were clad in the same marble as the WB. It is remarkable that Pei devised a symmetric fa�:ade, despite the triangu­ lar, asymmetric geometry behind it. In our exploration of the complex, we will see the triangle in general, and the isosceles triangle in particular, as a re­ curring theme. Moreover, wherever there are isosceles triangles, the ratio of their two nonequal sides is 1: 1 . 5 . This

N

t·

z

-

-

-

-

- -

-·

c.

Fig.2 ABCO - the trapezoid plan AC the dMding diagonal,

(not to scale),

GHJ atrium. DE �e of EB, TZ � of WB. XY central axis of WB , TEOZ the connecting plaza area Figure 2.

Sketch of developmental plan.

is Pei's "golden ratio"! His design sys­ tem evolved from the primary isosceles triangle whose sides measured 270 feet and 405 feet. The recurring theme of self-similar triangles can be considered as elements of a fractal sequence. An associated geometric theme is the tetra­ hedron (or pyramid) that is a 3-dimen­ sional analogue of the triangle. The tetrahedron provides sculptural forms for the complex. It seems likely that Pei's inspiration for the triangular geometry stemmed from the idea of the Federal Triangle, an area of Washington that is between Pennsylvania Avenue, Constitution Av­ enue, and 1 5th Street just north of the Mall (Fig. 1). This triangle contains most of the key federal offices and govern­ ment buildings, including the White House and the Capitol. It is the "heart" of Washington. In fact, the EB is on the eastern tip of this triangle. As for the pyramid, the inspiration might have been the Washington Mon­ ument, which is at the west end of the Mall. The Monument is an obelisk­ shaped building consisting of four trapezoidal sides, topped by a 55-foot pyramid. It is the tallest building in Washington at 550 feet. No taller build­ ing is allowed in Washington. The Mon­ ument is a signpost for the city. The atrium

The atrium is an isosceles triangle­ shaped structure positioned at the junc­ tion of the two primary triangular build-

© 2008 Springer Science+ Business Media, Inc., Vdume 30, Number 1, 2008

45

ble floor tiles and the beige concrete coffers of the ceiling. This amazing 16,000 square foot space unencum­ bered by columns or any structural sup­ ports was designed using the "space frame" technology. A space frame consists of linear bars connected or hinged at their nodes arranged in a tetrahedral or other poly­ hedral frame; the minimum number of bars b needed for a rigid space frame structure of n nodes, assuming exposed exclusively to compression or tension at the nodes, is

b = 3n - 6

Figure 3.

EB skylight of atrium with Calder mobile (photo by Gael Hammer).

ings. Two sides are each 225 feet long and the third side is 1 50 feet. It is 80 feet high, capped by a skylight made up of 25 tetrahedrons of equal sizes (Fig. 3). A novel glazing of the tetrahe­ dron frame contributed much sculptural value to the skylight. The standard pro­ cedure used to be that the horizontal face only was glazed, and the three sloping faces were covered by the glazed one. In the case of the Atrium, the glazing was done the other way

Figure 4.

46

around. The sloping faces are glazed and the horizontal face left open. This procedure creates an interchanging, concave-convex, diamondlike surface. Remarkably, the marble tiles cover­ ing the floor and the concrete coffers in the ceiling of the entrance lobby are all isosceles triangles with the same "golden ratio. " We experience a won­ derful rhythmic interplay between the triangles in the crystal-like glass of the skylight, the lavender pink of the mar-

Roofs by Andy Goldsmith (photo by Gael Hammer).

THE MATHEMATICAL INTELLIGENCER

A tetrahedral frame satisfies this equa­ tion; in fact, the tetrahedron is the ba­ sic element of a space frame. A space frame consisting of convex polyhedrons is rigid if and only if all its facets are tri­ angles. Today the space frame is one of the most important devices when designing a roof with a big expanse without ob­ structing columns (for more informa­ tion, consult, e.g., [2]). The galleries

In addition to its architectural beauty, the atrium serves several functions. It is the central junction to all places within the complex, uniting and providing ac­ cess to the two wings, the exhibition galleries, and the research center. The atrium itself provides space for special exhibitions and large sculptures. Two sculptures have become partic­ ular icons of the atrium. In a glass en­ veloped gallery on its northern side, fac­ ing Pennsylvania Avenue, British sculptor Andy Goldsworthy has in­ stalled Roofs, which consists of nine stacked slate hollow domes, each 5 . 5 feet high and 2 7 feet i n diameter, with centered oculi 2 feet in diameter (Fig. 4). Sections of some domes penetrate into the atrium underneath the glass, so it appears that intruding spherical ele­ ments are "gate crashing" the all-linear structure of the atrium. The sculpture also echoes the domes characteristic of the general architecture of the city, and the Mall in particular, where there are, mystically, also nine domes. The magnificent Calder mobile sus­ pended from the skylight ceiling (Fig. 3) is about 30 feet high and 80 feet across, with 1 3 biomorphic or honey­ comb-shaped aluminum blades of sev-

era! colors and sizes, contrasting with the crystal-like tetrahedrons. The mo­ bile is continuously moving in response to the slightest air currents. It appears as a huge flying bird with outstretched colored-feather flapping wings. Much of the 1 10,000 square foot gallery space is housed in the three rhombus-shaped towers positioned at the three corners of the primary isosce­ les triangle, the focus points of the com­ plex (Fig. 5). At 1 07 feet from the ground, they are the highest points of the building. We notice that the galleries inside the towers are hexagonal and not rhomboid. The reason is simple. A rhombus has two acute angles. It was not suitable to hang paintings on walls that enclose acute angles as they are too close to each other. Pei closed off the two acute angle corners with walls, thereby obtaining hexagon-shaped ar­ eas so all the angles became obtuse. In the cut-off triangular prism-like shafts, attractive spiral staircases were built. Apparently these are the only curved structures in the entire complex! How­ ever, their steps themselves are right­ angled triangles, similar to the primary right-angled triangle that houses the re­ search center. The principal feature of the research center is the reading room of the library. Its high ceiling comprises triangular concrete coffers similar to the entrance lobby ceiling). On its roof is a triangular pyramid skylight soaring 70 feet above a beautiful set of staircases.

Figure 5.

EB fac;:ade and the pyramids (photo by Gael Hammer).

cal. The source of the water is a row of 24 fountain jets on the plaza above, showering the circular "court" of the pyramids. This cascade is arguably the liveliest scene of the complex. A bird's eye view of the EB

Because of its pivotal position, the EB is one of the most exposed sites in the Mall. The east side is exposed to the National Capitol. The southern side faces the Mall and the northern side faces the historic thoroughfare, Penn­ sylvania Avenue. We have seen how much care was taken in designing the symmetric H-fa�:ade on the west side to harmonize with the WB. All the other facades were designed to respond to the neighborhood. However, the least

visible fifth fa�:ade, the roof-top of the building, is possibly the most interest­ ing, geometrically and aesthetically. On the roof we can find the basic elements of the floorplan of the com­ plex (Fig. 2). We can see the two pri­ mary triangles with the three towers as their focus points. We also have a closer vision of the "wrong" side of the trian­ gular atrium with the diamond shaped skylight. In addition, we have a view of seven other skylights over exhibition galleries and stairwells of different shapes and sizes. Interestingly, from the roof we can see reflecting the group of seven skylight pyramids downstairs on the plaza and a vista of the mall. It is remarkable that the surface of the roof itself is on several levels, frag-

The plaza

Linking the two buildings of the NGA is a plaza displaying a cluster of seven pyramids (tetrahedrons) of colored glass. All are of differing sizes and they appear to be randomly placed on the ground. However, this effect is achieved by aesthetic hands with a sense of space and sculpture. This abstract-appearing sculpture has an important practical purpose: the pyramids provide skylights for the cafeteria and the underground concourse linking the two buildings. They resemble the skylight of the atrium. Next to the skylights is a 37 foot wide waterfall with a 13 foot fall. The water cascades on a coarse ribbed granite sur­ face made up of two folded triangles. (Think of a rectangle folded along its diagonal. ) The upper triangle has 45 de­ gree slope and the lower one is verti-

Figure

6. NMAI fa�ade (photo by Katherine Fogden, NMAI).

© 2008 Springer Science+ Business Media, Inc., Volume 30, Number 1 , 2008

47

- -t - - poltlt

Figure 7.

Construction of circular arcs.

mented and fractured. Some sections have steeply angled, wedge-shaped forms with sculptural intention. It is an impressive display of fractal motives [consult e.g. Ref. 3l. It is rare to have a roof-top with such architectural sculp­ ture. Unfortunately there is no public access to the roof and there are no buildings in the vicinity from which the roof might be viewed. So for the pre­ sent it can be seen only by helicopter or from aerial photos. It would be nice to have a lookout (memorial?) tower on the mall. Now we begin to understand the title of Goldsworthy's domes as Roofs. He possibly wanted to show us the fifth fa�ade of a dome and give a sense of the oculus. But, lo, of the ocu­ lus we see just a "black hole"! (Fig. 4).

The National Museum of the American Indian Opposite the EB across the Mall stands the newest museum. The National Mu­ seum of the American Indian (NMAI), the 1 9th museum of the Smithsonian In­ stitute, opened in 2004. Its principal ar­ chitect was Douglas Cardinal of Canada. Like the EB, this site is also trapezoidal.

48

THE MATHEMATICAL INTELLIGENCER

Like Pei, Cardinal also was concerned with alignment, in his case with the fa�ade of the National Capitol. But he handled it differently. The geometry of the fa�ade

The topography of this complex is in striking contrast to the EB (compare Figs. 5 and 6) . The EB is linear and tri­ angulated, with sharp corners; the NMAI is a curvilinear undulating com­ plex. The EB, and for that matter, the majority of the Mall buildings, are cov­ ered with polished marble blocks; the NMAI fa�ade is made of rusticated or honey-colored dolomite limestone blocks of various sizes and textures, giving the appearance of a stratified stone mesa carved by wind and water from time immemorial. This look is most notice­ able on the northern side where cas­ cading water flows and washes over surrounding rocks and boulders that are sunk in pools winding around the fa­ cade. The spectacular eastern entrance fa�ade faces the Capitol with a five­ story curved cantilever and multilay­ ered caves and ridges (Fig. 6). It looks

like a "refuge" from the elements of the weather-from the burning sun or the pouring rain. The fa�ade provides fur­ ther drama: From the outside circular plaza there is a panoramic vista of the nearby Capitol, and if you take the ten­ minute walk to the Capitol, you will turn and see the building differently; it will appear as a rock-like sculpture. It is remarkable that the curveous sculptural exterior walls are made up only of circular arcs (Fig. 7). To deter­ mine the arcs, a polar grid system was used, which is an unusual procedure in architecture. The discrete center­ points of the arcs are defined relative to the (0,0,0) polar center that is the conceptual center of the building. Hundreds of centerpoints were needed for the project. A centerpoint may have defined several arcs in different loca­ tions. One difficult problem, among many others, was to confirm that each arc was "closed," interlocking with the adjacent joining arcs without geometric misses and kinks. This had to be done with meticulous precision to ensure the smooth flow of the surface. Obviously such laborious work could only be car­ ried out by the extensive use of spe­ cially designed computer programs and with the close collaboration of engi­ neers, contractors, and architects. The interior

We have seen that the leitmotif in the EB interior is the triangle. In the NMIA complex, the circle or arcs of circles is the repeating theme. The best place to observe this motif is in the atrium of the complex, "The Potomac," which is a ro­ tunda of 120 feet in diameter, capped with a hemispheric dome soaring 120 feet to its apex. The dome is crowned by a skylight: a glass oculus of about 15 feet in diameter. The inside of the dome resembles an inverted beehive built of ascending circular rings of de­ creasing radius (Fig. 8). The snow white color of the dome contrasts with the textured earthbound color outside. Di­ rectly below the oculus, a circular gran­ ite disc with celestial references is in­ laid in the ground. In the four cardinal directions, emanating from the center of the disc, the solstices and equinoxes are mapped on red and grey rings of gran­ ite along these axes. In a window facing due south of the Potomac, eight huge crystallic (glass)

nomical references appear throughout the museum. Most of the 25 ,000 square foot exhi­ bition space consists of curvelinear ar­ eas. However, this is not just an aes­ thetic preference of the architects. The nature of the exhibits and artifacts of the collection are most advantageously seen in such spaces. This is unlike the EB, where mostly planar paintings are shown, making curved walls unsuitable. The native botanic garden

Seventy-four percent of the 4.25-acre site is a native botanic garden, which is an integral part of the museum. This wonderful garden was designed by Donna House, together with other land­ scape architects. Approximately 30,000 trees, shrubs, and plants of 30 different species can be found here. Many of them have been used by Native Amer­ icans for livelihood, food, medicine, recreation, and shelter. It is well-known that several of the crops, such as potatos, com and squash, were unfa­ miliar to white settlers. This collection of native plants of the western hemi­ sphere is the largest in the world. Botanic tourists in a (not yet existing) Botanic Intelligencerwould write an ex­ citing article about this garden! ACKNOWLEDGMENTS

Many thanks are due to Bruce Condit and Marlene Justsen and the staff of the NGA, Chris Wood of Smith Group of Wash­ ington, Leonda Levchuk and the staff of the NMAI, and the staff of Smithsonian National Museums for providing valuable data related to the complexes. F igure 8.

Beehive (photo by Leonda Levchuk, NMAI).

REFERENCES AND LITERATURE

1 . Longstreth, R. (ed.}. The Mall in Washing­

prisms are installed, designed and fab­ ricated by a New York-based artist, Charles Ross. Each prism is about 44 inches long with sides of 14 inches. The prisms catch the changing rays of the sun and glitter according to the time of day and the season, and they reflect a spectacular color spectrum onto the in­ terior surfaces of the Potomac. For the occasional tourist, this is simply a fas­ cinating display, but it is more than that. By observing the light changes in the spectrum, these or similar devices can

provide an estimation of the time of day or a calendar of the seasons. The two main theaters are also cir­ cular and both have celestial or historic references. One of the theaters is crowned by a dome that is 40 feet in diameter, on which is mapped the na­ tives' image of the Arctic. Mapped on the deep blue ceiling of the other the­ ater is the night sky with its twinkling constellations. On the curved wall, glass sconces show the monthly phases of the moon. Many other celestial and astro-

ton, 1 79 1- 199 1 . New Haven, Yale Univer­

sity Press, 2003. 2. Gabriel, J. F. (ed.). Beyond the Cube: The Architecture of Space Frame and Polyhe­ dra. New York: John Wiley, 1 997.

3. Bovill, C. Fractal Geometry in Architecture and Design. Boston: Birkhauser, 1 996.

Joe Hammer School of Mathematics and Statistics University of Sydney NSW 2006 Email: [email protected]

© 2008 Springer Science+ Business Media, Inc., Volume 30, Number 1, 2008

49

[email protected]§ bhlfii@J§ifii.p.i§:id ..

E ncou nter at Far Point MICHAEL KLEBER

Ibis column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, suprising, or appealing that one has an urge to pass them on. Contributions are most welcome.

M i c hael Kleber and Ravi Vaki l , Ed itors

I

doubt I will ever tire of browsing through Richard Guy's Unsolved Prob­ lems in Number Tbeory [l]. I can still vividly recall my shock the first time I read UPINT problem Dl9, "Is there a point all of whose distances from the corners of the unit square are rational?" How is it possible that this is not known! But so it goes when you meet Q. Here is a grab-bag of results involv­ ing configurations of points in the plane separated by rational distances.

Integral Heptagons How many points can you find in the plane such that all pairwise distances are rational? Well, infinitely many along any line, of course, so we require sets of points with no three collinear. It turns out to be easy if you allow points that all lie on a circle, too. Ptolemy's Theorem says that if points A, B, C, and D all lie on a circle (in that order), then the lengths of the edges and diagonals of the quadrilateral are related by

IA ciiBDI = iAB ilcnl + IBcllnA

So if any five of these distances are ra­ tional, the sixth one must be as well. Now take A = (l,O) and B = (- 1,0); finding points C on the unit circle at rational distances from A and B is as

easy as rescaling Pythagorean triangles. Ptolemy then tells us that all possible Cs are at rational distances from one another. We are now picky enough to want sets of points with pairwise rational dis­ tances and with no three on a line and no four on a circle-that is, "in general position," the way some people use the phrase. How large can such a set of points be? We have no idea; it is still open whether the largest such set is fi­ nite or infinite. See UPINT D20 for an extended discussion. Paul Erdos once asked whether even five points was possible. But it is now known that you can find seven points. In a preprint posted in November 2006 [2] , Tobias Kreisel and Sascha Kurz of the University of Bayreuth present two integral hep­ tagons; they are depicted in Figure 1 . With a finite set of points, of course, we can clear denominators and require distances to be integers, so solutions come with a natural sense of scale. Kreisel and Kurz found the first hepta­ gon by an exhaustive computer search with a diameter up to 30,000; there was just one integral heptagon in general position, with a diameter of 22,270. The second heptagon was found by a more restricted search, so first some

Please send all submissions to the Mathematical Entertainments Editor,

Figure I.

The first two integral heptagons, by Kreisel and Kurz. Pictures by Ed Pegg. 1

Ravi Vakil, Stanford University, Department of Mathematics, Bldg. 380, Stanford, CA 94305-2125, USA

1The figures were produced by Ed's "Labeling the Integer Heptagon" from the Wolfram Demonstrations Proj­

e-m a i l : vakil@math .stanford .edu

ect, http://demonstrations.wolfram.com/LabelingThelntegerHeptagon/. Go play; it's worth it.

50

THE MATHEMATICAL INTELLIGENCER © 2008 Springer Science+ Business Media. Inc.

background. The area of a triangle with integer side-lengths can be written in the form A qVc where q is rational and c is a square-free integer, called the characteristic of the triangle. A re­ markable theorem says that given a set of points at pairwise integer distances, all (nondegenerate) triangles must have the same characteristic. Kurz has gen­ eralized this to higher dimensions, and cites Kemnitz's work [3] as the source, but comments that the planar version "can be traced back at least to Kummer." The characteristic of the first integral heptagon was 2002 2 7 1 1 · 13, and Kreisel and Kurz observed that many other integral-point sets likewise had round numbers as characteristics. So they searched only for point sets whose characteristic used primes up to 29; this let them push the search as far as diameter 70,000, and the second in­ tegral heptagon-again with character­ istic 2002, as it happens-was found. Octagons, anyone? =

=

·

·

Cubic Graphs We can give more structure to such questions by thinking about graphs in­ stead of point sets. Let G be a graph on n vertices, and define a rational draw­ ing of G to he a map cp : V(C) ----> 1Ri2 such that if u and v are adjacent ver­ tices in G, then the distance from cp( u) to cp(v) is rational. (You may, if you like, draw the rational-length lines that correspond to the edges of the graph as well. But we allow lines to cross freely, so this is not quite the usual def­ inition of a rational drawing. ) Figure 1 shows two rational drawings of K7 in general position. Here's an audacious thought: Per­ haps, for some graphs, rational draw­ ings are dense in the space of all draw­ ings. That is, perhaps for any cp : V{ G) ----> !Ri2, there always exists a ratio­ nal drawing If; : V(C) ----> !Ri2 such that cp ( v) and 1/J( v) are within E of each other. Now we don't need any "general position" caveat: if cp's points are in gen­ eral position, an arbitrarily small per­ turbation won't change that. In a November 2006 preprint [4], Jim Geelen, Anjie Guo, and David M'Xin­ non, all of the University of Waterloo, prove that rational drawings are indeed dense for any G with vertices of degree at most 3. (Actually they allow one ver-

tex of unbounded degree.) What's more, they produce a If; in which the coordi­ nates of the points, in addition to the relevant edge-lengths, are all rational. Much of the heavy lifting is done by a delightful theorem of Berry ['5] : If A, B, C E !Ri2 are noncollinear points such that AE:, 1Ac'2, and IBC I2 are all ratio­ nal, then the set of points that are at ra­ tional distances from all of A, B, and C is a dense subset of the plane. (The con­ dition that one edge of 6ABC be ra­ tional is indeed important, as Berry notes that there are no points in the plane at a rational distance from the ver­ tices of a right triangle with side-lengths \12, \(3, and Vs. ) As Geelen et al. point out, this raises the burning question of classifying which graphs G have the "dense ratio­ nal drawing" property-though perhaps the answer is that all do! It is an open conjecture that rational drawings of G are dense when G is pla­ nar. The conjecture could he proved, using the methods in this paper, if the following variant of Berry's theorem were true: Consider five points A, B1, . . . , B4 E Q2 with no three collinear, and such that all four distances lAB,! are rational. Does the set of points at ra­ tional distances from all five points form a dense subset of the plane'

