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Home , Alain Connes, Fields Medal, Michael Atiyah

Sir , a Knight A Tribute to Michael Atiyah, an Inspiration and a Friend and Joseph Kouneiher

Sir Michael Atiyah was considered one of the world’s fore- most . He is best known for his work in algebraic and the codevelopment of a branch of called topological 퐾-theory, together with the Atiyah–Singer index theorem, for which he received the (1966). He also received the (2004) along with Isadore M. Singer for their discovery and proof of the index theorem, bringing together topology, , 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 of mathematics at the Collège de , nated. Our experience with him is that, in the manner of I.H.E.S., and , 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 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 . Hodge’s greater being curious, interested, talking to people, stir- international standing swung the balance and, in 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 , 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 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 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 , , and , 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 in 1912 and from him faces” [4], for which he received the Smith’s Prize, was I acquired a love of , 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 , 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 ” [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 and Princeton for the 1955–56 session. On this occasion some articles on theory during his two-year National Atiyah met Jean-Pierre Serre, , Kuni- Service. Thanks to his exceptional talent, he came out hiko Kodaira, Donald Spencer, , , ranked first and wrote his first paper “A note on thetan- and others. During his stay at the IAS Atiyah attended gents of a ” (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 theory called 퐾-theory. Hodge as supervisor. He obtained his 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 .” 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 ’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 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 . This work induced his inter- complex projective space. From (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. I soon realized dex theory.9 Using 퐾-theory Atiyah was able to give a sim- that Grothendieck’s formulae led to rather strong ple and short proof of the problem. This in- results for the James problems. Moreover the Bott volved the 휓푘 operations that were introduced by Adams periodicity theorems fitted in with the in his subsequent work on the vector field problem. Grothendieck formalism, so that one could draw Algebraic 퐾-theory was successfully developed by genuine topological conclusions. It was this which using homotopy theory in 1969 and 1972, convinced me that a topological version of following a trend of ideas that can be traced back to the Grothendieck’s 퐾-theory, based on the Bott peri- work of Whitehead in 1939. A further extension was de- odicity theorem, would be a powerful tool in alge- veloped by Friedhelm Waldhausen in order to study the braic topology. ([1], Vol. 2) algebraic 퐾-theory of spaces, which is related to the study Using the same construction applied to vector bundles of pseudoisotopies. Much modern research on higher 퐾- 퐾(푋) in topology, Atiyah and Hirzebruch defined for a theory is related to algebraic geometry and the study of 6 푋 topological space in 1959 [7], and using the Bott peri- motivic cohomology. The corresponding constructions in- odicity theorem they made it the basis of an extraordinary volving an auxiliary quadratic form received the general cohomology theory [9]. It played a major role in the sec- 7 name 퐿-theory, a major tool of surgery theory. In ond proof of the index theorem. theory, the 퐾-theory classification of Ramond–Ramond The Atiyah–Hirzebruch relates the or- field strengths and the charges of stable 퐷- was first dinary cohomology of a space to its generalized cohomol- proposed in 1997. ogy theory. Atiyah applied the approach to finite groups 퐺 The index theorem. The concept of a linear operator, and showed [8] that for a finite group 퐺, the 퐾-theory of which together with the concept of a is funda- its , 퐵퐺, is isomorphic to the completion mental in linear algebra, plays a role in very diverse of its character ring: ∧ 8 훼 퐾(퐵퐺) ≅ 푅(퐺) . (1) The principal symbol of a linear differential operator Σ∣훼∣≤푚푎훼(푥)휕푥 is by 훼 definition the function Σ∣훼∣≤푚푎훼(푥)(푖휉) obtained from the differential op- The same year Atiyah and Hirzebruch established the re- erator (of a polynomial) by replacing each partial derivative by a new variable. lation between the 푅(퐺) of a compact At this point, the vector 휉 = (휉1, … , 휉푛) is merely a formal variable. connected and 퐾-theory of its classifying space An essential and interesting fact is that if one interprets (푥, 휉) as variables [9]. in the cotangent bundle, then the principal symbol is an invariantly defined 푇∗푋 푋 Later on the result could be extended to all compact function on , where is the on which the operator is initially defined, which is homogeneous of degree 푚 in the cotangent variables. The Lie groups using results from ’s thesis. How- symbol captures locally some very strong properties of the operator. For exam- ple, an operator is called elliptic if and only if the symbol is invertible (whenever 4 It takes its name from the German Klasse, meaning “class.” The work of J. H. 휉 ≠ 0). C. Whitehead or Whitehead torsion can be considered ultimately as the other 9Atiyah and Singer’s interest in the elliptic operator questions was dependent on historical origin. the fact that real 퐾-theory behaves differently from complex 퐾-theory and re- 5 퐾(푋) has cohomological behavior, and 퐺(푋) has homological behavior. quires a different notion of reality. The essential point is that the Fourier trans- 6Atiyah’s joint paper with Todd was the starting point of this work [11]. form of a real-valued function satisfies the relation 푓(−푥) = 푓( ̄푥). Their new 7This approach led to a noncommutative 퐾-theory for ℂ∗-algebras. version of real 퐾-theory involved spaces with involution.

