Memories of Sir Michael Atiyah Nigel Hitchin, Editor

Home , ADHM construction, Donaldson theory, Edward Witten, Frances Kirwan, George Lusztig, Michael Atiyah

lection of more personal recollections—what it was like to be his student, to work alongside him, to have avenues of exploration pointed out, or to be inspired and energized by his unique personality. The contributions cover a full fifty years in which his interests moved from mathematics to incursions into physics. They are arranged in chrono- logical order.

Michael Atiyah and Representation Theory George Lusztig I was very fortunate to have the opportunity to study with Michael for two months in Oxford in 1968 and for two years (1969–71) at the Institute for Advanced Study in Princeton, where he arranged for me to be invited. Michael Atiyah is famous for his work in algebraic geometry, algebraic topology, index theory, and, later, physics, but on the side he had important contributions to representation theory. I will try to explain some of them here. Michael’s 1967 paper with Bott contains a proof of the holomorphic Lefschetz fixed point formula that provides a wonderfully simple explanation for Weyl’s character formula for tr(푔, 푉) (푔 is a regular semisimple element, and 푉 is an irreducible rational representation of a complex Figure 1. semisimple group 퐺). The explanation was in terms of the (finite) fixed point set of the automorphism defined by 푔 Sir Michael Atiyah, a hugely influential figure in mathemat- on the flag manifold of 퐺. This was a model for me when ics, died on January 11,2019. A tribute appeared in the No- I later worked with Deligne on representation theory of vember 2019 issue of the Notices, but what follows is a col- finite reductive groups, where the holomorphic Lefschetz fixed point formula was replaced by a Lefschetz fixed point This is Part 2 of a two-part series on Sir Michael Atiyah. Part 1 was included formula in 푙-adic cohomology for certain automorphisms in the November issue. of finite order of a flag manifold and certain subvarieties Communicated by Notices Associate Editor Chikako Mese. For permission to reprint this article, please contact: George Lusztig is the Abdun Nor Professor of Mathematics at MIT. His email [email protected]. address is [email protected]. DOI: https://doi.org/10.1090/noti1997

1834 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 11 of it. Michael’s 1968 paper “The index of elliptic operators, III” with Singer contains an explicit formula (see 4.6) for tr (푔, 푉) in which 푔 ∈ 퐺 is of finite order but not neces- sarily regular. While tr(푔, 푉) could in principle be determined from the analogous traces with 푔 regular semisimple, this paper gives for the first time an explicit formula for tr(푔, 푉) as a sum of contributions of the connected components of the fixed point set of 푔 on the flag manifold of 퐺. This is a very interesting complement to Weyl’s character formula. Michael’s 1977 paper with Schmid provides a new per- spective on the discrete series of real semisimple groups based on the 퐿2-index theorem (a version of the index theorem for noncompact manifolds). When I was Michael’s student I was mainly working on Figure 2. Lusztig and Atiyah in Edinburgh, June 2018. problems in algebraic topology and index theory, but I also learned many things in representation theory from Michael, and eventually representation theory became my main focus. the affine Weyl group. Around 1971, I attended a talk by Quillen at the IAS, One of the things that I learned as Michael’s student was 퐾 where he explained his solution of a conjecture of F. Adams to use topological -theory and its equivariant version, a in algebraic topology, which surprisingly used the modu- theory very closely associated with Michael’s work. In the lar and complex representations of the finite group early 1980s I used this in connection with an affine Hecke algebra 퐻 by realizing the principal series representations GL푛(퐅푞). After the talk I asked Michael whether the com- of 퐻 in terms of suitable equivariant 퐾-theory of a flag plex irreducible representations of GL푛(퐅푞) were known. 푞 He told me that the characters were determined by J. A. manifold in which the parameter of the Hecke algebra Green, but the representations themselves were not under- becomes the standard generator of the representation ring stood. This seemed to be an interesting question, and thus of the circle group. This led me to a conjecture in which the 퐻 I became more and more involved in representation the- representations of are realized in terms of equivariant 퐾 ory (probably to the disappointment of Michael). My first -theory of certain Springer fibers. (This conjecture was job after Princeton, which I got, I believe, with some help later proved by Kazhdan and me, thus establishing the part 퐻 from Michael, was at the University of Warwick, where J. A. of the local Langlands conjecture which involves . Here 퐾 Green was teaching. Michael’s equivariant -homology played a key role.) After going to Warwick, I often visited Michael in Ox- I last saw Michael in June 2018 when, with my wife, I vis- ford, where he returned in 1972. On one of those visits he ited him at his apartment in Edinburgh. We had a wonder- told me about a beautiful formula in which the 푘th Adams ful time. I told him that we were staying in a hotel next to operation 휓푘(푉) applied to the standard representation 푉 the statue of Maxwell (the existence of this statue is largely due to Michael’s efforts). Michael told us about his ad- of GL푛(퐂) is expressed as an explicit alternating sum of ir- miration for Maxwell, whom he put on the same level as reducible representations of GL푛(퐂). Understanding this formula became one of the motivations for my work (in Newton and Einstein. He also explained some of his new 1973) with Roger Carter on the modular representations mathematical ideas. of GL푛. In our 1974 joint paper we give, among other things, a refinement of Michael’s formula in which 퐂 is replaced by 퐅푝, an algebraic closure of the field with 푝 elements, and 푘 = 푝. In this case 휓푘(푉) becomes the Frobe- nius twist of 푉, and our refinement gave a resolution of 휓푘(푉) by Weyl modules (predicted by Michael’s formula) that are linked with each other by an action of the affine Weyl group. This paper gave me later some support for stating a conjectural formula for the irreducible modular representations of a reductive group over 퐅푝 in terms of George Lusztig

DECEMBER 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1835 stone of Penrose’s twistor theory, trading differential ge- Instantons and Monopoles ometry in four dimensions for algebraic geometry in three. But everything was over the complex numbers, whereas the Nigel Hitchin Yang–Mills problem concerned the 4-sphere. So Atiyah first showed how twistor theory could be Atiyah’s most celebrated contribution to mathematics is adapted to this situation by regarding complex projective the index theorem. Indeed the papers occupy two whole 3-space as a real algebraic variety with no real points but volumes of his collected works covering the twenty years with real lines. With this viewpoint the projective space since the Bulletin of the AMS published the initial paper in fibered over the 4-sphere (and was even part of a moregen- 1963. Louis Nirenberg recalls one of the seminars Atiyah eral picture for even-dimensional spheres). He then used gave at NYU at the time, which he began with the words: the modern work in algebraic geometry of Horrocks, Barth, “Whenever I come here I seem to talk about the index the- and others to give an explicit construction of the holomor- orem.. .,” whereupon Nirenberg replied, “.. .and you are phic vector bundles defining instantons. This followed going to do it till you get it right!” (echoing the old story from studying the structure of a module over a ring of poly- about the young soprano wondering why she was asked nomials generated by some sheaf cohomology groups as- for so many encores). Yet, however powerful the index sociated to the vector bundle. If a certain sheaf cohomol- theorem was, it would be wrong to think of him carrying ogy group vanished, then the construction followed from around this huge weapon and firing it at every opportunity. essentially a single matrix of linear forms. But this was In fact, in what was probably his final mathematical paper where the twistor theory came into play again, as it trans- [1], he deliberately avoided the more obvious application lated back to a simple question about a differential oper- of the Atiyah–Patodi–Singer version for a more direct ap- ator on the sphere. The fertile common ground between proach. algebraic geometry and differential geometry that was in His choice of mathematical problems was more a recog- evidence here was dear to his own interests, not just clas- nition that they resonated with something he was familiar sical algebraic geometry but also going back to his link- with. This was apparent in the work on instantons and ing of the Riemann–Roch theorem with elliptic differential monopoles in the 1970s and ’80s when I became less of operators—the birth of the index theorem. a witness to his mathematical output and more of a par- Atiyah’s work on the related equations describing mag- ticipant. Singer brought the problem of solving the self- netic monopoles in 퐑3 was rather different. These gauge- dual Yang–Mills equations to Oxford in 1977 from his con- theoretic equations, the Bogomolny equations, were tact with physicists at MIT. The question was one of find- 퐑3 ing, up to gauge equivalence, all connections on a princi- close cousins of the instantons but defined on . They de- 4 2 scribe static configurations, and Atiyah’s engagement came pal SU(2)-bundle over 퐑 that minimize the 퐿 norm of from discussions with the mathematical physicist Nick the curvature. With appropriate decay at infinity (which Manton on the conjectural dynamics of slowly moving was confirmed later by Uhlenbeck’s work) this became by monopoles. Arguing that slow motion could be approxi- conformal invariance a problem on the 4-sphere, a com- mated by geodesic motion on the space of static solutions pact manifold where both manifold and bundle have non- presented the question of finding the natural metric on trivial characteristic classes. the moduli space. When I came back to Oxford from an The index theorem did then provide a starting point, academic year in Stony Brook in 1984 I explained to him giving the local structure of the moduli space [2] and show- how both well-known and new hyperkähler metrics could ing that there were more solutions than the ones provided be obtained by a quotient construction from flat space in- to us by the physicists. But Michael’s real interest was in volving an adaptation of the theory of moment maps, but, using the ideas of Roger Penrose and his student Richard spurred by his recall of the work with Bott on the Yang– Ward to find the solutions in concrete form. Here charac- Mills equations on a Riemann surface, he insisted on treat- teristically he saw an opening into the type of geometry ing the moduli space as an infinite-dimensional hyperkäh- that he had enjoyed since the beginning: both he and Pen- ler quotient and us working out the metric for two rose had been exposed as undergraduates in Cambridge monopoles. to the classical geometry of describing the lines in com- Working extensively that summer, I managed to use plex projective 3-space by the points in a four-dimensional twistor theory to do this, but this was to Michael too com- complex quadric, the Klein quadric. This was the corner- plicated, so he derived the formula it gave by essentially elementary means, carefully marshalling known proper- Nigel Hitchin is an Emeritus Savilian Professor of Geometry at the University of Oxford. His email address is [email protected]. ties of the metric involving symmetry, spectral curves, and Donaldson’s rational maps until the specific solution to

