Celebratio Mathematica

M. F. Atiyah: “A note on the tangents of a twisted cubic,” Proc. Cambridge Philos. Soc. 48 (1952), pp. 204–205. MR 0048079 Zbl 0046.14604 article

W. Hodge and M. Atiyah: “Formes de seconde espèce sur une variété algébrique” [Forms of the second kind on an algebraic variety], C. R. Acad. Sci. Paris 239 (1954), pp. 1333–1335. MR 0068869 article

M. F. Atiyah: “Complex fibre bundles and ruled surfaces,” Proc. London Math. Soc. (3) 5 (1955), pp. 407–434. MR 0076409 Zbl 0174.52804 article

W. V. D. Hodge and M. F. Atiyah: “Integrals of the second kind on an algebraic variety,” Ann. Math. (2) 62 (1955), pp. 56–91. MR 0074082 Zbl 0068.34401 article

M. Atiyah: “On the Krull–Schmidt theorem with application to sheaves,” Bull. Soc. Math. France 84 (1956), pp. 307–317. MR 0086358 Zbl 0072.18101 article

It is well-known that many standard algebraic results in the theory of groups, rings, modules, etc., can be proved more generally for suitable categories, in the sense of Eilenberg–Maclane [1945]. This has the usual advantages of abstraction. It singles out those features of a given algebraic structure which are essential to the results in question, and by so doing it extends the validity of these results to other domains. In this Note we shall be concerned with the Krull–Schmidt theorem for modules, which asserts under suitable conditions the existence and essential uniqueness of a direct decomposition into indecomposable factors. It is clear that if such a theorem is to have a meaning in some general category, then such notions as kernel, image and direct sum must be defined in the category, and must possess the usual properties. Such a category, called an exact category, has been considered by Buchsbaum [1955]. Basing ourselves on his paper we then have at our disposal all the necessary notions with the usual properties. Our purpose will be to investigate conditions under which the Krull–Schmidt theorem holds in an exact category. This categorical formulation will then enable us to obtain a Krull–Schmidt theorem for suitable categories of sheaves. This is of special interest in algebraic geometry, and it was this case of the theorem which provided our motivation.

M. F. Atiyah: “Vector bundles over an elliptic curve,” Proc. London Math. Soc. (3) 7 : 1 (1957), pp. 414–452. MR 0131423 Zbl 0084.17305 article

M. F. Atiyah: “Complex analytic connections in fibre bundles,” Trans. Amer. Math. Soc. 85 : 1 (1957), pp. 181–207. See also Symposium internacional de topología algebraica (1958). MR 0086359 Zbl 0078.16002 article

In the theory of differentiable fibre bundles, with a Lie group as structure group, the notion of a connection plays an important role. In this paper we shall consider complex analytic connections in complex analytic fibre bundles. The situation is then radically different from that in the differentiable case. In the differentiable case connections always exist, but may not be integrable; in the complex analytic case connections may not exist at all. In both cases we are led therefore to certain obstructions, an obstruction to the integrability of a connection in the differentiable case, an obstruction to the existence of a connection in the complex analytic case. It is a basic theorem that, if the structure group is compact, the obstruction in the differentiable case (the curvature) generates the characteristic cohomology ring of the bundle (with real coefficients). What we shall show is that, in a large class of important cases, the obstruction in the complex analytic case also generates the characteristic cohomology ring. Using this fact we can then give a purely cohomological definition of the characteristic ring. This has a number of advantages over the differentiable approach: in the first place the definition is a canonical one, not depending on an arbitrary choice of connection; secondly we remain throughout in the complex analytic domain, our characteristic classes being expressed as elements of cohomology groups with coefficients in certain analytic sheaves; finally the procedure can be carried through without change for algebraic fibre bundles.

M. F. Atiyah: “On analytic surfaces with double points,” Proc. Roy. Soc. London. Ser. A 247 : 1249 (1958), pp. 237–244. MR 0095974 Zbl 0135.21301 article

M. F. Atiyah: Some examples of complex manifolds. Bonner Mathematische Schriften 6. Universität Bonn, 1958. MR 0105718 Zbl 0080.37502 book

M. F. Atiyah: “Complex analytic connections in fibre bundles,” pp. 77–82 in Symposium internacional de topología algebraica (Universidad Nacional Autónoma de México). Universidad Nacional Autónoma de México and UNESCO (Mexico City and Paris), 1958. See also Trans. Amer. Math. Soc. 85:1 (1957). MR 0098194 Zbl 0123.16503 incollection

M. F. Atiyah and F. Hirzebruch: “Riemann–Roch theorems for differentiable manifolds,” Bull. Am. Math. Soc. 65 : 4 (1959), pp. 276–281. MR 110106 Zbl 0142.40901 article

The Riemann–Roch Theorem for an algebraic variety \( Y \) (see [Hirzebruch 1956]) led to certain divisibility conditions for the Chern classes of \( Y \). It was natural to ask whether these conditions held more generally for any compact almost complex manifold. This question, and various generalizations of it, were raised in [Hirzebruch 1954] and most of these have since been answered in the affirmative in [Borel and Hirzebruch 1958] and [Milnor 1960].

More recently Grothendieck has obtained [Borel and Serre 1958] a more general Riemann–Roch Theorem for a map \( f: Y\to X \) of algebraic varieties. This reduces to the previous Riemann–Roch Theorem on taking \( X \) to be a point. Grothendieck’s Theorem implies many conditions on characteristic classes, and again it is natural to ask if these conditions hold more generally for almost complex or even differentiable manifolds. The purpose of this note is to enunciate certain differentiable analogues of Grothendieck’s Theorem. These “differentiable Riemann–Roch Theorems” yield, as special cases, the divisibility conditions mentioned above and also certain new homotopy invariance properties of Pontrjagin classes. As an application of the latter we get a new proof (and slight improvement) of the result of Kervaire–Milnor [1960] on the stable \( J \)-homomorphism.

Another differentiable Riemann–Roch Theorem, with applications to embeddability problems of differentiable manifolds, will be found in [Atiyah and Hirzebruch 1959].

The proofs of our theorems rely heavily on the Bott periodicity of the classical groups [Bott 1957, 1958, 1959], and are altogether different from the earlier methods of [Borel and Hirzebruch 1958] and [Milnor 1960], which were based on Thom’s cobordism theory and Adams’ spectral sequence.

M. F. Atiyah and F. Hirzebruch: “Quelques théorèmes de non-plongement pour les variétés différentiables” [Some non-immersion theorems for differentiable manifolds], Bull. Soc. Math. France 87 (1959), pp. 383–396. MR 114231 Zbl 0196.55903 article

Nous avons montré dans [Atiyah and Hirzebruch 1959] que le théorème de Riemann–Roch [Borel and Serre 1958] a des analogues différentiables. Un exposé de ces résultats a été fait par l’un des auteurs au Séminaire Bourbaki [Hirzebruch 1958/59]. Les théorèmes de Riemann–Roch différentiables fournissent comme cas particulier certaines conditions de divisibilité pour les classes caractéristiques d’une variété différentiable que l’on peut considérer comme des analogues différentiables du théorème de Riemann–Roch de [Hirzebruch 1956].

La plupart de ces conditions de divisibilité ont été prouvées précédement dans [Borel and Hirzebruch 1958], [Borel and Hirzebruch 1960] et [Milnor 1960]. Dans ce qui suit nous démontrons à l’aide des méthodes de [Atiyah and Hirzebruch 1959] que les classes caractéristiques d’une variété différentiable compacte orientée de dimension \( d \) satisfont aux conditions de divisibilité supplémentaires si la variété peut être différentiablement plongée dans un espace euclidien (ou ce qui est équivalent, une sphère) de dimension \( 2d - q \). Ces conditions de divisibilité «non stables» nous permettent de prouver des théorèmes de non-plongement qui semblent beaucoup plus forts que ceux qui étaient connus avant (3.6). L’outil essentiel est encore le théorème de Bott [Borel and Hirzebruch 1958/59; Bott 1958] qui dit que la \( n \)-ième classe de Chern d’un fibré vectoriel complexe sur la sphère \( S_{2n} \) est divisible par \( (n-1)! \).

M. F. Atiyah and J. A. Todd: “On complex Stiefel manifolds,” Proc. Cambridge Philos. Soc. 56 : 4 (1960), pp. 342–353. MR 0132552 Zbl 0109.16102 article

M. F. Atiyah and F. Hirzebruch: “Quelques théoremes de non-plongement pour les variétés différentiables” [Some non-embedding theorems for differentiable manifolds], Colloques Int. Centre Nat. Rech. Sci. 89 (1960), pp. 383–396. See also Bull. Soc. Math. France 87 (1959). Zbl 0108.18202 article

Nous avons montré dans [Atiyah and Hirzebruch 1959] que le théorème de Riemann–Roch [Borel and Serre 1958] a des analogues différentiables. Un exposé de ces résultats a été fait par l’un des auteurs au Séminaire Bourbaki [Hirzebruch, 1958/59]. Les théorèmes de Riemann–Roch différentiables fournissent comme cas particulier certaines conditions de divisibilité pour les classes caractéristiques d’une variété différentiable que l’on peut considérer comme des analogues différentiables du théorème de Riemann–Roch de [Hirzebruch 1956].