1662 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 10 branches of mathematics and physics, above all in anal- not change the index. Indeed, a continuous, ellipticity- ysis and its applications. Up to the beginning of the twen- preserving deformation of the symbol should not affect tieth century the only linear operators that had been sys- the index,11 and so Gel’fand noted that the index should tematically studied were those between finite-dimensional depend only on a suitably defined homotopy class of the spaces over the fields ℝ and ℂ. For instance, the classical principal symbol. The hope was that the index of an ellip- theory of Fredholm integral operators goes back at least tic operator could be computed by means of a formula in- to the early 1900s. Fredholm essentially proved (and D. volving only the topology of the underlying domain (the Hilbert perceived this “geometrical” background) that a manifold), the bundles involved, and the symbol of the linear operator of the form 핀 + 픸, with compact 픸, has operator. index [푑푖푚 푘푒푟(픸) − 푑푖푚 푐표푘푒푟(픸)] equal zero. In early 1962, Atiyah and I. M. Singer discovered the (el- Special forms of index formulas were known even ear- liptic) in the context of Riemannian lier, for example, the Gauss–Bonnet theorem and its multi- geometry, and they were concerned by the proof of the fact dimensional variants. Indeed, the precursors of the that the 퐴-genuŝ of a spin manifold is the index of this index theorem were formulas for Euler characteristics— Dirac operator. They were concerned also by the geomet- alternating sums of of cohomology spaces. ric meaning of the 퐴-genus.̂ Hirzebruch, with Borel, had Putting the even ones on one side and the odd ones on the proved that for a spin manifold the index was an integer. other and using , this could be seen in more [The index theory of elliptic operators] had its ori- general terms as the difference between the dimensions of 퐾 10 gins in my work on -theory with Hirzebruch and the kernel and the cokernel of an elliptic operator. Sub- the attempt to extend the Hirzebruch–Riemann– sequently a number of generalizations of index formulas Roch theorem into differential geometry [7,9]. We were obtained for objects of a more complex nature. had already shown that the integrality of the Todd The first “infinite-dimensional” observations, concern- genus of an almost and the 퐴-̂ ed also with general fields, were made by O. Toeplitz [18]. genus of a spin-manifold could be elegantly ex- As a rule, linear operators between infinite-dimensional plained in terms of 퐾-theory. For an algebraic vari- spaces are studied under the assumption that they are con- ety the Hirzebruch–Riemann–Roch theorem went tinuous with respect to certain . Continuous lin- one step further and identified the Todd genus ear operators that act in various classes of topological vec- with the arithmetic genus or Euler characteristic of tor spaces, in the first place Banach and Hilbert spaces, are the . Also the 퐿-genus of a dif- the main object of study of linear . ferential manifold, as proved by Hirzebruch, gave Historically, the work of Fritz Noether in analysis in the signature of the quadratic form of middle di- 1921 exhibits already a case where the index is different mensional cohomology and, by Hodge theory, from zero. Noether gave a formula for the index in terms this was the difference between the dimensions of of a winding number constructed from data defining the the relevant spaces of harmonic forms. As Hirze- operator. This result was generalized by G. Hellwig, I. N. bruch himself had realized it was natural therefore Vekua, and others. to look for a similar analytical interpretation of Later Israel Gel’fand noticed in 1960 [14] that the in- the 퐴-genus.̂ The cohomological formula and the dex, the difference in these dimensions, is invariant under associated character formula clearly indicated that homotopy. This suggested that the index of a general ellip- one should use the spin representations. ([1], Vol. tic operator should be expressible in terms of topological 4) data, and Gel’fand asked for a formula for it by means of topological invariants. Gel’fand proposed that the index It was who turned their attention to of an elliptic differential operator (with suitable boundary Gel’fand’s general program: conditions in the presence of a boundary) should be ex- Once we had grasped the significance of pressible in terms of the coefficients of highest order part and the Dirac equation it became evident that the (i.e., the principal symbol) of the operator, since the lower 퐴-genuŝ had to be the difference of the dimen- order parts provide only compact perturbations that do sions of positive and negative harmonic spinors.

11In some sense index theory is about regularization; more precisely, the index 10 In general, the dimensions of the cokernel and kernel of an elliptic op- quantifies the defect of an equation, an operator, or a geometric configuration erator are extremely hard to evaluate individually—they even vary under from being regular. Index theory is also about perturbation invariance; i.e., the deformation—but the index theorem shows that we can usually at least evaluate index is a meaningful quantity stable under certain deformations and apt to their difference. In applications, it is sometimes possible to show that the coker- store certain topological or geometric information. Most important for many nel is zero, and then we have a formula for the of the null space with- mathematicians, the index interlinks quite diverse mathematical fields, each out solving the equation explicitly. with its own very distinct research tradition.