1836 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 11 the differential equation dropped out. We were then able decades-long process that led to the Standard Model had to analytically describe some geodesics and discuss the scat- been largely driven by experiment and by considerations tering in detail, a project he pursued enthusiastically. He of quantum theory that were rather far afield from the van- even convinced researchers at IBM’s lab in the UK to test tage point of most mathematicians. Conversely, during the power of their processors by producing a video of this this period the ideas of modern mathematics seemed phenomenon. largely irrelevant to physicists grappling with elementary This ability to fashion proofs in an understandable form particles. A physics graduate student of the period, for ex- was key to the appeal of many of his papers and also to his ample, would most probably never hear about a homol- lectures. In fact, the two topics above formed the basis of ogy group, let alone something more contemporary like his series of Fermi Lectures on Yang–Mills theory [3] and the Atiyah–Singer index theorem. the Porter Lectures on monopoles [4]. By the mid-1970s, the gauge theory revolution had created for physicists a new situation that would call for References greater mathematical sophistication. But this was under- [1] Atiyah M, Lebrun C. Curvature, cones and characteristic stood only gradually. Michael Atiyah and other mathe- numbers, Math. Proc. Cambridge Philos. Soc. 155 (2013), maticians who became interested in what physicists were 13–37. MR3065256 doing in quantum gauge theory played an important role [2] Atiyah M, Hitchin NJ, Singer IM. Deformations of instantons, Proc. Nat. Acad. Sci. USA 74 (1977), no. 7, 2662– in the process. 2663. MR0458424 An early turning point came in 1976. A puzzle about [3] Geometry of Yang-Mills Fields, Lezioni Fermiane, Ac- the Standard Model known as the U(1) problem, iden- cademia Nazionale dei Lincei & Scuola Normale Superiore, tified by Murray Gell-Mann and Steve Weinberg, among Pisa, 1979. others, was abruptly solved by Gerard ’t Hooft by studying [4] Atiyah M, Hitchin NJ. The geometry and dynamics of mag- the Dirac equation in the field of a gauge theory instanton. netic monopoles, M. B. Porter Lectures, Princeton University Soon, Albert Schwarz showed that the key facts were best Press, Princeton, NJ, 1988. MR934202 understood in the context of the Atiyah–Singer index theorem. Few physicists at the time knew what to make of this. I first met Atiyah when he visited MIT in the springof 1977, invited by Roman Jackiw. At the time, he was ex- plaining his work with Richard Ward, solving the instanton equation on ℝ4 by use of the Penrose twistor transform. His lectures had a big impact in the math and physics communities in the Cambridge (Massachusetts) area. Physicists at the time were very interested in solving the instanton equations because of speculation by Alexan- Nigel Hitchin der Polyakov about the dynamics of gauge theories. How- ever, the ingredients in the twistor transform of the instan- Michael Atiyah and Physics ton equation—complex manifolds, sheaf cohomology, fiber bundles—were quite unfamiliar to me and most other physicists. Edward Witten By January 1978, when Atiyah invited me to visit Ox- Michael Atiyah played a major role, starting in the mid- ford for a few weeks, he was lecturing at the Maths Institute 1970s, in redefining the relationship between mathematics about a more precise understanding of instantons—the and physics. Atiyah–Drinfeld–Hitchin–Manin (ADHM) construction. I By that time, theoretical physics had reached a major can well remember my perplexity in this period. Clearly, turning point with the emergence of the Standard Model Atiyah and his colleagues were saying interesting and re- of particle physics, based on nonabelian gauge theory or markable things about the nonlinear classical equations Yang–Mills theory. In a sense, theory had caught up with of nonabelian gauge theory. At the same time, it was very experiment, though it took a while for that to be clear. The hard to imagine how their results could be applied to the questions of quantum dynamics that most interested physicists. Physical applications of the ADHM construction Edward Witten is a professor in the School of Natural Sciences at the Institute for Advanced Study, Princeton. His email address is [email protected]. seemed far away. Toward the end of my visit, Atiyah showed me two

DECEMBER 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1837 physics papers that I had not seen before. By this time, panded the scope of interaction between physicists and it was known that nonabelian gauge theories can describe mathematicians. Among many other things, this led to un- magnetic monopoles as well as ordinary elementary parti- expected applications of the ADHM construction in cles. Ordinary particles arise by quantization of fields and physics. Unfortunately, to explain all that here would take are in representations of the gauge group 퐺; monopoles us too far afield. are classified by topology. But Peter Goddard, Jean Nuyts, In 1987, Atiyah twice visited the Institute for Advanced and David Olive (GNO) had suggested that magnetic Study, and he was more excited than I could remember. charge can be interpreted as a weight of a “dual gauge What he was excited about was Floer theory (of symplec- group” 퐺∨, and Olive with Claus Montonen had gone on tic manifolds or of flat connections on a three-manifold), to suggest a quantum duality between theories with gauge which he thought should be interpreted as the Hamilton- groups 퐺 and 퐺∨. Atiyah told me that the GNO dual ian formulation of a quantum field theory. This quantum gauge group was the same as the dual group introduced field theory was supposed to be, in language that wasin- by Robert Langlands in the Langlands program and that troduced later, a topological quantum field theory, which he thought there was something very deep here. He urged would be related to Gromov invariants of a symplectic me to go to London to discuss the matter with Olive. manifold or Donaldson invariants of a four-manifold. The By the time I got to London, I was skeptical. Like other idea of topological quantum field theory was mostly physicists of the time, I had never heard of the Langlands Atiyah’s conception. Atiyah set for me the task of trying to program and I had no idea what to make of Atiyah’s obser- interpret what he was saying in the language of physicists. vation that the Langlands and GNO dual groups were the At first, this was difficult, for a variety of technical reasons. same. But I could see that technically the Montonen–Olive For example, the fermionic symmetry used by Floer had proposal was not correct if taken literally. However, by the spin 0, as opposed to the half-integral spin of spacetime end of the day, Olive and I understood that the technical supersymmetries as studied by physicists. But eventually objections to Montonen–Olive duality are absent in the I realized that a simple “twisting” of supersymmetric field supersymmetric case, and we had formulated a number of theories could give a theory with the properties that Atiyah ideas that eventually (in the mid-1990s) were important wanted. This gave, at a formal level, a reformulation of the in understanding it more fully. Gromov and Donaldson invariants in a language that was Atiyah’s deeper idea—that the Langlands program natural to physicists. should somehow be tied up with electric-magnetic dual- The other problem that Atiyah recommended for physi- ity in four-dimensional gauge theory—remained in limbo cists in the years 1987–8 was to understand the Jones poly- for much longer. A concrete understanding depended on nomial of a knot via quantum field theory. I had never many intervening developments in both math and physics heard of the Jones polynomial before Atiyah recommend- and only emerged in the mid-2000s. The full scope of this ed this problem, and this certainly put me in the majority relationship is probably still far out of sight. among physicists. The challenge about the Jones polyno- I will mention just a few highlights of the following mial that Atiyah posed was specifically to find a descrip- decade. At the 1979 Carg´esesummer school, Atiyah and tion of it with manifest topological invariance. By 1987–8, Raoul Bott undertook to educate physicists about Morse a number of constructions of the Jones polynomial were theory. I and most (or all?) of the physicists there had cer- known, but topological invariance was never manifest a tainly never been exposed to Morse theory before. Another priori; it was always proved by checking generators and re- highlight was a conference in Texas where Atiyah and Is lations. Singer began to elucidate the topological meaning of what The following year brought many new clues about the physicists know as perturbative anomalies in gauge theory. Jones polynomial in work by, among others, Erik Verlinde, This helped introduce physicists to a deeper understand- Greg Moore and Nathan Seiberg, and Akihiro Tsuchiya and ing of fermion path integrals. Two papers by Atiyah and Yukihiro Kanie. Eventually, at a meeting in Swansea, where Bott in these years were ultimately influential for physi- I had the benefit of further discussions with Atiyah and cists. Their 1983 paper “The Yang–Mills equations over with Graeme Segal, I had the good fortune to put some of Riemann surfaces” introduced ideas that were important the pieces together and interpret the Jones polynomial in later in understanding quantum gauge theories in two di- terms of a three-dimensional gauge theory with the mensions. Their 1984 paper “The moment map and equi- Chern–Simons function as its action. variant cohomology” helped lead to the important tech- This answered some of the questions, but actually nique of “localization” in supersymmetric quantum field Atiyah’s vision about the Jones polynomial had two im- theory. Starting in the mid-1980s, the emergence of string portant aspects that were vindicated only long afterwards. theory greatly widened the horizons of physicists and ex- First, Atiyah predicted that the argument 푞 of the Jones

1838 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 11 polynomial should ultimately be understood as a Putting these two lines of thought together, Atiyah, parameter that counts instantons on a four-manifold. In Maldacena, and Vafa explored the significance in M-theory 1988–9, this idea looked to me like a bridge too far, but of isolated singularities of 퐺2 manifolds and their resolu- something along these lines was actually understood in tions. This was new at the time but is now regarded as an the last decade. Various things were needed first, includ- important direction. In the spring of 2001, Atiyah spent ing the invention of a refinement of the Jones polynomial, several months at Caltech, where I was on sabbatical. We known as Khovanov homology, and a greatly enriched had a memorable collaboration on this topic, leading to understanding by physicists of the consequences of electric- our paper “M-theory dynamics on a manifold of 퐺2 holo- magnetic duality. Also, Atiyah advocated that a natural ex- nomy.” planation of the three-dimensional invariance of the Jones Atiyah, along with colleagues such as Raoul Bott and Is polynomial should have an extension to explain the spec- Singer, played an enormous role in introducing new ideas tral parameter of integrable systems and the associated and encouraging and teaching physicists to study quan- Yang–Baxter equation. Something along these lines was tum field theory from new points of view. It took many understood only in the last few years in the work of Kevin twists and turns for these lessons to be really learned and Costello. absorbed in the physics world. Atiyah always believed that Actually, one facet of Atiyah’s vision is still unclear as the study of quantum field theory as a tool in geometry of 2019. Atiyah was always extremely interested in the had to be integrated with the study of more “physical” as- spectral parameter that appears in the twistor transform pects of quantum field theory. His vision and clairvoyance of instantons—as in the ADHM construction—and in the have had a truly far-reaching influence. construction of monopoles—as explored in a book that he wrote with Nigel Hitchin. He often expressed a suspicion that the monopole spectral parameter should be related to the spectral parameter of integrable spin systems and integrable models of lattice statistical mechanics. As of this writing, there is hope that something along these lines will emerge in further developments from the work of Costello. Going back to Donaldson theory, after formulating it in terms of a twisted version of supersymmetric gauge theory, I thought that this would lead to immediate progress, Edward Witten but that was not the case. In the period around 1990, Atiyah probably understood better than I did the following essential point: to contribute something new to Don- Recollections of Michael Atiyah aldson theory, physicists would need some sort of strong coupling methods, since anything that could be said for Simon Donaldson weak coupling would involve retracing the steps that math- My main interaction with Michael Atiyah fits neatly into ematicians had already taken. Eventually, Seiberg and I the decade of the 1980s, from the time when I arrived in were able to apply strong coupling methods to this prob- Oxford in 1980 as a new graduate student until his de- lem, leading to a relationship between Donaldson theory parture in 1990 to take up the Mastership of Trinity Col- and an abelian theory with monopoles (Seiberg–Witten lege, Cambridge. While we perhaps did not perceive so at theory). the time, it was an extraordinary decade. There were two Michael Atiyah worked with physicists on many occa- large groups in the Mathematical Institute in Oxford, one sions. I will describe the background to one of the papers led by Atiyah and the other by Penrose. These had come he wrote with physicists, “An M-theory flop as a large 푁 du- together, a few years before, through the Penrose “twistor” ality,” written in the year 2000 with Juan Maldacena and construction that had brought ideas from complex geom- Cumrun Vafa. Early in his career, Atiyah had explored the etry into mathematical physics. Atiyah’s interests centered “small resolutions” of certain complex threefold singular- on Yang–Mills theory, a theme that he developed in ities. By the mid-1990s, it was known that these small resolutions are important in the physics of Calabi–Yau manifolds, and it was also understood that nonperturba- Simon Donaldson is a permanent member at the Simons Center for Geometry and Physics in Stony Brook and a professor at Imperial College London. His tive “dualities” can relate string theory on a Calabi–Yau email address is [email protected]. threefold to M-theory on a manifold of 퐺2 holonomy.