Ambiguously Placed Points We could also take the integral n-gon problem and give it less structure. Sup­ pose you are told the set of rational pairwise distances between n points, hut not which distances correspond to which point pairs. Can you reconstruct the configuration, or are some sets of distances ambiguous? Without the rationality condition, this is the venerahk problem of "homomet­ ric sets," and the short answer is that multiple configurations with the same distance data are indeed possible. The question was posed as far back as the 1 930s in the context of x-ray crystallog­ raphy, and has reappeared in as distant a context as computational biology. For an overview of the subject, including pointers to the literature and results on the algorithmics of the reconstruction problem, see Skiena, Smith, and Lemke's paper [6] . I learned of the question only recently, when Stan Wagon of "Problem of the Week" fame (www.mathfo­ rum.org/wagon) and Ed Pegg of Math-

Puzzle (www .mathpuzzle.com) to­ gether posed some variants. They ap­ peared on Ed's web site in November 2006 [7], and inspired an enjoyable dis­ cussion among a group of colleagues. A great deal is known about the one­ dimensional version of this problem­ that is, the challenge of finding two sets of n real numbers A = {a1) and B { b1) with the same (multi)set of ( � ) pairwise differences, { j a, - atl l {lb, - htl l . In this case, adding the rationality con­ straint has no effect: The set of differ­ ences {d11 is characterized by its set of relations of the form d1 + d1 dk , so any solution can easily be converted to an equivalent solution using only small integers. Integer solutions, in turn, are amenable to generating function meth­ ods: The set of points A = {a11 corre­ sponds to the polynomial PA (x) = I 1 Xai with pairwise distances encoded hy � (x)/� (1/x) . Finding homometric sets on a line then reduces to questions of factoring polynomials. Solutions are rare in a technical sense, but they do exist for all n 2:: 6. I will explicitly mention the pair of sets !0, 1 , 4, 1 0 , 1 2 , 171 and { 0, 1 , 8, 1 1 , 13, 171, discovered by G. S. Bloom in 1 977, because they have the added pleasing property that the pairwise dis­ tances are all distinct. Evidently no larger such set is known. The problem of finding homometric sets in two dimensions turns out to he easy if you allow points to all lie on a circle, though it is easy for a reason quite different than in the integral n­ gon question. Take 2 n points equally spaced around a circle; any n-point sub­ set has the same pairwise distance data as its complement! In music, this is known as the Hexachordal Theorem: The (multi)set of intervals you can demonstrate using a given six notes of a twelve-tone scale is the same as when you use only the other six notes. Other constructions based on equally-spaced points generate many sets of points that are all homometric to one another. On a circle, though, the rationality constraint on straight-line distances does indeed change the problem. I do not know whether the dense set of points on a circle at mutual rational dis­ tances is rife with pairs of homometric subsets. When Stan Wagon and Ed Pegg brought up the topic, they presented an =

=

=

© 2008 Springer Science + Business Media, Inc. . Volume 30, Number 1 , 2008

51

ambiguous set of six pairwise distances among four points; four of the distances were integers, and two were given as decimal approximations. In later dis­ cussion, James Buddenhagen produced many more examples with five integer distances and one square-root of an in­ teger. All were instances of a nice con­ struction that Dan Asimov explained to me in general: CONSTRUCTION 1 Take three line segments AB, CC', and DD', which all

have the same midpoint 0, and such that CC' and DD' are perpendicular to one another. Tbe point sets {A, B, C, D l and {A, B, C', D') are congruent, as are {A, B, C ', D) and {A, B, C, D ' ), but they are not congruent to one another (as long as AB is not parallel to CC' or DD '). Tbe two configurations are homometric.

13 Figure 2. A homometric pair from an ambiguous set of six integer pairwise distances among four points, but not in general position.

CONSTRUCTION 2 Let M and N be the

midpoints of two sides of a triangle EFG, and consider the lines through M and N perpendicular to their respective medians EM and FN. Tbese lines intersect at a point H, and the set ofpairwise distances among the points {E, F, G, H) is un­ changed if we replace E by its reflection E' through M, or F by its rf!!lection F' through N.

-

D

� ��D' B

B

that all the distances be rational? Fred Lunnon ran a computer search for small values of the total distance, in which he viewed the six distances as the edge lengths of a tetrahedron. Planar config­ urations are just those tetrahedra with volume zero, and homometric pairs arise when there are two different flat tetrahedra among the 30 possible as­ signments of lengths to edges. Lunnon found many pairs of config­ urations similar to the one shown in Figure 2, with ambiguous distances 15, 8, 9, 1 1 , 13, 17}: In one configuration, three points are collinear (here with dis­ tances 5 + 8 1 3) . The solution turns out to be an instance of Construction 1 ; the collinearity arises because one end­ point of the length-1 1 AB happened to fall on length-1 3 C D. This seems to be a coincidence, but perhaps it is a symp­ tom that the same-characteristic con­ straint only applies to proper triangles. In any case, Fred and Ed Pegg both say their aesthetics demand examples with proper triangles and all lengths distinct. I ran a computer search using Con­ struction 1 but with the further require­ ment that the coordinates of the vertices be rational as well-this is not the deep use of rationality that appeared in the Cubic Graphs work mentioned earlier, but rather a simplistic computers-han­ dle-integers-better-than-reals use. To do this, we need to pick axes, of course; the natural choice is parallel to CC' and DD ' (or one could choose one axis par­ allel to AB). Rational coordinates allow us to build the configuration from a stack of Pythagorean triangles. My search was successful, but turned up only three homometric pairs with A's smaller coordinate having a value less than one million. The pairs are shown in Figure 3. The distances in sets (ii) and (iii) have GCDs of 73 and 4 1 , re­ spectively, and ought to be corre­ spondingly dilated, leaving them with =

Doubtless this construction has been described before, though I am unable to offer a citation. If all three line seg­ ments are the same length, and AB is at 45° to the other two, the resulting two configurations are the unique non­ trivial instance of the Hexachordal The­ orem for n = 4. According to [6], a 1 974 Bell Labs technical report by Gilbert and Shepp showed how to construct three 4-point configurations with the same set of six distances.

D =(O, Dy ) A =( A ., , A y)

B=-A

A, 462 58344 495040

D=- D

l AB I

l Ac I l sc' l

F

E'

If you squint, you can see two copies of Construction 1 placed on top of each other, with 0 at M and N Sticking to n = 4 points for the mo­ ment, can we also satisfy the constraint

coordinates Ay C,

1040 79695 676200

distances l Ac' I l AD I lE D' I lsc I

969 127896 647185

l AD' I I BD I

D

y

1480 373030 221892

IC*D * I

(i) (ii) (iii)

1 157 2562 1769 (i) 2276 1769 638 202575 456469 394346 (ii) 197538 1 05777 299081 1676080 693105 671908 1327375 1025492 684167 (iii) Figure 3. Three ambiguous sets of six-integer pairwise distances among four points in general position.

52

THE MATHEMATICAL INTELLIGENCER

noninteger coordinates (although the dilated set ( ii) does yield integers in AB­ parallel coordinates). This leaves a host of questions unan­ swered: • Can you find more ambiguous sets of six integer distances ( preferably all distinct )? Or better still, an infinite family? • Can you find an integer-distance re­ alization of Construction 2, or in some other way find a set of three (or more) homometric configura­ tions with integer distances? • Can you find any ambiguous set of integer distances which is not an in­ stance of Construction I ? • The homometric configurations in Construction 1 are quite similar: you can change one to the other ( in two ways) by moving only a single point, preserving a full triangle. Can you find an ambiguous set of six integer distances in which no triangle is pre­ served' Finally, what about more than n 4 points? With the rationality constraints, I do not know of any solutions. Note that there is no reason to expect a pair of homometric 5-point configurations to contain a 4-point pair as a subset; each n stands on its own. If we don't ask for rational distances, Bill Thurston offered the following (per­ sonal correspondence), which I've il­ lustrated as Figure 4. Here's a nice mental image for ghost symmetries giving multiple arrangements with all the same pair­ wise distance data. Imagine a figure in the plane-say a pine tree-that is symmetric by re­ flection in a vertical line, another fig­ ure-say a log-that is symmetric by reflection in a horizontal line, and a third figure-say a letter S-that is symmetric by 180° rotation about the intersection of these two lines. There is a "ghost'' symmetry group of order =

4 suggested by the picture, hut each of the three figures is invariant only by a subgroup of order 2, and there is no actual symmetry of the actual figure. Each of the three things has one other image under the ghost symmet1y group. Now if you replace any one of the three items by its other image under the ghost group--say, you replace the S by a mirror S-you get a different picture. yet any two of the three new items is isomorphic to the original. It's visually kind of intriguing.

Figure S. Girl

with

Ghost Symmetries of

Order Three, with apologies to Picasso.

bilized by various subgroups of a never­ fully-used symmetry group, opens a world of possibilities to explore. Other symmetry groups and other geometries await. REFERENCES

[1 ] Guy, Richard. Unsolved Problems in Num­ ber Theory, 3rd ed. Springer, New York,

2004.

()

Figure 4 .

Bi l l m e tri c hol id ay

[2] Kreisel, Tobias and Kurz, Sascha. There are integral heptagons, no three points on a Thurston's not-quite-sym­

line, no four on a circle. Discrete and Com­

card design.

putational

Geometry,

to

appear.

DOl

1 0. 1 007/s00454-007 -9038-6.

Construction 1 is the simplest possi­ ble realization of this not-quite-sym­ metric design: The vertically- and hor­ izontally-symmetric sets are the single points C and D, which fall on the lines of reflection, and the third set consists of points A and B, symmetric under lR0° rotation around 0. A set of n > 4 points realizing this solution with rational distances would be welcome. If we hew close to Con­ struction 1 hut let one (or both) of the mirror-symmetric sets consist instead of two points, we generically get two ho­ mometric sets of n 5 ( or 6) points with no repeated distances. More generally, this notion of ghost symmetries, with subsets of a figure sta=

[3] Kemnitz, A. Punktmengen mit ganzzahligen Abstanden. Habilitationsschrift, TU Braun­

schweig, 1 988. [4] Geelen, Jim, Guo, Anjie, and McKinnon, David. Straight line embeddings of cubic planar graphs with integer edge lengths. J. Graph Theory, to appear.

[5] Berry, T. G. Points at rational distance from the vertices of a triangle. Acta Arith. 62 (1 992) 391 -398. [6] Skiena, Steven, Smith, Warren, and Lemke, Paul. Reconstructing Sets From lnterpoint Distances. Sixth ACM Symposium on Com­

putational Geometry, June 1 990 (final ver­ sion, 1 995). [7] Pegg, Ed. Web site http://www.mathpuz­ zle.com,

material added 1 3 November

2006.

© 2008 Springer Scrence+Business Media, Inc, Volume 30, Number 1 , 2008

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I il§'j 1§'4J

Osmo Pekonen , E d itor

I

Classical and Quantum Orthogonal Polynomials in One Variable Mourad Ismail ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS, 98 CAMBRIDGE UNIVERSITY PRESS, CAMBRIDGE, 2005,

xviii + 706

ISBN 13-978-0-521-78201-2,

PP,

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REVIEWED BY J. J. FONCANNON

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Column Editor: Osmo Pekonen,

'LJ ichard Askey has said that when

he began working with orthogo-

1 \ nal polynomials he was informed

dismissively that the subject was "out of date. " No assessment could have been further from the truth. The subject of orthogonal polynomials attracted the at­ tention of many of the greatest analysts of the 19th century, and its scope and domain of applications have grown, not diminished, steadily since then. As this remarkable volume evidences, ortho­ gonal polynomials have applications in many other areas of mathematics, such as group representation theory, combi­ natorics, and many areas of physics as well. Researchers have obtained some of the most exciting results since the mid-1970s-the collective accomplish­ ment of many who are fervently work­ ing in a discipline that is "out of date." The field has experienced such a tempest of activity that anyone writing a treatise on the subject has to choose wisely what should be included. Since Gabor Szego's groundbreaking book "Orthogonal Polynomials" appeared in 1 939 [1], scores of books and confer­ ence proceedings on the subject have appeared. These works vary from those that are flagrantly theoretical and de­ pend on heavy doses of measure the­ ory and potential theory to achieve

their purposes, such as the book of Stahl and Totik [2], to those that are more or less handbooks, such as the Erdelyi volumes [3]. About a quarter of Szego's book dealt with theoretical mat­ ters, and the balance was devoted to the study of the classical orthogonal polynomials, with small diversions into the study of exotic families of more re­ cent polynomials, such as the Pollaczek polynomials. Geza Freud's similarly en­ titled 1971 book [4] was truly a high­ voltage theoretical affair, with almost no attention devoted to special families of orthogonal polynomials. 1 The author o f the present volume has chosen a middle course. He has in­ cluded enough theory to provide a re­ spectable foundation, but most of the book consists of particular results-for­ mulas that involve both novel and tra­ ditional special functions of mathemat­ ics and physics. In the battle between the theoretical and the formal, Ismail has come down soundly on the side of the formal. In fact, his book can be seen as a monumental updating of the Erdelyi volumes. In its scope and re­ cency, however, it dwarfs all similar ef­ forts. No one working in the area of or­ thogonal polynomials has been more creative or indefatigable than Ismail. In this discipline, both qualities are es­ sential to success. Furthermore, Ismail is a master synergizer. He consorts eas­ ily with workers in many areas of ap­ plied mathematics, and his ecumenism gives the book a delightfully eclectic flavor; many of the examples are taken from developments in modern physics and engineering. (In the interests of full disclosure, I admit to having coau­ thored several papers with Mourad.) For those who are unfamiliar with this subject, let me begin with some def­ initions. (Most of what I say can be ex­ tended and generalized, for instance, from the reals to the complex plane, from one to two variables, from inte-

Agora Centre, 40014 U niversity of Jyvaskyla, Rnland

1 My problem with Freud's book has to do with its legibility. The author's obsessive need to display the de­

e-mail: [email protected]

pendence of every quantity on all its parameters makes for a murky appearance.

54

THE MATHEMATICAL INTELLIGENCER © 2008 Springer Science+ Business Media, Inc.

grals to arbitrary linear functionals act­ ing on the space of polynomials. Gen­ eralizations, even when they are sig­ nificant, are usually straightforward in conception.) Let /.L be a measure on the real line whose moments

I.L n =

JIP. xn d/.L,

n = 0,1,2 . . . ,

exist. If /.L is not positive, it is called a signed measure. (Signed measures pre­ sent formidable challenges and may oc­ cur when different measures give rise to the same moments. More about this later.) If /.L is absolutely continuous, then = w (x) is called a weight func­ tion. The older mathematical literature emphasizes this case. All the traditional orthogonal polynomials involve weight functions, for instance the Legendre polynomials (weight function = 1 supported on [- 1 , 1]), the Chebyshev polynomials (weight function = ( 1 - x 2) - 112 , supported on ( - 1 , 1)), the Jacobi polynomials ((1 - x)a(l + x)f3, supported on ( - 1 , 1)), the Laguerre polynomials (exp( - x), supported in [O,oo)). Given the moments, one may con­ struct by means of the Gram determi­ nant a sequence of polynomials Pn(x) , n = 0 , 1 ,2, .

:

Pn(X) = /.Lo /.L l /.L2 /.Ll /.L3 /.L2 /.L n - 1 I.Ln 1 X n =

/.L3 /.L4 /.Ls

/.L2 /.L3 /.L4

/.L n + l I.L n + l x-

0, 1 ,2, . = kn x n

:xf3

+

/.L n I.Ln + l I.L n + 2 /.L2 n - l xn

(lower order terms) .

If w e assume, as w e shall, that the coefficient of x n, k11, is not zero, the polynomial will be a polynomial of ex­ act degree n. The polynomial is or­ thogonal to all inferior powers of x, that is, the integral with respect to the mea­ sure of the polynomial times inferior powers of x is zero. To see this, mul­ tiply the last row of the determinant by PnXj /.LdX, j = 0 , 1 ,2, . . . , n - 1 , n = 1 , 2 , . . . , and integrate. The result is a determinant with two equal rows, hence the determinant is zero. Thus the polynomials form an orthogonal set with respect to the measure /.L,

{

0,

m * n;

r PnPmdI.L JIP. h n * o, m = n . I t i s trivial t o verify that the polyno­ mials satisfy a three-term recurrence re­ lationship. Assume each polynomial is divided by k11 so its leading coefficient is 1 (the polynomials are then called monic). The recurrence relationship as­ sumes the form (*)

X Pn(X) = Pn+ l (x) + a nPn(X) + f3 nPn - l (x) , n = 0, 1 ,2, . . . , P- 1 Cx) = 0, Prix) = 1 .

We now have the background to pose some important theoretical ques­ tions, all of which are addressed in Is­ mail's book. The first question is: Given polynomials Pn and the three-term rela­ tionship (*) , is there a measure with re­ spect to which of the polynomials are orthogonal? The answer is, yes. Favard ( 1935) often gets credit for this remark­ able fact, but I first saw it in Stone's (1932) book on Hilbert spaces. Ismail, who calls it the spectral theorem, indi­ cates it can be found in even earlier writ­ ers. In its most useful form, the theorem states that if f3 n =F 0, n = 1 ,2,3, . . . , there is a (signed) measure of orthogo­ nality. If f3n > 0, the measure is positive. Whether the measure is unique depends on whether the associated moment problem is determinate, that is, whether a given set of moments corresponds to more than one measure. This is equiv­ alent to asking whether there are mea­ sures on the support of /.L that generate 0 moments. There are nontrivial mea­ sures that generate 0 moments. Stieltjes found several that are supported on [0, oo) , one of which I will need Iater. Let d!L S(x) = exp( - x1 14)sin(x114) . dx That this rather ordinary looking function on [O,oo) has zero moments is a result to die for, but it is easy to prove . Write the moment integral as =

I.L n =

r 0

X 11 exp( - x 1 14) sin(x114) dx = Im

A = V2e- i7TI4 ,

r

r

x n exp(- A:0 14) dx,

()

t 4 n + 3 exp(- A t)dt = 4(4 n + 3 )! Im(A - 4 n - 4) = 0.

/.Ln = 4 Im

0

The second issue of interest: From the recurrence relation (*) , can one

construct the measure? (Ismail calls this the inverse problem.) When the mea­ sure is positive, this construction often can be accomplished, but it usually in­ volves adult analysis. The process re­ quires detailed asymptotic information as n -.. oo , for x a complex variable, of two linearly independent solutions of the above recurrence relation consid­ ered as a difference equation, and it in­ volves inverting what may be a very hairy Stieltjes transform. Even then, ob­ stacles may remain. In my experience the most vexing task is to show the measure is free from mass points. (In the case where /.L is a function of bounded variation, for instance, mass points correspond to points where /.L has jumps.) However, there are theo­ retical guideposts that occasionally may be useful. One final duo of problems: Given the measure, can one determine the asymptotic behavior as n -'> oo of the polynomials? Can one determine the asymptotic behavior as n -.. oo of the coefficients a 11, f3 n in the recurrence re­ lation for the polynomials? Several re­ searchers have solved some difficult problems of the latter kind. Rakhmanov gained some local celebrity by his dif­ ficult analysis of the behavior of the co­ efficients in the case of the weight func­ tion w(x) = exp(- x
=

© 2008 Springer Science +Business Media. Inc . • Volume 30. Number 1. 2008

55

Poisson kernels, associated polynomi­ als, and mechanical quadrature. There is some attention-getting theoretical material. I liked the Cauchy interlace theorem, for instance. A welcome sec­ tion has to do with modifications of the measure. If we modify the measure by adding a finite discrete part or multi­ plying it by some function, can we characterize the resulting orthogonal polynomials? The most far-reaching re­ sult was accomplished by Uvarov. This result shows how the polynomials that are orthogonal to a rational function, multiplied by the original measure, can be expressed in terms of the original orthogonal polynomials. It is a treat to have all this material for the first time in one place. Chapter 3, "Differential Equations, Discriminants, and Electrostatics," is a potpourri of fascinating results. For me, this chapter justifies the cost of the book. Suppose the weight function is w(x) = exp(- v(x) ) with v(x) twice con­ tinuously differentiable. Then Pn satis­ fies the differential-difference relation

p�(x) =

-

Bn(X) Pn(X) - A n(X) Pn- l (x).