NOVEMBER 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1663 Proving this became our objective. By good for- Atiyah, Bott, and Patodi gave a second proof of the in- tune Smale passed through Oxford at this time dex theorem using the [17]. The heat kernel and, when we explained our ideas to him, he drew method had its origins in the work of McKean in the late our attention to a paper of Gel’fand on the general 1960s drawing on Minakshisundaram and Pleijel’s works problem of computing the index of elliptic opera- and was pioneered in the works of Patodi and Gilkey. The tors. ([1], Vol. 4) method can also largely apply for more general elliptic The generalization in 1962 of Hirzebruch’s proof of the pseudodifferential operators. Hirzebruch–Riemann–Roch theorem [10] and the proof of In the heat kernel approach, the invariance of the index the desired index formula by Atiyah and Singer were based under changes in the geometry of the operator is a conse- in fact on work by Agranovic, Dynin, and other analysts,12 quence of the index formula itself more than a means of particularly those involving pseudo-differential operators. proof. Moreover, it can be argued that in some respects the In their first published proof13 [15] they used 퐾-theory 퐾-theoretical embedding/ methods are more instead of cohomology and replaced the cobordism14 the- subtle and direct. For instance, in some circumstances the ory of the first proof with arguments akin to those of index theorem involves torsion elements in 퐾-theory that Grothendieck. The proof is based on embedding a man- are not detectable by cohomological means and hence are ifold in Euclidean space and then transferring the prob- not computable in terms of local densities produced by lem to one on Euclidean space by a suitable direct image heat asymptotics. construction. It worked purely in a 퐾-theory context and More importantly, the heat kernel approach exhibits the avoided rational cohomology. They used this then to give index as just one of a whole sequence of spectral invariants proofs of various generalizations in their papers [15]. appearing as coefficients of terms of the asymptotic expan- We can then verify that a general index function hav- sion of the trace of the relevant heat kernel. ing these properties is unique, subject to normalization. The third proof, in the early seventies, aimed for a more To deduce the Atiyah–Singer index theorem (i.e., analytic direct transition between the analysis and characteristic index = topological index), it then suffices to check that classes by expressing these as curvature integrals. This local the two indices are the same in the trivial case where the proof of the index theorem relies on a detailed asymptotic base manifold is just a single point. In other words, the in- analysis of the heat equation associated to a Dirac opera- dex problem concerns the computation of the (analytical) tor. index of 퐷 using only its highest order term, the symbol. This is analogous to certain intrinsic proofs of the The symbol defines a homomorphism between vector bun- Gauss–Bonnet theorem, but the argument doesn’t provide dles and naturally defines a class in 퐾-theory. Topological the same kind of intuition that the 퐾-theory argument data derived from the symbol and the underlying mani- does. The basic strategy of the local argument is to invent fold defines a rational number, the topological index, and a symbolic calculus for the Dirac operator which reduces we have the theorem [16] the theorem to a computation with a specific example. Many subsequent developments and applications have The analytical index of 퐷 is equal to its topological appeared since then in the work of Widom. Getzler draws index. on a version of the quantum-mechanical harmonic oscil- The key point is the fact that we can usually evaluate the lator operator and a coordinate calculation directly to pro- topological index explicitly, so this makes it possible to duce the 퐴-genuŝ (the appropriate “right-hand side” of the evaluate the analytical index. index theorem for the Dirac operator). The more recent proofs of the index theorem which are 12By investigating the particular case of the Dirac operator, Atiyah and Singer motivated by provide a rationale for the had an advantage over the analysts investigating the index problem and mostly −푥 because they knew already what had to be the answer, namely, the 퐴-genus.̂ role which functions like 푥/(1 − 푒 ) and 푥/tanh 푥 play In fact this case encompasses virtually all the global topological applications by in sorting out the combinations of characteristic classes twisting with an appropriate . which occur in the index formula. 13 The proof in [15] is based on the idea of the invariance of the index under In the first years the index theorem was applied inthe homotopy, which implies that the index (say, the analytic index) of an elliptic area of topology or number theory. A typical number theo- operator is stable under continuous changes of its principal symbol while main- taining ellipticity. Using this fact, one finds that the analytical index of an ellip- retical application would involve taking a on tic operator transforms predictably under various global operations such as em- a manifold that was so well analyzed that both the analyt- bedding and extension. Using 퐾-theory and Bott periodicity, we can construct, ical side and the topological side could be calculated inde- from the symbol of the elliptic operator, a topological invariant (the topological pendently. The index theorem then produced an identity, index) with the same transformation properties under these global operations. sometimes well known, at other times new. 14The details of this original proof involving cobordism actually first appeared in [15]. Today the index theorem plays an essential role in par-