DECEMBER 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1839 many directions. One prominent achievement was the ties for the eigenvalues of a Hermitian matrix, which were Atiyah–Drinfeld–Hitchin–Manin (1978) description of all soon embedded in symplectic geometry and a general the- finite-energy solutions of the Yang–Mills instanton equa- ory of convexity results for moment maps. The common tion on 퐑4, built on the twistorial construction that trans- feature of all these talks was that they were immensely en- lates the problem into one about holomorphic vector bun- joyable experiences for the audience, full of Atiyah’s excite- dles over CP3, thus connecting Yang–Mills theory to alge- ment, drawing together old and new mathematics in sur- braic geometry. Another 1978 paper of Atiyah, Hitchin, prising ways. and Singer took a more differential geometric direction, Atiyah occupied a large office with a fine view surveying setting up the foundations for the theory of self-dual Rie- the area of Oxford around St. Giles church. In his company mannian four-manifolds and the Yang–Mills instantons one felt that one likewise surveyed with him the mathe- over them. A third direction brought in topology: this was matical scene, and he would pour out a continual stream represented by the 1982 paper of Atiyah and Bott on the of ideas, suggestions, and observations. The fact that he Yang–Mills equations over Riemann surfaces and a 1978 travelled widely and talked to everybody meant that he al- paper of Atiyah and Jones proposing a stabilization conjec- ways had many strands of current thought in his hands, ture for the homology of instanton moduli spaces, partly which he would constantly weave into new patterns, seek- motivated by an analogy with results of Segal on spaces of ing connections between different developments. Just as rational maps. one example of a suggestion that was important to me, I Over that decade, Atiyah’s interests and activity moved remember him telling me about a variational point of view over a variety of other topics. One of his long-standing on symplectic quotients that he had heard about during a interests was in the Bogomolny equation for monopoles visit to Harvard. He suggested that perhaps there was some on 퐑3; others involved moment maps for group actions similar functional in the problem of the existence of Her- in symplectic geometry and equivariant cohomology, de- mitian Yang–Mills connections on stable vector bundles. veloping in part from the Atiyah–Bott paper mentioned With this prompt it was not hard to find the functional, above. One exciting application of localization techniques which, with a number of variants, has been important in came in the influential proposal of Witten to prove the complex differential geometry. Atiyah–Singer index theorem by applying (formally) local- I met Michael regularly after 1990, and he remained an ization in the loop space of a manifold. Later in the decade inspiration and source of wise advice and support in count- the Jones knot invariant and the general notion of a topo- less ways. His 2001 paper with Witten is an important logical field theory came into prominence (the latter in- reference for those of us attempting to understand seven- fluenced particularly by ideas of Segal). The close ofthe dimensional manifolds with exceptional holonomy. I last decade saw the arrival of “mirror symmetry” in geometry. saw him in August 2018 at the ICM in Rio. He was frail but One of the highlights of each year was the June trip to Bonn otherwise as ever, overflowing with ideas and enthusiasm for the Arbeitstagung, where by tradition Atiyah gave the for mathematics. opening lecture. His last lecture in this series (circa 1990) discussed a paper of Candelas et al. involving mirror symmetry for Calabi–Yau hypersurfaces in weighted projective space. Some of my most vivid memories from that time come from the wonderful seminars given by Atiyah, usually leading off his Monday afternoon seminar series with the first seminar of the term. The topics moved over many fields. In one example he discussed classical results on determi- nantal representations of plane curves (i.e., equations Simon Donaldson 푑푒푡(퐴0푧0 + 퐴1푧1 + 퐴2푧2) = 0) in connection with Hitchin’s work from the early 1980s on Nahm’s equation and spectral curves. In another lecture the topic was a connection between the Dedekind 휂-function and the 휂- invariant of Atiyah, Patodi, and Singer, emerging from work of Witten, Quillen, Bismut, Freed, and others in the mid-1980s. Another memorable seminar was based on Atiyah’s paper “Convexity and commuting Hamiltonians.” This began at a down-to-level level with Horn’s inequali-

1840 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 11 and Mumford’s geometric invariant theory (GIT). I was Moment Maps and Convexity: able to hear about all these ideas from Michael as soon as I Memories of Michael Atiyah started as his student, and he immediately gave me a beautiful problem to work on: to explore whether the same sorts of results he and Raoul had found for the Yang–Mills Frances Kirwan functional in an infinite-dimensional setting also hold for In the early 1980s, when I was Michael Atiyah’s student, the normsquare of a moment map in the situation of a moment maps, convexity, and their links with each other Hamiltonian action of a compact group 퐾 on a compact and with algebraic and differential geometry and mathe- symplectic manifold 푀, suggesting that his convexity re- matical physics formed an important focus of his interests sults for torus actions might be relevant to this, as indeed [2–5,7–9]. they turned out to be. I first met Michael around Christmas 1980 when Iwas A moment map for a Lie group 퐺 acting on a symplec- ∗ in my final year as a Cambridge undergraduate wonder- tic manifold (푋, 휔) is a smooth map 휇 ∶ 푋 → 픤 that ing what to do next and whether I should apply for PhD is 퐺-equivariant with respect to the coadjoint action of 퐺 places. I had summoned up the courage to look up his on the dual of its Lie algebra 픤 and is such that the compo- home phone number in the Oxford telephone directory nent 휇⋅푎 ∶ 푋 → ℝ of 휇 along any 푎 ∈ 픤 is a Hamiltonian and ring him. I was horribly embarassed when he an- function for the infinitesimal action of 푎 on 푋 (that is, the swered the phone sounding very sleepy, although it was 1-form 푑(휇 ⋅ 푎) corresponds under the duality defined by close to lunch time, and I discovered that he had just re- 휔 to the vector field on 푋 determined by 푎). The restric- −1 turned from the United States and I had woken him up. tion of the symplectic form 휔 to 휇 (0) is degenerate pre- Nonetheless he was very friendly and set up a meeting with cisely along the orbits of 퐺, and the “symplectic quotient” −1 me at which he gave me much helpful advice and told me 휇 (0)/퐺 inherits a (stratified) symplectic structure. to mention his name on my application if I decided to ap- An infinite-dimensional moment map plays a crucial ply to Oxford. I did that and had the huge good fortune role in the Atiyah–Bott paper [7] on the Yang–Mills equa- to end up in the Mathematical Institute nine months later, tions over a Riemann surface, which was motivated by the- joining other students in the geometry group, including oretical physics and brings together a wide range of dif- Simon Donaldson, John Roe (who very tragically died a ferent mathematical topics, including algebraic topology year before his former supervisor), Michael Murray, and and Morse theory, algebraic geometry and number theory, Jacques Hurtubise. gauge theory and analysis. The main object of study is the Nobody who ever met Michael Atiyah will be surprised moduli space 푀Σ(푛, 푑) of semistable holomorphic vector to be told that he was a wonderful supervisor to have: al- bundles of rank 푛 and degree 푑 over a compact Riemann ways full of an exciting mix of ideas and enthusiasm. He surface Σ of genus 푔 ⩾ 2 (and more generally moduli also had around him at the time a fantastic group of col- spaces of principal 퐺-bundles for a compact Lie group 퐺). leagues and visitors as well as students. Raoul and Phyl- Here a holomorphic vector bundle 퐸 over Σ is (semi)stable lis Bott spent the academic year 1981–2 in Oxford, and if it has no proper subbundle 퐹 with slope 휇(퐹) ∶= Cliff Taubes (who had recently completed his PhD) also degree(퐹)/rank(퐹) (strictly) greater than the slope 휇(퐸) visited. Michael and Raoul were writing their fundamen- = 푑/푛 of 퐸. Any semistable bundle 퐸 has a Jordan– tal papers on the Yang–Mills equations over a Riemann Hölder filtration surface [7] (submitted March 1982, based on a prelim- 0 = 퐸0 ⊂ 퐸1 ⊂ ⋯ ⊂ 퐸푚 = 퐸 inary account [6] written in 1980) and on the moment map and equivariant cohomology [8] (submitted Decem- by subbundles of the same slope such that each subquo- ber 1982). Michael had just completed his paper “Convex- tient 퐸푗/퐸푗−1 is stable for 1 ⩽ 푗 ⩽ 푚, and the associated ity and commuting Hamiltonians” [2] (submitted March graded bundle 1981); he had recently learnt that Victor Guillemin and 푚 Shlomo Sternberg had discovered similar results indepen- gr(퐸) = ⨁ 퐸푗/퐸푗−1 dently and essentially simultaneously [14]. He was also in- 푗=1 trigued by the observation (made to him by Sternberg and is independent up to isomorphism of the choice of filtra- David Mumford) of a relationship between moment maps tion. Two semistable bundles 퐸 and 퐸′ over Σ of rank 푛 and degree 푑 are said to be 푆-equivalent (after Seshadri) Frances Kirwan is the Savilian Professor of Geometry at the University of Oxford. and represent the same point in 푀Σ(푛, 푑) if and only if ′ Her email address is [email protected]. gr(퐸) ≅ gr(퐸 ). When 푛 and 푑 are coprime then semista- bility is equivalent to stability and 푀Σ(푛, 푑) is the moduli

