Interview with Raoul Bott, Volume 48, Number 4

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Interview with Raoul Bott

This is the edited text of two interviews with Raoul Bott, conducted by Allyn Jackson in October 2000.

aoul Bott is one of the outstanding re- searchers in geometry and topology in recent Rtimes. He has made important contributions to topology, Lie group theory, foliations and characteristic classes, K-theory and index theory, and many other areas of modern mathematics. One of his most spectacular successes was the application of Morse theory to the study of the homotopy groups of Lie groups, which yielded the Bott periodicity theorem. This central result has reappeared in many other contexts, including several versions of K-theory and noncommutative geometry. Raoul Bott was born on September 24, 1923, in Bott at McGill University, about 1942. Budapest, Hungary. At McGill University he earned a bachelor’s degree in 1945 and a master’s degree Notices: First let’s talk a little bit about your in 1946, both in engineering. He then switched to early background. You had an unconventional ed- mathematics and received his Sc.D. from the ucation, and when you were a youngster you were Carnegie Institute of Technology (now Carnegie not especially distinguished in mathematics. Mellon University) in 1949. He spent the next two Bott: That’s putting it mildly! years at the Institute for Advanced Study in Prince- Notices: But looking back now, do you see some ton. From 1951 to 1959 he was at the University experiences from that time that put you on the of Michigan, except for a stay at the Institute dur- path to becoming a mathematician? ing 1955–57. In 1959 he accepted a professorship Bott: Well, I’ve always thought my interest in at Harvard University. He retired from Harvard in electricity was a manifestation of trying to un- 1999. His honors include the AMS Oswald Veblen derstand something, but it certainly wasn’t math- Prize (1964), the National Medal of Science (1987), ematics. When I was about twelve to fourteen the AMS Steele Prize for Lifetime Achievement years old, a friend and I had fun working with elec- (1990), and the Wolf Prize (2000). tricity, and it was really a collaboration. We had a lab where we tried to make very primitive things, such as a microphone. We enjoyed creating Allyn Jackson is the senior writer and deputy editor of sparks, and we wanted to know how gadgets the Notices. Her e-mail address is [email protected]. The assistance of Dieter Kotschick, Ludwig-Maximilians- work. So I think this was closest to what really Universität München, who provided mathematical help makes a mathematician—someone who likes to with the interview, is gratefully acknowledged. get at the root of things.

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Notices: This was much more practical than your mathematics is. Bott: Yes, it was definitely practical, and, although I wasn’t so very good at it, I enjoyed working with my hands. I always said later on that I would have liked to live in the day of Marconi. I would have loved to invent things in a small lab, discover the basic properties of electromagnet- ism. I think that would have been wonderful. Notices: In mathematics you worked in pretty pure areas; you didn’t work in applied areas. Bott: Well, after I got my degree in engineering, I went into applied mathematics. I solved a quite famous problem with my thesis director, Richard Photograph courtesy of Hans Samelson. Duffin. The result is now called the Bott-Duffin the- E. Pitcher, Johnson (first name unknown), R. Bott, orem [1]. It was practical and was put to use by Bell H. Samelson, J. Nash, H. Rauch, 1956 AMS Confer- Labs for a while. ence in Seattle. Notices: What did this theorem allow them to do? Bott: This problem had to do with building fil- visited, and I am sure it was this theorem that ters. In those days one didn’t have transistors, so brought me to the Institute in 1949. if one made electric circuits, one had only very stan- Notices: Were you aware of the literature that dard objects: resistors, capacitors, and coils. If the engineers had written? these elements are hooked up in an arbitrary fash- Bott: No. Duffin and I didn’t like to search ion and placed into a “black box”—as it was called— through literature. We still don’t! We thought one so that one has only two terminals showing, then should be able to get rid of the ideal transform- the steady-state frequency response of such a net- ers, and we knew these other papers didn’t do work is determined by a rational function of the that. There is a tremendous literature in the math- frequency, called the “impedance” of the box. And ematical world about functions that map the upper because such a box contains no energy sources, this half-plane into itself—the moment problem—but nothing in those papers actually helped. The final impedance has the crucial property that it maps proof of the theorem was really quite easy once we the right half of the complex number plane into learned of a theorem of I. R. Richards in abstract itself. So the mathematical problem was: Given complex variable theory. I recently asked