1664 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 10 tial differential equations, stochastic processes, Riemann- means that there is a Hilbert space ℋ of fields 휓(푥) ian geometry, algebraic geometry, , and on 푀 (i.e., “sections of the spinor bundle”) on which both mathematical physics. the algebra of functions on 푀15 and the Dirac operator16 퐷 Since the appearance of the Atiyah–Singer index the- act. The “spectral triple” (풜, ℋ, 퐷) faithfully encodes the orem, generalizations have been obtained encompassing original and opens the door to the many new geometric situations. Among them some con- wide landscape of new that cover leaf spaces of structions involve non-Fredholm operators and use in a de- foliations, the fine structure of space-time, and the infinite- cisive way an extra ingredient consisting of von Neumann dimensional geometries of . algebras and the Murray–von Neumann dimension theory. Behind this concept of spectral triple is the realization While the numerical index of elliptic operators on non- by Atiyah [21] that the cycles in the theory, compact does not make much sense, some use- called 퐾-homology, which is dual to 퐾-theory, are natu- ful assumptions can enable us to associate them to appro- rally given by a Fredholm representation of the algebra priate type II von Neumann algebras and to give a sense to 퐴 = 퐶(푋) of continuous functions on the compact space the Murray–von Neumann index, particularly in the case 푋. Moreover, one owes to Atiyah and Singer the identifica- of a manifold with an infinite fundamental group. The tion of the 퐾푂-homology class of the Dirac operator of a use of von Neumann algebras was quite natural and led to spin manifold as the fundamental cycle realizing Poincar´e concrete nontrivial results. in 퐾푂-homology. This Poincar´eduality had been shown by Dennis Sullivan to be a key property of mani- Since I was not an expert on von Neumann alge- folds, much deeper than Poincar´eduality in ordinary ho- bras I attempted in my presentation (at the meet- mology, since for instance this fundamental cycle deter- ing in honour of ) to give a simple, el- mines the Pontryagin classes of the manifold. One conse- ementary and essentially self-contained treatment quence of the duality is that it reduces the index problem of the results. Later on in the hands of Alain to the computation of the index of twisted Dirac opera- Connes, the world expert on the subject, these sim- tors.17 In this case and for any hermitian vector bundle 퐸 ple ideas were enormously extended and devel- over an even-dimensional compact spin manifold 푀, the oped into a whole theory of linear analysis for fo- + ̂ index formula is Ind(휕퐸 ) = ⟨ch(퐸)퐴(푀), [푀]⟩, where liations. ([1], Vol. 4 and [19]) + 휕퐸 is the positive part of the self-adjoint Dirac operator The index theorem has had many variants, for example, twisted by the bundle 퐸, ch(퐸) is the Chern character of to the extension of Hodge theory on combinatorial and 퐸, and 퐴(푀)̂ is the 퐴-polynomial̂ of the tangent bundle Lipschitz manifolds and Atiyah–Singer’s signature opera- 푇푀 of 푀. Treating this index as a pairing between 퐸 and tor to Lipschitz manifolds. For any quasi-conformal man- the operator 휕 [21] defines the Chern character of the Dirac ifold there exists a local construction of the Hirzebruch– operator as the homology class Thom characteristic classes [20]. This work [20] provides ch(휕) = 퐴(푀)̂ ⋆ [푀]. local constructions for characteristic classes based on (2) higher-dimensional relatives of the measurable Riemann From equation (2) we can interpret the index theorem mapping theorem in dimension two and the Yang–Mills as a homological pairing theory in dimension four. At the same time, they provide Ind(휕+) = ⟨ch(퐸), ch(휕)⟩. (3) also an effective construction of the rational Pontryagin 퐸 classes on topological manifolds. In 1985 Teleman pro- The key fact is that this pairing between 퐾-theory and 퐾- vided in his paper a link between index theory and Thom’s homology makes sense in the general noncommutative original construction of the rational Pontryagin classes. case and can be computed using a natural map from 퐾- From the Atiyah–Singer index theorem to noncommuta- homology to cyclic cohomology as well as a “Chern char- tive geometry. Besides topological 퐾-theory, Michael acter” from 퐾-theory to . Then pairing Atiyah made two discoveries that played a major role in these two elements from cyclic cohomology and homol- the elaboration of the fundamental paradigm of noncom- ogy gives the same value as if we start from 퐾-theory and mutative geometry (NCG). This theory is an extension of 퐾-homology. Riemannian geometry to spaces whose coordinate algebra 15 풜 is no longer commutative. A geometric space is en- A function 푓(푥) acts by multiplication (푓.휓)(푥) ∶= 푓(푥)휓(푥) on spinors. 16 휇 coded spectrally by a representation of 풜 as operators in a In local coordinates 퐷 = 푖훾휇휕푥 , 훾휇 the Dirac 훾-matrices, which satisfy the well-known condition 훾 훾 + 훾 훾 = 2푔 . ℋ 휇 휈 휈 휇 휇휈 Hilbert space together with an unbounded self-adjoint 17Notice that a reduction is already worked out in [17]. The idea basically is operator 퐷. Consider an 푛-dimensional compact mani- that the classical operators are sufficiently numerous to generate, in a certain fold 푀 with Riemannian metric and spin structure. This topological sense, all elliptic operators.

NOVEMBER 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1665 Equation (3) represents the cists have been able to adopt and use sophisticated math- point of view of the index theorem. To make the compu- ematical ideas and techniques without which the elabo- tation effective, the idea is to obtain a local index formula. ration of the physical theories would be greatly impeded. Given a spectral triple, the Connes–Moscovici local index Conversely, mathematicians have used the insights derived theorem18 is then a complete solution of this general in- from a physical interpretation to break new ground in ge- dex problem. The first highly nontrivial spectral triple is ometry. the example that encodes the Diff-equivariant index the- Atiyah has been influential in stressing the role of topol- ory. The identification of the local terms appearing inthe ogy in quantum field theory and in bringing the workof index theorem for such a triple requires the definition of theoretical physicists to the attention of the mathematical an appropriate cyclic homology for Hopf algebras [20] en- community. coding the characteristic numbers. In his paper with Bott [17], they attempt to understand the results of V. K. Patodi on the heat equation approach A New Era in Mathematical Physics to the index theorem, a topic which will be developed The “recent” interaction between physics and mathemat- later in connection with . In their work ics, and particularly geometry, is mainly due to the fact that on the Lefschetz formula for elliptic complexes they had physicists started exploring complicated nonlinear models described the Zeta-function approach to the index theo- of a geometrical character as possible explanations for fun- rem. Singer and McKean took a step further, in the heat damental physical processes. Differential geometry had equation version, by concentrating on the Riemannian ge- been closely associated with physics ever since the intro- ometry. They suggest the possibility of cancellations lead- duction of Einstein’s theory of general relativity, but quan- ing directly to the Gauss–Bonnet from the Euler character- tum theory, the essential ingredient for the physics of ele- istic.20 mentary particles, had not impacted the type of geometry Another topic that would have an impact on theoreti- involving global topological features. cal physics concerned Atiyah’s work on spectral asymmetry. One of the first occurrences of topology19 in physics was Aspects of spectral asymmetry appear in differential geom- Dirac’s famous argument for the quantization of electric etry, topology, analysis, and number theory. These ap- charge, and what we are now witnessing is essentially the plications became popular, especially after Witten’s work nonabelian outgrowth of Dirac’s initial idea. on global anomalies brought the 휂-invariant into promi- This intimate relation between geometry and physics nence. The 휂-invariant introduced in a joint paper with has had beneficial consequences in both directions. Physi- Patodi and Singer was motivated, in part, by Hirzebruch’s