The author presents A n(X), Bn(x) ex­ plicitly in terms of integrals involving v and, paradoxically, Pn· The result, in its final form, is thanks to Chen and Ismail, and the consequences of the result are manifold. The author uses the result to show that Pn satisfies a second-order ho­ mogeneous linear differential equation whose coefficients can be computed in terms of A n (X) , Bn(X). There are intri­ cate and compelling nonlinear differ­ ence relations between A n (X), Bn (X) and the coefficients of the recurrence. In the case of exponential weights, these rela­ tions sometimes assist in resolving the Freud problem for the polynomials. The author gives applications to the explicit evaluation of A n(X), Bn(x) for some clas­ sical polynomials. The discriminant of a polynomial is the product of the differences of its roots. In the case of orthogonal poly­ nomials, the discriminant has an ex­ plicit evaluation, which involves An(x) . Chapter 4 treats the Jacobi polyno­ mials and their special cases, which in­ clude all the classical orthogonal poly-

nomials. The author introduces the term generating function which is ac­ tually a misnomer. Generatingfunction is just an expensive term for a series, a Taylor series in one variable, repre­ senting a function of two variables. For instance, 00

F(x, t)

=

l JnCx)t n,

n�o

is a generating function for the se­ quence fn · If there is more than one x variable, as in, 00

F(x,y, t)

=

L

n�o

fn(x)gn(y)t n ,

we call it a multilinear generating func­ tion. When fn , gn are the same ortho­ gonal polynomial, the bilinear formula is a Poisson kernel. Since the mid-1960s, a tsunami of ornately parameterized generating functions has issued from the East, threatening to engulf the simple, the beautiful, and the useful, and to cast the whole subject of special functions into disrepute. "Look at this!" one of my colleagues snorted, only half hu­ morously, thrusting at me a copy of a journal published in the Mideast. "Aren't you interested in THIS KIND of mathematics?" How wrong Keats was to equate truth with beauty. However, one of Ismail's attributes is his good taste: He refuses to be beguiled by truth for the sake of truth. The Kibble­ Slepian formula, a sum of the product of Hermite polynomials established by the sleek use of partial-differential op­ erators, is beautiful by anyone's stan­ dards. Theorems from Sarmanov, Berg, and others are beautiful and unex­ pected-an aesthetic ideal. In Section ( 4.8), the author lists the most useful asymptotic formulas for the classical polynomials. The chapter closes with a discussion of the Bessel polynomials, a subject to which I will return later. The exercises in the chap­ ter are exceptionally congenial. In Chapter 5, the author discusses the inverse problem: How do we ob­ tain the measure of orthogonality from the recurrence relation? Markov's The­ orem is the essential tool here, although to make such a statement is as reveal-

ing as saying that flour is an essential ingredient of bread. The artistry is in ap­ plying the theorem. Ismail shows how to obtain the measure for the Gegen­ bauer polynomials and for an intriguing set of polynomials called birth and death polynomials, which are somewhat of a cottage industry for the author. Ismail next defines and derives the properties of a species of integral, a Hadamard integral, which he intends to use to find the measure for the re­ doubtable Pollaczek polynomials. The Pollaczek polynomials, a fasci­ nating four parameter species of poly­ nomials, occupy a special place in my heart. It is natural to wonder what class of polynomials would result from the defining recurrence relation if one as­ sumes the coefficients in the relation are general linearfunctions of n? After some trivial scaling, the recurrence would be

(n +

c

+ =

l )pn + l(x) 2[(n + A + a + c)x + blpnCx) - ( n + 2A + c - l )Pn- I(x).

The resulting polynomials are the Pollaczek polynomials, PnA(x;a, b,c). Pollaczek's original work-some pa­ pers and a thin memoir-appeared in 1 949-1 950, and then Pollaczek van­ ished from the field. 2 To say his work here is opaque would be an under­ statement, and the problem isn 't the French. He derives a generating func­ tion for the polynomials, which is easy enough, but what he does with this function to obtain the measure is ar­

cane. In particular, I don't see how he restricts the parameters in a way that can rule out mass points. What Ismail does with Hadamard integrals is, as is typical with his writing, clear and rig­ orous. But he cheats a little: he exam­ ines only the three parameter family PnA(x;a, b) . (There is an intriguing sec­ tion in the chapter on the case where a > j b j is violated, when mass points may occur.) Pollaczek himself used Hadamard integrals in this case. I don't wish to denigrate an otherwise brilliant achievement. I wonder, though, whether Pollaczek, weary of the chase, realiz­ ing how short life was, and attending to the siren call of probability theory,

2His later work, up to 1 975, consists exclusively of statistics and probability. Born in Vienna in 1 892 into an illustrious Jewish family, Pollaczek fled from Nazi Germany

to France in 1 933 and became a citizen of that country by naturalization in 1 947. He died in Bous-le-Roi, France, in 1 981 . He secured his reputation by his brilliant work as a pioneering queuing-theorist.

56

THE MATHEMATICAL INTELLIGENCER

just issued his incondite workings for all of us to believe. At one time I imag­ ined I had the solution to the Pollaczek puzzle (I didn't), and I included my so­ lution in a chapter in my first book. Too bad. Given a recurrence relation for any set of polynomials, one may replace n by n + c, c > 0 a real parameter, to ob­ tain a new recurrence. The polynomi­ als satisfying this recurrence are the as­ sociated polynomials. Even ordinary polynomials in this fashion give rise to polynomials with bizarre and beautiful measures, as exotic as the inhabitants of a medieval bestiary. Ismail details the work done in the most general case­ that of the (shifted) Jacobi polynomi­ als. The associated polynomials he de­ notes, for some reason, by the whimsical name "Wimp polynomials . ., Their weight function involves the rec­ iprocal of the modulus squared of the sum of two (complex-valued) hyper­ geometric functions. When c = 0, the weight function reduces to xa( l - x)f3, as it should. (Ismail, with David Mas­ son, treated some related polynomials, whose theory Ismail also describes here.) Although the Jacobi polynomials have convenient explicit representa­ tions, the representations for the asso­ ciated polynomials are much more complicated, involving a sum of ( ter­ minating) 4F3 (1 ) hypergeometric func­ tions. By using the powerful Wilf-Zeil­ berger algorithms, one may derive an expression for the important associated Legendre polynomials that is succinct. Nothing simple is known for general

a,f3. Discrete orthogonal polynomials aren't really my cup of tea, although their importance is undeniable. They are a staple in many areas of engi­ neering and physics, and they enjoy a privileged role in statistics. Chapter 6 contains much up-to-date information about them. Chapter 7, "Zeros and Inequalities,'' is exceptional. It is the strongest, most contemporary, and fullest treatment of this subject I have seen. The author ob­ tains a number of results about the zeros of orthogonal polynomials by us­ ing two powerhouse theorems: a gen­ eralized theorem of Markov, and a version of the Hellman-Feynman the­ orem proved by Ismail and Zhang. Tlw latter is a result from Hilbert space the-

ory, but, with some ingenuity, one can adapt the theorem to the study of the zeros of orthogonal polynomials. The resulting theorems allow one to decide how the zeros of a parametric set of orthogonal polynomials vary as the pa­ rameter varies. We have the following beautiful result of Ismail and Muldoon:

The zeros of the associated Laguerre polynomials LnCa>(x:c) increase with a for a 2: 0 and c > - 1 . The zeros of a set of orthogonal polynomials are the eigenvalues of a matrix whose entries depend on the coefficients in the recurrence relation. Information about the location of the eigenvalues of such matrices, for in­ stance, assertions about the positive­ definiteness of the matrix, will yield information about the zeros of the polynomials. Chain sequences are use­ ful tools for examining such matrices. Wall and Wetzel in 1 94 1 used chain se­ quences to study positive-definite ]-fractions, and their adaptation to the locating of polynomial zeros entails considerable ingenuity. Ismail's treat­ ment of chain sequences is remarkably cogent and concise. Y. L. Geronimus, elaborating on work started by Szegc), has written an entire book on polynomials orthogonal on the unit circle, and Chapter 8 of this book summarizes the current state of affairs. If I.L is a probability measure supported on an infinite subset of [- 7T, 7T], the quantities used to define the polynomials are I.L n

=

f7T e� irz1Jdi.L(8), � 7T

for instance, the circular Jacobi polyno­ mials have the weight function 1 1 - eiOi za, a > - 1 . Polynomials orthogonal on the unit circle are more than just a labora­ tory curiosity. They occur in many con­ texts, and Ismail has been assiduous in tracking them down: random unitary matrix ensembles, electrostatics, lengths of subsequences of random words. Let P,z(x;a) be an �--parameter sys­ tem of orthogonal polynomials. Two significant expansion problems are as­ sociated with the polynomials. The first is to determine the con­ nection coefficients cll. k ( a, b) in the ex­ pansion P,( x ; b ) =

"

L

,,� o

c"·"'(a, h ) Pn(x;a),

and the second is to determine the lin­ earization coefficients Cm, n,i a) in the expansion Prn(x;a)Pn(x; a)

= kIO cm, ll, i a) PJx;a). m+ n

�

Results in this area, the subject of Chapter 9, have interesting implications for combinatorial and positivity ques­ tions. ForJacobi polynomials, the author demonstrates a closed-form expression for the connection coefficients. It is im­ portant in applications to know when the connection coefficients are positive. Ismail provides several theorems, due to Wilson, Askey, and Swarc, which can aid in making this judgment. Such the­ orems display a common theme: the techniques used to prove them may be drawn from diverse areas of mathe­ matics and may involve ideas that ap­ parently have little to do with special functions as such, for instance, Stieltjes matrices and boundary-value problems. Linearization problems emerged in the early 1 930s in a study by Friederichs and Lewy of the discretization of the time-dependent wave equation. It was necessary to show the nonegativity of the coefficients A( k, m, n) in the three­ fold Taylor series, ( 1 - r)( l - s) + ( 1 - r)( l - t) + (1 - s)( l - t) X

= I A( k, m, n) r ksm t n . k, m, n = O Szego proved that A(k, m, n) could be expressed as an integral of a product of three Laguerre polynomials, which implied that the original problem was equivalent to showing that the lin­ earization coefficients in X

=

L A(k,m, n)e�2xLk(x) ,

k�O

are positive. Sections ( 4.3) and ( 4 . 4) of Ismail's book describe this problem and some of its generalizations. The chapter next treats product for­ mulas and addition formulas for the Ja­ cobi polynomials. The product formulas express the product of two polynomi­ als as an integral, whereas the addition formulas express a polynomial of com­ pound argument as a sum of products of the polynomial whose arguments are the individual variables. The addition theorem for the special case, the Gegenbauer polynomials, had been

© 2008 Springer Science+ Business Media, Inc., Volume 30, Number I , 2008

57

known for a long time. For years, re­ searchers obsessively searched for the corresponding result for the Jacobi polynomials. Finally, in 1 972, Tom Kornwinder found it. He used group­ theoretic methods, and, in the acclaim and enthusiasm the result generated, there was a scramble to prove it by more conventional means. Kornwinder himself did this in 1 977, and Ismail in­ cludes his astonishing proof. It is both short and demanding. Better tum off the TV when decoding it. The chapter closes with the impor­ tant Askey-Gasper inequality, an es­ sential component of de Brange's proof of the Bieberbach conjecture. Rota introduced the umbra! calculus many years ago, and others developed the subject in a rather meagre series of papers. Umbra! calculus has been un­ justly neglected. It is a powerful ana­ lytical tool, and, when used judiciously, it can accomplish near miracles. I used the umbra! calculus in a paper several decades ago to derive explicit evalua­ tions of some hypergeometric func­ tions. Since the calculus is based on formal series, there is a general feeling that the calculus lacks rigor. It does not. The word "formal, " in this context does not mean nonrigorous. The Sheffer classification of polynomials, dating from Sheffer's work in 1 939, fits into the structure of the umbra! calculus beautifully-though the author has brought the Sheffer classification into the 2 1 st century. It is the subject of the too brief Chapter 10. The details of the classification are technical , but I can provide an interesting consequence of a special classification: a polynomial se­

quencefn is ofSheffer A -type zero ifand only if it has the generating function 00

2.JnCx)t n n�o

=

A(t)exp(x H(t)) .

Several authors have discussed the problem of procuring conditions on A(t) and H(t) that will ensure the or­ thogonality of the fn· The major part of the rest of the book, Chapters 1 1 through 20, deal with q-series. It is hopeless to attempt to summarize the wealth of material here. q-series is the author's home turf, and few have been as productive in this

3This reference is a galleon of misprints.

58

THE MATHEMATICAL INTELLIGENCER

area as he. The author draws much of the material from his own research. The word "quantum" in the title of the book refers to this very q, the author's at­ tempt, I suppose, to add some juice to the title, perhaps even securing cita­ tions in physics journals. Best of luck to him. The species called q-series grew out of research initiated by Heine in 1 878, and has effloresced almost unrecogniz­ ably since then. Heine considered the following series (I am simplifying a bit):

polynomials is indeterminate, there is more than one measure. This is the case, for example, with the Bessel poly­ nomials. The author has an apt and concise discussion of the Hamburger moment problem, where there is no re­ striction on the support of JL. (The Stieltjes and Hausdorff moment prob­ lems require the support to be in [O,oo) and [0, 1 ] , respectively.) He gives a num­ ber of applications to various families of orthogonal polynomials, such as the q-Hermite polynomials. Among the en­ ticements the author displays is a class of polynomials investigated by some Russian authors:

Un+ 1 (x) =

2xun(X) -

=

n Uo(X)

+ . . .,

which converges for jzj < 1 , j q j < 1 . Note when q � 1 - , the series reduces to the geometric series for (1 z) - a. Generalizing the above series presents no difficulties, and the result, called a basic series or q-series, generalizes even further the generalized hypergeo­ metric function. q-series enjoy beautiful and deep formal properties, with many implica­ tions for number theory, combinatorics, and especially for modem physical the­ ories. All the basic functions-the ex­ ponential function, the Gamma function, even orthogonal polynomials-have q-series analogues, sometimes more than one, with the property that when q � 1 - , the series reduces to the tradi­ tional function. Much of Ramanujan's most brilliant work was with q-series. The reader might wish to read Ismail's book in tandem with the book, "Spe­ cial Functions," by George Andrews, Richard Askey, and Ranjan Roy, pub­ lished in the same series [5]. This book provides a readable introductory treat­ ment of q-series. Another excellent ref­ erence (more detailed) is the book by Gasper and Rahman [6]. Indeterminate moment problems, Chapter 2 1 , is one of the most intrigu­ ing topics in the field of orthogonal polynomials. As noted earlier, when the moment problem associated with the

-

=

1,

1,2,

·

·

u1 (x)

(- )

n - qn Un- 1Cx), _1 q - q

q

· ,

=

2x,

0 < q < 1.

These polynomials arise in the study of the harmonic oscillator and generalize the Hermite polynomials in the sense that lim Un(X) Hn(X).

q---> 1 -

=

What else is known about Un(x)? The answer is: almost nothing (al­ though Dennis Stanton has established by combinatorial reasoning a mind­ bending linearization relationship). The corresponding moment problem is in­ determinate. However, no one has found an explicit formula for the poly­ nomials, nor a measure for them. Con­ templating this sad situation, one real­ izes how meagre are our resources for dealing with the inverse problem in the indeterminate case . (However, when a weight function satisfies a simple dif­ ferential equation, other options may be available. More about this later.) In­ deed, the subject is awash in unre­ solved issues, for example, .find a mea­

sure whose moments are the Bernoulli numbers. For this and related prob­

lems, see [7] . 3 With Chapter 22, "The Riemann­ Hilbert Problem for Orthogonal Poly­ nomials," the book switches gears, or more accurately, vehicles. Being inured to the disjunctions of conference pro­ ceedings, I don't mind this so much, though other readers may find them­ selves disoriented. Chapters 22 and 23 were written by Walter van Assche, and

the exposition has a flavor different from Ismail's writing. The fundamental idea behind the Riemann-Hilbert method in the study of orthogonal poly­ nomials is to characterize the polyno­ mials corresponding to a given weight function on the real line by means of a boundary-value problem for matrix­ valued analytic functions. Van Assche provides the solution for three great classes of orthogonal polynomials: the Hermite, the Laguerre, and the Jacobi polynomials. One of the payoffs of the Rie­ mann-Hilbert approach is that it can yield uniform asymptotic expansions valid in the entire complex plane. Van Assche outlines the necessary steps in a protocol that is mindbogglingly tortuous. A great deal of expertise will be required to carry out the program for any given set of polynomials. This is not an un­ dertaking for sissies. Nevertheless, P . De­ ift and his coworkers have found uni­ form expansions for poly-nomials orthogonal with respect to Freud type weights exp( - arm) , m = 1,2, · · · . The case m 1 gives uniform asymptotics that generalize the Plancherel-Rotach asymptotics for Hermite polynomials. Multiple orthogonal polynomials, the subject of Chapter 23, are polyno­ mials of one variable that are defined by orthogonality relations with respect to several different measures. The field has fairly recent origins. The author dis­ cusses the Riemann-Hilbert problem for some of these polynomials. The book concludes in Chapter 24 with a welcome description of feasible research problems. In a book this monumental, it is probably unfair to ask the author to do even more . However, there are other topics that I would like to have seen included in the book, perhaps in place of some of the material on q-series. I would like to have seen a more in­ depth discussion of combinatorial in­ terpretations of polynomial identities and of the use of uniqueness theorems for partial-differential equations for es­ tablishing identities. I wish the author had mentioned Luke's virtually un­ known finding [8], one that enables us to obtain high-precision approxima­ tions to the Fourier coefficients of a function: every set of orthogonal poly­ =

nomials is also orthogonal with respect to summation.

Especially, I would have liked to see more about the work of Lance Little­ john and his followers. Littlejohn is probably now the leading proponent of a field of study initiated by Alan Krall and furthered by Norrie Everitt: the in­ terrelationship of differential equations and orthogonal polynomials. Ismail has a few words about and several refer­ ences to this body of work, but it de­ serves more attention: it has significant ramifications. I want to explore one at length. The fabled Bessel polynomials arise when a confluence argument is used on the Jacobi polynomials shifted to the interval [0,1], Pn(a,f3)(2x - 1). Replace a by a {3, x by x{3, let f3 � oo, and then set a = 0. Admittedly, the procedure is untenable-the weight function for a < - 1 is not integrable-but flying in the face of reason and applying this process to the definition of the Jacobi polynomials as hypergeometric func­ tions and to their recurrence, we obtain a passel of plausible formulas. Denote by Bn(X) the resulting polynomials, the so-called Bessel polynomials: -

n

Bn(X)

=

�o

( - 1) k (n + k) ! xk ( n - k) ! k! ,

- 2x(2n + n = 1 ,2,3, . . .

Brz+l(x) =

1 )Bn(X) + Bn-l(x),

, Bo(x) = 1 , B1(x) = 1 - 2x. If the reader writes out these poly­ nomials for a few values of n, he or she may recognize them: they are the numerator and denominator Pade ap­ proximants for e -x. The Taylor series for the rational function ( - l) nxnBn(l/x) n x BnC - 1/x)

agrees with that of e- x through 2n + 1 terms. The polynomials occur in many areas of mathematics. (They are related to the modified Bessel function Kn+'-, 2 but that need not concern us here.) We even obtain a putative weight function in the process and a credible set of moments: w(x) = e- llx,

f-1- n =

(n + 1)!

(One can prove that the latter are in­ deed the correct moments for the Bessel polynomials by taking the limit in the Gram determinant for the Jacobi polynomials.)

But here our malfeasances have caught up with us. The theorem noted in the beginning of the book states that there is no positive measure associated with the last mentioned recurrence. There is a measure, but it is a signed one. The traditional inverse theorems, for example, Markov's theorem, are useless. (It is easy to show the Bessel polynomials are orthogonal along a contour in the complex plane, but that isn't what we want to know.) For many years, a frustrating open question was: What is a real measure for the Bessel polynomials? Duran once claimed to have dis­ covered a measure, but I find his work, which seems to have something to do with distributions, incomprehensible. Furthermore, all his exertions didn't as­ suage my craving for an honest-to-God function, something whose value for a given argument can be reasonably in­ ferred. At long last, Kil Kwon and two of his students, S. S. Kim and S. S. Han, obtained a real measure, in fact, a weight function [9] . Their argument was transcendently clever, and it was sim­ ple, too. Clearly w(x) = e- l/x can't be a weight function on [O,oo): Its moments don't exist. But it does provide a point of departure. It satisfies x2 w'(x) -

w(x)

=

0.

Write x2 W' (x) - W(x)

S(x) ,

=

where S(x) is the Stieltjes function, de­ fined at the beginning of this review. Solving this equation yields ellts(t) 2 dt. W(x) = - e- llx

foo X

t

Note W(O) 0. W(x) is continuous and decays exponentially, so its mo­ ments exist. Multiplying the previous differential equation by x n , integrating from 0 to oo, and using the fact that S(x) has zero moments shows that the mo­ ments of W(x) satisfy =

f-1- n + l

- p,n =

n+2

.