my friend such a rational function, can one build a black box and colleague Curt McMullen to provide a proof for for it? This was a very natural question, because this Richards theorem, and he produced a purely the frequency response in a filter is the important algebraic proof from the Schwarz lemma, very in- thing: one wants certain frequencies to go through geniously applied. The original version of the the- and others to be blocked. orem seemed more complicated. This problem had fascinated me when I was at What I like about this work with Duffin is that McGill, and I brought it to Carnegie Tech with me. it also brought about a wonderful moment of col- In my first interview with Duffin I immediately di- laboration. We had been working on our problem vulged it to him, and he became interested. Actu- with the Richards formula all afternoon, and it ally, the problem had been nearly solved by Brune, didn’t seem to work. We then went home, and on who was a South African engineer, many years be- the way I saw that of course it did work! So rush- fore. He had given an inductive procedure for ing home, I immediately called him up, but his building a black box, starting from such a rational phone was busy. He was calling me with the same function. Unfortunately, at one step in his proce- insight! dure he had to introduce an “ideal transformer”. Notices: What area of mathematics was Duffin His procedures were quite feasible, except for this working in? one step. In practice, your black box would become Bott: He was a jack of many trades. He was a as big as a house to accommodate an ideal trans- physicist to start with, and I liked to tease him that former! So my dream was to get rid of these ideal he did applied mathematics the wrong way around. transformers at the cost of making a more com- He would take his physical intuition and try to plicated network. And that is precisely what Duf- make it mathematics. fin and I managed to do. This work wasn’t my the- Notices: Why is that the “wrong way around”? sis, but it was much more interesting than my Bott: Well, ideally, the physicists would like to thesis, and it started my career, no question about have mathematics predict nature. There is some- it. The engineers were amazed, because they had thing more exciting if you predict, in terms of written wrong papers on the subject for twenty mathematical ideas, a phenomenon that was un- years. Hermann Weyl heard about it when he expected. So I used to think of it as the wrong way

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around. But Duf- opment turned Nash off. Eventually he went to Am- fin was definitely brose and asked for a “real problem”. And then of a master at it. course Nash proved his remarkable embedding And of course, as theorems. But I was at Michigan at that time. Un- I said, he worked fortunately, Nash’s great gifts were marred by his in many different terrible disease. areas. For in- Notices: After Carnegie Tech, you went to the In- stance, there is stitute in 1949. You had been doing things associ- something called ated with engineering up to that point. How did your the Duffin basis perspective on mathematics change when you went in physics for to the Institute? spinors. He had Bott: Well, I felt like a kid in a candy store. First also worked ex- Photograph courtesy of F. Hirzebruch. of all, the people around me were so outstanding! tensively in com- It was a sort of Valhalla, with all these semigods Raoul Bott lecturing at Universität Bonn, plex variable the- around. Amazingly enough, we mathematicians 1969. ory. He was an have a type of negative feedback built into us: If artist in a way. He we don’t understand something, it makes us want wrote beautifully written, short papers. He was to understand it all the more. So I went to lectures, not a specialist at all, and that impressed me most of them completely incomprehensible, and from the start. my gut reaction was: I want to understand this. Os- Notices: You wrote in a Notices article that you tensibly I was at the Institute to write a book on tried to emulate his way of being a “mathemati- network theory, but after I found out I didn’t have cal samurai”. to do that, I went to an incredible number of lec- Bott: Yes, that’s the point. It’s the problem you tures and just absorbed the atmosphere. I didn’t go after rather than the field. You have to trust write a single paper in my first year there. So I was your instincts and hope that sometimes you will very delighted when Marston Morse called me up hit upon a subject to which you can maybe make at the end of that year and said, “Do you want to a contribution. stay for another year?” And I said, “Of course, Notices: Did you come into contact with John yes!” He said, “Is your salary enough?” It was $300 Nash at Carnegie Tech? a month. I said, “Certainly!” because I was so de- Bott: Yes, indeed. He was in my class. In fact, lighted to be able to stay another year. My wife took in this class there was Nash and also Hans Wein- a dimmer view! But we managed. berger, a very good applied mathematician now Notices: So this was a big change for you, to go at Minnesota, and maybe two or three