18 work on cusp singularities and in particular his result ex- The term local refers to the fact that in the classical situation the index for- pressing the signature defect in terms of values of 퐿- mula would involve integration and could be computed from local data. An- other point of view is that “we are working in momentum space and local functions of real quadratic fields. means therefore asymptotic.” In the 1970s and 1980s, methods gleaned from the in- 19One of the earliest occurrences of some topological noteworthiness can be dex theorem unexpectedly played a role in the develop- found in Euler’s 1736 solution of the bridges of Königsberg problem. In solv- ment of theoretical physics. One of the earliest and most ing one of the first problems in combinatorial topology, Euler gave birth towhat 푈(1) we now call graph theory. A close connection between physics and graph theory important applications is ’t Hooft’s resolution of the occurs in the work of Kirchhoff in 1847 on his two famous laws for electric cir- problem [23]. This refers to the lack of a ninth pseudo- cuits. Goldstone boson (like the pions and kaons) in QCD that In the nineteenth century we encounter more substantial examples of physi- one would naively expect from chiral breaking. cal phenomena with topological content. In Gauss’s 1833 work on electromag- There are two parts to the resolution. The first is the fact netic theory, found in his Nachlass (estate), the topology concerned the link- ing number of two curves. Another work was in the theory of vortex atoms pro- that the chiral 푈(1) is anomalous. The second is the re- posed by Lord Kelvin (alias W. H. Thomson) in 1867. Kelvin was influenced by alization that there are configurations of finite action (in- an earlier fundamental paper by Helmholtz (1858) on vortices and a long and stantons) that contribute to correlation functions involv- seminal paper of Riemann (1857) on abelian functions. ing the divergence of the 푈(1) axial current. The analysis Maxwell on the state of the art in the nineteenth century: relies heavily on the index theorem for the Dirac opera- It was the discovery by Gauss of this very integral, expressing the tor coupled to the SU(3) gauge field of QCD, the main work done on a magnetic pole while describing a closed curve in presence of a closed electric current, and indicating the geometrical remaining law in the theory of the strong interactions. Al- connexion between the two closed curves, that led him to lament the small progress made in the Geometry of Position since the time of 20This was proved later by Patodi, who extended the result to deal with the Leibnitz, Euler and Vandermonde. We have now some progress to Riemann–Roch theorem on Kähler manifolds. On his side Gilkey produced an report, chiefly due to Riemann, Helmholtz, and Listing. alternative indirect approach depending on a simple characterization of the Pon- Poincar´e’s work on the Analysis Situs enlivened considerably the topologi- tryagin forms of a Riemannian manifold. Later on, Atiyah, Bott, and Patodi re- cal matters and its link to physics. alized that Gilkey’s results were a consequence of the Bianchi identities.

1666 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 10 bert Schwarz showed that some of the ingredients in the so- what mathematicians would call a vanishing theorem for lution were best understood in terms of the Atiyah–Singer 푐표푘푒푟퐷, provides. Naturally, the mathematical physicist index theorem. There were also important applications in is more familiar with eigenvalues of operators and the heat the 1990s to S-duality of 푁 = 4 Super Yang–Mills which kernel than with characteristic classes, but once the realiza- involve the index theorem for the Dirac operator on mono- tion came that each side had something to learn from the pole moduli spaces. other, the most recent phase of index theory was formed. Describing all those activities and the first uses of the A man of intuition. The term “” refers to a index theorem in physics, Atiyah said: quite general class of quantum field theories used for the description of elementary particles and their interactions. From 1977 onwards my interests moved in the di- These theories involve deep and interesting nonlinear dif- rection of gauge theories and the interaction be- ferential equations, in particular, the Yang–Mills equations, tween geometry and physics. I had for many years which have turned out to be particularly fruitful for math- had a mild interest in theoretical physics, stimu- ematicians. lated on many occasions by lengthy discussions with George Mackey. However, the stimulus in Meanwhile I had been engrossed with the Yang– 1977 came from two other sources. On the one Mills equations in dimension 4. I realized that hand Singer told me about the Yang–Mills equa- these questions were essentially trivial in dimen- tions, which through the influence of Yang were sion 2, but one day, walking across the University just beginning to percolate into mathematical cir- Parks with Bott it occurred to me that one might cles. During his stay in Oxford in early 1977 Singer, nevertheless be able to use the Yang–Mills equa- Hitchin and I took a serious look at the self-duality tions to study the moduli spaces. The essential equations. We found that a simple application of point was the theorem of Narasimhan and the index theorem gave the formula for the num- Seshadri stating that stable bundles arose from uni- ber of parameters.… At about the same tary representations of the fundamental group. time A. S. Schwarz in the Soviet Union had inde- ([1], Vol. 5) pendently made the same discovery. From this pe- Atiyah initiated much of the early work in this field, cul- riod the index theorem began to become increas- minated by ’s work in four-dimensional ingly familiar to theoretical physicists, with far- geometry. Indeed, Andreas Floer and Donaldson’s works reaching consequences. provided new insights into the topology of lower dimen- The second stimulus from theoretical physics sional manifolds. In three and four dimensions this was came from the presence in Oxford of Roger Pen- and the Donaldson polynomials based on 21 rose and his group.…at that time I knew nothing Yang–Mills theory. about twistors, but on Penrose’s arrival in Oxford Donaldson’s work showed that, in contrast to other di- we had lengthy discussions which were mutually mensions, there were some very subtle invariants in four educational. The geometry of twistors was of dimensions. These invariants were preserved under course easy for me to understand, since it was the smooth deformation but not under general continuous de- old Klein correspondence for lines in 푃3 on which formation [24]. Donaldson found his invariants by study- I had been brought up. The physical motivation ing special solutions of the Yang–Mills equations, the “in- and interpretation I had to learn. In those days stantons.” These solutions are essentially localized to a Penrose made great use of complex multiple inte- small region in space-time, thereby describing an approxi- grals and residues, but he was searching for some- mately instantaneous process. thing more natural and I realized that sheaf co- Donaldson’s work involved ideas from nonlinear homology groups provided the answer. He was analysis of partial differential equations, differential and quickly converted and this made subsequent dia- algebraic geometry, and topology, and a little less physics. logue that much easier, since we now had a com- Nevertheless, beside the influence of Atiyah–Hitchin– mon framework. ([1], Vol. 5) Singer’s paper, the whole idea of studying a