La plupart de ces conditions de divisibilité ont été prouvées précédement dans [Borel and Hirzebruch 1958; 1960] et [Milnor 1960]. Dans ce qui suit nous démontrons à l’aide des méthodes de [Atiyah and Hirzebruch 1959] que les classes caractéristiques d’une variété différentiable compacte orientée de dimension \( d \) satisfont aux conditions de divisibilité supplémentaires si la variété peut être différentiablement plongée dans un espance euclidien (ou ce qui est équivalent, une spère) de dimension \( 2d - q \). Ces conditions de divisibilité «non stables» nous permettent de prouver des théorèmes de non-plongement qui semblent beaucoup plus forts que ceux qui ’\etaient connus avant (3.6). L’outil essentiel est encore le théorème de Bott ([Borel and Hirzebruch, 1958/59] et [Bott 1958]) qui dit que la \( n \)-ième classe de Chern d’un fibré vectoriel complexe sur la sphère \( S_{2n} \) est divisible par \( (n-1)! \).

M. F. Atiyah and F. Hirzebruch: “Vector bundles and homogeneous spaces,” pp. 7–38 in Differential geometry (Tucson, AZ, 18–19 February 1960). Edited by C. B. Allendoerfer. Proceedings of Symposia in Pure Mathematics 3. American Mathematical Society (Providence, RI), 1961. Republished in Algebraic topology: A student’s guide (1972). MR 139181 Zbl 0108.17705 incollection

F. Hirzebruch	Related	Volume
Carl Barnett Allendoerfer	Related

M. F. Atiyah: “Characters and cohomology of finite groups,” Inst. Hautes Études Sci. Publ. Math. 9 : 1 (1961), pp. 23–64. MR 0148722 Zbl 0107.02303 article

M. F. Atiyah: “Thom complexes,” Proc. London Math. Soc. (3) 11 : 1 (1961), pp. 291–310. MR 0131880 Zbl 0124.16301 article

The spaces which form the title of this paper were introduced by Thom in [1954] as a tool in his study of differentiable manifolds. In addition certain special Thom complexes have been studied by James [1959] in connexion with Stiefel manifolds (cf. [James 1958a; 1958b]). The purpose of this paper is to prove a number of general results on Thom complexes, and to deduce the main theorems of James [1958b; 1959] as immediate consequences. Our main result (3.3) is a duality theorem (in the Whitehead–Spanier \( S \)-theory) for Thom complexes over differentiable manifolds. Besides its application this is a result of some independent interest, since it provides a satisfactory place for manifolds in \( S \)-theory.

M. F. Atiyah and F. Hirzebruch: “Cohomologie-Operationen und charakteristische Klassen” [Cohomology operations and characteristic classes], Math. Z. 77 : 1 (1961), pp. 149–187. Dedicated to Friedrich Karl Schmidt on the occasion of his sixtieth birthday. MR 156361 Zbl 0109.16002 article

F. Hirzebruch	Related	Volume
Friedrich Karl Schmidt	Related

M. F. Atiyah: “Bordism and cobordism,” Proc. Cambridge Philos. Soc. 57 (1961), pp. 200–208. MR 0126856 Zbl 0104.17405 article

In [1959], [1960] Wall determined the structure of the cobordism ring introduced by Thom in [1953]. Among Wall’s results is a certain exact sequence relating the oriented and unoriented cobordism groups. There is also another exact sequence, due to Rohlin [1953], [1958] and Dold [1959/60] which is closely connected with that of Wall. These exact sequences are established by ad hoc methods. The purpose of this paper is to show that both these sequences are “cohomology-type” exact sequences arising in the well-known way from mappings into a universal space. The appropriate “cohomology” theory is constructed by taking as universal space the Thom complex \( \mathit{MSO}(n) \), for \( n \) large. This gives rise to (oriented) cobordism groups \( \mathit{MSO}^*(X) \) of a space \( X \).

M. F. Atiyah and F. Hirzebruch: “Bott periodicity and the parallelizability of the spheres,” Proc. Camb. Philos. Soc. 57 : 2 (April 1961), pp. 223–226. MR 126282 Zbl 0108.35902 article

The theorems of Bott [1958, 1959a] on the stable homotopy of the classical groups imply that the sphere \( S^n \) is not parallelizable for \( n \neq 1 \), \( 3, 7 \). This was shown independently by Kervaire [1958] and Milnor [1958; Bott and Milnor 1958]. Another proof can be found in [Borel and Hirzebruch 1959, §26.11]. The work of J. F. Adams (on the non-existence of elements of Hopf invariant one) implies more strongly that \( S^n \) with any (perhaps extraordinary) differentiable structure is not parallelizable if \( n \neq 1 \), \( 3, 7 \). Thus there exist already four proofs for the non-parallelizability of the spheres, the first three mentioned relying on the Bott theory, as given in [Bott 1958, 1959a]. The purpose of this note is to show how the refined form of Bott’s results given in [Bott 1959b] leads to a very simple proof of the non-parallelizability (only for the usual differentiable structures of the spheres). We shall prove in fact the folowing theorem due to Milnor [1958] which implies the non-parallelizability.

There exists a real vector bundle \( \xi \) over the sphere \( S^n \) with \( w_n(\xi) \neq 0 \) only for \( n = 1 \), \( 2,4 \) or 8

M. F. Atiyah and F. Hirzebruch: “Charakteristische Klassen und Anwendungen” [Characteristic classes and applications], Enseignement Math. (2) 7 : 1 (1961), pp. 188–213. MR 154294 Zbl 0104.39801 article

M. F. Atiyah and F. Hirzebruch: “The Riemann–Roch theorem for analytic embeddings,” Topology 1 : 2 (April–June 1962), pp. 151–166. MR 148084 Zbl 0108.36402 article

In [Borel and Serre 1958] Grothendieck formulated and proved a generalization of the Riemann–Roch theorem which we shall refer to as GRR. This theorem is concerned with a proper morphism \( f:Y\to X \) of algebraic manifolds (any ground field) and reduces to the version (HRR) given in [Hirzebruch 1956] when \( X \) is a point (and the ground field is \( \mathbb{C} \)). It is not known whether GRR or even HRR holds for arbitrary complex manifolds. However the proof of GRR given in [Borel and Serre 1958] breaks up into two separate cases:

\( f \) is an embedding,
\( f \) is a projection \( X\times P_N \to X \), where \( P_N \) is a projective space,

and the main purpose of this paper is to give a proof of GRR in case (i) for arbitrary complex manifolds. This proof is quite different from, and in many ways simpler than, that of [Borel and Serre 1958] and, for the complex algebraic case, it gives a new proof of GRR.

M. F. Atiyah: “Vector bundles and the Künneth formula,” Topology 1 : 3 (1962), pp. 245–248. MR 0150780 article

The purpose of this note is to establish a Künneth formula for the ring \( K^*(X) \) introduced in [Atiyah and Hirzebruch 1960]. We shall prove the following:

Let \( X \), \( Y \) be finite CW-complexes, then we have a natural exact sequence \[ 0 \rightarrow K^*(X)\otimes K^*(Y) \stackrel{\alpha}{\longrightarrow} K^*(X\times Y)\stackrel{\beta}{\longrightarrow} \operatorname{Tor}(K^*(X),K^*(Y)) \rightarrow 0. \]

Here \( \otimes \), \( \operatorname{Tor} \) are applied to abelian groups, and \( \alpha \) is the natural map induced by the product in \( K^* \). Moreover the sequence is \( \mathbb{Z}_2 \)-graded with \( \deg\alpha = 0 \), \( \deg\beta = 1 \), where \( K^p\otimes K^q \) and \( \operatorname{Tor}(K^p,K^q) \) are given degree \( p+q \) (\( p,q \in \mathbb{Z}_2 \)).

M. F. Atiyah and F. Hirzebruch: “Analytic cycles on complex manifolds,” Topology 1 : 1 (January–March 1962), pp. 25–45. MR 145560 Zbl 0108.36401 article

Let \( X \) be a complex manifold, \( Y \) a closed irreducible \( k \)-dimensional complex analytic subspace of \( X \). Then \( Y \) defines or “carries” a \( 2k \)-dimensional integral homology class \( y \) of \( X \), although the precise definition of \( y \) presents technical difficu1ties. A finite formal linear combination \( \sum n_iY_i \) with \( n_i \) integers and \( Y_i \) as above is called a complex analytic cycle, and the corresponding homology class \( \sum n_iY_i \) is called a complex analytic homology class. If an integral cohomology class \( u \) corresponds under Poincaré duality to a complex analytic homology class we shall say that \( u \) is a complex analytic cohomology class. The purpose of this paper is to show that a complex analytic cohomology class \( u \) satisfies certain topological conditions, independent of the complex structure of \( X \). These conditions are that certain cohomology operations should vanish on \( u \), for example \( \mathrm{Sq}^3u = 0 \): they are all torsion conditions. We also produce examples to show that these conditions are not vacuous even in the restricted classes of (a) Stein manifolds and (b) projective algebraic manifolds.