others. from an environment where you had been work- Duffin was teaching us a very amusing course on Hilbert spaces. One of Duffin’s principles was ing on the engineering side to a place where there never to prepare a lecture! So we were allowed to was so much mathematics. see him get confused, and part of the fun was to Bott: I didn’t think of it that way. see whether we could fix things up. We were read- Notices: It wasn’t such a contrast for you? ing von Neumann’s book on quantum mechanics, Bott: No, because the actual work is just the which developed Hilbert spaces at the same time. same. When I worked with Duffin, it was mathe- And it soon became clear that Nash was ahead of matical thought; only the concepts were different. all of us in understanding the subtleties of infi- But the actual finding of something new seems to nite-dimensional phenomena. me the same. And you see, the algebraic aspects Notices: Was he an undergraduate? of network theory were an ideal introduction to dif- Bott: He was an undergraduate, yes, and the ferential geometry and the de Rham theory and to rest of us were graduate students. I was friends what Hermann Weyl was studying at the time, that with Nash; he didn’t have any close friends, re- is, harmonic theory. In effect, networks are a dis- ally, but we often talked about this and that. crete version of harmonic theory. So when I came When he later got sick and had a really bad bout, to the Institute, the main seminar I attended was he would sometimes send me a postcard with Hermann Weyl’s, and Kodaira and de Rham were some very strange associations, usually with re- lecturing on harmonic forms. Weyl wanted to have, ligious overtones. My closest contact with John finally in 1949, a proof of Hodge’s theorem that was at Carnegie Tech. When I came to the Insti- he could live with. Hodge’s theorem was proved in tute in Princeton, he came to Princeton as a grad- the 1930s, but in a somewhat sloppy way. The de- uate student, and then I only saw him casually. tails were cleaned up in this seminar by two quite Later when he came to MIT and started his work different people from different points of view. So in geometry, I unfortunately wasn’t at Harvard yet. this didn’t seem strange to me; it was within my I would have been glad to have been part of the domain of thinking. It then led to topology, and development of geometry by Ambrose and Singer there my course with Steenrod was the dominant at MIT at that time. However, this whole devel- experience for my future development.

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Notices: In your collected works Paul Baum wrote Notices: When you were some reminiscences about working with you. One in Princeton, was there any of the things he said he learned, or relearned, from activity in relativity, or was you is that there is a mainstream to mathematics Einstein working by him- and that certain mathematicians like you under- self? stand instinctively what that mainstream is. How Bott: When I came to the does one come to understand what that mainstream Institute, Oppenheimer had is? You’re born with it? You learn it? You pick it up taken over, and he was very from the environment? dominant in the physics Bott: A good point. I must say I always followed community. He had a sem- my taste. And sometimes my taste led me in di- inar that every physicist rections that weren’t fashionable but that luckily went to. We mathemati- turned out to be fashionable later on! But these cians always thought they things are dangerous, the fashions change, and it’s ran off like sheep, for we hard to tell in retrospect whether you were in the would pick and choose our mainstream. I was just very affected by the early seminars! I felt that Einstein was pretty isolated, yes. I’m

development of sheaf theory, and especially the Photograph courtesy of Harvard University Dept. Mathematics. combination of analysis with topology that then en- very surprised that in my Bott, 1972. sued. Suddenly complex variable theory fitted in own case I did not make a with topology and even certain aspects of number big effort to get close to theory. So I think that at that time it was very easy him, because he had always been my hero, and as a young boy I wanted nothing more than to un- to discern this development as a main road in derstand relativity. Also, we both liked the same mathematics. music, we spoke the same languages—it would But I’ve seen the mainstream change consider- have been too easy to become a groupie. We had ably over my lifetime. For instance, if I think of one or two exchanges, but they were always, “How Princeton before sheaf theory, the emphasis was do you do, the weather is nice....” However, at the very different. When I first came there, much of Institute, I was much more interested in topology. topology in those early years had to do with very And in a way, it’s just as well, because what he was abstract questions of pathological spaces, com- working on then has not been very helpful. paring fifteen different cohomology theories, and Notices: That’s interesting that you had so much such. This was what I would have said at first was in common with