The theoretical physicists working with gauge theories 21One can transpose the same ideas to maps from a surface into a symplec- were interested in calculating the number of zero modes tic manifold and express the Floer homology of 푋 and the symplectic homol- of a differential operator in terms of the topological charge ogy of 푋/Σ, for a special class of Hamiltonians, in terms of absolute and rela- tive Gromov–Witten invariants of the pair (푋, Σ) and some additional Morse- of the gauge field. Translating the language, they wanted theoretic information. The key point of the argument is a relation between so- the dimension of 푘푒푟퐷 in terms of characteristic classes, lutions of Floer’s equation and pseudoholomorphic curves, both defined on the and this is precisely what the index theorem, coupled with symplectization of a prequantization bundle over Σ.

NOVEMBER 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1667 in this context (a space of connections up to gauge equiv- polynomial and physics. [25] alence) had a physics origin in some sense. “In fact, the relation between knot invariants and parti- Witten later showed that Donaldson’s invariants could cles goes to the very beginning of relativistic quantum field be interpreted in terms of a quantum field theory and that theory as developed by Feynman and others in the 1940s. this would have profound consequences. Moreover, this The basic idea is that, if we think of a classical particle mov- field theory was a close cousin of the standard theories ing in space-time, it will move in the direction of increas- used by particle physicists, except that it had a “twist” that ing time. However, within quantum theory the rules are produced topological invariants not dependent on the in- more flexible. Now a particle is allowed to travel backin tricate details of the underlying geometry of space-time. time. Such a particle going backwards in time can be inter- Beside Seiberg’s previous work on dualities, this interpreta- preted as an antiparticle moving forwards in time. Once it tion of Witten motivated Seiberg and Witten’s work. They is allowed to turn around, the trajectory of a particle can showed that the corresponding physical theory could form, as it were, a complicated knot in space-time. The be solved in terms of a much simpler structure. Yang– rules of quantum theory will associate to each such trajec- Mills theory is fundamentally based on a choice of non- tory a probability amplitude that describes the likelihood abelian Lie group, usually taken to be the group SU(2). of this process actually taking place. The noncommutativity of this symmetry group leads to the “In order to establish contact between this formulation nonlinearities of the associated partial differential equa- of particle physics and , Witten had to replace tions. However, in physics it was known that in quan- the usual four dimensions of space-time with three dimen- tum theories these nonabelian often manifest sions and work with a special type of gauge theory based themselves only at very short distance scales. At large dis- on the Chern–Simons topological invariant. The full the- tances the symmetry can be broken into a much simpler ory allowed for many choices, such as the gauge group, the abelian group. For example, in the case of SU(2) only representation of the particles associated to the knot, and the group 푈(1) of electromagnetism would appear coupling constants. together with possibly some charged matter particles [22] “Relating quantum field theory to knot invariants along ([1], Vol. 7). As Witten writes: these lines had many advantages. First, it was fitted into In the spring of 1987 Atiyah visited the Institute a general framework familiar to physicists. Second, it was for Advanced Study and was more excited than I not restricted to knots in three-dimensional Euclidean could remember. What he was so excited about space and could be defined on general three-dimensional was Floer theory which he felt should be inter- manifolds. One could even dispense with the knot and get preted as the Hamiltonian formulation of a quan- an invariant of a closed three-dimensional manifold. Such tum field theory. Atiyah hoped that a quantum topological invariants are now called quantum invariants field theory with Donaldson polynomials asthe and have been extensively studied” [22] ([1], Vol. 7). correlation functions and Floer groups as the Atiyah related all of this to Witten’s work on quantum Hilbert spaces could somehow be constructed by mechanics and , arguing that there should physics methods.… I was skeptical and though the be a quantum field theory version of the story. This was idea was intriguing, I did not pursue it until I was the birth of topological quantum field theory (TQFT), with reminded of the question during another visit by Witten to work out a year later what the quantum field the- Atiyah to the Institute at the end of 1987. This ory conjectured by Atiyah actually was. Soon after the first time I dropped some of my prejudices and had topological TQFT, Witten was to find a quantum theory the good luck to notice that a simple twisting of with observables the Jones polynomial, now often called 푁 = 2 supersymmetric Yang–Mills theory would the Chern–Simons–Witten theory. give a theory with the properties that Atiyah had The theories of and and the wanted. [25] of fundamental particles, which involves the Another topic that was developed following Atiyah’s theory of Riemann surfaces in novel and unexpected ways, suggestions and indications was the connection with the were all areas of theoretical physics that developed using knot polynomial of . the ideas that Atiyah was introducing. Concerning the interaction between new mathematics The other problem that Atiyah recommended for and physics he said: physicists in the years was to understand the Jones knot polynomial via quantum field theory. It was Mathematics and physics are more like Siamese from him that I first heard of the Jones polyno- twins. A separation in their case usually leads mial. There followed other clues about the Jones to death of at least one twin, so it should not be