Since an orthogonal set is uniquely determined (up to a constant) by its mo­ ments, iterating this recurrence shows the f-1-n must be the moments of a weight function for the Bessel polynomials. If

© 2008 Springer Science+ Business Media, Inc., Volume 30, Number 1 , 2008

59

we can show JLo =1=- 0--the only difficult part of the analysis-we will have shown that W(x) is the explicit, computable weight function we seek. The reader should consult the aforementioned arti­ cle for this remaining detail [6]. Of course, the required measure is simply an integral of the weight function. Signed measures are not, generally, unique. There are several known func­ tions with 0 moments, and each will yield a different measure. Although the previous analysis is self-contained, it grew out of research the authors have conducted on or­ thogonalizing weights for differential equations, a field dominated by and large by Lance Littlejohn. The work has heady implications, because, as the reader may already have guessed, one can use the same procedure to obtain nonstandard measures for many fami­ lies of orthogonal polynomials. The reader might enjoy obtaining in a similar fashion a signed measure for the generalized Laguerre polynomials Ln(a>(x) , a < 1 and nonintegral. These topics, though, are probably destined for the book someone else should write. The author has produced a book that has grown out of his own rich and productive research career, as he has every right to do. My criticisms of the book's exposi­ tion, considering its girth, are few. Is­ mail is usually a clear writer. The book is occasionally plagued with notational confusion. For instance, the A n and the A n(X) in (3.4) aren't the same. I found the book difficult to read and devilish to dip into for a particular result. I wasn't always certain what recurrence the author was working with, and sometimes in navigating the book I felt adrift. It would have been preferable to do as Erdelyi does in defining the re­ currence for orthogonal polynomials. State the general recurrence as -

Pn + l (x) = A nPn(X) + Bn.xpn(X) - CnPn- l(x) ,

and then indicate the form the recur­ rence takes when (i) the polynomials are monic; (ii) the polynomials are or­ thonormal. Giving the connection be­ tween the coefficients for both sets of polynomials would have been useful, as would a little table, such as Erdelyi has, indicating the relations between the coefficients in the recursion and the

60

THE MATHEMATICAL INTELLIGENCER

quantities hn, kn, and k�, the coeffi­ cient of x n - l in Pn· The Erde!yi vol­ umes are niggardly for theory, but they set the standard for clear organization. The present book suffers from a vesti­ gial index. Important terms such as "orthonormality, " "Pollaczek polynomi­ als," "umbra! calculus," "Legendre poly­ nomials," are absent. (Is this a trend? The index of Andrews, et a!. [5] is even sparser, consisting only of names, and the index of Stahl and Totik [2] is an anorexic 2 pages.) A notational index would have been helpful. Some of the theoretical material has, unnecessarily, been dispersed through­ out the book, which makes it nearly inaccessible to the research worker. Ma­ terial on the measure of the polynomi­ als, for instance Nevai's theorem and a theorem discussing mass points, both given in Chapter 1 1 , should all have been placed in Chapter 2. Vitali's dou­ ble series theorem is stated twice, in dif­ ferent forms, on p. 4 and p. 294. A no­ tational index would have been helpful. The author has prepared a website for errata in the book: http://math. ucf. edu/ �Ismail/ My overall impression of the book, however, is overwhelmingly favorable. It is an ambitious and imposing testa­ ment to the author's eminence in and love for the subject. All research work­ ers in orthogonal polynomials will want to own this special work. I feel fortu­ nate to have a copy of it.

[6] George Gasper and Mizan Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics and Its Applications,

Cambridge University Press, Cambridge, 1 990, XX + 287 pp.

[7] Fader AI Muta-Qalifi, Hankel determinants and some polynomials arising in combina­ torial analysis, Num. Algorithms 24 (2000),

1 79-1 93. [8] Y. L. Luke, On the error in a certain inter­ polation formula and in the Gaussian inte­ gration formula, J. Amer. Math. Soc. 29

(1 975), 1 96-208. [9] Kil H. Kwon, Sung S. Kim, and Sung S. Han, Orthogonalizing weights of Tcheby­ chev sets of polynomials. Bull. London

Math. Soc. 24 (1 992), 361 -367. J. J. Foncannon Philadelphia, PA USA e-mail: [email protected]

G rothendieck-Serre Correspondence by Alexandre Grothendieck, Pierre Colmez (editor), and Jean-Pierre Serre SOCI ET E MATH E MATIQUE FRANCAISE 2001; BILINGUAL EDITION: AMERICAN MATHEMATICAL SOCIETY, 2004

REFERENCES

35,

ix

+

288 PP. US $69.00

ISBN: 0-8218-3424-X

[1 ] Gabor Szego, Orthogonal Polynomials, (4th ed.) American Mathematical Society, Collo­ quium Publications, Vol . XXIII. American Mathematical Society, Providence, Rl, 1 975, xiii + 432 pp. [2] Herbert Stahl and Vilmos Totik, General Or­ thogonal

Polynomials,

Encyclopedia

Mathematics and Its Applications,

of

43,

Cambridge University Press, Cambridge, 1 992, xii + 250 pp.

[3] A. Erdelyi, et al. , Higher Transcendental Functions, 3 Vol. , McGraw-Hill, New York,

1 953. [4] Geza Freud, Orthogonal Polynomials, Perg­ amon Press, New York, 1 97 1 , 294 pp. [5] George E. Andrews, Richard Askey, and Ranjan Roy, Special Functions, Encyclope­ dia of Mathematics and Its Applications, 7 1 ,

Cambridge University Press, Cambridge, 1 999, xvi + 653 pp.

REVIEWED BY LEILA SCHNEPS

he Grothendieck-Serre corre­ spondence is a very unusual book: one might call it a living math book. To retrace the contents and history of the rich plethora of mathe­ matical events discussed in these letters over many years in any complete man­ ner would require many more pages than permitted by the notion of a book review, and far more expertise than I possess. More modestly, what I hope to accomplish here is to render the flavour of the most important results and no­ tions via short and informal explana­ tions, while placing the letters in the context of the personalities and the lives of the two unforgettable epistolarians. The exchange of letters started at the

beginning of the year 1 955 and contin­ ued through to 1 969 (with a sudden burst in the 1 980s), mostly written on the occasion of the travels of one or the other of the writers. Every mathemati­ cian is familiar with the names of these two mathematicians, and has most probably studied at least some of their foundational papers-Grothendieck's "Tohoku" article on homological alge­ bra, Serre's FAC and GAGA, or the vol­ umes of EGA and SGA. It is well known that their work profoundly renewed the entire domain of algebraic geometry in its language, in its concepts, in its meth­ ods and of course in its results. The 1 950s, 1 960s, and early 1 970s saw a kind of heyday of algebraic geometry, in which the successive articles, semi­ nars, books, and of course the impor­ tant results proven by other mathe­ maticians as consequences of their foundational work-perhaps above all Deligne's finishing the proof of the Weil conjectures-fell like so many bomb­ shells into what had previously been a well-established classical domain, shat­ tering its concepts to reintroduce them in new and deeper forms. But the arti­ cles themselves do not reveal anything of the actual creative process that went into them. That, miraculously, is exactly what the correspondence does do: it sheds light on the development of this renewal in the minds of its creators. Here, unlike in any mathematics article, the reader will see how Grothendieck proceeds and what he does when he is stuck on a point of his proof (first step: ask Serre), share his difficulties with writing up his results, participate with Serre as he answers questions, provides counterexamples, shakes his finger, complains about his own writing tasks, and describes some of his theorems. The letters of the two men are very dif­ ferent in character. Grothendieck's are the more revealing of the actual creative process of mathematics, and the more surprising for the questions he asks and for their difference with the style of his articles. Serre's letters for the most part are finished products which closely re­ semble his other mathematical writings, a fact which in itself is almost as sur­ prising, for it seems that Serre reflects directly in final terms. Even when Grothendieck surprises him with a new result, Serre responds with an accurate explanation of what he had previously

known about the question and what Grothendieck's observation adds to it. They tell each other their results as they prove them, and the responses are of two types. If the result fits directly into their current thoughts, they absorb it instantly and usually add something as well. Otherwise, there is a polite ac­ knowledgement ("That sounds good"), sometimes joined to a confession that they have had no time to look more closely. The whole of the correspon­ dence yields an extraordinary impres­ sion of speed, depth, and incredible fer­ tility. Most of the letters, especially at first, are signed off with the accepted Bourbaki expression "Salut et frater­ nite". At the time the correspondence be­ gan, in early 1 955, Jean-Pierre Serre was twenty-eight years old. A young man from the countryside, the son of two pharmacists, he had come up to the Ecole Normale Superieure in 1945 at the age of 19, then defended an extraordi­ nary thesis under the direction of Henri Cartan in 1 95 1 , in which he applied Leray's spectral sequences, created as a tool to express the homology groups of a fibration in terms of those of its fibre space and base space, to study the re­ lations between homology groups and homotopy groups, in particular the ho­ motopy groups of the sphere. After his thesis, Serre held a position in the Cen­ tre National des Recherches Scientifiques (CNRS) in France before being ap­ pointed to the University of Nancy in 1 954, the same year in which he won the Fields Medal. He wrote many papers during this time, of which the most im­ portant, largely inspired by Cartan's work and the extraordinary atmosphere of his famous seminar, was the influential ''FAC" (Faisceaux Algebriques Coherents, published in 1955), developing the sheaf theoretic viewpoint in abstract algebraic geometry (sheaves had been introduced some years earlier by Leray in a very dif­ ferent context) . Married in 1948 to a bril­ liant chemist who had been a student at the Ecole Normale Superieure for girls, Serre was the father of a small daugh­ ter, Claudine, born in 1949. In January 1955, Alexandre Grothen­ clieck had just arrived in Kansas to spend a year on an NSF grant. Aged twenty­ six, his personal situation was chaotic and lawless, the opposite of Serre's in almost every possible way. His earliest

childhood was spent in inconceivable poverty with his anarchist parents in Berlin; he then spent five or six years with a foster family in Germany, but in 1939 the situation became too hot to hold a half-Jewish child, and he was sent to join his parents in France. The war broke out almost immediately and he spent the war years interned with his mother in a camp for "undesirables" in the south of France; his father, interned in a different camp, was deported to Auschwitz in 1942 and never returned. After the war, Grothendieck lived in a small village near Montpellier with his mother, who was already seriously ill with tuberculosis contracted in the camp; they lived on his modest university scholarship, complemented by his occa­ sional participation in the local grape harvest. He, too, was the father of a child: an illegitimate son from an older woman who had been his landlady. His family relations-with his mother, the child, the child's mother, and his half-sister who had come to France to join them after a twelve-year separation, were wracked with passion and conflict. He was state­ less, with no permanent job. As it was legally impossible to hold a university position in France, he was compelled to accept temporary positions in foreign countries while hoping that some suit­ able research position in France might eventually be created. After Montpellier, he spent a year at the Ecole Normale in Paris, where he met Cartan, Serre, and the group that surrounded them; then, on their advice, he went to do a thesis under Laurent Schwartz in Nancy. His friends from his time in Nancy and af­ ter, such as Paulo Ribenboim, remember a young man deeply concentrated on mathematics, spending his (very small amount of) spare time taking long walks or playing the piano, working and study­ ing all night long. Throughout his life, Grothendieck would keep his mathe­ matical activities sharply separate from his private affairs, about which next to nothing appears in his letters. He also had a lifelong habit of working and writ­ ing through the night. At the time of his visit to Kansas, Grothendieck already had his disserta­ tion and nearly twenty publications to his credit on the subject of topological vector spaces, their tensor products, and nuclear spaces, which constituted a real revolution in the theory. He had com-

© 2008 Springer Sc1ence +Business Media, Inc., Volume 30, Number 1 , 2008

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pleted his thesis in 1 953, and he then spent the years 1 953-1955 in Sao Paulo, where he continued to work on the sub­ ject. His move to Kansas marked the beginning of the first of several major shifts in his mathematical interests.

195S-1957: Two Mathematicians in Their Twenties From the very first letter of the corre­ spondence with Serre, dated January 1955, the words homology, cohomol­ ogy, and sheaf make their appearance, as well as a plethora of inductive and projective limits. These limits and their duality to each other, now a more-than­ familiar concept even for students, were extremely new at the time. Their intro­ duction into homological algebra, to­ gether with the notion that the two types of limit are dual to each other, dates to very shortly before the ex­ change of the earliest letters of the cor­ respondence. In those letters, Grothendieck ex­ plains that he is in the process of learn­ ing (as opposed to creating) homolog­ ical algebra: "For my own sake, I have made a systematic (as yet unfinished) review of my ideas of homological al­ gebra. I find it very agreeable to stick all sorts of things, which are not much fun when taken individually, together under the heading of derived functors." This remark is the first reference to a text which will grow into his famous Tohoku article, which established the basis of many of the notions of mod­ ern homological algebra. In fact, he wanted to teach a course on Cartan and Eilenberg's new book, but he couldn't get hold of a copy, and so he was com­ pelled to work everything out for him­ self, following what he "presumed" to be their outline. The Tohoku paper introduced abelian categories, extracting the main defining features of some much-studied categories such as abelian groups or modules, introducing notions such as having "enough injectives," and ex­ tending Cartan-Eilenberg's notion of de­ rived functors of functors of the cate­ gory of modules to a completely general notion of derived functors. It is really striking to see how some of the most typical features of Grothendieck's style over the coming decade and a half are already totally visible in the early work

62

THE MATHEMATICAL INTELLIGENCER

discussed here: his view of the most general situations, explaining the many "special cases" others have worked on, his independence from (and sometimes ignorance oO other people's written work, and above all, his visionary apti­ tude for rephrasing classical problems on varieties or other objects in terms of morphisms between them, thus obtain­ ing incredible generalizations and sim­ plifications of various theories. Six months later, in December, Grothendieck was back in Paris with a temporary job at the CNRS, obtained thanks to help of Serre and others, who were always searching for a way to al­ low the homeless maverick to remain permanently in France. Serre, at this time, was on leave from the University of Nancy, spending some time in Princeton and working on his "ana­ lytic = algebraic diplodocus," which would become the famous GAGA, in which he proved the equivalence of the categories of algebraic and analytic co­ herent sheaves, obtaining as applica­ tions several general comparison theo­ rems englobing earlier partial results such as Chow's theorem (a closed an­ alytic subspace of projective space is al­ gebraic). The comparison between al­ gebraic and analytic structures in any or every context is at this point one of the richest topics of reflection for both Grothendieck and Serre. During the period covered by these early letters, the notion of a scheme was just beginning to make its appearance. It does not seem that Grothendieck paid particular attention to it at the time, but a scattering of early remarks turns up here and there. Already at the begin­ ning of 1 955 Grothendieck wrote of FAC: "You wrote that the theory of co­ herent sheaves on affine varieties also works for spectra of commutative rings for which any prime ideal is an inter­ section of maximal ideals. Is the sheaf of local rings thus obtained automati­ cally coherent? If this works well, I hope that for the pleasure of the reader, you will present the results of your paper which are special cases of this as such; it cannot but help in understanding the whole mess." Later, of course, he would be the one to explain that one can and should consider spectra of all commu­ tative rings. A year later, in January 1956, Grothendieck mentions "Cartier­ Serre type ring spectra, " which are noth-

ing other than affine schemes, and just one month after that he is cheerfully proving results for "arithmetic varieties obtained by gluing together spectra of commutative Noetherian rings"­ schemes! A chatty letter from Novem­ ber 1 956 gives a brief description of the goings-on on the Paris mathematical scene, containing the casual remark "Cartier has made the link between schemes and varieties," referring to Cartier's formulation of an idea then only just beginning to make the rounds:

The proper generalization of the notion of a classical algebraic variety is that of a ringed space (X, fJx) locally isomor­ phic to spectra of rings. Over the com­ ing years, Grothendieck would make this notion his own.

1957-1958: Riemann-Roch, Hirzebruch . . . and Grothendieck The classical Riemann-Roch theorem, stated as the well-known formula g + 1, €(D) - €(K - D) deg(D) concerns a non-singular projective curve over the complex numbers equipped with a divisor D, the formula computes a difference of the dimen­ sions of two vector spaces of mero­ morphic functions on the curve with prescribed behavior at the points of the divisor D (the left-hand side) in terms of an expression in integers associated topologically with the curve (the right­ hand side). In the early 1 950s, Serre reinter­ preted the left-hand side of the Rie­ mann-Roch formula as a difference of the dimensions of the zero-th and first cohomology groups associated to the curve, and he generalized this expres­ sion to any n-dimensional non-singular projective variety X equipped with a vector bundle E as the alternating sum .2: (- 1) ; dim If (X, E) . In 1 953, Hirzebruch gave a general­ ization of the classical Riemann-Roch theorem to this situation, by proving that Serre's alternating sum was equal to an integer which could be expressed in terms of topological invariants of the variety. It seems that the idea of trying to prove a general algebraic version of Rie­ mann-Roch was in Grothendieck's mind from the time he first heard about Hirze­ bruch's proof. In the end, what Grothendieck brought to the Riemann=

-

Roch theorem is one of the basic fea­ tures of all of his mathematics, and it was already visible in his Tohoku arti­ cle: the transformation of statements on objects (here, varieties) into more gen­ eral statements on morphisms between those objects. He reinterpreted both sides of the formula that Hirzebruch proved in the framework of morphisms f : X ---'? Y between varieties. Grothen­ dieck did this work between 1 954 and 1957. He wrote up something (RRR­ "rapport Riemann-Roch") which he con­ sidered a mere preliminary and sent it to Serre, then in Princeton; Serre orga­ nized a seminar around it, and then, as Grothendieck was clearly onto other things and not going to publish, Serre wrote the proof up, together with Borel and published it in the Bulletin de la Societe Mathematique Franr.;aise in 1958 [BS]. Grothendieck finally included his original RRR at the beginning of SGA 6, held in 1 966-67 and published only in 1 97 1 , at the very end of his established mathematical career. What is not revealed in the letters is that Grothendieck's mother was dying at the very time of these exchanges. He does add as a postscriptum to the let­ ter of November 1 , "You are moving out of your apartment; do you think it might be possible for me to inherit it? As the rent is not very high, if I re­ member rightly, I would then be able to buy some furniture (on credit). I am interested in it for my mother, who is­ n't very happy in Bois-Colombes, and is terribly isolated. " But Hanka Grothen­ dieck was suffering from more than iso­ lation. She had been nearly bedridden for several years, a victim of tuberculo­ sis and severe depression. After their five-year separation during his child­ hood, she and Alexander had grown in­ separable in the war and post-war years, but during the last months of her life, she was so ill and so bitter that his life had become extremely difficult. She died in December 1 957. Shortly before her death, Grothendieck encountered, through a mutual friend, a young woman named Mireille who helped him care for his mother during her last months. Fascinated and overwhelmed by his powerful personality, she fell in love with him. At the same time, the Grothendieck-Riemann-Roch theorem propelled him to instant stardom in the world of mathematics.

1958-1960: Schemes and EGA The idea of schemes, or more gener­ ally, the idea of generalizing the classi­ cal study of coordinate rings of alge­ braic varieties defined over a field to larger classes of rings, appeared in the work and in the conversation of vari­ ous people-Nagata, Serre, Chevalley, Cartier-starting around 1 954. It does not appear, either from his articles or from his letters to Serre, that Grothen­ dieck paid overmuch attention to this idea at first. However, by the time he gave his famous talk at the ICM in Ed­ inburgh in August 1 958, the theory of schemes, past, present, and future, was already astonishingly complete in his head. In that talk, he presents his plan for the complete reformulation of clas­ sical algebraic geometry in these new terms: I would like, however, to emphasize one point [ . . . ], namely, that the natural range of the notions dealt with, and the methods used, are not really algebraic varieties . . . it ap­ pears that most statements make sense, and are true, if we assume onlyAto be a commutative ring with unit . . . It is believed that a better insight in any part of even the most classical Algebraic Geometry will be obtained by trying to re-state all known facts and problems in the context of schemata. This work is now begun, and will be carried on in a treatise on Algebraic Geometry which, it is hoped, will be written in the following years by ]. Dieudonne and myself. . . . By October 1 958, the work is un­ derway, with Grothendieck sending masses of rough-and not so rough­ notes to Dieudonne for the final writ­ ing-up. In this period, the exchanges between Serre and Grothendieck be­ come less intense as their interests di­ verge, yet they continue writing to each other frequently, with accounts of their newest ideas-fundamental groups, in particular-inspiring each other with­ out actually collaborating on the same topic. In the fall of 1 958, Zariski invited Grothendieck to visit Harvard. He was pleased to go but made it clear to Zariski that he refused to sign the pledge not to work to overthrow the American government which was nec­ essary at that time to obtain a visa. Zariski warned him that he might find

himself in prison; Grothendieck, per­ haps mindful of the impressive amount of French mathematics done in prisons (think of Galois, Weil, Leray . . . ) re­ sponded that that would be fine, as long as he could have books and stu­ dents would be allowed to visit. A break of several months in the let­ ters, due no doubt to the presence of both the correspondents in Paris, brings us to the summer of 1 959. During the gap, Grothendieck's job problem had been solved once and for all when he accepted the offer of a permanent re­ search position at the IHES (Institut des Hautes Etudes Scientifiques), newly cre­ ated in June 1 958 by the Russian im­ migrant Leon Motchane as the French answer to Princeton's Institute for Ad­ vanced Study. He and Mireille had also become the parents of a little girl, Jo­ hanna, born in February 1 959. The let­ ters from this period show that Grothen­ dieck was already thinking about a general formulation of Wei! cohomol­ ogy (planned for chapter XIII of EGA, now familiarly referred to as the Multi­ plodocus), while still working on the fundamental group and on writing the early chapters, whose progress contin­ ues to be seriously overestimated.