Einstein, and you could have got- the mainstream. Then topology moved more to ten to know him more, but you didn’t do that in part what I felt was the real world: the study of com- because you had so much in common with him. You pact manifolds and their invariants. Lower-di- wanted something different? mensional topology was not emphasized then, but in the 1990s it came to the fore again. So there is Bott: Einstein had an assistant before I arrived really a tremendous difference in perspective over there, John Kemeny, who later became president the years. of Dartmouth. Kemeny was Hungarian, like me. In fact, once Hermann Weyl mistook me for Kemeny, Notices: But isn’t there a core of mathematics that and I didn’t want to become the second Kemeny! is vital and lively, independent of fashion, and there Maybe I am not very well cut out to be a disciple. are other fields that are more outlying; and one And also, as I said, I was fascinated by topology. needs a sense of what is central and what perhaps At the Institute I had a marvelous tutor in topol- is not so central? ogy in Ernst Specker. He was and is quite a salty Bott: I don’t know to what extent I believe this. character, and we got along famously. Ernst was I think, for instance, that Bourbaki had that feel- a student of Heinz Hopf, and unfortunately—from ing, and I was always a little skeptical of Bourbaki. my point of view—he eventually moved into logic. The subject is just too big. It doesn’t just have one Reidemeister was lecturing to a small group, in- main road. There are too many unsuspected cluding me and Specker, on new things that Car- branches. So although I was in a sense very much tan was doing at the time. Reidemeister spoke in influenced by Bourbaki, I don’t really subscribe to a fluent mixture of half English and half German, the belief that there is just one way of looking at but for Specker and me this was not a problem, and things. An example is what’s happening in physics those sessions were an inspiration to me. and mathematics right now. Physicists with a com- From the Institute I went to Michigan and met pletely different intuition come up with things Hans Samelson, who was also a student of Hopf. that we now find very fascinating. I believe in the Samelson was a real master of geometry and Lie virtue of quite different cultures affecting mathe- group theory. I learned a lot from him during the matics. If you had one really good main highway, years we worked together. But again, it was a par- it would be dangerous, because then everybody ticular problem that brought me into Lie group the- would be marching along it! ory rather than wanting to learn an area.

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Notices: If problems get you to learn work with Samelson, where we extended this in- about an area of mathematics, what guides sight on the loop space of a group to the larger class you in your choice of problems? of symmetric spaces [22]. The techniques we Bott: It’s hard to say. As in music, one learned there were all I needed for the periodicity falls in love with different things at dif- theorem. But it took a few more years for the ap- ferent times. Right now for me it’s the 4th propriate context to develop. This occurred in partita of Bach. What brings on these im- 1955–57, when I returned to the Institute. pulses is hard to say. During that period there was a controversy in Notices: Is there impetus in the opposite homotopy theory. The question concerned the direction, that is, things you don’t like in homotopy group of the unitary group in mathematics? Photograph courtesy of F. Hirzebruch. dimension 10. The homotopy theorists said it Bott: Often I don’t like the way mathe- was Z . The results of Borel and Hirzebruch Bott and Michael 3 matics is presented. I like the old way of predicted it to be 0. This contradiction intrigued Atiyah on the presenting things with an example that me, and I thought I should be able to say some- Rhine River, 1984. gives away the secret of the proof rather thing about it using the Morse-theoretic techniques than dazzling the audience. I can’t say that there that Samelson and I had discovered. Finally I is any mathematics that I don’t like. But on the hit upon a very complicated method involving the whole I like the problems to be concrete. I’m a bit exceptional group G2 to check the conundrum of an engineer. For instance, in topology early on independently. My good friend Arnold Shapiro and the questions were very concrete—we wanted to I spent all weekend computing. At the end we find a number! came out on the side of Borel and Hirzebruch, so Notices: Are you a geometric thinker? Do you vi- I was convinced that they were right. And if they sualize things a lot when you do mathematics? were right, the table of homotopy groups started Bott: My memory is definitely visual, but I also to look periodic for a long stretch. In the odd like formulas. I like the magical aspects of classi- dimensions they were Z up to nine dimensions, and cal mathematics. My instinct is always to get as ex- in the even dimensions they were 0. So I thought, plicit as possible. “Maybe they are periodic all the