DECEMBER 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1841 space of stable rank 푛 and degree 푑 bundles over Σ up to identified with the space of holomorphic structures on ℰ0, isomorphism. and thus 풞(푛, 푑) can be stratified using the Harder– There are many different ways to think about the geom- Narasimhan type of a holomorphic vector bundle defined etry of the moduli space 푀Σ(푛, 푑). It is a complex pro- as follows. Any holomorphic vector bundle 퐸 has a canon- jective variety of dimension 푛2(푔 − 1) + 1, and when ical filtration (its Harder–Narasimhan filtration) 푛 and 푑 are coprime it is a compact Kähler manifold. It 0 = 퐸0 ⊂ 퐸1 ⊂ ⋯ ⊂ 퐸푠 = 퐸 can be constructed as an infinite-dimensional analogue of a quotient in the sense of Mumford’s geometric invari- such that each subquotient 퐸푗/퐸푗−1 is semistable and ant theory of an infinite-dimensional complex affine space their slopes 휇(퐸푗/퐸푗−1) are strictly decreasing for 1 ⩽ 풞(푛, 푑) of unitary connections on a fixed 퐶∞ hermitian 푗 ⩽ 푠. The Harder–Narasimhan type of 퐸 is then deter- vector bundle ℰ0 of rank 푛 and degree 푑 by the action mined by the ranks and degrees of the semistable bundles 퐸 /퐸 of the complexified gauge group 풢ℂ(푛, 푑) of complex au- 푗 푗−1. In order to describe how bundles of one Harder– tomorphisms of ℰ0. Atiyah and Bott observed that this Narasimhan type can degenerate to another (using work of infinite-dimensional affine space 풞(푛, 푑) has a natural Shatz), Atiyah and Bott encode the type 휇 of 퐸 as the vec- Kähler structure that is invariant under the gauge group tor (휇1, … , 휇푛) = (푑1/푛1, … , 푑푠/푛푠) in which the slope 푑 /푛 퐸 /퐸 푛 = dim(퐸 /퐸 ) 풢(푛, 푑) of unitary automorphisms of ℰ0 and that asso- 푗 푗 of 푗 푗−1 appears 푗 푗 푗−1 successive ciating to a connection its curvature can be regarded as a times or, equivalently, as the convex polygon 푃 with ver- moment map for the action of 풢(푛, 푑) on 풞(푛, 푑). They tices (0, 0), (rank퐸1, degree퐸1), … , (rank퐸푠, degree퐸푠) = showed that adding a suitable central constant determined (푛, 푑). The subset 풞휇(푛, 푑) of 풞(푛, 푑) consisting of holo- by the topological invariants 푛 and 푑 gives a moment map morphic structures on ℰ0 of Harder–Narasimhan type 휇 is a locally closed complex submanifold of 풞(푛, 푑) with ∗ 휇 ∶ 풞(푛, 푑) → Lie 풢(푛, 푑) finite codimension −1 whose corresponding symplectic quotient 휇 (0)/풢(푛,푑) 푑휇 = ∑ (휇푖 − 휇푗 + 푔 − 1), 휇 >휇 can be identified with the moduli space 푀Σ(푛, 푑) via the 푖 푗 theorem of Narasimhan and Seshadri relating (semi)sta- and its closure is contained in the union of the strata la- bility of bundles to unitary representations of the funda- belled by 휇′ with 휇′ ⩾ 휇 in the sense that the polygon mental group 휋1(Σ). The normsquare of this moment 푃′ associated to 휇′ lies above the polygon 푃. In particular, 푠푠 map is (up to the addition of a constant) the Yang–Mills when 휇 = (푑/푛, … , 푑/푛) then 풞휇(푛, 푑) = 풞 (푛, 푑) is functional for the group 퐺 = 푈(푛) over Σ, and its crit- the open subset of semistable holomorphic structures on ical points are the solutions to the Yang–Mills equations ℰ0. 4 whose analogues over Minkowski space and ℝ were well An elegant argument (which has become known as the known to be important in physics. The aim of [7] was to Atiyah–Bott lemma) shows that the Thom–Gysin long ex- apply Morse theory to the Yang–Mills functional to show act sequence that it is equivariantly perfect (in the sense that it induces ∗−2푑휇 풢(푛, 푑)-equivariant Morse inequalities that are actually ⋯ → 퐻풢(푛,푑) (풞휇(푛, 푑); ℚ) equalities) and to use this to study the cohomology of the ∗ → 퐻 (풞\ 풞 ′ (푛, 푑); ℚ) 푀 (푛, 푑) 풢(푛,푑) ⋃ 휇 quotient Σ of its minimum by the gauge group 휇′>휇 풢(푛, 푑). ∗ In fact, serious analytic difficulties arise with this pro- → 퐻풢(푛,푑)(풞\ ⋃ 풞휇′ (푛, 푑); ℚ) → ⋯ 휇′⩾휇 gram, not only because of the infinite-dimensionality of the picture but also because the Yang–Mills functional is breaks up into short exact sequences by proving ∗−2푑휇 a long way from being a Morse function in the traditional that the composition of 퐻풢(푛,푑) (풞휇(푛, 푑); ℚ) → ∗ sense, or even a Morse–Bott function, as in general the con- ′ 퐻풢(푛,푑)(풞\ ⋃휇′>휇 풞휇 (푛, 푑); ℚ) with the restriction map nected components of its critical locus have singularities. ∗ ∗ The required analysis showing that Yang–Mills paths of 퐻풢(푛,푑)(풞\ ⋃ 풞휇′ (푛, 푑); ℚ) → 퐻풢(푛,푑)(풞휇(푛, 푑); ℚ) ′ steepest descent converge appropriately to critical points 휇 >휇 was later carried out by Daskalopoulos [12], but Atiyah is injective. Indeed this composition is multiplication by and Bott avoided its use by constructing directly the associ- the equivariant Euler class ated “Morse stratification” of the space 풞(푛, 푑). 2푑휇 푒 ∈ 퐻 (풞 (푛, 푑); ℚ) In complex dimension one the Newlander–Nirenberg 휇 풢(푛,푑) 휇 integrability condition for 휕-operators̄ holds vacuously, so of the normal bundle to 풞휇(푛, 푑) in 풞(푛, 푑), and so it suf- the space 풞(푛, 푑) of unitary connections of ℰ0 can be fices to show that 푒휇 is not a zero divisor in

1842 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 11 ∗ 푠 퐻풢(푛,푑)(풞휇(푛, 푑); ℚ), which is isomorphic to 풞 (푛, 푑), and the moduli space 푀Σ(푛, 푑) can be identi- 푠푠 fied with the quotient 풞 (푛, 푑)/풢ℂ̄ (푛, 푑) where 푠 ̄ ∗ 푠푠 퐻∗ (풞푠푠(푛, 푑); ℚ), 풢ℂ(푛, 푑) = 풢ℂ(푛, 푑)/ℂ acts freely on 풞 (푛, 푑). This ⨂ 풢(푛푗,푑푗) 푗=1 means that