M. F. Atiyah: “Immersions and embeddings of manifolds,” Topology 1 : 2 (April–June 1962), pp. 125–132. MR 0145549 Zbl 0109.41101 article

M. F. Atiyah: “The Grothendieck ring in geometry and topology,” pp. 442–446 in Proceedings of the International Congress of Mathematicians 1962 (Stockholm, 15–22 August 1962), vol. 1. Inst. Mittag-Leffler (Djursholm), 1963. MR 0180975 Zbl 0121.39702 incollection

M. F. Atiyah and I. M. Singer: “The index of elliptic operators on compact manifolds,” Bull. Amer. Math. Soc. 69 : 3 (1963), pp. 422–433. MR 157392 Zbl 0118.31203 article

In [1960] Gel’fand posed the general problem of investigating the relationship between topological and analytical invariants of elliptic differential operators. In particular he suggested that it should be possible to express the index of an elliptic operator in topological terms. This problem has been taken up by Agranovič [1962a; 1962b], Dynin [1962b; 1961a; 1961b], Seeley [1961; 1963] and Vol’pert [1962] who have solved it in special cases. The purpose of this paper is to give a general formula for the index of an elliptic operator on any compact oriented differentiable manifold. As a special case of this formula we get the Hirzebruch–Riemann–Roch theorem for any compact complex manifold. This was previously known only for projective algebraic manifolds. Some other special cases, of interest in differential topology, are discussed.

M. F. Atiyah, R. Bott, and A. Shapiro: “Clifford modules,” Topology 3 : Supplement 1 (July 1964), pp. 3–38. MR 0167985 Zbl 0146.19001 article

Raoul H. Bott	Related	Volume
Arnold Samuel Shapiro	Related

M. F. Atiyah and R. Bott: “The index problem for manifolds with boundary,” pp. 175–186 in Differential Analysis: Papers presented at the international colloquium (Bombay, 7–14 January 1964). Tata Institute of Fundamental Research Studies in Mathematics 2. Oxford University Press (London), 1964. MR 0185606 Zbl 0163.34603 incollection

M. Atiyah and R. Bott: “On the periodicity theorem for complex vector bundles,” Acta Math. 112 : 1 (1964), pp. 229–247. MR 0178470 Zbl 0131.38201 article

The periodicity theorem for the infinite unitary group [Bott 1959] can be interpreted as a statement about complex vector bundles. As such it describes the relation between vector bundles over \( X \) and \( X\times S^2 \), where \( X \) is a compact space and \( S^2 \) is the 2-sphere. This relation is most succinctly expressed by the formula \[ K(X\times S^2) \simeq K(X)\otimes K(S^2), \] where \( K(X) \) is the Grothendieck group of complex vector bundles over \( X \). The general theory of these \( K \)-groups, as developed in [Atiyah and Hirzebruch 1961], has found many applications in topology and related fields. Since the periodicity theorem is the foundation stone of all this theory it seems desirable to have an elementary proof of it, and it is the purpose of this paper to present such a proof.

M. F. Atiyah and R. Bott: Notes on the Lefschetz fixed point theorem for elliptic complexes. Harvard University (Cambridge, MA), 1964. Zbl 0161.43101 book

M. F. Atiyah: “The index of elliptic operators on compact manifolds” in Séminaire Bourbaki. 15e année: 1962/63. Séminaire Bourbaki. Secrétariat Mathématique (Paris), 1964. Exposé no. 253. Reprint of article in Bull. Am. Math. Soc. 69 (1963). See also Séminaire Bourbaki 8 (1995). Zbl 0124.31102 incollection

M. F. Atiyah: “On the \( K \)-theory of compact Lie groups,” Topology 4 : 1 (1965), pp. 95–99. MR 0178092 Zbl 0136.21001 article

M. F. Atiyah: “The index theorem for manifolds with boundary,” pp. 337–351 in Seminar on the Atiyah–Singer index theorem. Edited by R. S. Palais. Annals of Mathematics Studies 57. Princeton University Press, 1965. Appendix I. Atiyah’s sole contribution to Seminar on the Atiyah–Singer index theorem (1965). Republished in Atiyah’s Collected works, vol. 3. See also similarly-titled article in Differential analysis (1965). incollection

M. F. Atiyah: “The role of algebraic topology in mathematics,” J. London Math. Soc. 41 : 1 (1966), pp. 63–69. See also Geometrie (1972). MR 0187231 Zbl 0137.17301 article

M. F. Atiyah: “\( K \)-theory and reality,” Quart. J. Math. Oxford Ser. (2) 17 : 1 (1966), pp. 367–386. MR 0206940 Zbl 0146.19101 article

The \( K \)-theory of complex vector bundles [Atiyah 1965; Atiyah and Hirzebruch 1961] has many variants and refinements. Thus there are:

\( K \)-theory of real vector bundles, denoted by \( \mathit{KO} \),
\( K \)-theory of self-conjugate bundles, denoted by \( \mathit{KC} \) [Anderson 1964] or \( \mathit{KSC} \) [Green 1964],
\( K \)-theory of \( G \)-vector bundles over \( G \)-spaces [Atiyah and Segal 1965], denoted by \( K_G \).

In this paper we introduce a new \( K \)-theory denoted by \( \mathit{KR} \) which is, in a sense, a mixture of these three. Our definition is motivated partly by analogy with real algebraic geometry and partly by the theory of real elliptic operators. In fact, for a thorough treatment of the index problem for real elliptic operators, our \( \mathit{KR} \)-theory is essential. On the other hand, from the purely topological point of view, \( \mathit{KR} \)-theory has a number of advantages and there is a strong case for regarding it as the primary theory and obtaining all the others from it. One of the main purposes of this paper is in fact to show how \( \mathit{KR} \)-theory leads to an elegant proof of the periodicity theorem for \( \mathit{KO} \)-theory, starting essentially from the periodicity theorem for \( K \)-theory as proved in [Atiyah and Bott 1964].

J. F. Adams and M. F. Atiyah: “\( K \)-theory and the Hopf invariant,” Quart. J. Math. Oxford Ser. (2) 17 (1966), pp. 31–38. MR 0198460 Zbl 0136.43903 article

M. F. Atiyah and R. Bott: “A Lefschetz fixed point formula for elliptic differential operators,” Bull. Am. Math. Soc. 72 : 2 (1966), pp. 245–250. MR 0190950 Zbl 0151.31801 article

The classical Lefschetz fixed point formula expresses, under suitable circumstances, the number of fixed points of a continuous map \( f:X\to X \) in terms of the transformation induced by \( f \) on the cohomology of \( X \). If \( X \) is not just a topological space but has some further structure, and if this structure is preserved by \( f \), one would expect to be able to refine the Lefschetz formula and to say more about the nature of the fixed points. The purpose of this note is to present such a refinement (Theorem 1) when \( X \) is a compact differentiable manifold endowed with an elliptic differential operator (or more generally an elliptic complex). Taking essentially the classical operators of complex and Riemannian geometry we obtain a number of important special cases (Theorem 2, 3). The first of these was conjectured to us by Shimura and was proved by Eichler for dimension one.

M. F. Atiyah: “Power operations in \( K \)-theory,” Quart. J. Math. Oxford Ser. (2) 17 : 1 (1966), pp. 165–193. Russian translation published in Mathematika 14:2 (1970). MR 0202130 Zbl 0144.44901 article

M. F. Atiyah: “A Lefschetz fixed point formula for holomorphic mappings,” pp. 28–32 in Contemporary problems in the theory of analytic functions (Yerevan, Armenia, 1965). Nauka (Moscow), 1966. MR 0206989 Zbl 0161.43202 incollection

M. F. Atiyah and R. Bott: “A Lefschetz fixed point formula for elliptic complexes, I,” Ann. Math. (2) 86 : 2 (1967), pp. 374–407. MR 0212836 Zbl 0161.43201 article

M. F. Atiyah and D. W. Anderson: \( K \)-theory. Mathematics Lecture Notes 7. W. A. Benjamin (New York and Amsterdam), 1967. Lectures by Atiyah (Fall 1964), notes by Anderson. Russian translation published as Lekcii po \( K \)-teorii (1967). 2nd edition published in 1989. MR 0224083 book

M. F. Atiyah: “A Lefschetz fixed-point formula for elliptic differential operators,” pp. 38–39 in Simposio internazionale di geometria algebrica (Rome, 30 September–5 October 1965), published as Rend. Mat. Appl. V : 25. Issue edited by G. Castelnuovo. Cremonese (Rome), 1967. See also Bull. Amer. Math. Soc. 72:2 (1966). Zbl 0149.41201 incollection

M. F. Atiyah: “Algebraic topology and elliptic operators,” Comm. Pure Appl. Math. 20 (1967), pp. 237–249. MR 0211418 Zbl 0145.43804 article