way.” I remember In most of my papers with Atiyah he would suggesting this to Milnor. Well, Milnor doesn’t like write the final drafts and his tendency was to make bombastic conjectures! He likes to be on firmer them more abstract. But when I worked with Chern, ground. And fairly soon after I saw that my old I wrote the final draft. Chern actually wrote a more ideas would actually do the job. down to earth version of our joint paper. In this way the unitary group was then settled. Notices: Is Chern even more of a “formulas man” Then I started to think about the orthogonal group, than you are? and that was much harder. But I do remember pre- Bott: Oh, yes. I’m pretty bad, but he is even cisely when I suddenly saw how to deal with it. That worse! It’s strange that in some sense it was he who occurred after we had left the Institute and were taught us to work conceptually, but in his own work moving house. You know how it is in mathemat- his first proofs are nearly always computational. ics: one suddenly understands something while one Notices: Can you talk about some of your favorite is unpacking one’s books or doing something results, things that you have a special fondness equally innocuous. In a flash I saw how it all fit- for? Bott: I told you already about the first one, the ted together [24]. work with Duffin. That was, I think, a nice piece Notices: From what you said, the periodicity the- of work and great fun to do. Later I was very for- orem was hidden from everybody because of those tunate to be the first to notice that the loop space mistakes in the original calculations. Nobody could of a Lie group is very easily attacked with Morse- have conjectured it. theoretic methods [8]. It turns out that if you look Bott: Yes, especially the topologists and homo- at the loop space rather than at the group, then the topy theorists who were led in a quite different di- so-called diagram of the group on the universal cov- rection by attacking the problem with the gener- ering of its maximal torus plays an essential role. ally accepted method. On the other hand, I had the So you can read off topological properties of the good luck of doing homotopy theory via Morse the- loop space much more easily from the diagram of ory, which provided a quite different approach. the group than you can read off things about the So that was really a high point, but it was a group itself. This insight was exciting. I found this purely homotopy-theoretic result. By that time, relation sometime in the early 1950s at Michigan, the latter 1950s, I’d been invited to Bonn, and I had and it is still one of my favorite formulas. met Hirzebruch and learned all this wonderful Now, the sad part of that story is that, if I had stuff with the Riemann-Roch theorem, and those been as gifted and as thorough as Serre or some- ideas started to fascinate me very much. Actually, body like that, I would have immediately discov- that same year at the Institute I wrote a paper that ered the periodicity theorem there and then. Well, I also like and that has been influential. It’s called not right then, but certainly during my subsequent “Homogeneous vector bundles” [15], and it

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computes the holomorphic cohomology of certain homogeneous spaces in a nice way. This paper was clearly influenced by what I learned at the In- stitute in 1956 from Borel, Hirzebruch, Serre, and Singer. The Riemann-Roch theorem conjectured something that I then proved on the actual cohomology level. Notices: The Riemann-Roch theorem just gives the alternating sum of the dimensions. You com- puted each one individually in that case. Bott: Yes. I related first of all the cohomology to some Lie algebra cohomology, and that was Photograph courtesy of M. Atiyah. then very much more thoroughly investigated by 60th birthday conference of M. Atiyah, Oxford, 1989. Kostant later on. So there are various versions of With Bott on left: Lily Atiyah, on right: Rosemary Zeeman. this theory. Notices: That is what’s called the Bott-Borel-Weil suddenly from the index point of view there theorem. seemed to be yet another approach to the same Bott: Yes. Then the next development was that problem. Before, we had taken complex analysis Grothendieck came on the scene and influenced all or algebraic geometry as a given, so that the dif- of us tremendously. One day I received a paper ferential operator was hidden. Whereas here, sud- from Atiyah and Hirzebruch about a generalized denly the topological twisting of the differential op- cohomology theory, now called topological erator came into the equation. Of course, Atiyah K-theory. That paper was a revelation. Their ap- and Singer immediately realized that this twisting proach had never occurred to me. It fitted in with is measured with the homotopy groups of the clas- the periodicity theorem but gave a completely new sical groups, by the so-called symbol. Eventually way of interpreting my computations. This was the the whole development of index theory fitted the start of my long and wonderful collaboration with periodicity theorem into the subject as an integral Michael Atiyah. We first of