∗ 푠푠 where 휇 = (푑1/푛1, … , 푑푠/푛푠). Since the central sub- 퐻풢(푛,푑)(풞 (푛, 푑); ℚ) 1 group 푆 of 풢(푛푗, 푑푗) given by multiplication by scalars ∗ 1 ∗ 푠푠 ≅ 퐻 (퐵푆 ; ℚ) ⊗ 퐻 (푀Σ(푛, 푑); ℚ), acts trivially on 풞 (푛푗, 푑푗), we have and we obtain a formula for the Betti numbers of the mod- 퐻∗ (풞푠푠(푛, 푑); ℚ) 풢(푛푗,푑푗) uli space 푀Σ(푛, 푑) of semistable holomorphic vector bun- ∗ 1 ∗ 푠푠 dles of rank 푛 and degree 푑 over Σ. We also obtain mul- ≅ 퐻 (퐵푆 ; ℚ) ⊗ 퐻풢(푛̄ ,푑 )(풞 (푛, 푑); ℚ), 푗 푗 tiplicative generators for its rational cohomology, since by 1 where 풢(푛̄ 푗, 푑푗) = 풢(푛푗, 푑푗)/푆 and the rational coho- equivariant perfection of the stratification the restriction mology 퐻∗(퐵푆1; ℚ) of the classifying space of the circle map 1 group 푆 is a polynomial ring ℚ[푢푗] on one variable 푢푗 ∗ ∗ 푠푠 퐻풢(푛,푑)(풞(푛, 푑); ℚ) → 퐻풢(푛,푑)(풞 (푛, 푑); ℚ) in degree 2. Atiyah and Bott show that the component of ∗ ∗ the equivariant Euler class 푒휇 in is surjective, and 퐻풢(푛,푑)(풞(푛, 푑); ℚ)≅퐻 (퐵풢(푛, 푑); ℚ) 푠 has generators 0 푠푠 ⨂ ℚ[푢푗] ⊗ 퐻 (풞 (푛, 푑); ℚ) ≅ ℚ[푢1, … , 푢푠] 풢(푛̄ 푗,푑푗) 2푟 푗 2푟−1 푗=1 푎푟 ∈ 퐻 (퐵풢(푛, 푑); ℚ), 푏푟 ∈ 퐻 (퐵풢(푛, 푑); ℚ) 2푟−2 is a nonzero polynomial (the product of the weights of the for 1 ⩽ 푗 ⩽ 2푔 and 푓푟 ∈ 퐻 (퐵풢(푛, 푑); ℚ) (1) 1 푠 torus (푆 ) on the normal bundle to 풞휇(푛, 푑) in 풞(푛, 푑)) given by decomposing the Chern classes of the universal and therefore that 푒휇 is not a zero divisor in bundle over 퐵풢(푛, 푑) × Σ using the Künneth decomposi- ∗ ∗ 퐻풢(푛,푑)(풞휇(푛, 푑); ℚ), as required. Removing each stra- tion of 퐻 (퐵풢(푛, 푑) × Σ; ℚ). tum 풞휇(푛, 푑) from 풞(푛, 푑) one by one, leaving an open Equivalent formulas for the Betti numbers of 푀Σ(푛, 푑), subset of 풞(푛, 푑) each time, and using the Atiyah–Bott when 푛 and 푑 are coprime, had already been obtained lemma give us the equivariant perfection of the stratifica- by Harder–Narasimhan and Desale–Ramanan using arith- tion: metic geometry and the Weil conjectures. Although the methods used looked very different in some ways, Atiyah dim 퐻푖 (풞(푛, 푑); ℚ) 풢(푛,푑) and Bott observed that there was a formal correspondence 푖−2푑휇 = ∑ dim 퐻풢(푛,푑)(풞휇(푛, 푑); ℚ), between them. Both methods are based on the stratifi- 휇 cation by Harder–Narasimhan-type of the moduli stack where ℳΣ(푛, 푑) = [풞(푛, 푑)/풢ℂ(푛, 푑)] of bundles over Σ (cf. 푠 [1, 13, 15]), although Atiyah and Bott did not use the lan- 퐻∗ (풞 (푛, 푑); ℚ) ≅ 퐻∗ (풞푠푠(푛, 푑); ℚ). guage of stacks, together with inductive descriptions of the 풢(푛,푑) 휇 ⨂ 풢(푛푗,푑푗) 푗=1 unstable strata in terms of moduli of semistable bundles of smaller rank, and the fact that when 푛 and 푑 are co- Since 풞(푛, 푑) is an affine space and so homotopically triv- prime the Betti numbers of the moduli space 푀 (푛, 푑) are ial, we have Σ given by the dimensions of the cohomology groups of the 푖 푖 퐻풢(푛,푑)(풞(푛, 푑); ℚ) ≅ 퐻 (퐵풢(푛, 푑); ℚ), semistable stratum of this stack. The methods are different, however, in how they use the stratification to deduce which Atiyah and Bott showed has dimension given by the a version of the inductive formula coefficient of 푡푖 in the Poincar´eseries 푠푠 푃푡([풞 (푛, 푑)/풢ℂ(푛, 푑)])=푃푡([풞(푛, 푑)/풢ℂ(푛, 푑)]) 푃푡(퐵풢(푛, 푑)) 푠 푛 푛−1 2푑휇 푠푠 − ∑ 푡 ∏ 푃푡([풞 (푛푗, 푑푗)/풢ℂ(푛푗, 푑푗)]) = (1 + 푡2푘−1)2푔/(1 − 푡2푛) (1 − 푡2푘)2. ∏ ∏ 휇=(푑1/푛1,…,푑푠/푛푠) 푗=1 푘=1 푘=1 ≠(푑/푛,…푑/푛) (2) This gave them an inductive formula (later made explicit 푠푠 by Zagier) for calculating the equivariant Betti numbers for the Poincar´eseries of the moduli stack ℳΣ (푛, 푑) = 푖 푠푠 푠푠 dim 퐻풢(푛,푑)(풞 (푛, 푑); ℚ) for the semistable stratum [풞 (푛, 푑)/풢ℂ(푛, 푑)] of semistable holomorphic 풞푠푠(푛, 푑). When 푛 and 푑 are coprime then 풞푠푠(푛, 푑) = bundles of rank 푛 and degree 푑 over Σ and to calculate

DECEMBER 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1843 the crucial ingredient given by the Poincar´e series with (3) and (4) gives 푃푡([풞(푛, 푑)/풢ℂ(푛, 푑)]) of the moduli stack ℳΣ(푛, 푑) = (푛2−1)(푔−1) −2 −푛 [풞(푛, 푑)/풢ℂ(푛, 푑)] of all holomorphic bundles of rank 푞 푍퐶(푞 ) ⋯ 푍퐶(푞 ) 푛 and degree 푑 over Σ. − ∑ ∑ 1/|Aut(ℰ)| In the approach motivated by gauge theory, the Atiyah– Λ,휇 휇≠(푑/푛,…,푑/푛) ℰ∈Bun (픽 ) Bott lemma gives the equivariant perfection of the Morse SL푛 푞 stratification of the Yang–Mills functional and thus as the number of isomorphism classes defined over 픽푞 of the inductive formula (2), while algebraic topology is semistable bundles on 퐶 with rank 푛 and degree 푑. The ob- used to calculate the Poincar´e series 푃푡(퐵풢(푛, 푑)) = servation of Atiyah and Bott that these inductive methods 푃푡([풞(푛, 푑)/풢ℂ(푛, 푑)]) of the classifying space of the for obtaining the Betti numbers of 푀Σ(푛, 푑) (when 푛 and gauge group. The earlier arithmetic approach was based 푑 are coprime) are parallel, with the Siegel mass formula on the Weil conjectures, which can be used to calculate (derived from the additivity of counting points over finite the Betti numbers of nonsingular complex projective vari- fields) playing the role of the equivariant perfection ofthe eties (or more generally projective ind-varieties) by count- Yang–Mills functional (derived from the Atiyah–Bott ing points in associated varieties defined over finite fields. lemma) and (3) playing the role of the contractibility of From this viewpoint we need to assume (as we may by a the infinite-dimensional affine space 풞(푛, 푑), has been ex- small perturbation) that the compact Riemann surface Σ plored further in the decades since (see e.g. [1,13] and the is a nonsingular complex projective curve defined over the references therein). rationals, and indeed over ℤ, with good reduction modulo When I became Michael Atiyah’s student he told me 푝 for all but finitely many primes 푝. Let 퐶 be a nonsingu- about these beautiful ideas and about his suspicion that lar projective curve over a finite field 픽푞 of characteristic 푝. a vital ingredient in the Yang–Mills picture was that the Using his theory of matrix divisor classes, Weil constructed Yang–Mills functional can be regarded as the normsquare a canonical bijection from the set of isomorphism classes of a moment map for the action of the gauge group 풢(푛, 푑) of vector bundles on 퐶 defined over 픽푞 to a double coset on the space 풞(푛, 푑). His hope (which turned out to be space correct) was that a similar picture should emerge in finite- 픎\GL푛(픸퐾)/GL푛(퐾), dimensional situations when a compact Lie group 퐾 acts where 퐾 = 픽푞(퐶) is the function field of 퐶 and 픸퐾 is on a compact symplectic manifold (푋, 휔) with moment ∗ its adèle ring, while 픎 is a maximal compact subgroup of map 휇 ∶ 푋 → 픨 where 픨 = Lie 퐾. The arithmetic aspects GL푛(픸퐾). It is more convenient to fix determinants; re- should apply (and a close link with Mumford’s geometric placing GL푛 with SL푛, the double coset space invariant theory should exist) when (푋, 휔) is a nonsingular complex projective variety with Fubini–Study Käh- 픎 \SL (픸 )/SL (퐾) Λ 푛 퐾 푛 ler form associated to a very ample vector bundle 퐿 on 푋 Λ (픽 ) is in bijection with the set BunSL푛 푞 of isomorphism and the complexification 퐺 = 퐾ℂ of 퐾 acts linearly on classes of bundles 퐸 on 퐶 equipped with an isomorphism 푋 with respect to 퐿. An important piece of evidence for det 퐸 ≅ Λ defined over 픽푞, where the maximal compact this was the appearance of convexity in the Yang–Mills pic- subgroup 픎Λ of SL푛(픸푘) depends on the line bundle Λ. ture, together with his recent paper on convexity and com- Siegel’s mass formula gives us muting Hamiltonians. In this he studied the case when 1 퐾 = 푇 ≅ (푆1)푟 is a compact torus acting on a compact 휏(SL푛) = vol(픎Λ) ∑ , connected symplectic manifold 푋 preserving the symplec- Λ |Aut(ℰ)| ℰ∈Bun (픽푞) ∗ SL푛 tic form 휔 and with a moment map 휇푇 ∶ 푋 → 픱 where where the Tamagawa number 휏(SL푛) is the total measure 픱 = Lie푇. He showed that 휇푇(푋) is a convex polytope, ∗ 푇 푇 of SL푛(픸퐾)/SL푛(퐾) with respect to a right invariant Haar the convex hull in 픱 of the finite set 휇푇(푋 ), where 푋 measure on SL푛(픸퐾) and vol(픎Λ) is calculated with re- is the set of 푇-fixed points in 푋. If moreover (푋, 휔) is spect to the same measure. We have Kähler, then the action of 푇 extends to its complexification ∗ 푟 푇ℂ ≅ (ℂ ) , and for any 푥 ∈ 푋 the image 휇푇(푇ℂ푥) of the 휏(SL푛) = 1, (3) closure of the 푇ℂ-orbit of 푥 is the convex hull of the finite 푇 while vol(픎Λ) is given in terms of the zeta function 푍퐶 of set 휇푇(푇ℂ푥 ∩ 푋 ), with 휇푇(푇ℂ푥) = 휇푇(푋) for generic 퐶 as 푥 ∈ 푋. The application of this to the maximal torus of −(푛2−1)(푔−1) −2 −푛 an arbitrary compact Lie group with a Hamiltonian action vol(픎Λ) = 푞 /푍퐶(푞 ) ⋯ 푍퐶(푞 ). (4) on a compact connected symplectic manifold allows us to Λ (픽 ) Partitioning BunSL푛 푞 by Harder–Narasimhan type into describe the Morse stratification of the normsquare of the Λ,휇(픽 ) subsets BunSL푛 푞 and combining Siegel’s mass formula moment map and show that it is equivariantly perfect by