M. F. Atiyah and C. T. C. Wall: “Cohomology of groups,” pp. 94–115 in Algebraic number theory (University of Sussex, Brighton, 1–17 September 1965). Edited by J. W. S. Cassels and A. Fröhlich. Thompson (Washington, DC), 1967. MR 0219512 incollection

Albrecht Fröhlich	Related
C. T. C. Wall	Related
John W. S. Cassels	Related

M. F. Atiyah and D. W. Anderson: Lekcii po \( K \)-teorii [Lectures on \( K \)-theory]. Mir (Moscow), 1967. Russian translation of \( K \)-theory (1967). Zbl 0159.53401 book

M. F. Atiyah and G. B. Segal: “The index of elliptic operators, II,” Uspehi Mat. Nauk 23 : 6 (144) (1968), pp. 135–149. Russian translation of article in Ann. Math. 87:3 (1968). MR 0236953 article

M. F. Atiyah and G. B. Segal: “The index of elliptic operators, II,” Ann. Math. (2) 87 : 3 (1968), pp. 531–545. Russian translation published in Uspehi Mat. Nauk 23:6(144) (1968). MR 0236951 Zbl 0164.24201 article

The purpose of this paper is to show how the index theorem of [Atiyah and Singer 1963] can be reformulated as a general “Lefschetz fixed-point theorem” on the lines of [Atiyah and Bott 1967]. In this way we shall obtain the main theorem of [Atiyah and Bott 1967], generalized to deal with arbitrary fixed-point sets, but only for transformations belonging to a compact group.

The content of this paper is essentially topological, and it should be viewed as a paper on the equivariant \( K \)-theory of manifolds. The analysis has all been done in [Atiyah and Singer 1968], and what we do here is simply to express the topological index in terms of fixed-point sets. This is quite independent of the main theorem of [Atiyah and Singer 1968] asserting the equality of the topological and analytical indices.

As in [Atiyah and Singer 1968], we avoid cohomology and use only \( K \)-theory. In paper III of this series, we shall pass over to cohomology obtaining explicit formulas in terms of characteristic classes.

M. F. Atiyah and I. M. Singer: “The index of elliptic operators, I,” Ann. Math. (2) 87 : 3 (May 1968), pp. 484–530. A Russian translation was published in Uspehi Mat. Nauk 23:5(143). MR 236950 Zbl 0164.24001 article

This is the first of a series of papers which will be devoted to a study of the index of elliptic operators on compact manifolds. The main result was announced in [Atiyah and Singer 1963] and, for manifolds with boundary, in [Atiyah 1964]. The long delay between these announcements and the present paper is due to several factors. On the one hand, a fairly detailed exposition has already appeared in [Palais 1965]. On the other hand, our original proof, reproduced with minor modifications in [Palais 1965], had a number of drawbacks. In the first place the use of cobordism, and the computational checking associated with this, were not very enlightening. More seriously, however, the method of proof did not lend itself to certain natural generalizations of the problem where appropriate cobordism groups were not known. The reader who is familiar with the Riemann–Roch theorem will realize that our original proof of the index theorem was modelled closely on Hirzebruch’s proof of the Riemann–Roch theorem. Naturally enough we were led to look for a proof modelled more on that of Grothendieck. While we have not completely succeeded in this aim, we have at least found a proof which is much more natural, does not use cobordism, and lends itself therefore to generalization. In spirit, at least, it has much in common with Grothendieck’s approach.

M. F. Atiyah and I. M. Singer: “The index of elliptic operators, III,” Ann. Math. (2) 87 : 3 (May 1968), pp. 546–604. A Russian translation was published in Uspehi Mat. Nauk 24:1(145). MR 236952 Zbl 0164.24301 article

In [1968a], paper I of this series, the index of an elliptic operator was computed in terms of \( K \)-theory. In this paper, we carry out what is essentially a routine exercise by passing from \( K \)-theory to cohomology. In this way, we end up with the explicit cohomological formula for the index announced in [1963].

In [1968a] we also considered elliptic operators (or complexes) compatible with a compact group \( G \) of transformations. The index in this case is a character of \( G \), and the main theorem of [1968a] gave a construction for this in \( KG \)-theory. In [1968b], paper II of this series, the value of this index-character at an element \( g\in G \) was expressed as the index of a new “virtual operator” on the fixed point set of \( g \). This was referred to as a Lefschetz fixed-point formula. By combining this formula with the cohomological formula for the index, we obtain finally an explicit cohomological formula for the index-character. We shall describe this formula in detail for a number of important operators. In particular we draw attention to the “integrality theorems” obtained in this way for actions of finite groups on manifolds. Most of these do not depend on the analysis in [1968a], but are a consequence of combining the purely topological results of [1968b] and the present paper.

M. F. Atiyah and R. Bott: “A Lefschetz fixed point formula for elliptic complexes, II: Applications,” Ann. Math. (2) 88 : 3 (November 1968), pp. 451–491. MR 0232406 Zbl 0167.21703 article

M. F. At’ja and I. M. Zinger: “The index of elliptic operators, I,” Uspehi Mat. Nauk 23 : 5(143) (1968), pp. 99–142. Russian translation of an article published in Ann. Math. (2) 87:3 (1968). MR 232402 article

M. F. Atiyah: “Bott periodicity and the index of elliptic operators,” Quart. J. Math. Oxford Ser. (2) 19 (1968), pp. 113–140. MR 0228000 Zbl 0159.53501 article

M. F. Atiyah: “Global aspects of the theory of elliptic differential operators,” pp. 57–64 in Proceedings of the International Congress of Mathematics (Moscow, 16–26 August 1966). Mir (Moscow), 1968. MR 0233378 Zbl 0204.41902 incollection

M. F. Atiyah and I. M. Singer: “Index theory for skew-adjoint Fredholm operators,” Inst. Hautes Études Sci. Publ. Math. 37 (1969), pp. 5–26. MR 285033 Zbl 0194.55503 article

M. F. Atiyah and I. G. Macdonald: Introduction to commutative algebra. Addison-Wesley Series in Mathematics 361. Addison-Wesley (Reading, MA), 1969. Russian translation published as Vvedenie v kommutativnuju algebru (1972). MR 0242802 Zbl 0175.03601 book

M. F. Atiyah: “Wandel und Fortschritt in der Mathematik,” Bild der Wissenschaft 4 (1969), pp. 315–323. Republished in Mathematiker über die Mathematik (1974) and Atiyah’s Collected works, vol. 1. article

M. F. Atiyah: “The signature of fibre-bundles,” pp. 73–84 in Global analysis: Papers in honor of K. Kodaira. Edited by S. Iyanaga and D. C. Spencer. Princeton Mathematical Series 29. University of Tokyo Press, 1969. MR 0254864 Zbl 0193.52302 incollection

For a compact oriented differentiable manifold \( X \) of dimension \( 4k \) the signature (or index) of \( X \) is defined as the signature of the quadratic form in \( H^{2k}(X;\mathbb{R}) \) given by the cup product. Thus \[\operatorname{Sign}(X) = p - q \] where \( p \) is the number of \( + \) signs in a diagonalization of the given quadratic form and \( q \) is the number of \( - \) signs. If \( \operatorname{dim} X \) is not by divisible by 4 one defines \( \operatorname{Sign}(X) \) to be zero. Then one has the the multiplicative formula \[\operatorname{Sign}(X \times Y) = \operatorname{Sign}(X)\cdot\operatorname{Sign}(Y)\] In [Chern, Hirzebruch and Serre 1957] it was proved that this multiplicative formula continues to hold when \( X \times Y \) is replaced by a fibre bundle with base \( X \) and fibre \( Y \) provided that the fundamental group of \( X \) acts trivially us on the cohomology of \( Y \).

In this paper we exhibit examples which show that this restriction on the action of \( \pi_1(X) \) is necessary, and that the signature is not multiplicative in general fibre-bundles.

Shōkichi Iyanaga	Related
Donald Clayton Spencer	Related
Kunihiko Kodaira	Related

M. F. Atiyah: “Algebraic topology and operators in Hilbert space,” pp. 101–121 in Lectures in modern analysis and applications, vol. I. Edited by C. T. Taam. Lecture Notes in Mathematics 103. Springer (Berlin), 1969. MR 0248803 Zbl 0177.51701 incollection

M. F. Atiyah and G. B. Segal: “Equivariant \( K \)-theory and completion,” J. Differential Geometry 3 (1969), pp. 1–18. MR 0259946 Zbl 0215.24403 article

It was shown in [Atiyah 1961] that, for any finite group \( G \), the completed character ring \( R(G)^{\wedge} \) was isomorphic to \( K^*(B_G) \) where \( B_G \) denotes a classifying space for \( G \). The corresponding result for compact connected Lie groups was established in [Atiyah and Hirzebruch 1961], and a combination of the methods of [Atiyah and Hirzebruch 1961] and [Atiyah 1961] (together with certain basic properties of \( R(G) \) given in [Segal 1968b]) can be used to derive the theorem for general compact Lie groups. Such a proof however would be extremely lengthy, the worst part being in fact the treatment for finite groups where one climbs up via cyclic and Sylow subgroups.