all gave a new proof of part. Atiyah very rightly chose Singer to collabo- the periodicity theorem which fitted into th rate on this project. I was working in a very dif- K-theory framework [35]. Over the years he and var- ferent direction. I wanted to look at local funda- ious people have found more and more ways of mental solutions of differential equations and use doing this, completely different from my Morse the- them as the tool for proving the index theorem, as ory way. K-theory then took off, and it was great it was called, by patching these together in the fun to be involved in its development. Many famous “Cˇech manner”. old problems that had been difficult could be In 1964 Michael and I were together again in solved easily in K-theory. You see, in most coho- Woods Hole, at an algebraic geometry conference. mology theories, natural operations are hard to find By that time, we had learned to define an elliptic and difficult to compute. But in K-theory, because complex, and we now saw the old de Rham theory you are dealing with vector bundles, exterior pow- in a new light: namely, that it satisfied the natural ers are very natural, and computing with them in extension to vector bundles of the classical notion this new setting turned out to be very effective. of ellipticity for a system of partial differential In the early 1960s Atiyah and I were at Stanford, equations. During that conference we discovered and we went to a cocktail party. Hörmander was our fixed point theorem, the Lefschetz fixed point there too. That was when I first heard the term theorem in this new context [42], [44]. This was a index used in the sense that it’s generally used in very pleasant insight. The number theorists at first analysis, that is, as the index of an operator. I re- told us we must be wrong, but then we turned out member Michael was very, very interested in dis- to be right. So we enjoyed that! cussing the index with Hörmander. He stopped In a way, I always thought of the Lefschetz the- drinking and just talked to Hörmander. (But I con- orem as a natural first step on the way to the index tinued my drinking. In fact, I was very nearly ar- theorem. You see, in the index theorem you com- rested that evening by a police officer! Luckily, I pute the Lefschetz number of the identity map. The was able to squirm out of it.) Suddenly the Rie- identity map has a very large fixed point set. So if mann-Roch theorem had taken a new turn. Hirze- you have the idea that the Lefschetz number has bruch’s first run at it involved cobordism theory to do with fixed points, then of course it’s much and all this beautiful algebraic geometry, and the easier to first try and prove the Lefschetz theorem index theorem of Hodge was the link between the for a lower-dimensional fixed point set. The fixed topology and the analysis. That was very beauti- point theorem we proved in Woods Hole dealt pre- ful. Then Grothendieck, in the purely algebraic cisely with the case in which the fixed point set was context, gave a completely different proof using zero dimensional. Over the years I’ve encouraged his K-theory in the formal, algebraic way. Now people to study it over bigger and bigger fixed

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point sets and approach the final answer in this implication of that integrability condition seemed way. The analysis needed for the Lefschetz theo- to me to be a very fascinating subject, and it still rem in the case that we studied is very simple seems so today. In the late 1960s I was giving a compared to the analysis needed for the true index course on characteristic classes, and, as is usual theorem. Nevertheless, this special case fitted in with me, I started from scratch, because I don’t have nicely with many things, and we could use it also notes and I don’t like to read books. I did it slightly to prove some theorems about actions of finite differently that time, because I was very influ- groups on spheres and so on. enced by the ideas of Haefliger. And then I soon The next piece of work that I enjoyed very much noticed that integrability has a topological conse- was around 1977, when I came back from India and quence. If you have a vector bundle that is a sub- visited Atiyah in Oxford. During that visit I became bundle to the tangent bundle, then in its isomor- aware of a new and exciting relationship between phism class there’s a definite obstruction to mathematics and physics. In this atmosphere we deforming it into an integrable one. A certain van- started to think about the problem of stable bun- ishing condition has to be satisfied by its charac- dles over Riemann surfaces in terms of gauge the- teristic classes. This work [52] then naturally led ory. We had two ideas: first, that one had to use to the exotic characteristic classes of foliations, that an equivariant version of Morse theory to tackle is, generalizations of the Godbillon-Vey invariant, this problem. And second, that one then had to get which were also discovered independently by at the final answer by a subtraction process. The Joseph Bernstein at the same time. I worked in this salient feature of this work was that