1844 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 11 applying the Atiyah–Bott lemma, thus obtaining results 푢 ∈ 픨∗ regarded as in Sym 픨∗. Then 푊(픨) is freely gener- similar to those of the Yang–Mills situation about the co- ated as a commutative graded algebra by a basis {휃훼} of ∗ homology of the symplectic quotient. Michael Atiyah and 픨 in degree 1 and the same basis denoted {푢훼} regarded Andrew Pressley also extended the convexity result in an as in degree 2. There is a differential operator 퐷 on 푊(픨) 훼 infinite-dimensional setting involving the loop group of defined on the generators {휃 } and {푢훼} in terms of the 퐾 [9]. structure constants for the Lie algebra 픨. When 푋 is a point, ∗ Moment maps and equivariant cohomology provide Ω픨 (푋) is the basic subcomplex of 푊(픨), which is given the title of another fundamental paper [8] by Atiyah and by polynomials on 픨 invariant under the adjoint action. A Bott written around the same time as the Yang–Mills paper symplectic form 휔 ∈ Ω2(푋) on 푋 is closed but is not [7]. This is closely related to independent work of Berline in general equivariantly closed for a compatible group ac- and Vergne [10] and was motivated by the Duistermaat– tion on 푋. However, when a compact torus 푇 = (푆1)푟 ∗ Heckman formula for the pushforward by a moment acts on (푋, 휔) with moment map 휇푇 ∶ 푋 → 픱 then 휇 ∶ 푋 → 픱∗ map 푇 for a torus action on a compact 퐷(휔 − 휇푇) = 0, and so 휔 − 휇푇 represents an equivari- connected symplectic manifold (푋, 휔) of the symplectic ant cohomology class. measure given by 휔푛/푛!. This pushforward is always ∗ Atiyah and Bott consider 퐻푇 (푋; ℂ) as a module over ∗ piecewise polynomial. Equivalently the stationary phase 퐻 (퐵푇; ℂ) = ℂ[푢1, … , 푢푟] and show that the kernel and approximation for its Fourier transform is exact, so that cokernel of the restriction map 퐹 휔dim 푖∗ ∶ 퐻∗(푋; ℂ) → ⨁ 퐻∗(퐹; ℂ) ∫ 푒−푖⟨휉,휇푇⟩휔푛/푛! = 푒−푖⟨휉,휇푇(퐹)⟩ ∫ 푇 푇 ∑ 퐹∈ℱ 푋 퐹∈ℱ 퐹 (dim퐹)! 푒퐹(푖휉) ∗ (5) (with components 푖퐹 for 퐹 ∈ ℱ) and the pushforward for any 휉 ∈ 픱, where ⟨,⟩ is the canonical pairing between map ∗ 픱 and 픱 , while ℱ is the set of connected components of ∗ ∗ 푇 푖∗ ∶ ⨁ 퐻푇 (퐹; ℂ) → 퐻푇 (푋; ℂ) the fixed point set 푋 for the 푇-action on 푋 (so that 휇푇|퐹 퐹∈ℱ is constant for each 퐹 ∈ ℱ) and 푒 is the equivariant Euler 퐹 (with components 푖퐹 for 퐹 ∈ ℱ) are torsion modules over class for the normal bundle to 퐹 in 푋. If 퐹 ∈ ℱ, then ∗ 퐻∗(퐵푇; ℂ), so their composition 푖∗푖 is an isomorphism 푇 ≅ (푆1)푟 acts trivially on 퐹, and so there is a canonical ∗ modulo torsion. Indeed 푖∗푖 is given on 퐻∗(퐹; ℂ) for isomorphism ∗ 푇 퐹 ∈ ℱ by multiplication by the equivariant Euler class 푒퐹 ∗ ∗ ∗ 퐹 푋 푒 퐻푇 (퐹; ℂ) ≅ 퐻 (퐹; ℂ) ⊗ 퐻 (퐵푇; ℂ), of the normal bundle to in , and the component of 퐹 in ∗ ∗ where 퐻 (퐵푇; ℂ) = Sym(픱 ) ≅ ℂ[푢1, … , 푢푟] is a poly- 0 ∗ ℂ 퐻 (퐹; ℂ) ⊗ 퐻 (퐵푇; ℂ) ≅ ℂ[푢1, … , 푢푟] nomial ring on 푟 generators of degree 2. As in the proof of ∗ is the Atiyah–Bott lemma, the component of 푒퐹 ∈ 퐻푇 (퐹; ℂ) codimℝ(퐹)/2 in 푒퐹,0 = ∏ (푎푗1푢1 + ⋯ + 푎푗푟푢푟), 0 ∗ 퐻 (퐹; ℂ) ⊗ 퐻 (퐵푇; ℂ) ≅ ℂ[푢1, … , 푢푟] 푗=1 ∗ where the representation of 푇 = (푆1)푟 on the normal to is nonzero, and hence 푒퐹 is not a zero divisor in 퐻푇 (퐹; ℂ), because terms of positive degree in 퐻∗(퐹) are nilpotent. 퐹 in 푋 at any point of 퐹 has weights given by 푎푗1푢1 + ∗ ⋯ + 푎 푢 1 ⩽ 푗 ⩽ (퐹)/2 Thus 푒퐹 is invertible in the localization 퐻 (퐹; ℂ) ⊗ 푗푟 푟 for codimℝ . If we localize ∗ with respect to the nonzero polynomial ∏퐹∈ℱ 푒퐹,0 in ℂ(푢1, … , 푢푟) of 퐻푇 (퐹; ℂ). This enables us to interpret dim퐹 ℂ[푢1, … 푢푟], then it follows that ∫퐹 휔 /(dim퐹)! 푒퐹(푖휉) as a rational function on 픱 and ∗ make sense of the identity (5). ∑ 푖퐹 /푒퐹 Atiyah and Bott observed that the formula (5) can be in- 퐹∈ℱ terpreted as a consequence of well-known localization for- is inverse to 푖 , and that if 휙 ∈ 퐻∗(푋; ℂ), then mulas in equivariant cohomology. For a compact group 퐾 ∗ 푇 퐹 ∗ acting on a manifold 푋 they describe a de Rham version 휙 = ∑ 푖∗푖퐹 휙/푒퐹. ∗ 퐹∈ℱ of equivariant cohomology as the basic complex Ω픨 (푋) ∗ ∗ Ω (푋) ⊗ 푊(픨) Ω (푋) ̄휔 of , where is the usual de Rham Pushing forward to a point and replacing 휙 with 휙푒 complex, 푊(픨) = Λ픨∗ ⊗ Sym 픨∗ is the Weil algebra, and where ̄휔= 휔 − 휇푇 gives us the integration formula elements 휙 of Ω∗(푋)⊗푊(픨) are basic if 휄(푎)휙 = 0 and ℒ(푎)휙 = 0 for all 푎 ∈ 픨. Here 푊(픨) is graded by giv- ∫ 휙푒 ̄휔 = ∫ 푖∗(휙푒 ̄휔)/푒 ∗ ∗ ∑ 퐹 퐹 ing degree 1 to elements 휃 ∈ 픨 in Λ픨 and degree 2 to 푋 퐹∈ℱ 퐹

DECEMBER 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1845 with the Duistermaat–Heckman formula as the special [11] Bott R. Lectures on Morse theory, old and new, Bull. Amer. case when 휙 = 1. Math. Soc. (NS) 7 (1982), 331–358. MR663786 Atiyah and Bott also related these ideas to (the second [12] Daskalopoulos G. The topology of the space of stable half of) the paper [18] of Witten, who later produced a bundles on a Riemann surface, J. Diff. Geom. 36 (1992), 699–746. MR1189501 nonabelian version of localization using the normsquare 2 [13] Gaitsgory D, Lurie J. Weil’s conjecture for function fields, ||휇|| of a moment map for a Hamiltonian action of a com- Annals of Math. Studies, 199, Princeton Univ. Press, 2019. pact group 퐾 on 푋 to study the integrals MR3887650 [14] Guillemin V, Sternberg S. Convexity properties of the ̄휔 ∫ (휂푒 )0 moment mapping, Invent. Math. 67 (1982), 491–513. −1 휇 (0)/퐾 MR664117 over a symplectic quotient 휇−1(0)/퐾 by localizing near [15] Halpern-Leistner D. Θ-stratifications, Θ-reductive stacks the critical points of ||휇||2. Here we assume that 0 is a and applications, Algebraic geometry: Salt Lake City 2015, ∗ Proc. Sympos. Pure Math., 97.1, Amer. Math. Soc., Prov- regular value of 휇, and if 훼 ∈ 퐻퐾 (푋; ℂ), then 훼0 ∈ ∗ −1 ∗ −1 idence, RI, 2018, 349–379. MR3821155 퐻 (휇 (0)/퐾; ℂ) ≅ 퐻퐾 (휇 (0); ℂ) is its image under ∗ ∗ −1 [16] Jeffrey L, Kirwan F. Localization for nonabelian group the restriction map 퐻퐾 (푋; ℂ) → 퐻퐾 (휇 (0); ℂ). Wit- actions, Topology 34 (1995), 291–327. MR1318878 ten used these ideas applied to the Yang–Mills functional [17] Jeffrey L, Kirwan F. Intersection theory on moduli to find formulas for the evaluation on the moduli space spaces of holomorphic bundles of arbitrary rank on a 푀Σ(푛, 푑) of any product of the generators Riemann surface, Annals of Math. 148 (1998), 109–196. MR1652987 2푟 푗 2푟−1 푎푟 ∈ 퐻 (푀Σ(푛, 푑); ℂ), 푏푟 ∈ 퐻 (푀Σ(푛, 푑); ℂ) [18] Witten E. Supersymmetry and Morse theory, J. Diff. 2푟−2 Geom. 17 (1982), 661–692. MR683171 for 1 ⩽ 푗 ⩽ 2푔 and 푓푟 ∈ 퐻 (푀Σ(푛, 푑); ℂ) [19] Witten E. Two-dimensional gauge theories revisited, J. ∗ Geom. Phys. 9 (1992), 303–368. MR1185834 for 퐻 (푀Σ(푛, 푑); ℂ) as at (1) above, and thus to extend Atiyah and Bott’s description of the cohomology of 푀Σ(푛, 푑) to include a complete set of relations among these generators (cf. [16, 17, 19]).