The purpose of this paper is to give a new and much simpler proof of the theorem about \( K^*(B_G) \) which applies directly to all compact Lie groups \( G \). The main feature of our new proof is that we generalize the whole problem in a rather natural way by working with the equivariant \( K \)-theory developed in [Segal 1968a]. We shall formulate and prove a general theorem about the completion \( K_G^*(X)^{\wedge} \) for any compact \( G \)-space \( X \). The theorem about \( R(G) \) then follows by taking \( X \) to be a point.

M. F. At’ja and I. M. Zinger: “The index of elliptic operators, III,” Uspehi Mat. Nauk 24 : 1(145) (1969), pp. 127–182. The English original was published in Ann. Math. (2) 87:3 (1968). MR 256417 article

M. F. Atiyah and D. O. Tall: “Group representations, \( \lambda \)-rings and the \( J \)-homomorphism,” Topology 8 : 3 (July 1969), pp. 253–297. MR 0244387 Zbl 0159.53301 article

M. F. Atiyah: “Topology of elliptic operators,” pp. 101–119 in Global analysis (Berkeley, CA, 1–26 July 1968). Edited by S.-S. Chern and S. Smale. Proceedings of Symposia in Pure Mathematics 16. American Mathematical Society (Providence, RI), 1970. MR 0264700 Zbl 0207.22601 incollection

Stephen Smale	Related
S. S. Chern	Related	Volume

M. F. Atiyah: Vector fields on manifolds. Arbeitsgemeinschaft für Forschung des Landes Nordrhein-Westfalen 200. Westdeutscher Verlag (Cologne), 1970. MR 0263102 Zbl 0193.52303 book

M. F. Atiyah: “Power operations in \( K \)-theory,” Matematika 14 : 2 (1970), pp. 35–65. Russian translation of article in Q. J. Math., Oxf. 17:1 (1966). Zbl 0208.51503 article

M. Atiyah and F. Hirzebruch: “Spin-manifolds and group actions,” pp. 18–28 in Essays on topology and related topics (Mémoires dédiés à Georges de Rham) [Essays on topology and related topics (Memoirs dedicated to Georges de Rham)] (Geneva, 26–28 March 1969). Edited by R. Narasimhan and A. Haefliger. Springer (New York), 1970. MR 278334 Zbl 0193.52401 incollection

F. Hirzebruch	Related	Volume
Georges de Rham	Related
Raghavan Narasimhan	Related
André Haefliger	Related

M. F. Atiyah: “Global theory of elliptic operators,” pp. 21–30 in Functional analysis and related topics (Tokyo, 1969). Edited by S. T. Kuroda. University of Tokyo Press, 1970. MR 0266247 Zbl 0193.43601 incollection

M. F. Atiyah: “Resolution of singularities and division of distributions,” Comm. Pure Appl. Math. 23 : 2 (1970), pp. 145–150. MR 0256156 Zbl 0188.19405 article

In this note I shall show how Hironaka’s theorem [1964] on the resolution of singularities leads very quickly to a new proof of the Hörmander–Lojasiewicz theorem [Hörmander 1958; Lojasiewicz 1959] on the division of distributions and hence to the existence of temperate fundamental solutions for constant-coefficient differential operators. Since most of the difficulties in the general theory of partial differential operators arise from the singularities of the characteristic variety, it is quite natural to expect Hironaka’s theorem to be relevant. In fact, this note is primarily intended to draw the attention of analysts to the power of this theorem. It seems likely that its application in the field of partial differential equations may yield many results besides those described here.

M. F. Atiyah, R. Bott, and L. Gårding: “Lacunas for hyperbolic differential operators with constant coefficients, I,” Acta Math. 124 : 1 (July 1970), pp. 109–189. A Russian translation was published in Usp. Mat. Nauk 26:2(158). MR 0470499 Zbl 0191.11203 article

Raoul H. Bott	Related	Volume
Lars Gårding	Related

M. F. Atiyah: “Elliptic operators and singularities of vector fields,” pp. 207–209 in Actes du Congrès International des Mathématiciens (Nice, 1–10 September 1970), vol. 2. Gauthier-Villars (Paris), 1971. MR 0415688 Zbl 0222.58004 incollection

M. F. Atiyah: “Riemann surfaces and spin structures,” Ann. Sci. École Norm. Sup. (4) 4 : 1 (1971), pp. 47–62. MR 0286136 Zbl 0212.56402 article

M. F. Atiyah: “On the work of Serge Novikov,” pp. 11–13 in Actes du Congrès International des Mathématiciens (Nice, 1–10 September 1970). Gauthier-Villars (Paris), 1971. MR 0414280 incollection

M. F. Atiyah and G. B. Segal: “Exponential isomorphisms for \( \lambda \)-rings,” Quart. J. Math. Oxford Ser. (2) 22 : 3 (1971), pp. 371–378. MR 0291250 Zbl 0226.13008 article

M. F. Atiyah and I. M. Singer: “The index of elliptic operators, V,” Ann. Math. (2) 93 : 1 (January 1971), pp. 139–149. A Russian translation was published in Uspehi Mat. Nauk 27:4(166) (1972). MR 279834 article

M. F. Atiyah and I. M. Singer: “The index of elliptic operators, IV,” Ann. Math. (2) 93 : 1 (January 1971), pp. 119–138. A Russian translation was published in Uspehi Mat. Nauk 27:4(166) (1972). MR 279833 Zbl 0212.28603 article

M. F. At’ja, R. Bott, and L. Gording: “Lacunas for hyperbolic differential operators with constant coefficients, I,” Usp. Mat. Nauk 26 : 2(158) (1971), pp. 25–100. Russian translation of an article in Acta Math. 124:1 (1970). MR 0606062 Zbl 0208.13201 article

Raoul H. Bott	Related	Volume
Lars Gårding	Related

M. F. Atiyah and I. G. Macdonald: Vvedenie v kommutativnuyu algebru [Introduction to commutative algebra]. Mir (Moscow), 1972. Russian translation of Introduction to commutative algebra (1969). MR 0349645 Zbl 0238.13001 book

M. F. At’ja and I. M. Zinger: “The index of elliptic operators, IV,” Uspehi Mat. Nauk 27 : 4(166) (1972), pp. 161–178. Russian translation of an article published in Ann. Math. (2) 93:1 (1971). MR 385933 Zbl 0237.58017 article

M. F. Atiyah and I. M. Singer: “The index of elliptic operators, V,” Usp. Mat. Nauk 27 : 4(166) (1972). Russian translation of an article published in Ann. Math. (2) 93:1 (1971). Zbl 0237.58018 article

M. F. Atiyah and J. L. Dupont: “Vector fields with finite singularities,” Acta Math. 128 : 1 (1972), pp. 1–40. MR 0451256 Zbl 0233.57010 article

In this paper we give some generalizations of the famous theorem of H. Hopf which states that the number of singularities of a tangent vector field on a compact smooth manifold is equal to the Euler characteristic. Instead of a single vector field we consider \( r \) vector fields \( u_1,\dots,u_r \) and we are interested in their “singularities”, that is, the set \( \Sigma \) of points on the manifold at which they become linearly dependent. In general \( \Sigma \) will have dimension \( r-1 \), it is a cycle and its homology class is the \( (n-r+1) \)-th Stiefel–Whitney class of the manifold. This is the standard primary obstruction theory and it provides one way of generalizing the classical Hopf Theorem. However, this theory says nothing about \( \Sigma \) if \( \dim\Sigma < r-1 \). In this paper following E. Thomas [1967] we shall generalize the Hopf theorem by considering the other extreme case in which \( \Sigma \) is finite, so that \( \dim\Sigma = 0 \).