area with André Haefliger [56] for many years, and in the equivariant Morse theory the this was also a wonderful collabo- absolute minimum plays a very spe- ration. cial role, in the sense that the higher Notices: Had you encountered him critical points tend to be “self-com- at the Institute? pleting”. This Bott: I did meet him in Princeton, paper [81] had but we were never both in residence connections to there at same time. By the way, this various other whole area is also related to the so- fields. On the one called Gelfand-Fuks cohomology. hand, it related to Graeme Segal, with a little help from the stability the- me, proved that actually this ory of Mumford, Gelfand-Fuks cohomology turns out and on the other, to be a homotopy-invariant functor. it had relations to Independent proofs were also given the moment map by others, including André. An ex- and work of citing development in this area Guillemin and turned out to be the examples of Sternberg. It was even related to Thurston, which showed that you could have patho- some work of Harder and logical foliations. My work at that time was very Photographs courtesy of Harvard University Dept. Narasimhan in number theory. much influenced by Graeme Segal and his ideas on of Mathematics. In recent times some- simplicial spaces. thing that Atiyah and I Notices: In your work with Duffin there were a discovered in the 1980s lot of engineering papers that were wrong, but you has been put to a lot of were at first not aware of them. Later, the homo- good use. It is called the topy theorists made mistakes in their calculations, equivariant fixed point but this did not prevent you from finding the right theorem. Just recently it answer. Do you think that, for example, had you led to proofs of the so- known about the engineering papers, they would called mirror conjecture have stopped you from proving your theorem with in certain instances in the Duffin? work of both the Russian Bott: That could very well have been. If either and the Chinese schools. Duffin or I had researched the literature well enough, But let me brag about we would have found insurmountable problems! Al- another theorem! The though I think of myself as a rather sloppy guy, I have question here had to do found errors quite often. So I’m skeptical. I do like with foliations. A foliation to see the nitty-gritty of the proof. I like to under- is a subbundle of the tan- stand things very much in detail. Sometimes my stu- gent bundle that satisfies dents get mad at me. A thesis has to be rewritten until Bott at his 70th birthday celebration an integrability condition. it’s an open book, so to speak. Otherwise I’m too stu- at Harvard, 1993. To see the topological pid to understand it!

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Notices: Your work has of mathematics that are touched on a lot of different not at all appreciated at areas: topology, geometry, the moment and that I Lie groups, PDEs, analysis. think will come back at But not number theory. some point. Have you ever been inter- Notices: What are you ested in number theory? thinking of there? Bott: Secretly, yes! In fact, I’m leaning in that di- Bott: That’s hard to Photograph by Gabriella Bollobàs. rection right now. I’m in- predict. But often some terested in the papers of new development will at Candelas, who is a physi- the same time resurrect cist. For example, he wrote old questions. However, one paper called “Calabi- there are also more pes- Yau manifolds over finite simistic points of view. fields”. It really fascinated My friend Samelson al- me this summer, so maybe ways said, “Eventually in my old age…! mathematics will run out. We have been using Notices: Will you try the same ideas, the your hand at the Millen- same basic things, for so long, eventually the oil nium Prize Problems? You could win a million in the will be gone.” For example, Lie groups: you can trace process. them back to very early origins, and we’ve cer- Bott: No, I prefer doing the problems I dream tainly mined them tremendously in this century. up myself. But I think there will always be some new slant that Notices: What do you make of offering these big will keep us going. prizes? Do you think it’s good for mathematics? For the truth of the matter is that there are Bott: Well, we are to a certain extent snobs and tremendous mysteries out there, and their solution feel that there is something demeaning about will lead us in quite new and unexpected directions. bringing huge sums of money into the game. But There was a show on TV yesterday about geysers on the other hand, it might bring some very gifted in Yellowstone Park in Wyoming. There are thou- people into mathematics. For instance, during the sands of these hot springs, where steam and water Sputnik era the whole preoccupation with the Rus- escape. Biologists found that things live in this sians made theoretical subjects more exciting in boiling water! They found living things in a geyser— America. At that time a group of very brilliant peo- in a very deep, black hole, without any light, at tem- ple went into mathematics. Today they might go peratures and in chemical solutions that were