1959-1961: The Weil Conjectures: First Efforts The Wei! conjectures, first formulated by Andre Wei! in 1 949, were very pres­ ent in the minds of both Serre and Grothendieck, at least from the early 1950s. Wei! himself proved his conjec­ tures for curves and abelian varieties, and he reformulated them in terms of an as yet non-existent cohomology the­ ory which, if defined, would yield his conjectures as natural consequences of its properties. This was the approach that attracted both Serre and Grothen­ dieck; as the latter explained at the very beginning of his 1 958 ICM talk, the pre­ cise goal that initially inspired the work on schemes was to define, for algebraic varieties defined over a field of charac­ teristic p > 0, a 'Wei! cohomology', i.e., a system of cohomology groups with coefficients in a field of characteristic 0 possessing all the properties listed by Weil that would be necessary to prove his conjectures. Serre used Zariski topology and tried cohomology over the field of definition of the variety; even though this field

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was in characteristicp, he hoped at least to find the right Betti numbers, but didn't. Then he tried working with the ring of Witt vectors, so that he was at least in characteristic zero, but this too failed to yield results. He writes some of his ideas to Grothendieck, but the re­ sponse is less than enthusiastic: "I have no comments on your attempts . . . be­ sides, as you know, I have a sketch of a proof of the Wei! conjectures based on the curves case, which means I am not that excited about your idea. " His mind still running on several simulta­ neous tracks, he adds: "By the way, did you receive a letter from me two months ago in which I told you about the fundamental group and its infini­ tesimal part? You probably have noth­ ing to say about that either!" The im­ pression is that the two friends are thinking along different lines, with an intensity that precludes their looking ac­ tively at each other's ideas. Yet it is only a question of time. Just a few years later, Serre's short note Analogues kiihleriens, an outcome of those same "attempts" which left Grothendieck cold at the time, was to play a fundamental role in his reflections aiming at a vast general­ ization of the Wei! conjectures. On November 1 5 , 1 959, came the news that Michel Raynaud, a 2 1-year­ old student at the time, describes as a thunderclap. Serre writes to Grothen­ dieck: "First of all, a surprising piece of news: Dwork phoned Tate the evening of the day before yesterday to say he had proved the rational ity of zeta ftmc­ tions (in the most general case: arbitrary singularities). He did not say how he did it (Karin took the call, not Tate) . . . It is rather surprising that Dwork was able to do it. Let us wait for confirma­ tion!" To quote Katz and Tate's memo­ rial article on Dwork in the March 1 999 Notices of the AMS: "In 1 959 he electri­ fied the mathematical community when he proved the first part of the Wei! con­ jecture in a strong form, namely, that the zeta function of any algebraic vari­ ety over a finite field was a rational function. What's more, his proof did not at all conform to the then widespread idea that the Wei! conjectures would, and should, be solved by the construc­ tion of a suitable cohomology theory for varieties over finite fields (a 'Wei! cohomology' in later terminology) with a plethora of marvelous properties. "

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THE MATHEMATICAL INTELLIGENCER

Dwork did, however, make use of the Frobenius morphism and detailed jr adic analysis in a large jradic field. It is hard to assess the effect this an­ nouncement had on Grothendieck, be­ cause he did not respond (or his re­ sponse is missing). However, one thing is absolutely clear: Dwork's work had little or no effect on his own vast re­ search plan to create an algebraic­ geometric framework in which a Wei! cohomology would appear naturally. He continues to discuss this in the summer of 1 960, the very year in which he be­ gan running his famous SGA (Seminaire de Geometrie Algebrique). The first year of the seminar, 1 960--6 1 , was devoted to the study of the fundamental group and eventually was published as SGA 1 .

1961: Valuations-and War October 1 961 finds Grothendieck hap­ pily ensconced at Harvard-married, now, to Mireille, as this made it easier for the couple to travel to the US to­ gether, and the father of a tiny son born in July, named Alexander and called Sasha after Grothendieck's father. His letters show him to be full of ideas and surrounded by outstanding students and colleagues: John Tate, Mike Artin, Robin Hartshorne, David Mumford. "The mathematical atmosphere at Har­ vard is absolutely terrific, a real breath of fresh air compared with Paris which becomes gloomier every year." By this time, Grothendieck's vision of the right way to do mathematics is strong and clear, and he is intolerant of

other views. Valuations, for some rea­ son, provoke intense annoyance, and lead to a tense discussion with Serre about their inclusion in the Bourbaki draft for Commutative Algebra. Serre defends them for various reasons in­ cluding the fact that several people had "sweated" over them: "I am much less 'fundamentalist' than you on such ques­ tions (I have no pretension to know 'the essence' of things) and this does not shock me at all . " This i s the first time that a pinch of annoyance can be felt in Serre's tone, underlying the real divergence between the two approaches to doing mathe­ matics. Serre was the more open­ minded of the two; any proof of a good theorem, whatever the style, was liable to enchant him, whereas obtaining even good results 'the wrong way'-using

clever tricks to get around deep theoretical obstacles--could infuriate Grothendieck. These features became more pronounced in both mathemati­ cians over the years; I still recall Serre's unexpected reaction of spontaneous de­ light upon being shown a very modest lemma on obstructions to the construc­ tion of the cyclic group of order 8 as a Galois group, simply because he had never spotted it himself, whereas Grothendieck could not prevent him­ self, later, from expressing bitter disap­ proval of Deligne's method for finish­ ing the proof of the Wei! conjecture, which did not follow his own grander and more difficult plan. Grothendieck, ever the idealist, fires back a response also tinged with irrita­ tion and again making use of his fa­ vorite word 'right' as well as the pic­ turesque style he uses when he really wants to get a point across. "The right point of view for this is not commuta­ tive algebra at all, but absolute values of fields (archimedean or not). The jr adic analysts do not care any more than the algebraic geometers (or even Zariski himself, I have the impression, as he seems disenchanted with his former loves, who still cause Our Master to swoon) for endless scales and arpeg­ gios on compositions of valuations, baroque ordered groups, full subgroups of the above and whatever . . . " These very same letters, as well as a famous one dated October 22, 1 96 1 , and adressed to Cartan, contain a fas­ cinating exchange of views on the sit­ uation in France connected with the Al­ gerian war and the necessity of military service. By October 1 96 1 , the end of the Algerian war of independence was thought to be in sight, but while the two factions awaited a cease-fire, hos­ tilities continued, with violent terrorist acts on the part of Algerian indepen­ dence factions, and even more violent repression from the French police and anti-independence groups such as the OAS (Secret Army Organization). On October 5, a curfew on all "French Mus­ lims from Algeria" was announced. On October 17, thousands of Algerians poured into the streets of Paris to protest. The massacre that occurred on that day left dozens of bloody bodies piled in the streets or floating down the Seine, where they were still to be seen days later.

Grothendieck's letter to Cartan was written from Harvard just four days af­ ter this event Surprisingly for a man whose extreme antimilitarist, ecological views were to become his preoccupa­ tion ten years later, when he left the IHES after a fracas because he discov­ ered that a small percentage of its fund­ ing was of military origin, the tone he adopts in criticizing the effect of the mandatory two years' military service on budding mathematicians is quite moderate. Rather than lambasting mili­ tary service on principle, he emits more of a lament at its effect on mathemat­ ics students. Cartan's response is not included in the Correspondence, but Cartan showed this letter to Serre, who responded to Grothendieck directly, in very typical , simple and pragmatic terms. which probably resonate with the majority: "What is certainly [ . j serious is the rather low level of the current genera­ tion ( 'orphans', etc . ) and I agree with you that the military service is largely responsible. But it is almost certain we will get nowhere with this as long as the war in Algeria continues: an ex­ emption for scientists would he a truly shocking inequality when lives are at stake. The only reasonable action at the moment-we always come hack to this-is campaigning against the war in Algeria itself (and secondarily, against a military government). It is impossible to 'stay out of politics'.'' It is not certain whether Serre himself took any kind of action against the war in Algeria, hut other mathematicians, above all Laurent Schwartz-whose apartment building was plastic bombed by the GAS-cer­ tainly did. Grothendieck replied to Serre, gen­ tly insisting that mathematicians should make some effort to avoid military ser­ vice, not because they should he treated specially, hut because each group of people can he responsible for organiz­ ing its own exemptions. A true pacifist, he writes: "The more people there are who, by whatever means, be it consci­ entious objection, desertion, fraud or even knowing the right people, man­ age to extricate themselves from this id­ iocy, the better.'' Few if any of his French colleagues shared his views, however, and even after the Algerian war wound down, military service re­ mained mandatory in France until 200 1 . .

.

1962-1964: Weil Conjectures More than Ever The letters of 1 962 are reduced to a cou­ ple of short exchanges in September; they are rather amusing to read, as the questions and answers go so quickly that letters containing the same ideas cross. The next letters date from April 1 963. By this time, Grothendieck had already developed many of the main properties of etale and £-adic coho­ mology, which he would explain com­ pletely in his SGA lectures of 1 963-64 ( etale, SGA 4) and 1 964-6'i <£-adic, SGA 'i ) . The £-aclic cohomology was devel­ oped on purpose as a Wei! cohomol­ ogy, and indeed, in his Bourbaki sem­ inar of December 1 964, Grothendieck stated that using it, he was able to prove the first and the fourth Wei! conjectures in April 1963, although he published nothing on the subject at that time. Serre must have been aware of this result, so that it is never explicitly men­ tioned in the letters of April 1 963 or in any others, leaving one with the same disappointed feeling an archeologist might have when there is a hole in a newly discovered ancient parchment document which must have contained essential words. But it was one of Grothendieck's distinguishing features as a mathematician that he was never in a hurry to publish, whether for rea­ sons of priority. or credit, or simply to get the word out . Each one of his new results fitted, in his mind, into an exact and proper position in his vast vision, and it would he written up only when the write-up of the vision had reached that point and not before ( as for actual publishing, this often had to wait for several more years, as there was far too much for Grothendieck to write up him­ self and he was dependent on the help of a large number of more or less will­ ing and able students and colleagues). Instead of writing explicitly about his results on the Wei! conjectures, Grothenclieck's letters ti·om April 1 963 are concerned with recasting the Ogg­ Shafarevitch formula expressing the Euler characteristic of an algebraic curve in his own language, and generalizing it to the case of wild ramification. To do this, he looks for "local invariants·· gen­ eralizing the terms of the Ogg-Shafare­ vitch formula, but although he sees what properties they should have, he doesn't know how to define them. His letter ask-

ing Serre this question bears fruit just days later, as Serre recognized that the desired local invariants can be obtained using the Swan representation, allowing Grothendieck to establish his Euler­ Poincare formula for torsion sheaves on an algebraic curve. Grothendieck did not get around to publishing this result either; it eventually appeared in his stu­ dent Michel Raynaud's Bourbaki semi­ nar of February 1965 (Raynaud recalls a slight feeling of panic the clay before the seminar, when Grothendieck lighthealt­ edly suggested that he talk about a grand generalization of what he had al­ ready carefully prepared.) Grothendieck continued to work on the second and third Wei! conjectures throughout 1963 and 1 964, probably proving the functional equation by the end of 1 966, when the SGA 'i lectures were completed. Rather than attacking the one remaining conjecture directly, he sketched out a vast generalization of the Wei! conjectures and stated the dif­ ficult 'standard conjectures' which re­ main unproven to this day. However, in 197'i, Deligne managed to get around the standard conjectures and prove the remaining Wei! conjecture by using deep and subtle properties of the £-adic cohomology and original, far-from-ob­ vious techniques.

1964-1965: Reduction of Abelian Varieties Over Local Fields-and Motives The Wei! conjectures on algebraic vari­ eties over finite fields, and all of the mathematics that grew up around them, stimulated great interest in the study of algebraic varieties defined over local fields, and consequently also the study of the different types of reduction mod­ ulo the prime ideal of the local field. The sudden tlush of letters exchanged during the fall of 1 964 very largely con­ cerns this theme, concentrating espe­ cially on elliptic curves and abelian varieties ( to Serre's delight and Grothen­ clieck"s annoyance: "It might perhaps be possible to get at least the abelian va­ riety case by this method [ . . . ] This would at least get us a bit further than the sempiternal elliptic curves via Tate's sempiternal construction . . . The irri­ tating thing is that one never seems to he able to get past abelian varieties!") It all began in August 1 964 at the Woods Hole Summer Institute, which

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Serre attended but Grothendieck didn't. On his return, Serre went off on vaca­ tion to the south of France, and from there, sent Grothendieck a very long let­ ter describing, in detail, the main new ideas from what must have been a very lively meeting. Reading over the inter­ ests, conjectures, and recent results of the mathematicians he names-Shimura, Atiyah, Bott, Verdier, Mumford, Ogg, Bombieri, Tate-makes it abundantly clear how the Wei! conjectures moti­ vated much of the work in number the­ ory and algebraic geometry at that time, with local fields playing a major role. In the months following this report, the exchange of letters between Serre and Grothendieck is exceptionally rich, with almost twenty letters exchanged over the summer and fall of 1 964. Even though both epistolarians were in France, the ideas they wanted to share were too complex to discuss only over the telephone, and the twenty or so kilometers separating the College de France from the IHES in Bures pre­ vented them from seeing each other on a daily basis. These letters contain "independence from .f" type results (for instance, the statement that an open subgroup of the inertia group acts unipotently on Te(A) for any given e prime to the residue characteristic if and only if A has a semi­ stable model over a finite extension of its field of definition), reminiscent of the original proof of the fourth Wei! con­ jecture (on Betti numbers) saying that the dimensions of the .f-adic cohomol­ ogy groups are given by the degrees of the factors of the rational function Z(X, t), and thus implying that these di­ mensions are independent of C. This cir­ cle of ideas was the initial stimulation for the idea of motives. Motives made their appearance dur­ ing the same exceptionally active (in terms of letter-writing) period, the fall of 1 964. The first mention of motives in the letters-the first ever written occur­ rence of the word in this context--De­ curs in Grothendieck's letter from Au­ gust 16: "I will say that something is a 'motive' over k if it looks like the .f-adic cohomology group of an algebraic scheme overk, but is considered as be­ ing independent of e, with its 'integral structure', or let us say for the moment its 'iQ' structure, coming from the the­ ory of algebraic cycles. The sad truth is

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THE MATHEMATICAL INTELLIGENCER

that for the moment I do not know how to define the abelian category of mo­ tives, even though I am beginning to have a rather precise yoga for this cat­ egory, let us call it M(k) . " He is quite hopeful about doing this shortly: "I sim­ ply hope to arrive at an actual con­ struction of the category of motives via this kind of heuristic consideration, and this seems to me to be an essential part of my 'long run program' [sic: the words 'long run program' are in (a sort of) English in the original). On the other hand, I have not refrained from mak­ ing a mass of other conjectures in or­ der to help the yoga take shape . " Serre's answer is unenthusiastic: " I have received your long letter. Unfor­ tunately, I have few (or no) comments to make on the idea of a 'motive' and the underlying metaphysics; roughly speaking, I think as you do that zeta functions (or cohomology with Galois action) reflect the scheme one is study­ ing very faithfully. From there to pre­ cise conjectures . . . " But Grothenclieck was not deterred from thinking directly in terms of motives in order to motivate and formulate his statements. The let­ ters of early September constitute the first technical discussion as to whether something is or is not a motive, here taken in the simple sense to mean that a family of .f-aclic objects forms (or comes from) a motive if each member of the family is obtained by tensoring a fixed object defined over iQ with iQe. Several more letters are exchanged, with all of the previous subjects still be­ ing touched upon: functional equations, good reduction of abelian varieties, es­ pecially elliptic curves. Then there is a silence of several months, interrupted only by a short letter from Serre in May 1 965, responding to a phone call and giving an elegant two-page exposition of the theory of the Brauer group and fac­ tor systems. Silence again until August 1965, when Grothenclieck aclclressecl to Serre one of the key letters in the his­ tory of motives: the one containing the standard conjectures. This letter-writ­ ten four months after the birth of Math­ ieu, his third child with Mireille-exucles an atmosphere of intense creativity in a totally new direction. This is the period in which Mireille described him as work­ ing at mathematics all night, by the light of a desk lamp, while she slept on the sofa in the study so as to be near him,

and woke occasionally to see him slap­ ping his head with his hand, trying to get the ideas out faster. Grothenclieck termed the first of these conjectures, called conjecture A, "the 'minimum minimorum' to be able to give a usable rigorous definition of the con­ cept of motive over a field." He also makes some initial attempts at sketching out proofs or directions of proofs of the conjectures, which, however, resisted his attempts and all other attempts to prove them. The letter ends with what consti­ tutes the major obstacle: "For the mo­ ment, what is needed is to invent a process for deforming a cycle whose di­ mension is not too large, in order to push it to infinity. Perhaps you would like to think about this yourself? I have only just started on it today, and am writing to you because I have no ideas." Although not the last letter, this let­ ter represents the end of the Grothen­ dieck-Serre correspondence in a rather significant way, expressing as it does the mathematical obstacle which pre­ vented Grothendieck from developing the theory of motives further; the stan­ dard conjectures are still open today. Of course he remained incredibly intense and hardworking for several more years, continuing the SGA seminar un­ til 1 969, the writing of the EGAs and ever more research. Yet this letter has a final feel to it. Only two more letters elate from before the great rupture of 1970: then fourteen years of silence.