M. F. Atiyah, V. K. Patodi, and I. M. Singer: “Spectral asymmetry and Riemannian geometry,” Bull. London Math. Soc. 5 (July 1973), pp. 229–234. MR 331443 Zbl 0268.58010 article

Isadore M. Singer	Related	Volume
Vijay Kumar Patodi	Related

M. Atiyah, R. Bott, and V. K. Patodi: “On the heat equation and the index theorem,” Matematika, Moskva 17 : 6 (1973), pp. 3–48. Russian translation of an article in Invent. Math. 19:4 (1973). Zbl 0364.58016 article

Raoul H. Bott	Related	Volume
Vijay Kumar Patodi	Related

M. Atiyah, R. Bott, and V. K. Patodi: “On the heat equation and the index theorem,” Invent. Math. 19 : 4 (1973), pp. 279–330. Dedicated to Sir William Hodge on his 70th birthday. Errata were published in Invent. Math. 28:3 (1975). A Russian translation was published in Matematika 17:6 (1973). MR 0650828 Zbl 0257.58008 article

Raoul H. Bott	Related	Volume
William V. D. Hodge	Related
Vijay Kumar Patodi	Related

M. F. Atiyah, R. Bott, and L. Gårding: “Lacunas for hyperbolic differential operators with constant coefficients, II,” Acta Math. 131 : 1 (December 1973), pp. 145–206. A Russian translation was published in Usp. Mat. Nauk 39:3(237). MR 0470500 Zbl 0266.35045 article

Raoul H. Bott	Related	Volume
Lars Gårding	Related

M. F. Atiyah: Elliptic operators and compact groups. Lecture Notes in Mathematics 401. Springer (Berlin), 1974. MR 0482866 Zbl 0297.58009 book

M. F. Atiyah: “How research is carried out,” Bull. IMA 10 (1974), pp. 232–234. Republished in Atiyah’s Collected works, vol. 1. article

M. F. Atiyah and L. Smith: “Compact Lie groups and the stable homotopy of spheres,” Topology 13 : 2 (1974), pp. 135–142. MR 0343269 Zbl 0282.55008 article

M. F. Atiyah: “Invariant theory and Riemannian geometry,” pp. 19–24 in Prospects in mathematics (Kyoto University, 18–23 April 1973). Kyoto University, 1974. MR 0482867 Zbl 0335.58014 incollection

M. F. Atiyah, V. K. Patodi, and I. M. Singer: “Spectral asymmetry and Riemannian geometry, II,” Math. Proc. Cambridge Philos. Soc. 78 : 3 (November 1975), pp. 405–432. MR 397798 Zbl 0314.58016 article

Isadore M. Singer	Related	Volume
Vijay Kumar Patodi	Related

M. F. Atiyah: “The heat equation in Riemannian geometry (after Patodi, Gilkey, etc.),” pp. 1–11 in Séminaire Bourbaki 1973/1974. Lecture Notes in Mathematics 431. Springer (Berlin), 1975. Exposé no. 436. MR 0431286 incollection

Vijay Kumar Patodi	Related
Peter Belden Gilkey	Related

M. F. Atiyah, V. K. Patodi, and I. M. Singer: “Spectral asymmetry and Riemannian geometry, I,” Math. Proc. Cambridge Philos. Soc. 77 (1975), pp. 43–69. MR 397797 Zbl 0297.58008 article

Isadore M. Singer	Related	Volume
Vijay Kumar Patodi	Related

M. F. Atiyah: “Eigenvalues and Riemannian geometry,” pp. 5–9 in Manifolds (Tokyo, 1973). Edited by A. Hattori. University of Tokyo Press, 1975. MR 0372928 Zbl 0319.53030 incollection

M. F. Atiyah: “Classical groups and classical differential operators on manifolds,” pp. 5–48 in Differential operators on manifolds (Varenna, Italy, 24 August–2 September 1975). Edited by E. Vesentini. Cremonese (Rome), 1975. MR 0650830 incollection

M. Atiyah, R. Bott, and V. K. Patodi: “Errata to: ‘On the heat equation and the index theorem’,” Invent. Math. 28 : 3 (1975), pp. 277–280. Errata for article in Invent. Math. 19:4 (1973). MR 0650829 Zbl 0301.58018 article

Raoul H. Bott	Related	Volume
Vijay Kumar Patodi	Related

M. F. Atiyah and E. Rees: “Vector bundles on projective 3-space,” Invent. Math. 35 : 1 (1976), pp. 131–153. MR 0419852 article

On a compact complex manifold \( X \) it is an interesting problem to compare the continuous and holomorphic vector bundles. The case of line-bundles is classical and is well understood in the framework of sheaf theory. On the other hand for bundles \( E \) with \( \dim E \geq \dim X \) we are in the stable topological range and one can use \( K \)-theory. Much is known in this direction, for example the topological and holomorphic \( K \)-groups of all complex projective spaces are isomorphic.

This paper deals with what is perhaps the simplest case not covered by the methods indicated above. We shall consider 2-dimensional complex vector bundles over the 3-dimensional complex projective space \( P_3 \). Our aim is to prove

Every continuous 2-dimensional vector bundle over \( P_3 \) admits a holomorphic structure.

M. F. Atiyah, V. K. Patodi, and I. M. Singer: “Spectral asymmetry and Riemannian geometry, III,” Math. Proc. Cambridge Philos. Soc. 79 : 1 (1976), pp. 71–99. MR 397799 Zbl 0325.58015 article

In Parts I and II of this paper [1975a; 1975b] we studied the ‘spectral asymmetry’ of certain elliptic self-adjoint operators arising in Riemannian geometry. More precisely, for any elliptic self-adjoint operator \( A \) on a compact manifold we defined \[ \eta_A(s) = \sum_{\lambda\neq 0}\operatorname{sign} \lambda \,|\lambda|^{-s}, \] where \( \lambda \) runs over the eigenvalues of \( A \). For the particular operators of interest in Riemannian geometry we showed that \( \eta_A(s) \) had an analytic continuation to the whole complex \( s \)-plane, with simple poles, and that \( s = 0 \) was not a pole. The real number \( \eta_A(0) \), which is a measure of ‘spectral asymmetry’, was studied in detail particularly in relation to representations of the fundamental group.

Isadore M. Singer	Related	Volume
Vijay Kumar Patodi	Related

M. F. Atiyah: “Singularities of functions,” Bull. Inst. Math. Appl. 12 : 7 (1976), pp. 203–206. Paper presented at the Symposium on Excitement in Mathematics. MR 0590054 article

M. F. Atiyah: “Elliptic operators, discrete groups and von Neumann algebras,” pp. 43–72 in Colloque “Analyse et Topologie” en l’honneur de Henri Cartan (Orsay, 1974). Astérisque 32–33. Société Mathématique de France (Paris), 1976. MR 0420729 Zbl 0323.58015 incollection

M. F. Atiyah: “Bakerian Lecture, 1975: Global geometry,” Proc. Roy. Soc. London Ser. A 347 : 1650 (1976), pp. 291–299. Republished in Amer. Math. Mon. 111:8 (2004). MR 0462821 article

M. F. Atiyah: “Trends in pure mathematics,” pp. 71–74 in Proceedings of the Third International Congress on Mathematical Education (Karlsruhe, 16–21 August 1976). Edited by H. Athen and H. Kunle. Universität Karlsruhe, 1977. Reprinted in Atiyah’s Collected works, vol. 1. incollection

Heinz Kunle	Related
Hermann Athen	Related

M. F. Atiyah and R. S. Ward: “Instantons and algebraic geometry,” Comm. Math. Phys. 55 : 2 (1977), pp. 117–124. MR 0494098 Zbl 0362.14004 article

M. F. Atiyah, N. J. Hitchin, and I. M. Singer: “Deformations of instantons,” Proc. Nat. Acad. Sci. U.S.A. 74 : 7 (July 1977), pp. 2662–2663. MR 458424 Zbl 0356.58011 article

Isadore M. Singer	Related	Volume
Nigel J. Hitchin	Related

M. Atiyah and W. Schmid: “A geometric construction of the discrete series for semisimple Lie groups,” Invent. Math. 42 : 1 (1977), pp. 1–62. Republished in Harmonic analysis and representations of semisimple Lie groups (1980). Errata published in Invent. Math. 54:2 (1979). MR 0463358 Zbl 0373.22001 article

M. F. Atiyah: “William Vallance Douglas Hodge,” Bull. London Math. Soc. 9 : 1 (1977), pp. 99–118. Republished with modifications from Biog. Mem. Fellows Roy. Soc. Lond. 22 (1976). MR 0427007 Zbl 0343.01010 article

M. F. Atiyah: “A survey of \( K \)-theory,” pp. 1–9 in \( K \)-theory and operator algebras (University of Georgia, Athens, GA, 21–25 April 1975). Edited by B. B. Morrel and I. M. Singer. Springer (Berlin), 1977. MR 0474299 Zbl 0345.55005 incollection

Isadore M. Singer	Related	Volume
Bernard B. Morrel	Related

M. F. Atiyah: “Geometry of Yang–Mills fields,” pp. 216–221 in Mathematical problems in theoretical physics (Rome, 6–15 June 1977). Edited by G. Dell’Antonio. Lecture Notes in Physics 80. Springer (Berlin), 1978. See also Geometry of Yang–Mills fields (1979). MR 518436 incollection

M. F. Atiyah, N. J. Hitchin, and I. M. Singer: “Self-duality in four-dimensional Riemannian geometry,” Proc. Roy. Soc. London Ser. A 362 : 1711 (September 1978), pp. 425–461. MR 506229 Zbl 0389.53011 article

Isadore M. Singer	Related	Volume
Nigel J. Hitchin	Related

M. F. Atiyah: “Geometry and physics,” Sûgaku 30 : 2 (1978), pp. 128–131. Special issue on the 100th anniversary of the Mathematical Society of Japan. MR 509876 article

M. F. Atiyah: “The unity of mathematics,” Bull. London Math. Soc. 10 : 1 (1978), pp. 69–76. Bulgarian translation published in Fiz.-Mat. Spis. 22(55):1 (1979). MR 0476223 Zbl 0376.00001 article