1668 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 10 contemplated. Some people sometimes oversim- derstanding. A proof by itself doesn’t give you un- plify by regarding mathematics and physics as sep- derstanding. You can have a long proof and no arate organisms, and by implication that mathe- idea at the end of why it works. But to understand maticians and physicists are different people. How why it works, you have to have a kind of gut reac- does one decide which of the great figures of his- tion to the thing. You have got to feel it. tory was a mathematician or a physicist: Archime- As for the beauty, as Dirac and Einstein and mostly Weyl des, Newton, Gauss, Hamilton, Maxwell, Rie- (his hero along with Maxwell), Atiyah places the “Beauty” mann, Poincar´e, Weyl? at the heart of what we can call the essence of mathematics. Before Dirac’s delta there was Heaviside (an en- It is not the kind of beauty that you can point to—it is gineer). Spinors are really due to Hamilton (both beauty in a much more abstract sense. mathematician and physicist); Elie´ Cartan built on Sophus Lie; fibre bundles and U(1) gauge theory It’s been known for a long time that some part of are due to Clifford. Grassmann used Maxwell’s un- the brain lights up when you listen to nice music, or read nice poetry, or look at nice pictures—and derstanding of Hamilton’s quaternions. Maxwell 22 derived all the notation of div, grad and curl from all of those reactions happen in the same place. Hamilton, who first wrote down the equation And the question was: Is the appreciation of math- ematical beauty the same, or is it different? And 2 푑 푑 푑 the conclusion was, it is the same. The same bit (푖 + 푗 + 푘 ) = −Δ 푑푥 푑푦 푑푧 of the brain that appreciates beauty in music, art and poetry is also involved in the appreciation of and said that this must have deep physical mean- . And that was a big discov- ing. This was many decades before Dirac. The di- ery. [3] alogues continue and we now have a better under- 푖휋 standing of fermions, supersymmetry and Morse For Atiyah Euler’s equation 푒 + 1 = 0 embodies this theory. Also of topological quantum field theory beauty: in three-dimensions and Jones knot invariants; Euler’s equation involves 휋; the mathematical Donaldson invariants in four dimensions, and the constant Euler’s number 푒 (2.71828..); 푖, the imag- unique role of dimension four, both in physics inary unit; 1; and 0. It combines all the most and geometry. Also moduli spaces in Yang–Mills important things in mathematics in one formula, theory; monopoles and ; anomalies, co- and that formula is really quite deep. So every- homology and index theory; holography and ge- body agreed that was the most beautiful equation. ometry, etc. I used to say it was the mathematical equivalent of Hamlet’s phrase “To be, or not to be”—very short, Understanding as a Way of Life very succinct, but at the same time very deep. Eu- In recent years, Atiyah concerned himself with relating our ler’s equation uses only five symbols, but it also state of mind and interest in mathematics to some cogni- encapsulates beautifully deep ideas, and brevity is tive notions like creativity, beauty, imagination, and an important part of beauty. dreams [3, 26]. Concerning the place of dreams in mathematics where For Atiyah creativity comes before any other act in math- “the ideas appear fast and furious, intuitive, imaginative, ematics; it is the triggering element. It is deeply embedded vague and clumsy commodities”: in our mind. People think mathematics begins when you write Dreams happen during the daytime, they happen down a theorem followed by a proof. That’s not at night. You can call them a vision or intuition. the beginning, that’s the end. For me the creative But basically they are a state of mind—without place in mathematics comes before you start to put words, pictures, formulas or statements. It is pre things down on paper, before you try to write a for- all that. It’s pre-Plato. It is a very primordial feel- mula. You picture various things, you turn them ing. And again, if you try to grasp it, it always dies. over in your mind. You are trying to create, just So when you wake up in the morning, some vague as a musician is trying to create music, or a poet. residue lingers, the ghost of an idea. You try to re- There are no rules laid down. You have to do it member what it was and you only get half of it your own way. But at the end, just as a composer right, and maybe that is the best you can do. [3] has to put it down on paper, you have to write things down. But the most important stage is un- 22The “emotional brain,” specifically the medial orbitofrontal cortex.