con- into biology. So I do feel that publicity for mathe- sidered anathema to life! So I do believe that the matics is a good thing, but I wish it could be done universe will have enough for us to work on for a in a less materialistic way. However, America is a long time. pragmatic country and likes to look at the bottom I’m very glad I went into mathematics, and I’m line. certainly surprised it worked out so pleasantly. Notices: Going back to physics, it seems as though What’s so wonderful in our field is the tremendous in mathematics, compared to physics, people are collaboration that goes on, that we enjoy showing more individualistic. In physics, there are “tastemak- our wares to each other and that we by and large ers”; in mathematics, it’s not like that. It’s more di- don’t fight as much about it as in most other fields. verse. This is very rare, really; I think you don’t find it in Bott: Thank God there are very good people in literature, or biology, or history. They don’t spend so many diverse areas that we have many more half their time in other people’s lectures. We are branches we can develop. This is true to a certain allowed to learn from each other, and although we extent in physics too. There are the solid state do give credit, we also often learn much more than people who don’t care about fancy new stuff; they can be easily credited. With one offhand remark are fascinated by different aspects of physics. But we give away our insight of years of thinking, and physics is still much more circumscribed than such a remark might illuminate a whole field or fit mathematics. Physicists are in close contact with into one’s brain just right to unlock some new in- experiments, and we don’t have this discipline. sight. We do this very generously with each other. Some people have found it disconcerting that we are allowed to go so much in our own direction. References They think we have too much license! And I must The reference numbers in this article corre- admit that my basic reaction to some mathemat- spond to the numbering in Raoul Bott: Collected ics lectures is, “Why in heaven’s name are they Papers, volumes 1–4, Robert D. MacPherson, doing this?!” But there are also very beautiful parts Editor, Birkhäuser, 1994.

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[1] R. BOTT and R. J. DUFFIN, Impedance synthesis without use of transformers, J. Appl. Phys. 20 (1949), 816. About the Cover [8] RAOUL BOTT, On torsion in Lie groups, Proc. Nat. The cover photograph captures Raoul Bott Acad. Sci. U.S.A. 40 (1954), 586–588. at a characteristic moment, doing what he has [15] ——— , Homogeneous vector bundles, Ann. of Math. (2) 66 (1957), 203–248. been very, very good at all his professional [22] RAOUL BOTT and HANS SAMELSON, Applications of the life—explaining mathematics. It was Friedrich theory of Morse to symmetric spaces, Amer. J. Math. Hirzebruch who brought the photograph to 80 (1958), 964–1029. our attention, and who sent along also a copy [24] RAOUL BOTT, The stable homotopy of the classical of the article in the Unabhängige Westdeutsche groups, Ann. of Math. (2) 70 (1959), 313–337. Landeszeitung of June 25, 1969, in which this [35] MICHAEL ATIYAH and RAOUL BOTT, On the periodicity picture first appeared. According to that arti- theorem for complex vector bundles, Acta Math. cle, Bott is lecturing to a group of undergrad- 112 (1964), 229–247. uates about vector fields on manifolds, which [42] , A Lefschetz fixed point formula for elliptic ——— is not apparent from the picture itself. complexes. I, Ann. of Math. (2) 86 (1967), 374–407. The photographer was Wolfgang Vollrath, [44] ——— , A Lefschetz fixed point formula for elliptic complexes. II. Applications, Ann. of Math. (2) 88 now working at Leica Microsystems and then (1968), 451–491. in his third term as a physics student at the [52] RAOUL BOTT, On topological obstructions to integra- University of Bonn. Dr. Vollrath writes, “At bility. Actes du Congrès International des Mathé- that time I was attending a lecture course in maticiens (Nice, 1970), Tome 1, Gauthier-Villars, linear algebra given by Prof. Hirzebruch. He Paris, 1971, pp. 27–36. used to organize once a year a symposium of [56] R. BOTT and A. HAEFLIGER, On characteristic classes very high reputation at the Mathematisches In- of -foliations, Bull. Amer. Math. Soc. 78 (1972), stitut of the University of Bonn. In 1969 Prof. 1039–1044. Hirzebruch had the great idea to ask some of [81] M. F. ATIYAH and R. BOTT, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. Lon- the symposium lecturers…to give readily com- don Ser. A 308 (1983), no. 1505, 523–615. prehensible talks to the younger students. One of the lecturers was Raoul Bott. Most fascinating for the German students, however, was not the lecture itself, but that he was smoking…while he was talking and at the same time writing and wiping on the black board. This was inconceivable for German students. We enjoyed it very much. I was sitting in the audience with my camera and a tele- photo lens on it and could hardly believe what I was seeing.” —Bill Casselman ([email protected])

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