1984-1987: The Last Chapter The six letters from these years included in the Correspondence--a selection from a much larger collection of exist­ ing letters-are intriguing and revealing, yet at the same time somewhat mis­ leading. From the tone of some of Grothenclieck's comments ("As you probably know, I no longer leave my home for any mathematical meeting, whatever it may be," or "I realize from your letter that beautiful work is being clone in math, but also and especially that such letters and the work they dis­ cuss deserve readers and commentators who are more available than I am,") it may seem as though by the 1980s, he had completely abandoned mathemat­ ics. Quite the contrary, although he did stop working in mathematics for months at a time, there were other months dur­ ing which he succumbed to a mathe-

matical fever, in the course of which he filled thousands of manuscript pages with "grand sketches" for future direc­ tions, finally letting his imagination roam, no longer reining himself in with the necessity of advancing slowly and steadily, proving and writing up every detail. A famous text ("Sketch of a Pro­ gram"), three enormous informally writ­ ten but more or less complete manu­ scripts and thousands of unread handwritten pages from his hand date from the 1980s and 1990s, describing more or less visionary ways of renew­ ing various subjects as concrete as the study of the absolute Galois group over the rationals, or as abstract as the the­ ory of n-categories. And this does not count the many thousands of non­ mathematical pages he wrote and still writes. At the time of the exchange of letters included in the published Corre­ spondence, Grothendieck had just com­ pleted his monumental mathemati­ co-autobiographical work Recoltes et Se­ mailles (Reaping and Sowing), retracing his life and his work as a mathemati­ cian and, over many hundreds of pages, his feelings about the destiny of the mathematical ideas that he had created and then left to others for completion. He sent the successive volumes of this work to his former colleagues and stu­ dents. The exchanges between Serre and Grothendieck on the topic of this text underscore all of the differences in their personalities already so clearly vis­ ible in their different approaches to mathematics. Serre, a lover as always of all that is pretty ("les jolies choses" is one of his favorite expressions), clean-cut, attrac­ tive and economical, viscerally repelled by the darker, messier underside of things, reacts negatively to the negative ("I am sad that you should be so bitter about Deligne . . . " ) , positively to the positive (" . . . we complemented each other so well for ten or fifteen years, as you say very nicely in your first chap­ ter . . . On the topic of nice things, I very much liked what you say about the Bourbaki of your beginnings, about Cartan, Wei!, and myself, and particu­ larly about Dieudonne . . . ") and un­ comprehendingly to the ironic ("There

must be about a hundred pages on this subject, containing the curious expres­ sion 'the Good Lord's theorem' which I had great difficulty understanding; I fi­ nally realized that 'Good Lord's' meant it was a beautiful theorem."*) Grothendieck picks up on this at once, and having known Serre for twenty years, is not in the least sur­ prised: "As I might have expected, you rejected everything in the testimony which could be unpleasant for you, but that did not prevent you from reading it (partially, at least) or from 'taking' the parts you find pleasant (those that are 'nice', as you write!)" After all, "One thing that had already struck me about you in the sixties was that the very idea of examining oneself gave you the creeps." It is true enough that self-analysis in any form strikes Serre as a pursuit fraught with the danger of involuntar­ ily expressing a self-love which to him appears in the poorest of taste. Grothen­ dieck, trying in all honesty to take a closer look at his acts and feelings dur­ ing the time of his most intense math­ ematical involvement, speaks of his "ab­ sence of complacency with respect to , myself. . Serre, disbelieving in the very possibility of self-analysis without com­ placency, and already struggling with the embarrassment of chapter after chapter of self-observation, writhes at this phrase which-worse than ever­ analyzes the analysis, and wonders how Grothendieck could have typed it at all without laughing: "How can you?" But where, exactly, does he perceive complacency, Grothendieck asks in some surprise. There is no need for Serre to answer. It is obvious that for him, the act of looking at oneself im­ plies self-absorption, which as a corol­ lary necessarily implies a secret self-sat­ isfaction, something which perhaps exists in everyone, but should remain hidden at all costs. And then, if one is going to do the thing at all, should one not do it com­ pletely? Pages and pages of self-exam­ ination, of railing because the beautiful mathematical work accomplished in the fifties and sixties met a fate of neglect after the departure of its creator­ mainly because basically no one, apart

from perhaps Deligne, was able to grasp Grothendieck's vision in its en­ tirety, and therefore perceive how to ad­ vance it in the direction it was meant to go. But Serre reproaches him for the fact that the major question, "the one every reader expects you to answer," is neither posed nor answered: "W'by did

you yourself abandon the work in ques­ tion?" Clearly annoyed by this, he goes on to formulate his own guesses as to the answer: "despite your well-known energy, you were quite simply tired of the enormous job you had taken on . . . " or "one might ask oneself if there is not a deeper explanation than sim­ ply being tired of having to bear the burden of so many thousands of pages. Somewhere, you describe your ap­ proach to mathematics, in which one does not attack a problem head-on, but one envelopes and dissolves it in a ris­ ing tide of general theories. Very good: this is your way of working, and what you have done proves that it does in­ deed work, for topological vector spaces or algebraic geometry, at least . . . It is not so clear for number the­ ory . . . whence this question: did you not come, in fact, around 1968-1970, to realize that the 'rising tide' method was powerless against this type of question, and that a different style would be nec­ essary-which you did not like?" Grothendieck's answer to this letter and the subsequent exchanges are not included in the present publication, but he did answer in fact, referring to a pas­ sage in Recoltes et Semailles in which he powerfully expresses the feeling of spir­ itual stagnation he underwent while de­ voting twenty years of his life exclu­ sively to mathematics, the growing feeling of suffocation, and the desper­ ate need for complete renewal which drove him to leave everything and strike out in new directions. Reading Recoltes et Semailles, it is impossible to believe that Grothendieck felt that his mathe­ matical methods were running into a dead end, whatever their efficacity on certain types of number theoretic prob­ lems might or might not have been. His visions both for the continuation of his former program and for new and vast programs are as exuberant as ever; what changed was his desire to devote him-

'A misunderstanding! Grothendieck's sarcastic references to the 'Good Lord 's theorem' meant that this theorem was not attributed by name to its author, whom he felt to have been neglected and mistreated by the mathematical establishment.

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self to them entirely. Recoltes et Se­ mailles explains much more clearly than his letters how he came to feel that do­ ing mathematics, while in itself a pur­ suit of extraordinary richness and cre­ ativity, was less important than turning towards aspects of the world which he had neglected all his life: the outer world, with all of what he perceived as the dangers of modern life, subject as it is to society's exploitation and vio­ lence, and the inner world, with all its layers of infinite complexity to be ex­ plored and discovered. And, apart from the sporadic bursts of mathematics of the 1980s and early 1990s, he chose to devote the rest of his life to these mat­ ters, while Serre continued to work on mathematics, always sensitive to the ex­ citement of new ideas, new areas, and new results. In some sense, the differ­ ence between them might be expressed by saying that Serre devoted his life to the pursuit of beauty, Grothendieck to the pursuit of truth. Equipe Analyse Algebrique Universite de Paris 6 1 75 rue du Chiraleret 750 1 3 Paris France e-mail: [email protected]

Google's Page Ranl< and Beyond: The Science of Search Engine Ranl
REVIEWED BY PABLO FERNANDEZ

-;:-' irst Scene: California,

1998. ]on

/--, Kleinberg, a young scientist

working at IBM's Almaden Research Center in Silicon Valley, is pre­ senting his HITS (Hypertext Induced Topic Search) algorithm. Almost simul­ taneously, at nearby Stanford Univer-

68

THE MATHEMATICAL INTELLIGENCER

sity, two Computer Science doctoral stu­ dents, Sergey Brin and Larry Page, are putting the finishing touches on their PageRank algorithm, a future core com­ ponent of Google's search engine. Both projects are based on an innovative idea: using the hyperlink structure of the web to enhance a search engine's results.

Second Scene: Madrid-Mountain View, California, 2006 (same charac­ ters). Kleinberg, now a professor at Cornell University, is receiving the Rolf Nevanlinna Prize at the International Congress of Mathematicians in Madrid. Brin and Page, sitting at their offices in Google's headquarters, are plotting some new google-gadgets to delight the immense host of googlemaniacs. Not a lot of time separates these two scenes, but it was enough to com­ pletely revolutionize the way we use the web. Google has become the stan­ dard for web searching. Its impact has moved far beyond the world of t ech­ nology; it pervades our daily lives. While Kleinberg's work has not been so successfully commercially devel­ oped, it has earned him the recogni­ tion of the mathematical community with the aforementioned Nevanlinna Prize, given "for outstanding contribu­ tions in mathematical aspects of Infor­ mation Science. " The book under review gives a com­ prehensive overview of the state-of-the­ art technology of web search engines, putting special emphasis, as the title suggests, on ranking procedures. A search engine is designed to perform several tasks. First, it collects the information contained in the myr­ iad pages of the web ("crawling, " in the jargon). Next, all of this information is stored, compressed, and processed to build content indexes. Finally comes the interaction with the user. When a user types a query the engine must find in the indexes the pages that contain relevant information and must show this outcome as an ordered list. Now comes the key point, perhaps the most important ingredient of the search process: In which order should the in­ formation be displayed? It is desirable, even essential, that most of the time the user be able to find the most relevant information in, say, the first 10 or 20 displayed pages.

The scale of the problem is enor­ mous. There are billions of pages on the web, with an estimated average page size of 500 Kb, and all of these figures are increasing day by day. The web is also dynamic: most pages change their contents on a regular ba­ sis and millions of pages are added (or disappear) each year. The engine, of course, must respond to queries in real time! Considering this, it is remarkable that the mathematical ideas behind the ranking algorithms that make effective searching possible require only some basic tools from linear algebra. Remember our basic question: after the relevant pages for a query have been selected, we need to assign a score to them to determine the order in which to display them. Part of this score is related to the position of the query term within the document (in the title, in the body) or, for combined searches, to the distance between the terms in the text. Each search engine has dif­ ferent rules for assigning this "content score . " But there is another score, to be combined with the former score, to make up the overall score, which should not be query dependent, but should reflect the global relevance of each page. This is the "popularity score, " given by the ranking schemes that are the focus of our book. The idea is simple: just regard the hyperlinks as recommendations, with two extra comments: ( 1 ) the status of the recommender is important, and (2) the recommendation should drop in weight if the recommender is too gen­ erous giving them. In short, a web page is important (gets a high popularity score) if it is pointed to by other im­ portant (high ranked) pages. All these features can be formalized with a fairly simple mathematical model. Let us imagine that there are n pages in the web, and let us consider the nxn adjacency matrix H for the internet, viewed as a directed graph. This inter­ net graph has web pages as vertices and hyperlinks as edges joining these ver­ tices. The ( i,j) entry of H is 1 , if there is a directed edge from vertex i to ver­ tex }, and 0 otherwise. To fulfil demand (2) above, PageRank considers a (row) normalized version, H ' , of this matrix, in which all the entries of each row are

divided by the total sum of the row (the number of out-links of the page). The resulting matrix is (almost) stochastic, so its entries may be viewed as proba­ bilities. A nice interpretation of this for­ mulation is a random surfer traveling along the web, following the links be­ tween nodes, and uniformly choosing his next destination from among the links in the current node. It turns out that the ranking list we need is just the stationary vector of the Markov chain associated to matrix H ' . In fact, some adjustment must be done in order to have stochasticity, because H could have zero rows (pages with no out­ links). In these cases we can replace these 0-rows with ll n entries. Call this adjusted and normalized matrix j But now we face a computational problem: How can we determine this vector' Remember the matrix is ex­ tremely large. The idea is now to make a second adjustment, to guarantee that the new matrix also be primitive. Then the stationary vector exists, is unique, and, as it is the eigenvector associated to the dominant eigenvalue of the ma­ trix, can be calculated with a simple and fast numerical procedure such as the power method. The reader should be aware that the Perron-Frobenius Theorem on positive ( or non-negative) matrices plays a key role in all these arguments. The precise adjustment is given by

G = a } + ( 1 - a)

1 --­ T e · e

n

where e stands for the vector of all ones and a is a number between 0 and 1 . G is known as Google's matrix. Page­ Rank output is just its dominant eigen­ vector, which can be obtained with an iterative method such as ak+ 1 = ak G. In terms of the random surfer, this ad­ justment brings in a new possibility, namely that with probability I - a the surfer gets bored of following the links and "teleports" to any page of the web. The reader could argue that this is an "artificial" matrix (for instance, Google's choice of the value of the teleportation constant, a, is around O.RS). And in­ deed it is, but it allows effective com­ putation of the ranking vector and, above all, it works! The resulting out­ put is incredibly good at assigning rel­ evance to web pages .

So this is the model. Not too com­ plicated, is it? Of course, there are many details to complete: computational as­ pects (such as convergence rates of the iterative scheme, sensitivities to the pa­ rameters), possible improvements of the numerical procedures (or the model itself) . . . Most of this can be found in the fif­ teen chapters of the book under re­ view. The first three chapters introduce the reader to the main features of web searching, including some review of the traditional methods of information re­ trieval. Chapters 4 to 10 deal with the mathematics behind Google's algo­ rithm: the Markov chain model, nu­ merical procedures, sensitivities to pa­ rameters, convergence issues, methods for updating the rankings, etc . All the mathematical concepts used in the book are treated in detail in the "Math­ ematical Guide" of Chapter 1 '5 : linear algebra, Markov chains, Perron-Frobe­ nius Theory, etc. Chapter 1 1 includes a brief review of Kleinberg's HITS algo­ rithm; other ranking methods are men­ tioned in Chapter 1 2 . Chapter 13 dis­ cusses some questions related to the future of web information retrieval, including spam and personalized searches. I would have liked to see a more comprehensive discussion of eth­ ical issues such as privacy and censor­ ship. Considering that Google has be­ come the standard source for information (you appear in Google or you are nothing! ), these are really dis­ turbing topics. But probably a whole new book could be written on this. The book under review is excel­ lently written, with a fresh and engag­ ing style. The reader will particularly enjoy the "Asides" interspersed throughout the text. They contain all kind of entertaining stories, practical tips, and amusing quotes. "How do search engines make money?,,. "Google bombs , '' ''The Google dance, " and "The ghosts of search" are some of these stimulating asides. The hook also con­ tains some useful resources for com­ putation: Pieces of Matlab code are scattered throughout the hook, and Chapter 14 contains a guide to web re­ sources related to search engines. Despite the technical sophistication of the subject, a general science reader can enjoy much of the book-certainly

Chapters 1 to 3, and also Chapter 1 3. With some basic knowledge of linear al­ gebra, the description of the model (Chapters 4 and 5) can be followed with­ out problem. Chapters 6 to 1 2 are more technical, and they are intended for ex­ perts. The authors provide a webpage (http:!/pagerankandbeyond.com/) that includes a list of errata for this edition. Departamento de Matematicas Facultad de Ciencias Universidad Aut6noma de Madrid Ciudad Universitaria de Cantoblanco 28049 Madrid, Spain e-mail: [email protected]

Mathematical Form : John Pickering and the Architecture of the Inversion Principle hy

Pamela Johnston (ed.)

With contributions

hy

Mohsen

Mostafavi, George L. Legendre, John Pickering, Chris Wise, John Silver, and John Sharp LONDON: ARCHITECTURAL ASSOCIATION, 2006, 96 PP., £15.00, ISBN 978-1-902902-37-1

REVIEWED BY KIM WILLIAMS

1

ohn Pickering's models--or build­ ings-of geometrical forms, derived through the process of inversion, are the subject of this little book. The book is essentially a catalogue of the 2002 exhibit of hb works in the gallery of London's Architectural k;sociation. Pickering is an artist (the exhibit in­ cluded his earlier studies of the human form, hut this book does not), who at a certain point, in his own words, "be­ came anti-nature" and dedicated him­ self to the derivation and visualization of complex three-dimensional geomet­ rical forms. To create his forms, Pick­ ering employs the "inversion principle," transformations of either plane or solid

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Ellipsoid consisting of one half of the hyperboloid of two sheets and one half of the hyperboloid of one sheet. 1971-1973. Card. 14 em X 27 em X 19 em. Photograph by Sue Barr. Reproduced by permission of AA Publications, http://www . aaschool.ac.uk/ publications.

figures first studied intensively in the 1820s through the 1840s by mathemati­ cians such as Jakob Steiner and Jean­ Victor Poncelet. Mathematician John Sharp's concise and clear essay that concludes the book should have opened it. His explanation (p. 87) of the inversion principle makes clear the basis of Pickering's work as well as its process: Inversion is simple. In the plane it is a kind of reflection in a circle, which maps all points inside a cir­ cle to points outside and vice versa; with the centre of the circle being a special point where all points at in­ finity map. In space, the mapping is of the inside and outside of a sphere. . . . The distances of the points from the centre are related by the formula: MP MQ r2 , where r is the ra­ dius of the circle. The term inversion arises be­ cause, to find the length MQ in or­ der to find the inverse point to [a given] point P, the forllJula is re­ arranged so that MQ �p· Understanding the inversion principle on which Pickering's forms are based, ·

=

=

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and placing them in an historic con­ text, our appreciation of both the metic­ ulous process of derivation and the fi­ nal product are greatly enhanced. I hope in any case that architects are bet­ ter at designing buildings than they are at designing hooks (don't get me wrong; those of you who know me will know that this is the pot calling the ket­ tle black: I am an architect who designs books). The page numbers and some captions in this one are lost in the bind­ ing rather than being on the outside margins for easy reading. It would be helpful if the book came with a mag­ nifying glass, as the OED does, for reading the captions and viewing the figures that are such a necessary part of John Sharp's essay. (I have heard it said, as far as required courses in math­ ematics in the architecture curriculum are concerned, that the architects would like to throw out the mathe­ maticians but keep the math, and per­ haps this attitude is reflected here.) The photographs of the objects, mostly by Sue Barr with some by Arthur Picker­ ing, are very good. That Pickering's work-a very de-

liberate and rigorous exercise in de­ velopment and realization of geomet­ ric form-should be the subject of an exhibit aimed primarily at architects says more about the present state of ar­ chitecture than about the artist. To his credit, Pickering himself does not say that his models are intended to be seen as architecture. None of the essays in the book says exactly how the archi­ tect might use the inversion principle as a design tool. But much space in the essays, by architects Mohsen Mostafavi, George Legendre, John Silver, and en­ gineer Chris Wise, is dedicated to the possible relationship between Picker­ ing's models and architecture. The models are "prototypes of giant struc­ tures waiting to be realised" (p. 7) and "look like buildings" (p. 22). Silver is the most unequivocally enthusiastic: "If ever conceived on a civic scale, John's sculptures would be incredible!" (p. 83). Only the dour engineer-always the architect's party-pooper--expresses skepticism: "As a basis for fledgling buildings, I feel Pickering's inversion principle is not really so good" (p. 80) . Why do Pickering's models look like buildings? For one thing, they are mass­ ings of geometric volumes that appear as though they could be containers of space. For another, they look-and are--engineered, in the sense that be­ hind the built form is an equally im­ pressive mass of manually executed calculations. For still another, the sur­ faces are only intimated; what we ac­ tually see is the framework that sup­ ports the surface, and that framework looks like floors, columns, and beams. But perhaps the biggest reason of all is that contemporary architects-first of all Frank Gehty-have taught us that buildings can have weird shapes; any­ thing you can imagine can become a building. (Is this why Pickering envi­ sions one of his models "as a large built structure 'perhaps in Spain' " (p. 7), the country that brought us Gehry's Guggenheim, where obviously any­ thing can be built?) As Sydney Pollack's 2005 documen­ tary "Sketches of Frank Gehry" showed, Gehry's process of form generation is the exact opposite of Pickering's. Gehry starts with the model, subjects it to his very personal tastes and instincts ("I don't like this, cut it, bend it, fold it,

corrugate it . . . there!"), then lets his technicians scan it and digitalize it in order to allow the computer to gener­ ate the necessary instructions for its full-scale construction. Pickering, in contrast, hegins with an equation, man­ ually calculates a series of vector lengths, maps the coordinates in space, then painstakingly cuts out and assem­ bles the required pieces to construct his final model. What Gehry and Pickering share is a fascination and predilection for form. Indeed, most of the architectural pro­ fession focuses on the issue of form these days (or so it seems from the pop­ ular architectural press, and from the buildings that most capture the public's attention). Such a preference is proper to the artist, hut improper to the ar­ chitect, because if formal questions predominate, they do so to the detri­ ment of function; that is, the building's program, or functional requirements, are made to fit inside the form con­ ceived by the architect rather than con­ tributing to its determination. It is in­ teresting to hear Geh1y describe the rectangular rooms where art is hung in the Bilbao Guggenheim-presumably the function of an art museum-as "ba­ nal space, " whereas his much more heroic spaces tend to overwhelm the art displayed in them. This is not a new dilemma in museum design: Wright's Guggenheim Museum in New York City, a top-heavy, descending spiral, was-and is-severely criticized he­ cause the downward force of gravity along the ramp tends to pull the spec­ tator past the art without giving him time to linger over it. If the architect adopts a subjective method of design, he may willfully subjugate function to form, but he at least has the opportu­ nity to let the building's form morph if function requires it. Adopting a rigor­ ous method of form generation only ex­ acerbates the problem, for if the form is to remain true to the generating prin­ ciple, the functional requirements are necessarily overpowered. This is why engineer Chris Wise writes (p. 80) Mutation, good or bad, has no place in Pickering's world. Deliberate in­ tervention does not feature either. There's no chance to "add a bit here" if the volume ends up a hit mean or looks a hit wobbly: if one little piece

is tweaked, everything else has to follow the inversion principle and change too, whether it wants to or not. The inversion principle is not the only form generator that imposes the im­ possibility of invariance; a rigorous sys­ tem of proportions does the same thing. Although rigor is a healthy prin­ ciple for mathematics, it is not always healthy for architecture. The obsession with form in archi­ tecture may arise out of the fact that today's architects have such powerful tools to work with-rendering tools such as AutoCAD that allow architects almost instantly to see their designs in three dimensions, and other more mathematical tools such as CATIA (originally developed for the French aerospace industry) , which models forms. Pickering, however, uses no such software. so his fascination with form is not derived from the possibilities pre­ sented by the tools, but rather out of a love of form for form's sake. As an artist, he has no building program to respect; his forms don't have to house specific functions. But architects' willingness to con­ centrate on form for form's sake may in itself explain why "the status of ar­ chitectural composition is presently at an all-time low" ( p. 25); that is, by con­ centrating on form alone, architects are shirking their other responsibilities, those of providing buildings that are functional and meaningful for the users. This is not to imply that mathematical principles shouldn't be used in the gen­ eration of architecture, hut simply that in themselves they are not sufficient. I think this presentation of Picker­ ing's work is misdirected. Is Pickering's work architecture, as the subtitle of this hook suggests' It is not. (I wonder if it is even art, hut I'm not opening that can of worms here . ) It is geometry: beautiful, complex, spine-tingling, never­ before-seen geometry. Trying to make it what it is not diminishes his accom­ plishment, which is to allow us to see forms that we perhaps cannot even imagine. In this he follows the footsteps of Leonardo cia Vinci when illustrating the solids for Luca Pacioli's De divina proportione. Unfortunately, we are never told how Pickering discovered

the inversion principle nor why he chose to explore it. In his own essay on "Music and the Inversion Principle, " h e says that h e i s now moving toward fractals, so we await the results of this new mathematical exploration by the artist. The main value of the book is this: Pickering shows that advanced mathe­ matics does not depend on advanced tools, only on advanced thinking. Mathematicians, enjoy. Architects, beware. Kim Williams Books-P.I.05056220485 Via Cavour 8- 1 01 23 Turin, Torino Italy e-mail: [email protected]

