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Sir Michael Atiyah, a Knight Mathematician a Tribute to Michael Atiyah, an Inspiration and a Friend Alain Connes and Joseph Kouneiher

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Sir Michael Atiyah, a Knight Mathematician a Tribute to Michael Atiyah, an Inspiration and a Friend Alain Connes and Joseph Kouneiher Sir Michael Atiyah, a Knight Mathematician A Tribute to Michael Atiyah, an Inspiration and a Friend Alain Connes and Joseph Kouneiher Sir Michael Atiyah was considered one of the world’s fore- most mathematicians. He is best known for his work in algebraic topology and the codevelopment of a branch of mathematics called topological 퐾-theory, together with the Atiyah–Singer index theorem, for which he received the Fields Medal (1966). He also received the Abel Prize (2004) along with Isadore M. Singer for their discovery and proof of the index theorem, bringing together topology, geometry, and analysis, and for their outstanding role in building new bridges between mathematics and theoreti- cal physics. Indeed, his work has helped theoretical physi- cists to advance their understanding of quantum field the- ory and general relativity. Michael’s approach to mathematics was based primar- ily on the idea of finding new horizons and opening up new perspectives. Even if the idea was not validated by the mathematical criterion of proof at the beginning, “the idea would become rigorous in due course, as happened in the past when Riemann used analytic continuation to justify Euler’s brilliant theorems.” For him an idea was justified by the new links between different problems that it illumi- Atiyah in Marseille, June 2018. Alain Connes is a professor of mathematics at the Collège de France, nated. Our experience with him is that, in the manner of I.H.E.S., and Ohio State University, Columbus. His email address is an explorer, he adapted to the landscape he encountered [email protected]. on the way until he conceived a global vision of the setting Joseph Kouneiher is a professor of mathematical physics and engineering of the problem. sciences at Nice SA University, France. His email address is Joseph 1 2 [email protected]. Atiyah describes here his way of doing mathematics: This is Part 1 of a two-part series on Sir Michael Atiyah. Part 2 will be included 1 in the December issue of Notices. The quotations of Michael Atiyah presented in this paper have as sources personal exchanges with Michael or interviews that Michael gave on various For permission to reprint this article, please contact: occasions. [email protected] . 2To learn more about Atiyah’s approach to mathematics, we refer to the general DOI: https://doi.org/10.1090/noti1981 section of Atiyah’s collected works [1]. 1660 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 10 Some people may sit back and say, I want to solve When I came to decide on my graduate work I this problem and they sit down and say, “How do oscillated between Todd and Professor W. V. D. I solve this problem?” I don’t. I just move around Hodge who represented a more modern approach in the mathematical waters, thinking about things, based on differential geometry. Hodge’s greater being curious, interested, talking to people, stir- international standing swung the balance and, in ring up ideas; things emerge and I follow them up. 1952, I became his research student. ([1], Vol. I) Or I see something which connects up with some- Speaking of the work for his thesis, Atiyah said [1]: thing else I know about, and I try to put them to- gether and things develop. I have practically never I’d come up to Cambridge at a time when the em- started off with any idea of what I’m going to be phasis in geometry was on classical projective alge- doing or where it’s going to go. I’m interested in braic geometry of the old-fashioned type, which I mathematics; I talk, I learn, I discuss and then in- thoroughly enjoyed. I would have gone on work- teresting questions simply emerge. I have never ing in that area except that Hodge represented a started off with a particular goal, except the goal more modern point of view—differential geome- of understanding mathematics. try in relation to topology; I recognized that. It was a very important decision for me. I could have We could describe Atiyah’s journey in mathematics by worked in more traditional things, but I think that saying he spent the first half of his career connecting math- it was a wise choice, and by working with him I got ematics to mathematics and the second half connecting much more involved with modern ideas. He gave mathematics to physics. me good advice and at one stage we collaborated Building Bridges together. There was some recent work in France at the time on sheaf theory. I got interested in it, he Michael Atiyah’s early education took place in English