References [1] Asok A, Doran B, Kirwan F. Yang–Mills theory and Tam- agawa numbers, Bull. London Math. Soc. 40 (2008), 533– 567. MR2438072 [2] Atiyah M. Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), 1–15. MR642416 [3] Atiyah M. Gauge theory and algebraic geometry, Proc. Bei- Frances Kirwan jing Symposium on differential geometry and partial differential equations, Science Press, China, 1982, 1–20. [4] Atiyah M. Angular momentum, convex polyhedra and al- Recollections of Michael Atiyah gebraic geometry, Proc. Edin. Math. Soc. 26 (1983), 121–38. MR705256 [5] Atiyah M. The moment map in symplectic geome- Peter Kronheimer try, Global Riemannian Geometry, Horwood, 1984, 43–51. MR757204 Somewhere among my files, I still have the short note that [6] Atiyah M, Bott R. Yang–Mills and bundles over algebraic Michael sent me in 1984. I had just completed my under- curves, Proc. Indian Acad. Sci. 90 (1981), 11–20. MR653942 graduate degree at Oxford, and he was writing to let me [7] Atiyah M, Bott R. The Yang–Mills equations over Riemann know that he would be happy to accept me as a student, surfaces, Phil. Trans. Royal Soc. Series A 308 (1983), 523– beginning in the autumn. 615. MR702806 Michael’s handwriting sloped upward across the page. [8] Atiyah M, Bott R. The moment map and equivariant co- The speed and energy of those rising lines was a clear re- homology, Topology 23 (1984), 1–28. MR721448 flection of his personality, which was mirrored in allsorts [9] Atiyah M, Pressley A. Convexity and loop groups, Prog. of other ways: a briefly raised brow as he talked; an abrupt Math. 36 (1983), 33–64. MR717605 [10] Berline N, Vergne M. Classes caracteristiques equivari- antes, formule de localisation en cohomologie equivari- Peter Kronheimer is the William Caspar Graustein Professor of Mathematics at ante, C. R. Acad. Sci. Paris S´er. I Math. 295 (1982), 539–541. Harvard University. His email address is [email protected]. MR685019

1846 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 11 clearing of the throat as he walked down the corridor at the Mathematical Institute; or his forward-leaning stance and searching look, under a slightly asymmetrical frown, as he asked a question, putting a student or colleague on the spot. It was an exciting time to be working in the closely tied areas of geometry and low-dimensional topology. On the topological side, the three years during which I was a student of Michael’s saw the development of Donaldson’s polynomial invariants for smooth four-manifolds, Floer’s Peter Kronheimer instanton homology for three-manifolds, and the Jones polynomial for knots. On the geometric side, the inter- actions between Michael Atiyah and Roger Penrose had re- Recollections of Michael Atiyah cently opened up a new area. In particular, the twistor interpretation of “instantons,” discovered by Richard Ward, Ruth Lawrence-Naimark had led to the “ADHM” description of instantons on the four-sphere, named after Atiyah, Drinfeld, Hitchin, and Over the three years that I was Atiyah’s student, I came Manin. By the time I was a student, Nigel Hitchin had for a one-hour meeting regularly once a week during term- adapted Ward’s work to begin the study of the Bogomolny time (apart from the semester when he took a sabbatical equations in three-space, solutions of which are at IAS and I worked with Nigel Hitchin). To each meeting Bogomolny monopoles. I would bring a written summary of what I had achieved The first research paper that Michael asked me toread since the last meeting and would explain it along with ob- “in earnest” was Nigel’s earliest paper on monopoles, in stacles to progress. Despite the fact that he didn’t seem to which he showed how to assign to each solution of the Bo- check the details, he was always quick to spot any error. I gomolny equations a complex-algebraic curve lying on a remember in my first year with him, I worked on aprob- singular quadric surface. My first-year dissertation was in- lem that he suggested, a local index theorem. For several tended to extend part of this story to monopole solutions weeks, I would come back each week with essentially the having simple singularities at prescribed points in three- same calculation, but with different signs! Finally I got it space. This project introduced me to the rich connections right, feeling rather embarrassed about the many changes between gauge theory and hyper-Kähler geometry, and it and resolved never to forget the lesson; he was very posi- formed a starting point for my doctoral thesis, which estab- tive and encouraging, recalling the role of signs in his own lished the existence of hyper-Kähler Riemannian metrics work many years earlier. on the complex surfaces obtained as resolutions of the Du Atiyah would then usually launch into a private whirl- Val singularities. There was no gauge theory visible on the wind tour expounding some relevant topic. Sometimes he pages of my thesis, but the ADHM construction lay behind would intersperse it with anecdotes from his work with it. My research interests soon swung more towards low- Singer or Bott or his own supervisor, Hodge. One would dimensional topology, but the Du Val singularities have then feel especially privileged to be hearing about history never stopped appearing on my blackboard, a constant ref- firsthand. He very rarely directly suggested reading inthe erence point. literature (although of course one was expected to do back- I was fortunate to maintain a close connection with ground reading); much more often he preferred to give Michael after my degree, first as a postdoc at the Institute the basics directly and then expect (and recommend) that for Advanced Study, then as a colleague in Oxford, and you worked it out yourself, even if it was already in the then as a regular visitor to Trinity. Meeting with Michael literature. Rarely, our discussions moved into areas (like always brought something special. If I parked my small quantum groups) where he didn’t feel as comfortable, and car in the spot marked “Reserved for the Master” and then then he wouldn’t hesitate to suggest talking to an expert in spent the morning talking mathematics, I knew that I that area. would return home with renewed energy for research and His Monday afternoon Geometry and Analysis seminar a renewed appreciation for the connectedness of my field. functioned like a departmental colloquium, being the

Ruth Lawrence-Naimark is a professor of mathematics at the Einstein Institute of Mathematics, Hebrew University, Jerusalem. Her email address is ruthel @math.huji.ac.il.

DECEMBER 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1847 highlight of the week with often around one hundred at- tending, some coming from far afield. It was a lecture in this seminar in my second year, by Vaughan Jones, that caught my interest and launched me away from index theory and into knot theory, quantum groups, etc. The following semester Sir Michael was on sabbatical at the In- stitute in Princeton, which was where Jones–Witten theory was conceived. What is now called Witten–Chern– Simons theory had its birth at a dinner at an International Congress in Mathematical Physics in Swansea in July 1988, Ruth Lawrence- where Atiyah was pushing the idea of the Jones polyno- Naimark mial arising from a topological quantum field theory, and finally Witten realized that the action needed was precisely Memories of Professor Chern–Simons and how it all fit together. (I attended the conference but not that dinner and heard about the event Michael Atiyah the next day.) Atiyah devoted a whole eight-week term to Jones–Witten theory in the Geometry and Analysis seminar at the start of the following year, with lectures by Segal, Lisa Jeffrey Hitchin, and him. I was the last research student who completed the doctor- Anyone who has attended a lecture of his can attest to ate under Professor Michael Atiyah’s supervision. One of his mastery of subject and audience and his ability to trans- the earliest such students was Graeme Segal, who finished port you on a beautiful and magical tour of mathemat- in the sixties and retired more than fifteen years ago from ics that would leave you walking out of the lecture feeling his position as an Oxford mathematics professor. In be- that you now understood precisely the interrelations of all tween, Professor Atiyah supervised more than twenty doc- these ideas, only to find half an hour later, upon trying to toral students to completion. Most members of the Oxford reconstruct the same picture alone, that without his con- mathematical community would agree with my belief that tinuing magic touch, the pieces didn’t continue to fit so if Professor Atiyah had chosen a different profession, many neatly without a lot of extra work! parts of modern mathematics might not have developed. However, this elusive quality was not there in the per- In the late eighties, Professor Atiyah had encouraged Ed- sonal lectures I had as a student. Sir Michael Atiyah was ward Witten to find an interpretation of the Jones polyno- very practical and down-to-earth, whether about mathe- mial (an invariant of knots) in terms of quantum field the- matics or logistics, and never left one feeling completely ory. In September 1988, Witten’s paper “Quantum field at a loose end. Sure, there was work to be done and un- theory and the Jones polynomial” appeared as an IAS pre- known factors, but he was a masterful teacher and under- print. It was published in Communications in Mathematical stood exactly where you were and what you were capable Physics the following year. The Jones polynomial is associ- of doing. I remember once visiting Cambridge for a con- ated to a knot in three-space, or in the three-dimensional ference, and despite having many demands on his time, he sphere or a three-dimensional manifold. Witten consid- checked that everything was fine with my accommodation ered the Chern–Simons functional, which associates a real and even gave me a private tour of Trinity College (this was number to a connection on a three-manifold. He used the a short time before he became Master). He was especially Chern–Simons functional as the Lagrangian in a quantum proud showing me the original manuscripts of Sir Isaac field theory. When a knot in the manifold was specified, Newton in the library. one could define the “Wilson loop” (the integral ofthe For him, mathematics had to be unifying and beautiful. connection around the knot) and use this to define a path I feel very privileged to have been his student and to have integral over all connections on the three-manifold. This had a peek into the view of the glory of mathematics, some construction provided the quantum field theory interpre- fractional part of which I have carried ever since. tation of the Jones polynomial. In autumn 1988 the Oxford geometry seminar was devoted to the Oxford seminar on Jones–Witten theory. Ruth

Lisa Jeffrey is a professor of mathematics at the University of Toronto. Her email address is [email protected].

1848 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 11 Lawrence (at the time a student of Professor Atiyah, com- plectic manifold, the Grothendieck–Riemann–Roch theo- pleting the final year of her doctorate) wrote notes ofthe rem, the Hirzebruch–Riemann–Roch theorem, derived seminar. These were typed (by his extremely efficient secre- functors, anomalies, determinant line bundles, Ray–Singer tary, Jane Cox, using an electric typewriter) and made avail- torsion, zeta function regularization, and stationary phase able to participants. The first seminar was given by Pro- approximation. All of this material was background for fessor Atiyah, “An introduction to Jones–Witten theory.” the geometry seminar on Jones–Witten theory. My notes Seminar 3 (“Moduli spaces of vector bundles”) and Sem- for this last meeting filled about ten pages. Professor inar 6 (“The path integral formulation”) were also given Atiyah not only listed these topics, he explicated them. by Professor Atiyah. Two of the remaining seminars were After my first year, Professor Atiyah encouraged meto given by Graeme Segal (“The abelian theory” and “Fusion attend conferences. He told me it might be easier for his rules and the Verlinde algebra”) and one by Nigel Hitchin students to find an opportunity to talk with him at con- (“Reduction to the Abelian case”). Ruth Lawrence gave the ferences than back in Oxford, where he had so many re- final seminar (“Computing the invariants”). sponsibilities and preoccupations. I remember the Bonn In his preface, Professor Atiyah wrote: “No serious at- Arbeitstagung, where Professor Atiyah was always the first tempt has been made to integrate [the notes of the semi- speaker. I have a vivid memory of finding a gentian on a nars] or produce a coherent polished account. These notes German hillside with other mathematicians such as Nigel therefore have limited value and are mainly designed for Hitchin and Graeme Segal. the audience who attended.” He is not doing himself jus- At one point in my first year I had told Professor Atiyah tice here. All participants in this seminar felt that the sem- that I was waiting for an idea to strike. He looked at me inar was covering groundbreaking material. The Oxford rather sternly and told me that ideas do not strike. Per- seminar notes roughly formed the basis for Professor haps he was referring to Pasteur’s statement that chance Atiyah’s book The Geometry and Physics of Knots (Lezioni favors only the well-prepared mind. In any case, it was an Lincee), which was published in 1990. The subject of immense privilege to work under Professor Atiyah’s super- Jones–Witten theory has had spectacular importance over vision. I cannot possibly repay the debt of gratitude that I the more than thirty intervening years, and Professor owe him. Atiyah’s account has been particularly influential. I began my doctorate in Oxford in 1988 under Professor Atiyah’s supervision. My generation of students was greatly influenced by Professor Atiyah’s survey article “New invariants of three- and four-dimensional manifolds,” which was published in the proceedings of the 1987 Durham conference The Mathematical Heritage of Hermann Weyl. This article outlined the axioms for topological field theories, which had been formulated by Professors Atiyah and Se- gal. It was natural for me to choose a doctoral thesis topic Lisa Jeffrey about Jones–Witten theory and its asymptotics (as a perturbative theory with an asymptotic expansion as a sum over contributions coming from flat connections on three- Michael Atiyah and Physics: manifolds). However, I was not the only student greatly in- The Later Years fluenced by the Jones–Witten theory seminar. Joergen An- dersen, Andrew Dancer, Oscar Garcia-Prada, Ruth Lawrence, and Michael Thaddeus had the directions of Bernd Schroers their thesis work altered by this seminar. The research stu- The inadvertent and the intentional physicist. Michael dents also spent time on the “junior geometry seminar” (a Atiyah frequently expressed surprise at the extent to which learning seminar where research students were the speak- his work on index theory and his early work in gauge the- ers) on topics related to the Jones–Witten seminar. ory turned out to be important in physics. At the end of his In my notes from an early meeting with Professor Atiyah commentary in Volume 5 of his collected works he writes: in early October 1988, the topics covered included flag manifolds, the Peter–Weyl theorem, the Bott–Borel–Weil Bernd Schroers is a professor of mathematics at Heriot-Watt University in Ed- theorem, and Jones–Witten theory. A subsequent meeting inburgh. His email address is [email protected]. three weeks later covered complex polarization of a sym-