Fear less Symmetry: Exposi ng the H idden Patterns of N umbers by Avner Ash and Robert Gross PRINCETON, NEW JERSEY, PRINCETON U N IVERSITY PRESS, 2006, 302 PP., US $24.95, ISBN-10: 0-691-12492-2; ISBN-13: 978-0-691-12492·6

l

REVIEWED BY PAMELA GORKIN

n an article in the Bulletin ofthe Lon­ don Mathematical Society entitled "Generalized non-Abelian Reciproc­ ity Laws: A Context for Wiles' Proof, " A vner Ash and Robert Gross explain generalized reciprocity to mathemati­ cians who know only "basic algebra" and the definition of homology groups. The hook Fearless Symmetry: Exposing the Hidden Patterns of Numbers is Ash and Gross's attempt to reach an even broader audience. Fearless Symmetry, as we discern from the cover of the book, is aimed at "math buffs . " The au­ thors' goal is not to explain the history of Fermat's Last Theorem-that has been done, and done well, before. Though the authors often refer to Si­ mon Singh's book, Fermat 's Hnigma, they do not repeat parts of Singh's hook in Fearless Symmetry. Instead, Ash and Gross focus on describing the proof of Fermat's Last Theorem and, in doing

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so, they intend to help the nonmathe­ matician understand what the research mathematician does and why he or she does it. They do not explain why peo­ ple "do math" by resorting to a list of applications. Instead we learn that pure mathematicians are drawn to the beauty of the subject and "the excite­ ment of the chase, " and "like all risk­ takers, mathematicians labor months or years for these moments of success. " The authors act a s tour guides for the uninitiated and, they never lose sight of their audience. The authors' trip begins with "Al­ gebraic Preliminaries . " They introduce groups, modular arithmetic, complex numbers, and the idea of a one-to-one correspondence. This section con­ cludes with a gentle introduction to equations, varieties , and quadratic rec­ iprocity. At this point it is difficult to get lost in the mathematics: each chap­ ter begins with a "Road Map" that shows readers where the authors are headed. Though the material sounds as if it might be inaccessible to some­ one with little background in mathe­ matics, the authors have taken all pos­ sible precautions to ensure that the reader will follow along to the end of the chapter. There are plenty of ex­ amples and exercises. In addition, one of the authors' most impressive skills is their ability to mind-read; they an­ ticipate the ways in which someone might misread the text, and the au­ thors try to correct potential errors be­ fore they occur. In the final chapter of this part of the book, for example, they introduce the Legendre symbol, de­ noted (�) , followed by a stern "WARN­ ING: The Legendre symbol . . . is not a divided by p. " The authors proceed to a partial proof of the Law of Qua­ dratic Reciprocity, but they inform readers that they should "feel free to skip it" if they so choose. When the Legendre symbol reappears, more than one hundred pages later, it does so with a reminder. " Remember that (]f) is called the Legendre �ymhol. It does not refer to dividing w by p. " If at this point you have forgotten what the Legendre symbol is, look it up in the index. It will he there, because everything you need is in the index of this user-friendly book. In Part Two, titled Galois Theory and

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Representations, the authors pick up the pace a bit. They begin with a chap­ ter titled Galois Theory, which is sig­ nificantly more elementary than one might imagine: It includes an introduc­ tion to polynomials and their zeros, the field of algebraic numbers, the absolute Galois group of Q, and a discussion of symmetry. A brief and dense discussion of elliptic curves and matrices follows, before the authors turn to "the heart of this hook, or at least the pericardium"­ group representations. The authors are headed toward a discussion of Frobe­ nius elements-a discussion they know will tax the reader: " If you choose to skip the rest of this discussion, you need to know only . . . " three facts (or, three increasingly complicated ver­ sions of one fact), presented in a bul­ leted list for the reader's convenience. The nonmathematician would he well­ advised to follow the authors' sugges­ tion to skip the rest of the section as well as the appendix. (It should be noted, however, that while the authors often indicate where to begin skipping sections, they rarely say where to stop skipping sections, and skipping too much is problematic. ) But with-re­ markably-only this brief exception, the first two parts of the book should he accessible to a relatively wide audi­ ence. That brings us to the third part of the hook, reciprocity laws. If some­ one has made it this far, it would be a pity to stop now. At this point in Fearless Symmetry, the authors' first, in-depth description of what Wiles did appears: [T]he proof went like this: Suppose you have two nonzero nth powers that add up to another nth power. From this equation, you can deduce the existence of a certain Galois rep­ resentation. The strong reciprocity law you have already proved implies the existence of a black box with a certain label. You work out the la­ bel of the black box. You go to your inventory of those black boxes-and there are not any with that label! Contradiction. At this point, the number of theo­ rems, definitions, and detailed expla­ nations increases. There are more skip­ pahle details too, but even at this late point in the hook, the authors will not

let their audience flounder. They end the detailed chapter on reciprocity (ap­ proximately) the same way they began the article in the Bulletin qf the London Mathematical Society-with a brief dis­ cussion of the Langlands program. This leads to their most difficult chapter, "Fermat's Last Theorem and General­ ized Fermat Equations. " There is n o question that the third part of the book will be hard on the mathematically uninitiated. Neverthe­ less, the reader is in such good hands through most of the text that he or she is likely to be willing to plow through the toughest parts of the book in or­ der to see what Ash and Gross want to show. And this brings us to the in­ evitable question: for whom is this book intended? F-earless Symmetry is not intended for someone who is bothered by a definition such as "the variezy S defined by a Z-equation (or a system of Z-equations) is the func­ tion that assigns to any number sys­ tem A the set of solutions S(A) of the equation (or system of equations) . " So, it's probably not for the seasoned mathematician. That's not to say that mathematicians won't learn something from this book; they may well learn quite a bit from it. The book is in­ tended for the neurologist who loved mathematics, but just didn't have time to continue studying it; for the musi­ cian who studied mathematics for a few years before deciding music was her calling; and for the sociologist who bought Godel, Escher, Bach when it was fashionable to do so and then ac­ tually read it. This is not just a book about the proof of Fermat's Last Theorem or the reciprocity laws. Recognizing that "many textbooks make no effort to tell about directions that are still to be ex­ plored, conjectures that are unproven, nor, of course, of ideas that are yet to be formulated, " the authors want to "give you a little insight into what mathematicians do . " That is precisely what Fearless Symmetry does so well.

Department of Mathematics Bucknell University Lewisburg, PA 1 7837 USA e-mail: [email protected]

The Life of N umbers: From an Idea by Antonio J . Duran by A ntonio ]. Duran, GeorMes {frah, and Alberto Manguel A. K. PETERS, WELLESLEY, MASSACHUSETIS, 2006, 180 PP. ISBN-10: 1568813252. ISBN-13: 978-1568813257, $38.00

l

REVIEWED BY VAGN LUNDSGAARD HANSEN

"""\. umbers have a secret life of their \ own-if you believe the authors

I

of this marvellous book. Numbers make themselves visible when we need them for counting, measuring, calcula­ tions, accounting, predictions, and so forth, and over the millennia they have matured to be our best friends. Our dis­ tant ancestors in prehistoric times were already aware of small integers. One early mathematician became so fasci­ nated by the number of his or her fin­ gers that he or she dearly displayed a handful of fingers in a cave-wall paint­ ing in Perch-Merle, France, about 30,000 years ago. Bigger numbers were more difficult to visualise, and it was a long struggle for humankind to represent them adequately. The decimal system, which developed from modest begin­ nings in Babylonia about l HOO Be to fundamental new ideas in India about 600 AD and its final shaping by the Arabs about HOO AD, was not introduced in Eu­ rope hefore the Renaissance. In fact, the decimal system really superseded the Roman number system only after the French Revolution in 1 789. The book under review is a fasci­ nating mixture of mystery, specula­ tions, and concrete facts. To reassure potential readers, let me say that I never felt any doubts as to the category in which each part of the book should be located. The structure, the contents, the tone, and the spirit of the hook are probably best described in these words of Duran from the Preface: Tbe L!fe ol Numbers is divided into three parts, each written by a dif­ ferent author. Alberto Manguel has written a sort of introduction in

which he descrihes the landscape where the numbers will run free; as the reader will have guessed, this landscape is none other than the written page. Georges Ifrah will tell us, among other things, how we learnt to count, the different cos­ tumes that numbers have worn throughout history in different parts of the world, how the Hindus dis­ covered what would ultimately be­ come the numbering system we use today, how the Arabs brought it to the West-first to Spain-and how it gradually spread throughout Eu­ rope in the late Middle Ages. My own contribution will be to search for the numbers behind some of the most important events and circum­ stances in the history of mankind, whether these are in our own phys­ iology, in the birth of writing, as­ tronomy, and the measurement of time. or in the changing fortunes of trade. My story begins in the caves of the Paleolithic, and brings us to the Renaissance to tell the love af­ fair between numbers and the print­ ing press; it even includes a special homage to the Vigilanus Codex, that mythical manuscript from the late l Oth century, which is today in the library of the monaste1y of El Esco­ rial. The texts in this book are in­ tended more to suggest than to ex­ plain something which because it is staring us in the face, may be hid­ den behind the life of numbers and the objects-cave walls, day tablets, papyri, coins, pre-Romanesque manuscripts, Mayan codices, printed books, engravings, typographical designs. . These have provided the narrative thread for a tale that encompasses many centuries and topics that in our day would be viewed as incompatible, although perhaps they are every bit as com­ patible as the many cultures, both extinct and living, which gave birth to numbers and the artefacts on which they appear. The title is well put. Reading the book, one gradually begins to think of numbers as "living creatures," having a life of their own in the written word or in illustrations of all kinds. One may begin to understand what Manguel means when, on page 1 o, he discusses

the "double nature of a page in a book," and the relation between a page and its page number. And I am sure that you will appreciate his words on page 25 about Descartes "whose scrib­ bling habits [in the page margins] amounted to a conversation [with the book] " . You really feel sympathy with the numbers when Duran explains that letters in medieval times often were used as stand-ins for numbers and on page 1 58 goes on to say "Not that it is any dishonour for numbers to be dressed up as letters, but l can't imag­ ine they were very happy with those disguises when it was not even carni­ val time. " Numbers started their life i n all mod­ esty on fingers and toes, and grew to power with trade and accountancy al­ ready used in old Babylonia. In Europe, their vital role for the development of science and technology became evi­ dent only during the Renaissance. Georges Ifrah gives a beautiful and well written account of all this in his chap­ ter The way people learnt how to count and calculate, which he adapted from his hook Histoire Uniz •erselle des Ch?fJirs from 1 994, translated into Eng­ lish in 1 998. In connection with the International Congress of Mathematicians in Madrid, 2006, the Spanish National Library ran an exhibition organized by Antonio Duran. which aimed to illustrate through manuscripts, hooks, and other objects, the life of numbers to the gen­ eral public. 1be L[/e of Numbers was conceived as an extended catalogue for this exhibition. Among the hooks on display was the famous Vigilanus Codex from 976, which is normally placed in the famous library of the monastery El Escorial . Many partici­ pants in the International Congress of Mathematicians enjoyed being guided by Duran to a beautiful visit of El Es­ corial . And now we all can take plea­ sure in reading his chapter on "The printing press and numbers . " The illustrations in this book are magnificent, and most of them were made explicitly for the occasion, which adds to the great value of this hook. The Lfle of Numbers is a remarkable book, and it would make a good gift for any lay person with an interest in mathematics and its cultural impact on

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society. Mathematicians will most cer­ tainly also find new information and will be inspired by the saga of one of the most important discoveries of mankind: numbers! Department of Mathematics Technical University of Denmark Matematiktorvet, Building 303 DK-2800 Kgs. Lyngby Denmark e-mail: [email protected]

Shadows of Real ity: The Fourth Dimension i n Relativity, C ubism, and M odern Thought by Tony Robbin YALE U NIVERSITY PRESS, 2002, 160 PP, US $40, ISBN: 0300110391

REVIEWED BY THOMAS BANCHOFF

---, ony Robbin is an artist who is fascinated by the geometry of the fourth dimension. His paintings, sculptures, and installations have been featured at many exhibitions and mu­ seums, and he has written articles and books on art and geometry. He inter­ acts with a great many mathematicians and scientists, as well as artists and art historians, and he can write well about his insights. He has a unique perspec­ tive on some subjects that are familiar and on other ideas that are new. Robbin's new book, Shadows of Re­ ality, explores the fourth dimension in a number of different ways. Geometers who want to understand a complicated object in ordinary three-dimensional space ordinarily study various views of the object, either in silhouette or with interior detail, or, alternatively, they cut the object by parallel planes or con­ centric spheres to reveal the interior structure as sequential slices or layers. Both of these methods have tradition-

1

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THE MATHEMATICAL INTELLIGENCER

ally been extended to study mathe­ matical phenomena in a space of four dimensions either by projecting them into our three-dimensional world or by displaying a sequence of three-dimen­ sional slices in time. The former model generalizes notions from projective geometry, whereas the second model finds expression in the space-time of relativity theory. This book explores the relationships between these two mod­ els. The author has a definite prefer­ ence for the first over the second. Before I go further, I would like to acknowledge that I am not an unbiased reviewer since, by request of the book review editor for the Mathematical In­ telligencer, I am reviewing a book that is codedicated to me: "To Linda Hen­ derson and Tom Banchoff, who climbed up first and set the rope." Linda Henderson is an art history professor at the University of Texas at Austin whose 1 975 Ph.D. thesis at Yale University led to her volume The

Fourth Dimension and Non-Euclidean Geometry in Modem Art, published in 1 983 and slated for republication in 2007. Her thesis made the case that artists in the early part of the twentieth century were inspired much more by the tradition of higher-dimensional geometry, already well established in the latter half of the previous century, than they were by the slowly awaken­ ing public interest in the theory of rel­ ativity. Robbin is climbing the same rope, and he is making new observations along the way. His major claim in this book is that the primary influence of the geometry of higher dimensions on art is much more a result of the pro­ jection model from projective geome­ try than from the slicing model, which he relates to Einstein's four-dimen­ sional space-time as a sequence of three-dimensional slices of four-dimen­ sional phenomena. In the author's view, the projection model is superior to the slicing model. Shadows of Reality begins with an informative first chapter on "The Ori­ gins of Four-Dimensional Geometry," citing contributions by Hermann Grass­ mann, August Mobius, Arthur Cayley, and Ludwig Schlaf1i. Particularly effec­ tive are the stories Robbin tells about the Johns Hopkins mathematician, William Stringham, whose projected

images of four-dimensional regular fig­ ures significantly influenced the way people tried to visualize objects in four­ space. The author traces the impor­ tance of classical techniques of engi­ neering by drawing on descriptive geometry (Darstellende Geometrie in German) for representing three-dimen­ sional phenomena in the plane. The generalizations of these techniques to deal with four-dimensional phenomena projected into lower dimensions were pioneered by Pieter Schoute and Vik­ tor Schlegel and were further devel­ oped by Esprit)ouffret at the beginning of the twentieth century. Chapter 2 presents a brief treatment of "Fantasies of Four-Dimensional Space. " Edwin Abbott Abbott's 1884 classic Flatland is identified as the source of the slicing model, portraying a higher-dimensional phenomenon as a sequence of slices by a moving lower­ dimensional space. Charles Howard Hinton's 1884 Scientific Romances and 1 904 The Fourth Dimension provide other models for a two-dimensional universe. In 1910, Henry Parker Man­ ning's The Fourth Dimension Simply Explained presented an edited collec­ tion of articles from a Scientific Amer­ ican contest on the topic. Noteworthy is that none of the essays mentions space-time or relativity, which had not yet reached the general public. The au­ thor also gives accounts of connections between the fourth dimension, spiritu­ alism, and magic tricks. He ends the chapter with the observation that "writ­ ers outside the mathematical commu­ nity largely ignored projection tech­ niques to study four-dimensional figures or four-dimensional space. " In Chapter 3, "The Fourth Dimension in Painting," Robbin makes the case that jouffret's 1903 volume " Traite elemen­

taire de Geometrie a Quatre Dimen­ sions' was a direct inspiration of several of Pablo Picasso's paintings in the early days of Cubism. He discusses at length the portraits of Ambroise Vollard and Henry Kahnweiler and their relationship with illustrations from this treatise on projection methods in geometry. He puts forward a "bold interpretation" about Picasso's Seated Woman with a Book "It seems probable," says Robbin, "that the painting is of Alice Derain read­ ing Jouffret's text and visualizing the fourth dimension." He sums up the

chapter, "The projection method of modeling four-dimensional space puts more than one three-dimensional space in the same place at the same time . . . . Picasso used the technical drawing of four-dimensional geometry to show his audience the reality they knew existed but could not otherwise see . " After his major chapter o n Cubism, the author turns to physics. Chapter 4 , entitled "The Truth, " presents a n ac­ count of the Michelson-Morley exper­ iment and Einstein's theory of special relativity from 1905. Robbin asserts that the description of this theory, given by Hermann Minkowski, uses geometric ideas that are closely connected with projective geometry from the previous century. ( Robbin worked out much of this material in collaboration with mathematician Charles Scheim from Hartwick College, misidentified as "Charles Strauss" in the Acknowledg­ ments section .) The final paragraph of the fourth chapter underscores the au­ thor's main point: "That the slicing model is still fixed in the popular imag­ ination as the only truth about four-di­ mensional geometry and space-time is a testament to how powerful the hy­ perspace philosophers were, and how slow culture as a whole is to change such deep structure intuitions as the na­ ture of space. " Robbin asserts that projective geom­ etry occupies a central explanatory role in the history of art and in the history of science. His Chapter 5 is "A Very Short Course in Projective Geometry. " For the mathematician reader it will serve as a reminder of more or less fa­ miliar ideas in projective geometry. For the nonmathematician reader, it can il­ lustrate a progression of abstraction from the theory of classical perspective to the techniques of mechanical draw­ ing and ultimately to a formal algebraic system. Chapter 6 proposes that much of crystal geometry can be viewed as pro­ jections into three-space of configura­ tions in dimensions six and higher. It is noteworthy that one of Robbin's largest and most impressive sculptures was based on such elaborate crystal structures. ( Installed at the Technical University of Denmark at Lyngby, this structure unfortunately has since been dismantled. ) I n Chapter 7 , the author presents an

exposition of the Twistor Theory of Sir Roger Penrose. He makes the case that this theory is also a manifestation of projective geometry through the use of complex numbers to describe geomet­ ric phenomena on the three-sphere. "Entanglement and Quantum Geom­ etry" in Chapter 8 is yet another use of projective geometry to help understand physics. The guru of this investigation is a physicist, Padmanabhan Aravind from Worcester Polytechnic Institute, a correspondent of the author. Chapter 9 provides a view of "Cat­ egory Theory, Higher-Dimensional Al­ gebra, and the Dimension Ladder. " Tour guides for this chapter are math­ ematicians Scott Carter from the Uni­ versity of South Alabama, his col­ laborator Masahico Saito from the Uni­ versity of South Florida, and Lou Kauff­ man from the University of Illinois at Chicago. A welcome addition to the literature of computer graphics is Chapter 10, "The Computer Revolution in Four-Di­ mensional Geometry, " which describes the pioneering work of A. Michael Noll, Kenneth Knowlton, and Heinz von Forster, as well as the contributions of the reviewer and his colleagues Charles Strauss and Davide Cervone. A more recent development is the CAVE virtual environment of Tom Defanti and oth­ ers at the University of Illinois in Chicago in 1992 and its use for math­ ematics and for art by George Francis and Donna Cox at the National Super­ computing Center at the University of Illinois, Urbana-Champaign. Other cur­ rent contributions include the work of Michael D'Zmura at the University of California, Irvine, and Andrew Hanson and his students at the University of In­ diana, Bloomington. The final chapter, "Conclusion: Art, Math, and Technical Drawing, " reiter­ ates the claim that projective geometry deserves a place of honor in modern thinking in mathematics and physics as well as in art. To emphasize his point, the author states, "Consider this book a modest proposal to rid our thinking of the slicing model of four-dimen­ sional figures and space-time in favor of the projection model. ,. Tony Robbin privately acknowl­ edges that his modest proposal is di­ rected more toward physics than math­ ematics. Personally I don't feel I have

to make a choice between the two models. As a mathematician, I need to use both projections and slicing in my teaching as well as my research, and some of the most effective computer graphics renderings of phenomena in four-space combine the two models by displaying projections of sequences of slices, or sequences of slices of pro­ jections. (There is also a third tradi­ tional way of describing properties of higher-dimensional objects in lower di­ mensions by means of fold-out mod­ els, not treated in this book. ) Never­ theless I still find it fascinating to read the stories he has uncovered and to see the examples he puts forward. Whether or not one accepts the full force of his arguments, reading Tony Robbin's account of shadows of four­ dimensional reality is well worth the experience. Brown University Department of Mathematics Box 1 9 1 7 1 51 Thayer Street Providence, Rhode Island 0291 2 USA e-mail: Thomas_Banchoff@brown. edu

The Math Behind the M usic Edited by Leon Harkleroad CAMBRIDGE UNIVERSITY PRESS, 2006, 142 PP., CD INCLUDED, US $24.99, ISBN-13: 9780521009355.