got interested in it, and we worked together and schools in Khartoum, Cairo, and Alexandria, but his love wrote a joint paper which was part of my thesis. and preference for geometry began as early as his two years That was very beneficial for me. at Manchester Grammar School, which he spent preparing to take the Cambridge scholarship examinations [2]: Atiyah became interested in analytic fiber bundles fol- lowing his encounter with Newton Hawley, who was visit- I found that I had to work very hard to keep up ing Cambridge. Reading the Comptes Rendus allowed him with the class and the competition was stiff. We to follow the development of sheaf theory in France. His had an old-fashioned but inspiring teacher who single-author paper “Complex fibre bundles and ruled sur- had graduated from Oxford in 1912 and from him faces” [4], for which he received the Smith’s Prize, was I acquired a love of projective geometry, with its el- motivated by the treatment of Riemann–Roch from the egant synthetic proofs, which has never left me. I sheaf-theory3 point of view and the classification of fiber became, and remained, primarily a geometer bundles over curves: “it looked at the old results on ruled sur- though that word has been reinterpreted in differ- faces from the new point of view.” ent ways at different levels. I was also introduced Atiyah published two joint papers with Hodge, “Formes to Hamilton’s work on quaternions, whose beauty de seconde espèce sur une vari´et´ealg´ebrique”and “Inte- fascinated me, and still does. (The Abel Prize 2004 grals of the second kind on an algebraic variety” [5]. Those lecture) papers and [4] granted him the award of a Commonwealth He won a scholarship to Trinity College, Cambridge, Fellowship to visit the Institute for Advanced Study in in 1947. He read Hardy and Wright’s Number Theory and Princeton for the 1955–56 session. On this occasion some articles on group theory during his two-year National Atiyah met Jean-Pierre Serre, Friedrich Hirzebruch, Kuni- Service. Thanks to his exceptional talent, he came out hiko Kodaira, Donald Spencer, Raoul Bott, Isadore Singer, ranked first and wrote his first paper “A note on thetan- and others. During his stay at the IAS Atiyah attended gents of a twisted cubic” (1952) while still an undergradu- Serre’s seminars, which were a source of inspiration and ate. motivated his subsequent papers, in which he showed how From 1952 to 1955 Atiyah undertook three years of re- the study of vector bundles on spaces could be regarded as search at Trinity College, Cambridge, with William V. D. the study of a cohomology theory called 퐾-theory. Hodge as supervisor. He obtained his doctorate in 1955 with his thesis “Some Applications of Topological Meth- 3It seems that Serre’s letter to Andr´eWeil that gave the sheaf-theory treatment ods in Algebraic Geometry.” In fact, Atiyah hesitated be- of Riemann–Roch for an algebraic curve, presented to him by Peter Hilton, mo- tween J. A. Todd and Hodge: tivated Atiyah’s paper [4]. NOVEMBER 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1661 퐾-theory. The historical origin of 퐾-theory4 can be found ever a simpler and more general proof was produced by in Alexander Grothendieck’s work on coherent sheaves on introducing equivariant 퐾-theory, i.e., equivalence classes an algebraic variety 푋 [6]. Grothendieck used it to formu- of 퐺-vector bundles over a compact 퐺-space 푋 [12]. In this late his Grothendieck–Riemann–Roch theorem. The idea paper Atiyah and Segal showed that under suitable condi- is to define a group using isomorphism classes of sheaves tions the completion of the equivariant 퐾-theory of 푋 is as generators of the group, subject to a relation that iden- isomorphic to the ordinary 퐾-theory of a space, 푋퐺, which tifies any extension of two sheaves with their sum. There- fibers over 퐵퐺 with fiber 푋: sulting group is called 퐾(푋) when only locally free sheaves ∧ 퐾퐺(푋) ≅ 퐾(푋퐺). are used or 퐺(푋) when all are coherent sheaves.5 The original result (1) can be recovered as a corollary by Having been exposed, at the Arbeitstagung, to many taking 푋 to be a point. In this case the left-hand side re- hours of Grothendieck expounding his generaliza- duced to the completion of 푅(퐺) and the right to 퐾(퐵퐺). tion of the Grothendieck–Riemann–Roch theo- In [13] Atiyah gave a 퐾-theory interpretation of the sym- rem, I was playing about with his formulae for bol8 of an elliptic operator. This work induced his inter- complex projective space. From Ioan James (at est in elliptic operators, which then became a dominant that time a colleague in Cambridge) I had heard theme in his work, so that his later papers on 퐾-theory ap- about the Bott periodicity theorems and also peared interspersed chronologically with his papers on in- about stunted projective spaces.
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