M. F. Atiyah, N. J. Hitchin, V. G. Drinfel’d, and Yu. I. Manin: “Construction of instantons,” Phys. Lett. A 65 : 3 (1978), pp. 185–187. MR 598562 article

Vladimir Gershonovich Drinfel’d	Related
Nigel J. Hitchin	Related
Yuri Ivanovich Manin	Related

M. F. Atiyah and J. D. S. Jones: “Topological aspects of Yang–Mills theory,” Comm. Math. Phys. 61 : 2 (1978), pp. 97–118. MR 503187 Zbl 0387.55009 article

M. Atiyah and W. Schmid: “Erratum: ‘A geometric construction of the discrete series for semisimple Lie groups’,” Invent. Math. 54 : 2 (1979), pp. 189–192. Erratum for article in Invent. Math. 42:1 (1977). See also Harmonic analysis and representations of semisimple Lie groups (1980). MR 550183 article

In the above paper [Atiyah and Schmid 1977] a key role is played by a result of Borel [1963], concerning discrete subgroups \( \Gamma \) of semisimple Lie groups \( G \). He proves that if \( G \) is linear, one can find a torsion-free \( \Gamma \) with \( \Gamma\backslash G \) compact. Unfortunately we applied this result in [Atiyah and Schmid 1977] even for non-linear \( G \), in which case the existence of such \( \Gamma \) is seriously in doubt, as pointed out to us by P. Deligne and J.-P. Serre. The difficulty is that a torsion-free subgroup of the adjoint group lifts to a cocompact subgroup \( \Gamma \subset G \) which contains the (finite) center \( Z \) of \( G \), and there may be an obstruction to removing this torsion subgroup. As it stands, [Atiyah and Schmid 1977] is correct only for linear \( G \), and we shall now indicate how to extend the proof to cover all \( G \).

M. F. Atiyah: “Utviklingslinjer innen ren matematikk” [Lines of development in pure mathematics], Normat 1 (1979), pp. 10–20, 48. In Norwegian. MR 527874 article

M. F. Atiyah: “The Harish-Chandra character,” pp. 176–181 in Representation theory of Lie groups (SRC/LMS Research Symposium, Oxford, 28 June–15 July 1977). Edited by G. L. Luke. London Mathematical Society Lecture Note Series 34. 1979. Zbl 0426.22014 incollection

M. F. Atiyah: “The unity of mathematics,” Fiz.-Mat. Spis. 22(55) : 1 (1979), pp. 11–18. Bulgarian translation of article in Bull. London Math. Soc 10:1 (1979). MR 543366 Zbl 0464.00034 article

M. F. Atiyah: Geometry of Yang–Mills fields. Lezioni Fermiane. Accademia Nazionale dei Lincei & Scuola Normale Superiore Pisa (Pisa), 1979. Reprinted in Atiyah’s Collected works, vol. 5 (1988). See also article in Mathematical problems in theoretical physics (1978). MR 554924 Zbl 0435.58001 book

M. F. Atiyah: “Real and complex geometry in four dimensions,” pp. 1–10 in The Chern Symposium 1979 (Berkeley, CA, June 1979). Edited by W. Y. Hsiang. Springer (New York), 1980. MR 609554 Zbl 0454.53043 incollection

Some fifty years ago Einstein’s theory of general relativity provided a great stimulus for different geometry, but after a period of fruitful interaction the interests of geometers and physicists diverged. Within the past few years there has been a resurgence of geometrical ideas in physics, arising partly from the popularity of gauge theories in elementary-particle physics and partly from the work of Hawking and Penrose on black holes. A characteristic feature in both cases has been the significance of global or topological properties, and it is this feature which has particularly attracted mathematicians.

The ideas and problems arising in this way have had a considerable impact on geometry, especially the differential geometry of four dimensions. Through the remarkable twistor theory of Penrose [1977] there is also an intimate connection with complex analytic geometry. In this lecture I shall attempt to survey these mathematical developments with especial emphasis on Penrose’s ideas and their application.

S. S. Chern	Related	Volume
Wu-yi Hsiang	Related

M. F. Atiyah and R. Bott: “Yang–Mills and bundles over algebraic curves,” pp. 11–20 in Geometry and analysis: Papers dedicated to the memory of V. K. Patodi. Indian Academy of Sciences (Bangalore), 1980. Republished in Proc. Indian Acad. Sci. Math. Sci. 90:1 (1981). MR 592249 Zbl 0482.14007 incollection

Raoul H. Bott	Related	Volume
Vijay Kumar Patodi	Related

M. Atiyah and W. Schmid: “A geometric construction of the discrete series for semisimple Lie groups,” pp. 317–378 in Harmonic analysis and representations of semisimple Lie groups (Liège, 5–17 September 1977). Edited by J. A. Wolf, M. Cahen, and M. de Wilde. Mathematical Physics and Applied Mathematics 5. D. Reidel (Dordrecht), 1980. Republished from Invent. Math. 42:1 (1977). Zbl 0466.22012 incollection

Joseph Albert Wolf	Related
Wilfried Schmid	Related
Michel Cahen	Related
M. de Wilde	Related

M. Atiyah and W. Schmid: “Erratum to the article ‘A geometric construction of the discrete series for semisimple Lie groups’” in Harmonic analysis and representations of semisimple Lie groups (Liège, 5–17 September 1977). Edited by J. A. Wolf, M. Cahen, and M. de Wilde. Mathematical Physics and Applied Mathematics 5. D. Reidel (Dordrecht), 1980. Erratum for article in same volume. Republished from Invent. Math. 54:2 (1979). Zbl 0466.22013 incollection

In the above paper [Atiyah and Schmid 1977] a key role is played by a result of Borel [1963], concerning discrete subgroups \( \Gamma \) of semisimple Lie groups \( G \). He proves that if \( G \) is linear, one can find a torsion-free \( \Gamma \) with \( \Gamma\backslash G \) compact. Unfortunately we applied this result in [Atiyah and Schmid 1977] even for non-linear \( G \), in which case the existence of such \( \Gamma \) is seriously in doubt, as pointed out to us by P. Deligne and J.-P. Serre. The difficulty is that a torsion-free subgroup of the adjoint group lifts to a cocompact subgroup \( \Gamma \subset G \) which contains the (finite) center \( Z \) of \( G \), and there may be an obstruction to removing this torsion subgroup. As it stands, [Atiyah and Schmid 1977] is correct only for linear \( G \), and we shall now indicate how to extend the proof to cover all \( G \).

Joseph Albert Wolf	Related
Wilfried Schmid	Related
Michel Cahen	Related
M. de Wilde	Related

M. F. Atiyah: “Geometrical aspects of gauge theories,” pp. 881–885 in Proceedings of the International Congress of Mathematicians (Helsinki, 15–23 August 1978), vol. 2. Edited by O. Lehto. Academia Scientiarum Fennica (Helsinki), 1980. MR 562703 Zbl 0428.58018 incollection

M. F. Atiyah: “Remarks on Morse theory,” pp. 1–5 in Recent developments in gauge theories. Edited by G. ’t Hooft, C. Itzykson, A. Jaffe, H. Lehmann, P. K. Mitter, I. M. Singer, and R. Stora. NATO Advanced Study Institutes Series B. Physics. 59. Plenum Press (New York and London), 1980. Republished in Atiyah’s Collected works, vol. 5. incollection

Morse theory is a topological approach to the Calculus of Variations. It aims to relate the critical points of a functional to the topology of the function space on which the functional is defined. It is only directly applicable in special rather restrictive conditions, notably for problems involving one independent variable. However I will discuss a number of special examples, in some of which the Morse theory really works, and others in which it clearly fails but where nevertheless some aspects appear still to survive. These examples include those of physical interest and it would be interesting to investigate these further. One can make a number of speculations in this direction.

Pronob K. Mitter	Related
Arthur Jaffe	Related
Raymond Stora	Related
Gerardus ’t Hooft	Related
Isadore M. Singer	Related	Volume
Claude Itzykson	Related
Harry Paul Lehmann	Related

M. F. Atiyah: “Green’s functions for self-dual four-manifolds,” pp. 129–158 in Mathematical analysis and applications, Part A, published as Adv. in Math. Suppl. Stud. 7A. Issue edited by L. Nachbin. Academic Press (New York), 1981. MR 634238 Zbl 0484.53036 incollection

The Penrose twistor theory converts problems of four-dimensional Riemannian geometry into complex analytic geometry of three dimensions. It applies whenever the Riemannian curvature is self-dual, in the sense that the Weyl tensor (the conformally invariant part of the Riemannian curvature) is self-dual. For a general exposition of this theory (in the positive definite case) see [Atiyah, Hitchin and Singer 1978]. Now on a Riemannian four-manifold there is a conformally invariant version of the Laplace operator (obtained by adding one-sixth of the scalar curvature of the usual Laplacian) and this has therefore a conformally invariant Green’s function (under appropriate global hypotheses). The purpose of this chapter is to show what corresponds to this Green’s function in the twistor theory. It turns out that we get a very well-known object in complex analytic geometry associated to the diagonal in \( Z \times Z \), where \( Z \) is the twistor space.