NOVEMBER 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1669 In the “D´echiffreurs”published in the Notices of the AMS, References Atiyah wrote about the place of dreams in our life: [1] Atiyah M. Collected works. Vol. 1, Early papers: general papers, Oxford Science Publications, The Clarendon Press, Oxford In the broad light of day mathematicians check University Press, New York, 1988. MR951892 their equations and their proofs, leaving no stone [2] Hitchin N. Sir Michael Atiyah: A brief biography, in unturned in their search for rigour. But, at night, The founders of index theory: Reminiscences of and about Sir under the full moon, they dream, they float among Michael Atiyah, Raoul Bott, Friedrich Hirzebruch, and I. M. the stars and wonder at the miracle of the heavens. Singer, edited by S.-T. Yau, International Press, Somerville, They are inspired. Without dreams there is no art, MA, 2003. no mathematics, no life.23 [3] Minio R. An interview with Michael Atiyah, Math. Intel- ligencer, no. 1 (6):9–19, 1984, DOI 10.1007/BF03024202. Atiyah was a man in a hurry, his mind crossed by an un- MR735359 interrupted flow of teeming reflections and new ideas. He [4] Atiyah MF. Complex fibre bundles and ruled surfaces, was a man who cogitates. Any exchange or discussion was Proc. Math. Soc. (3) (5):407–434, 1955, DOI a good opportunity to bring out mathematics and mathe- 10.1112/plms/s3-5.4.407. MR0076409 matical physical thinking. He enjoyed being around and [5] Hodge WVD, Atiyah MF. Integrals of the second kind on an algebraic variety, Ann. of Math. (2) (62):56–91, 1955, talking (even under the rain and for hours!) with col- DOI 10.2307/2007100. MR0074082 leagues and young researchers. He was a great conversa- [6] Grothendieck A (1957). Classes de faisceaux et th´eorème de tionalist. Riemann-Roch. Published in SGA 6, Springer-Verlag, 1971, However, despite the tornado that could be released by 20–71. his presence and his willingness to communicate, interro- [7] Atiyah MF, Hirzebruch F. Riemann-Roch theorems for gate, and especially to be heard, Atiyah was paradoxically differentiable manifolds, Bull. Amer. Math. Soc. (65):276– intrinsically a solitary man who loved to be surrounded. 281, 1959, DOI 10.1090/S0002-9904-1959-10344-X. Atiyah was a man for whom friendship mattered. He did MR110106 not have a superiority complex, but knew how to step back [8] Atiyah MF. Characters and cohomology of finite groups, Inst. Hautes Etudes´ Sci. Publ. Math. (9):23–64, 1961. and advance people who played a role in his mathematical MR0148722 life. [9] Atiyah MF, Hirzebruch F. Vector bundles and homoge- Atiyah fondly acknowledged the place of his wife, Lily, neous spaces, Proc. Sympos. Pure Math., Vol. III, 7–38, 1961. in his life and how she allowed him the luxury of realiz- MR0139181 ing his journey. She was a mathematician and left the aca- [10] Hirzebruch F. Topological methods in algebraic geometry, demic path so that he could go forward in his own career. Third enlarged edition. New appendix and translation But if there is one word that describes his life well, a from the second German edition by R. L. E. Schwarzen- constant feature of his story is to “understand.” Atiyah was berger, with an additional section by A. Borel. Die always guided by the desire to understand. Grundlehren der Mathematischen Wissenschaften, Band 131, Springer-Verlag New York, Inc., New York, 1966. I always want to try to understand why things work. MR0202713 I’m not interested in getting a formula without [11] Atiyah MF, Todd JA. On complex Stiefel manifolds, knowing what it means. I always try to dig behind Proc. Cambridge Philos. Soc. (56):342–353, 1960, DOI the scenes, so if I have a formula, I understand why 10.1017/s0305004100034642. MR132552 [12] Atiyah MF, Segal GB. Equivariant 퐾-theory and comple- it’s there. tion, J. Differential Geometry (3):1–18, 1969. MR0259946 Atiyah was always a visionary and he was less interested [13] Atiyah MF. The Grothendieck ring in geometry and in the past than in new ideas about the future. topology, Proc. Internat. Congr. Mathematicians (, 1962), 442–446, 1963. MR0180975 [14] Gel’fand IM. On elliptic equations, Russ. Math. Surv., 15 ACKNOWLEDGMENT. We want to warmly thank (3):113–123, 1960. Reprinted in volume 1 of his collected for his reading and suggestions during works, pp. 65–75. the preparation of the tribute, Erica Flapan for her sup- [15] Atiyah MF, Singer IM. The index of elliptic opera- port, and Chika Mese for handling the tribute. We want tors. I, Ann. of Math. (2) (87):484–530, 1968, DOI to acknowledge the referee’s careful reading, remarks, 10.2307/1970715. MR0236950 and suggestions. [16] Atiyah MF, Singer IM. The index of elliptic operators on compact manifolds, Bull. Amer. Math. Soc. (69):422– 433, 1963, DOI 10.1090/S0002-9904-1963-10957-X. MR157392 [17] Atiyah M, Bott R, Patodi VK. On the heat equation 23Atiyah MF. Les D´echiffreurs (2008), Notices of the AMS, 2010.

1670 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 10 and the index theorem, Invent. Math. (19):279–330, 1973, DOI 10.1007/BF01425417. MR0650828 [18] Toeplitz O. Über die Auflösung unendlichvieler linearer Gleichungen mit unendlichvielen Unbekannten Rend. Cir- colo Mat. Palermo, 28, 88–96, 1909. [19] Connes A. A survey of foliations and operator algebras, Operator algebras and applications, Part I (Kingston, Ont., 1980), 521–628, 1982. MR679730 [20] Connes A, Moscovici H. Hopf algebras, cyclic cohomology and the transverse index theorem, Comm. Math. Phys., no. 1 (198):199–246, 1998, DOI 10.1007/s002200050477. MR1657389 [21] Atiyah MF. Global theory of elliptic operators, Proc. In- ternat. Conf. on Functional Analysis and Related Topics (Tokyo, 1969), 21–30, 1970. MR0266247 [22] Kouneiher J. Where we stand today, in Foundations of Mathematics and Physics, one century after Hilbert, ed. J. Kouneiher, Springer, 1–73, 2018. MR3822546 [23] ’t Hooft G. Phys. Rev. D (14):3432, 1976 [Erratum:ibid. D (18):2199, 1978]. [24] Donaldson SK, Kronheimer PB. The geometry of four- manifolds, Oxford Mathematical Monographs, The Claren- don Press, , New York, 1990. MR1079726 [25] Witten E. Michael Atiyah and the physics/geometry in- terface, Sir Michael Atiyah: a great mathematician of the twentieth century, Asian J. Math., no. 1 (3):lxi–lxiv, 1999. MR1701919 [26] Atiyah M, Kouneiher J. Preface in Foundations of Mathe- matics and Physics, one century after Hilbert, ed. J. Kouneiher, Springer, vii–xi, 2018.

Alain Connes Joseph Kouneiher

Credits Photo of Michael Atiyah is courtesy of Joseph Kouneiher. Author photos are courtesy of the authors.

See https://arxiv.org/abs/1910.07851 for an ex- tended version with a complete list of references.

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