DECEMBER 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1849 “...I am really struck by the way most of the work which fields or wave functions can be computed from values ona Singer and I did in the 60s and 70s has become relevant particular time slice. The difference-differential equation to physics.” Reading this work with hindsight one can in [1] abandons that assumption and introduces an ele- indeed only marvel at the contrast between the absence ment of nonlocality. In his lectures on the subject, Michael of any physical motivation and the ultimate importance highlighted these features as important motivations. in theoretical physics. It seems that, at least during the The central role of the Dirac operator. The paper [1] pro- first half of his mathematical career, Michael’s impact on vides an elegant solution to the problem of defining a fi- physics was entirely inadvertent. This is also borne out by nite time-shift operator that is also relativistic by exponen- a remark in his 1998 lecture “The Dirac equation and ge- tiating the Dirac operator. The use of the Dirac operator ometry,” where he writes: “When Singer and I were inves- is no accident. As we already saw, Michael had “rediscov- tigating these questions we ‘rediscovered’ for ourselves the ered” this operator in a purely mathematical context, but Dirac operator. Had we been better educated in physics, or in his later thinking about physics it always played a cen- had there been the kind of dialogue with physicists which tral role, often as a possible bridge between the nonlinear is now so common, we would have got there much sooner.” world of geometry and the linear world of quantum me- When Michael moved to Edinburgh in 1997, his own chanics. This was also true in the work on geometric mod- physics education had benefited hugely from the dialogue els of matter that I discuss next. between mathematicians and physicists that he had done The inadequacy of gauge theory and the importance of so much to initiate and develop. He was now interested in four-dimensional geometry. Michael’s ideas for purely tackling fundamental questions in physics head-on, fully geometric model of particles can be traced back to a 1989 aware of their importance and with the actual intention of paper with Nick Manton on the Skyrme model in nuclear revolutionizing the foundations of physics. The inadver- physics. The paper is based on the observation that one tent physicist had become an intentional physicist. can obtain three-dimensional Skyrme fields by comput- In this brief contribution I will collect some memories ing the holonomy of instantons in four dimensions, i.e., and impressions from Michael’s second decade in Edin- by integrating along the fourth dimension. Fifteen years burgh, from about 2009 until his death in 2019. During later, work by Sakai and Sugimoto suggested that this ho- this time we met regularly, sometimes weekly, for discus- lonomy can in fact be viewed as the constant term in a sions related to the three main projects in physics that Fourier series expansion of an instanton and that the other Michael pursued: namely, his conjecture regarding the con- terms in this expansion can be attributed to further three- figurations of points in three-dimensional Euclidean space, dimensional meson fields. The message was that three- the role of difference-differential equations in physics, and dimensional physics could be described by an SU(2) geometric models of matter. gauge theory in four (Euclidean) dimensions. Our discussions had started because the difference- Michael took this result as an invitation to go one step differential equations that Michael had introduced in the further. He always felt that gauge theory, the mathemati- paper [1] with Greg Moore arise naturally in three-dimen- cal language in which the standard model of elementary sional quantum gravity, which I was working on at the particles is written, required too many arbitrary choices: time. However, our focus soon shifted to geometric mod- of a gauge group, of an associated vector bundle, of cou- els of matter, which became the topic of two joint papers plings, and so on. His instinct was therefore to treat the 1 [2,3] and a jointly held EPSRC research grant. I will focus appearance of a fourth dimension as an indication that on the geometric models here, but my goal is to identify static, three-dimensional particles could be modelled in general themes in Michael’s thinking about physics. terms of the geometry of four-manifolds. Formulating this The inadequacy of conventional quantum mechanics. in detail and studying examples took several years and led Michael expressed his dislike of conventional quantum to several publications, including [2–4]. The basic chal- mechanics on several occasions. He did not believe that lenge, only partly met, was to interpret the (integer) quan- any linear theory could be truly fundamental, and he tum numbers of particle and nuclear physics—like baryon shared Einstein’s dislike of the collapse of the wave func- number, lepton number, and electric charge—in terms of tion induced by observation. He felt that in their efforts topological invariants of four-manifolds. The Dirac opera- to go beyond quantum theory, physicists had tried many tor featured too, its kernel providing the linear space that avenues but had not critically questioned the paradigm of encodes a particle’s spin degrees of freedom. initial value problems, i.e., the assumption that the future There is no room here to discuss the geometric models of matter in any detail, but a few general observations 1I believe this is Michael’s only EPSRC grant for personal research, i.e., not about Michael’s style of working during the collaboration counting the ones he held as director of the Isaac Newton Institute.

1850 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 11 on this topic may be of interest. Even when he was well mathematical considerations, had been so unexpectedly into his eighties, he worked hard, coming to his office in and powerfully relevant to physics boosted his confidence, the School of Mathematics at the University of Edinburgh but I should stress that I never discussed this with Michael. almost every day that he was not travelling and putting in Michael trusted his own instincts, but he also trusted additional shifts at home. He generated new ideas at a great minds. He had his heroes, and in physics these were, prodigious rate, announcing them with infectious enthu- above all, Maxwell, Einstein, and Dirac. He thought that siasm, but abandoning them casually if a new and more all their ideas would ultimately prove right in some sense. promising avenue presented itself. These ideas were es- With Einstein and Dirac he was aware that the work they sentially mathematical even though the questions we dis- had done as young men was celebrated, whereas aspects of cussed came from physics. Michael needed discussion part- their later work (e.g., Einstein on unified field theory and ners, or at least listeners, to develop his ideas, and he was Dirac on large numbers) were viewed critically. He felt this usually juggling several projects and associated conversa- was not justified and saw parallels in his own biography. tions at any point in time. Remarkably, he always took a In an interview for the Newsletter of the European Math- personal interest in his discussion partners and created ad- ematical Society in November 2018, the last one he gave, ditional connections where he could. For example, I was Michael offered the following advice to young mathemati- travelling to Africa regularly during our collaboration to cians: “What you need is passion, persistence and risking teach at one of the African Institutes for Mathematical Sci- things for searching for the beauty.” He certainly lived that ences (AIMS). Michael immediately offered his help and advice. Michael’s courage to address fundamental ques- joined the international advisory board for the AIMS cen- tions in science head-on, to follow his own instincts and ter in Ghana. dreams in seeking answers, and to use beauty as a guide Division algebras and Bohmian mechanics. In our dis- meant that discussions with him transcended the cussions, Michael frequently mentioned two other themes humdrum of scientific activity. Michael sought and en- that he wanted to incorporate in his description of matter, couraged a similar ambition and independence of mind namely, division algebras and Bohmian mechanics. They in others, too. In doing so, he took a risk and sometimes never entered his published work on physics in any detail, attracted criticism, particularly in later years. However, he but I think they are worth recording. Michael felt that the also created a space for both deep and lateral thinking that four division algebras—real and complex numbers, quater- is rare and that many treasured. nions, and octonions—provided essentially the only math- References ematically natural way to account for the number of funda- [1] Atiyah M, Moore G. A shifted view of fundamental mental forces (four) or the number of generations (three) physics, Surv. Differ. Geom., 15, Int. Press, Somerville, MA, in the Standard Model. However, he never settled on a 2011, pp. 1–15. MR2815723 definite proposal of how this matching could work. [2] Atiyah M, Manton NS, Schroers BJ. Geometric models Michael liked David Bohm’s formulation of quantum of matter, Proc. Roy. Soc. Lond. A468 (2012), 1252–1279. mechanics in terms of nonlocal classical variables that fol- MR2910348 low trajectories determined by the wave function. He felt [3] Atiyah M, Franchetti G, Schroers BJ. Time evolution in a they clarified some of the foundational issues in quantum geometric model of a particle, J. High Energy Phys. (2015), mechanics by making the nonlocality explicit, and he han- 062. MR3321401 kered after a role for them in his ideas about particles. We [4] Atiyah M, Manton NS. Complex geometry of nuclei and atoms, Internat. J. Modern Phys. A33 (2018), no. 24, discussed a possible use related to his configuration space 1830022. MR3888339 conjecture but never came up with a convincing proposal. Passion and beauty. Anybody who interacted with Michael soon noticed that he had a strong personal taste in life and in science and that his instincts were not eas- ily discouraged. Sometimes I felt Michael did not take on board the painstaking work required to match a theoretical model in physics to the experimental data, and occasionally our discussions would turn into arguments. On one such occasion, having listened to Michael’s latest ideas, I asked, “But this is just a gut feeling, right?” to which Michael shot back, “Yes, but it is my gut!” Presum- Bernd Schroers ably his awareness that his early work, motivated by strictly

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