7

REVIEWED BY EHRHARD BEHRENDS

/ill//

athematics and Music: Sev-

era! interesting connections and a number of books that discuss various aspects of this interplay have been published during the first few years of the twenty-first century. Whereas [1] and [2] are collections of ar­ ticles written by different people, the book under review is the author's sum­ mary of a number of courses he has taught at Cornell University. The problem of constructing scales is chosen as a natural starting point. How is the Pythagorean scale defined, .

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what are just intonations, what is the equal temperament, and how does the twelfth root of two come into play? Harkleroad also uses the opportunity to explain some fundamental facts of Fourier analysis. In Chapter 4, some elementary pro­ cedures for varying a musical theme are explained in the language of group the­ ory. The transposition of 7 semitones, or " T7," has T- 7 as its inverse, and so forth. Here one has to calculate mod­ ulo 1 2 , and the reader can learn how arithmetic modulo a fixed integer is de­ fined. As illustrations of these theoret­ ical facts, many examples are included. For example, in Pomp and Circum­ stance (by E. Elgar), the main theme is varied by using transpositions, whereas inversion plays a crucial role in Tbe Musical Offering (by ]. S. Bach): It helps that a compact disc is included, which allows one to compare the the­ oretical facts with the real sound. The "art of bell ringing" is described in Chapter 5. The interest in this aspect of "mathematics and music" is recent. The presentation here is more elemen­ tary than in [2) (see the article of Roaf and White) and [3), and the emphasis is on the connections with group the­ ory. "Subgroups" and "cosets" enter the stage, and a modestly complicated ex­ ample of a bell tower rendition can be found on the CD. The next chapter is devoted to "Mu­ sic by chance" (an aspect which, sur­ prisingly, is not covered in many other books on this subject). The idea came up in the eighteenth century, and many composers-among them Haydn and Mozart-have presented such compo­ sitions. The idea is simple. First write a finite number of bars that can serve as the first bar of a minuet, for example. Then continue with several suggestions for bar two, and so on. Randomness comes into play when producing a con­ crete minuet from these suggestions. For example, if there are 1 1 suggest­ ions for bar number one, numbered 2, 3, . . . 1 2 , throw two dice and select that bar that corresponds to the total sum. Find bar number two in a similar way, and continue; in the end you will have a complete "random minuet." The num­ ber of different pieces is gigantic and, although masterpieces are not to be ex­ pected, it is interesting to hear that some of them sound melodious. (Admittedly,

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after listening to the first few examples, one has the impression of knowing them all.) This, however, is not the end of the story. The book also explains how more elaborate stochastical procedures, such as Markov chains, are used to pro­ duce musical pieces. The reader can learn some basic notions from proba­ bility theory, for example, conditional probability. The step from random selection to deterministic procedures that use geo­ metrical forms or are seemingly sto­ chastic is not very large. The author re­ ports on compositions that convert the shape of the mountains close to Rio de Janeiro into music, or that use Linden­ mayer systems (these are recursively defined; after some steps one arrives at structures that are self-similar and "chaotic"). The title of the final chapter is "How not to mix mathematics and music. " One learns that numerical relations in musi­ cal pieces have been overemphasized occasionally. For example, it is not hard to find the Fibonacci numbers or (more or less convincing approximations oO the golden ratio in any given composi­ tion if one juggles with the number of bars, notes, themes, and so on. The book can be recommended as a first survey. Those who want to know more can find many suggestions for fur­ ther study in the extensive bibliogra­ phy. The book addresses readers who do not have much mathematical knowl­ edge. Everything is explained clearly and at great length, and the theoretical facts are illustrated by many pictures, scores, and the tracks on the accom­ panying CD. It should be noted, however, that this book cannot give a substantial an­ swer to the question of why music is such a fascinating part of our lives. Mathematics only helps to explain some superficial aspects of music or to understand how certain contemporary composers use mathematical methods. But it would probably be naive to ex­ pect more.

REFERENCES

[1 ] G. Assayag, H. G. Feichtinger, and J. F. Ro­ driguez (eds.), Mathematics and music. New York: Springer Verlag (2002).

[2] J. Fauvel, R. Flood, and R. Wilson (eds.),

Music and mathematics. Oxford, Oxford

University Press (2003). [3] B. Polster, The mathematics of juggling. New York, Springer Verlag (2003).

Department of Mathematics and Computer Science of the Free University of Berlin Arnimallee 2-6 D-1 4 1 95 Berlin Germany e-mail: [email protected]

M usic: A Mathematical Offering by David]. Benson CAMBRIDGE U NIVERSITY PRESS, CAMBRIDGE, NEW YORK, 2007, 412 PP. PAPERBACK U S $48.00, HARDBACK U S $120.00, ISBN 0-521-61999-8 PAPERBACK, ISBN 0-521-85387-7 HARDBACK

l

REVIEWED BY PETER KRAVANJA

n his book, Music: A Mathematical Offering, David ]. Benson offers a clear and interesting account of the interplay between music and mathe­ matics. The first chapter, "Waves and harmonics, " starts by recalling a few facts from physics. It also describes in some detail the anatomy of the human ear; the ear's various limitations are dis­ cussed, such as sound intensity, deci­ bels, the threshold of hearing, the Fletcher-Munson curves, the just no­ ticeable difference, and the limit of dis­ crimination. Sine waves and the fre­ quency spectrum are then introduced and applied to vibrating strings, damped harmonic motion, and reso­ nance. The second chapter is devoted to Fourier's theory of harmonic analysis. The author explains the Gibbs phe­ nomenon, proves Fejer's theorem, and analyzes Bessel functions. He also in­ troduces some concepts from the the­ ory of signal processing: the Fourier transform and its inverse, the spectrum, the Poisson summation formula, the Dirac delta function, convolution, the cepstrum, and the Hilbert transform. Chapter Three, "A mathematician's

guide to the orchestra," considers par­ tial-differential equations ( the wave equation) in the context of musical in­ struments, such as a bowed string, wind instruments, a circular drum, a horn, xylophones and tubular bells, an mbira ( a popular melodic instrument of Africa, especially used by the Shona people of Zimbabwe), a gong, and a bell. The next three chapters present a mathematician's reading of some major topics in the history of music theory. More specifically, Chapter Four dis­ cusses at length the history of different explanations of consonance and disso­ nance. The chapter also briefly de­ scribes a few paradoxes of musical per­ ception, such as the Shepard scale. Listening to this scale, one has the im­ pression of an ever-ascending scale where the end joins with the beginning, just like Escher's famous ever-ascend­ ing staircases. Chapter Five and Six present a de­ tailed account of scales and tempera­ ments: Just scales, tempered scales, meantone scales, well-tempered scales, Harry Partch's forty-three tone scale, the fifty-three tone equally tempered scale, the Bohlen-Pierce scale, among many others, are explained. The link between some of these scales and con­ tinued fractions is fascinating. Chapter Seven, "Digital music" , briefly recalls a few definitions and re­ sults from the theory of signal process­ ing, including sampling, Nyquist's the­ orem, the z-transform, digital filters, the discrete Fourier transform, and, of course, the famous Cooley-Tukey al­ gorithm. Chapter Eight then builds upon these mathematical foundations to investigate the synthesis of musical sounds. Special attention is paid to Fre­ quency Modulation (FM) synthesis, the Karplus-Strong algorithm, which yields very good plucked strings and percus­ sion instruments, the Yamaha DX7 syn­ thesizer, which came onto the market in 1 983 and was the first affordable commercially available digital synthe­ sizer, and CSound, a public domain syn­ thesis program written at the MIT Me­ dia Lab in the C programing language. The Ninth and final chapter relates the symmetries that appear in music to group theory. Fragments from Beethoven, Bach, or Mozart meet Cay­ ley's theorem, dihedral groups, Burn-

side's lemma, and P6lya's enumeration theorem. I enjoyed reading and exploring this book, which is an excellent introduc­ tion to the interdisciplinary subject of music and mathematics (which also involves physics, biology, psycho­ acoustics, and the history of science and digital technology). The book can easily be used as the text for under­ graduate courses. Department of Art, Culture, and Media University of Groningen The Netherlands Department of Communication Science University of Antwerp Belgium e-mail: [email protected]

Mathematicians in Love hy Rudy Rucker TOM DOHERTY ASSOCIATES BOOK, NEW YORK, 2006, 364 PP. , ISBN-13: 978-0-765-31584-7, HARDBACK, US $24.95

REVIEWED BY TOM PETSINIS

---,1

his intricate science-fiction novel is narrated by Bela Kis, a young postgraduate mathematician and musiCian who plays guitar in a rock­ band called "e to the i pi. " Bela and his colleague, Paul Bridge, have been re­ searching the Morphic Classification Theorem: mathematics that seeks to represent complex physical processes by simpler entities, such as the repre­ sentation of the stock market by, say, a fish in a tea-pot. Aided by their erratic hut gifted supervisor, Roland Haut, they complete their theses, gain Ph. D . 's, and both fall in love with Alma, a student of the humanities. Here I was surprised by how quickly their theses were ex­ amined and passed; a process that usu­ ally takes months was dispatched in what appeared to be a few days. Their theorem is found to have use­ ful applications when combined with the latest ( in a slightly futuristic sense) information technology. They soon dis­ cover a program for reading minds and predicting the future, and, through a

paradox in Haut's research, they dis­ cover a way of creating a tunnel through space-time, which allows them to enter different worlds. Of course, their discoveries are sought by nefari­ ous politicians intent on using them to win an upcoming election. This leads to the death of Bela's friend and co­ musician Cammy. The three young pro­ tagonists embark on an adventure through space-time in the hope of re­ turning to the past to avert Cammy's death. They enter a tropical alien world inhabited by aquatic creatures capable of advanced mathematical thinking. These creatures have already proved Goldbach's conjecture and Riemann's hypothesis. At one point, discussing Mersenne primes, Paul mentions that 2 to the power of 25,964,951 - 1 is the largest such prime he has come across, to which a lizard replies that this 1 4million digit perfect number is nothing compared to what their Number The­ ory has produced. The characters are soon transported to Earth-2, a slightly different place from Earth, where they find Cammy alive, but in which Alma dies. Here Bela and Paul release countless bub­ bles, each a microcomputer for pre­ dicting the future, with humorous and eventually chaotic consequences for Earth-2. In the novel's bloody climax, Paul and several other characters are killed, while Bela just manages to es­ cape through another space-time tun­ nel, eventually finding himself in Earth3-a place where Paul and Cammy are still alive, where Bela marries Alma, and where peace and happiness pre­ vail. The interplay between the various incarnations of Earth is neatly ex­ pressed by Bela in the following terms: "If I reached back in time to keep my parents from meeting, then I wouldn't have been born, so I wouldn't be around to reach back and keep them from meeting, so I'd be born the same as before and end up wanting to reach back in time to keep them from meet­ ing-like that." The author overcomes the potency of this circuitous logic by using a God-like sea creature that has the ability to create new worlds and an ultimate Earth. A mathematician as well as an es­ teemed science-fiction writer, Rucker shows his considerable knowledge of various branches of mathematics and

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displays imaginative flair in moving the complicated narrative, with its many twists, turns, alternative realities, and plethora of characters. His writing is en­ ergetic, often humorous, and sprinkled with striking metaphors. At times, though, I felt that too much was hap­ pening in the novel-that it contained too many subplots and minor charac­ ters. Also, some of the dialogues in­ volving secondary characters, together with their detailed descriptions, could have been more succinct. Rucker's novel is inventive in its plot construction, intelligent in its use of mathematics and information technol­ ogy, and written under the influence of a high dose of imagination. The action is fast, almost frenetic at times, which occasionally makes it difficult for the reader to keep up. The author evokes a sense of place with a deft touch, par­ ticularly places on Earth, and depicts the protagonists in an engaging man­ ner. His speculative use of fact is clev­ erly done, offering the reader interest­ ing ideas and food for thought. In a decade when mathematics has found its way into fiction and theatre, this novel uses mathematical ideas to push the boundaries of science-fiction, and so make an important contribution to this genre. 55, Glenair Street Lower Templestowe Melbourne 3 1 07 Australia e-mail: [email protected]

Crossing the Equal Sign by Marion Deutsche Cohen PLAIN VIEW PRESS, AUSTIN, TEXAS, 2007, 112 PP., US $17.95 ISBN: 978-1-891386-69-5

REVIEWED BY JOANNE GROWNEY

--, or her e-mail address and web­ L, I site, Marion Deutsche Cohen uses the designator "MathWoman"­ and this label aptly describes the speaker in Crossing the Equal Sign. This, Cohen's most recent poetry collection, is a series of self-snapshots-sometimes provocative, sometimes meditative and

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THE MATHEMATICAL INTELLIGENCER

poetic-of a woman who loves mathe­ matics. I first found poems by Marion Co­ hen several years ago on the Internet, whose search engines unfailingly find connections among seemingly dis­ parate combinations of words (such as "mathematics" and "poetry"). Marion Cohen and I share this unusual pair of interests. Formerly a mathematics pro­ fessor at Pennsylvania's Bloomsburg University, I now devote my time to writing-and some of my poems are about mathematics. Cohen, who wanted from girlhood to become a mathemati­ cian, and who in youth developed the habit of keeping a journal and has con­ tinued to write, is revealed by Crossing the Equal Sign to be engagingly "off­ beat" and "one-of-a-kind." She is also articulate about her experiences as a woman, parent, teacher, writer, and mathematician. There are many circumstances in which it is difficult to be truthful. The v1s1onary feminist poet, Muriel Rukeyser 0913-1980), ends the third section of her poem, Kathe Kottwitz, with these words: What would happen if one woman told the truth about her life? The world would split open. Rukeyser's lines are quoted often as a reminder of how rare it is to find certain types of truth-telling. Who is able to tell unwanted truth to those who have power over them? Pioneers (such as women in mathematics), who seek to forge new pathways, have the double difficulty of first div­ ing beneath veneers of expectations to discover the truth and then resur­ facing to choose the best words for saying it. Marion Cohen has had years of practice-she discovers her truth and tells it. Crossing the Equal Sign is a se­ quence of wanderings-at times poetic in their language and their ironies, and warmly human in their unashamed love of people with their frailties and their love of mathematics with its conun­ drums. During my first reading of the book, I easily selected page 82 as my favorite. For me, these lines vividly cap­ ture the essence of living in today's scattered society in which it takes great stamina to resist starting off in all di­ rections, subdividing life into mere points.

A

mathematician should never watch action films. She has already swum through iron, run without roads, flown without sky has already known too many direc­ tions has already been reduced to a point. She has had enough of thinking hard enough of hoping that thinking will save her. (An unusual aspect of the style of Co­ hen's hook is that, although there is a table of "Contents, " there are not titles on the poetry-pages themselves. I ini­ tially disliked this feature, but I have come to appreciate the lack of titles as an opportunity for the reader; I may create my own titles.) Some of Cohen's poems use math­ ematical terminology and the mathe­ matician-reader will interpret particular phrases more variously than the non­ mathematician. But Crossing the Equal Sign emphasizes the person who inter­ acts with mathematics, and thereby the hook speaks to nonmathematical read­ ers as well. Cohen's musings speak of both the delights and the dilemmas of having an active mind, a mind that won't stop­ a mind that ever wants to grab an idea even at the most unlikely times. On page 58, for example, she shares some thoughts of a mathematician in labor: (The latest math dream) I love one over n minus one over n-plus-one equals one over n times n-plus-one along with its proof. But tonight I revere it so that I write it, for the case n 3 on the birth blanket just before birthing. Some of her verses, as the example above, resemble prose that's been bro­ ken up into short lines-and I wished for more poems that (like the "action films" example) offer vivid imagery and application of poetic craft. Some of her musings left me feeling lost: were these lines simply playful, or was there an important point that I completely missed? Still others are wonderfully subtle and stay with you long after the book has been closed. One of these (from page 1 4) reads thus: (The One-Dimensional Man: Some Questions) =

Is his face along or across his body? Is his mouth along or across his face? Can his lips part? Can the corners turn? What I mean is, can he smile? In summary, I found Crossing the Equal Sign worth reading. For a few bright verses and, more so, for its forth­ rightness and the insight it offers into the heart and mind of a woman math­ ematician, I recommend it. REFERENCE

Jan Heller Levi (ed.), A Muriel Rukeyser Reader, Introduction by Adrienne Rich, New York, W. W. Norton, 1 994, p. 21 7 . 7981 Eastern Avenue, #207 Silver Spring, Maryland 209 1 0 USA e-rnail: [email protected]

� Springer

the langu g of sd nee

S p ri n g r a nd Soci ety P bl i s h i ng The selection of a publisher is a critical decision for scholarly and professional Societies. Our flexible approach, our record of mnovation, and our long history as publisher of respected journals make Springer the preferred publishing pa rtner of some of the most renowned scholarly Societ1es in the world.

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© 2008 Springer Science+ Business Media, Inc., Volume 30, Number 1 , 2008

79

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R o b i n W i l so n

The Philamath's A lphabet: P Pacioli One of the earliest printed texts was the 1494 Summa de arithmetica, geomet­ n·ca, proportioni et proportionalita of Luca Pacioli ( 1 445-1517) , a 600-page compilation of the mathematics known :H the time; it included the first pub­ lished account of double-ent1y book­ keeping. One of Pacioli's students was Leonardo da Vinci.

Pascal's Triangle Blaise Pascal (1623-1662) contributed to many areas of mathematics. He built a calculating machine, investigated the theory of probability, wrote about at­ mospheric pressure, and stated his "hexagon theorem" about six points on

I

a conic. He wrote an important treatise on binomial coefficients, and the trian­ gular pattern of these coefficients, shown here in a Chinese text of 1 303, is now known as "Pascal's triangle. "

Plato's Academy Around 387 BC, Plato (c. 427-347 nc) founded a school in a part of Athens called Academy. Here he wrote and di­ rected studies, and the Academy be­ came the focal point for mathematical study and philosophic research, de­ signed to provide the finest training for those who would hold positions of re­ sponsibility in the state. Over the en­ trance appeared the inscription: "Let no­ one ignorant of geometry enter here."

Poincare Henri Poincare ( 1854-1912) wrote on the "three-body problem" of determin­ ing the motion of the sun, earth, and moon. Arguably the most brilliant math­ ematician of his generation , and a gifted popularizer of the subject, Poincare contributed to many areas of mathe­ matics and physics, including celestial

mechanics, differential equations, and algebraic topology.

Punched Card A punched card has holes punched in specific locations to convey informa­ tion, and was used in the Jacquard loom to mechanize the weaving of compli­ cated patterns. Data processing with punched cards was developed by Her­ man Hollerith for the US population census of 1890, and such cards were in widespread use for many years after­ wards.

Pythagoras Pythagoras (c. 572-497 Be) was a semi­ legendary figure. Born on the Aegean island of Samos, he later emigrated to the Greek seaport of Crotona (now in Italy), where he founded the Pytha­ gorean school, a close-knit brotherhood formed to further the study of mathe­ matics and philosophy. It is not known whether there is any connection be­ tween Pythagoras and the famous the­ orem on right-angled triangles named after him.

Plato'• Academy

Please send all submissions to the Stamp Corner Editor, Robin Wilson, Faculty of Mathematics, The Open Un iversity, Mi lton Keynes, MK7 6AA, England e-mail: [email protected] . u k

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Punched card

THE MATHEMATICAL INTELLIGENCER © 2008 Springer Science+ Business Media, lrx;.

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