M. F. Atiyah and R. Bott: “Yang–Mills and bundles over algebraic curves,” pp. 11–20 in Geometry and analysis: Papers dedicated to the memory of V. K. Patodi, published as Proc. Indian Acad. Sci., Math. Sci. 90 : 1. Indian Academy of Sciences (Bangalore), 1981. MR 653942 Zbl 0499.14005 incollection

Raoul H. Bott	Related	Volume
Vijay Kumar Patodi	Related

M. F. Atiyah: “Convexity and commuting Hamiltonians,” Bull. London Math. Soc. 14 : 1 (1982), pp. 1–15. MR 642416 Zbl 0482.58013 article

M. F. Atiyah: “Gauge theory and algebraic geometry,” pp. 1–20 in Differential geometry and differential equations (Beijing, 1980), vol. 1. Edited by S.-S. Chern and W.-t. Wu. Science Press, Gordon and Breach (Beijing, New York), 1982. Zbl 0544.14012 incollection

Wen-tsün Wu	Related
S. S. Chern	Related	Volume

M. Atiyah: “What is geometry?,” Math. Gaz. 66 : 437 (1982), pp. 179–184. Norwegian translation published in Normat 31:2 (1983). A Czech translation published in Pokroky Mat. Fyz. Astronom. 29:4 (1984). MR 677460 Zbl 0549.51001 article

M. F. Atiyah: “Solutions of classical equations,” pp. 207–219 in Gauge theories: Fundamental interactions and rigorous results (Poiana Braşov, Romania, summer 1981). Edited by P. Dita, V. Georgescu, and R. Purice. Progress in Physics 5. Birkhä user (Boston), 1982. Zbl 0539.70028 incollection

Radu Purice	Related
Petre Dita	Related
Vladimir Georgescu	Related

M. F. Atiyah: “Geometry of monopoles,” pp. 3–20 in Monopoles in quantum field theory (Trieste, December 1981). Edited by N. S. Craigie, P. Goddard, and W. Nahm. World Scientific (Singapore), 1982. MR 766750 incollection

Werner Nahm	Related
Peter Goddard	Related
N. S. Craigie	Related

M. F. Atiyah, H. Donnelly, and I. M. Singer: “Geometry and analysis of Shimizu \( L \)-functions,” Proc. Nat. Acad. Sci. U.S.A. 79 : 18 (September 1982), pp. 5751. MR 674920 Zbl 0503.12007 article

Isadore M. Singer	Related	Volume
Harold Gerard Donnelly	Related

M. F. Atiyah and A. N. Pressley: “Convexity and loop groups,” pp. 33–63 in Arithmetic and geometry: Papers dedicated to I. R. Shafarevich, vol. 2. Edited by M. Artin and I. R. Šafarevič. Progress in Mathematics 36. Birkhäuser (Boston), 1983. MR 717605 Zbl 0529.22013 incollection

A. N. Pressley	Related
Michael Artin	Related
Igor Rostislavovich Shafarevich	Related

M. Atiyah: “Hva er geometri?” [What is geometry?], Normat 31 : 2 (1983), pp. 70–74. Norwegian translation of article in Math. Gaz. 66:437 (1982). See also Pokroky Mat. Fyz. Astronom. 29:4 (1984). MR 702261 article

M. F. Atiyah and R. Bott: “The Yang–Mills equations over Riemann surfaces,” Philos. Trans. R. Soc. Lond., A 308 : 1505 (1983), pp. 523–615. MR 702806 Zbl 0509.14014 article

M. F. Atiyah, H. Donnelly, and I. M. Singer: “Eta invariants, signature defects of cusps, and values of \( L \)-functions,” Ann. Math. (2) 118 : 1 (July 1983), pp. 131–177. An addendum to this article was published in Ann. Math. (2) 119:3 (1984). MR 707164 Zbl 0531.58048 article

Isadore M. Singer	Related	Volume
Harold Gerard Donnelly	Related

M. F. Atiyah: “Angular momentum, convex polyhedra and algebraic geometry,” Proc. Edinburgh Math. Soc. (2) 26 : 2 (1983), pp. 121–133. MR 705256 Zbl 0521.58026 article

The three families of classical groups of linear transformations (complex, orthogonal, symplectic) give rise to the three great branches of differential geometry (complex analytic, Riemannian and symplectic). Complex analytic geometry derives most of its interest from complex algebraic geometry, while symplectic geometry provides the general framework for Hamiltonian mechanics.

These three classical groups “intersect” in the unitary group and the three branches of differential geometry correspondingly “intersect” in Kähler geometry, which includes the study of algebraic varieties in projective space. This is the basic reason why Hodge was successful in applying Riemannian methods to algebraic geometry in his theory of harmonic forms.

M. Atiyah: “Anomalies and index theory,” pp. 313–322 in Supersymmetry and supergravity/nonperturbative QCD (Mahabaleshwar, India, 5–19 January 1984). Edited by P. Roy and V. Singh. Lecture Notes in Mathematics 208. Springer (Berlin), 1984. MR 774599 incollection

Probir Roy	Related
Virendra Singh	Related

M. F. Atiyah: “The moment map in symplectic geometry,” pp. 43–51 in Global Riemannian geometry (Durham University, UK, July 1983). Edited by T. Willmore and N. J. Hitchin. Mathematics and its Applications. Ellis Horwood (Chichester, UK), 1984. MR 757204 Zbl 0615.53023 incollection

Nigel J. Hitchin	Related
Thomas Willmore	Related

M. F. Atiyah and R. Bott: “The moment map and equivariant cohomology,” Topology 23 : 1 (1984), pp. 1–28. MR 721448 Zbl 0521.58025 article

M. F. Atiyah: “Geometry and analysis of the Yang–Mills equations,” pp. 3–8 in Mathematics in the Gulf area (Riyadh, Saudi Arabia, 17–21 October 1982). Edited by Y. Al-Khamees. Arab Bureau of Education for the Gulf States (Riyadh), 1984. Zbl 0595.53072 incollection

M. F. Atiyah and I. M. Singer: “Dirac operators coupled to vector potentials,” Proc. Nat. Acad. Sci. U.S.A. 81 : 8 (April 1984), pp. 2597–2600. MR 742394 Zbl 0547.58033 article

M. F. Atiyah: “Instantons in two and four dimensions,” Comm. Math. Phys. 93 : 4 (1984), pp. 437–451. MR 763752 Zbl 0564.58040 article

M. F. Atiyah, H. Donnelly, and I. M. Singer: “Signature defects of cusps and values of \( L \)-functions: The nonsplit case,” Ann. Math. (2) 119 : 3 (1984), pp. 635–637. Addendum to an article published in Ann. Math. (2) 118:1 (1983). MR 744866 Zbl 0577.58030 article

Isadore M. Singer	Related	Volume
Harold Gerard Donnelly	Related

M. F. At’ya, R. Bott, and L. Gårding: “Lacunas for hyperbolic differential operators with constant coefficients, II,” Usp. Mat. Nauk 39 : 3(237) (1984), pp. 171–224. Russian translation of an article in Acta Math. 131:1 (1973). MR 747794 Zbl 0568.35058 article

Raoul H. Bott	Related	Volume
Lars Gårding	Related

M. F. Atiyah: “The Yang–Mills equations and the structure of 4-manifolds,” pp. 11–17 in Global Riemannian geometry (Durham University, UK, July 1983). Edited by T. Willmore and N. J. Hitchin. Series in Mathematics and its Applications. Ellis Horwood (Chichester, UK), 1984. MR 757200 Zbl 0614.57007 incollection

Nigel J. Hitchin	Related
Thomas Willmore	Related

M. F. Atiyah: “Co je geometrie?” [What is geometry?], Pokroky Mat. Fyz. Astronom. 29 : 4 (1984), pp. 213–217. Czech translation of article in Math. Gaz. 66:437 (1982). See also Normat 31:2 (1983). MR 762935 article

M. F. Atiyah and N. J. Hitchin: “Low energy scattering of nonabelian monopoles,” Phys. Lett. A 107 : 1 (1985), pp. 21–25. MR 778313 Zbl 1177.53069 article

M. Atiyah: “Eigenvalues of the Dirac operator,” pp. 251–260 in Arbeitstagung Bonn 1984 (Max-Planck-Institut für Mathematik, Bonn, 15–22 June 1984). Edited by F. Hirzebruch, J. Schwermer, and S. Suter. Lecture Notes in Mathematics 1111. Springer (Berlin), 1985. MR 797424 Zbl 0568.53022 incollection

F. Hirzebruch	Related	Volume
Joachim Schwermer	Related
Silke Suter	Related

M. F. Atiyah: “Circular symmetry and stationary-phase approximation,” pp. 43–59 in Colloque en l’honneur de Laurent Schwartz (École Polytechnique, Palaiseau, 30 May–3 June 1983). Astérisque 131. Société mathématique de France (Paris), 1985. MR 816738 Zbl 0578.58039 incollection

M. Atiyah: “Topological aspects of anomalies,” pp. 22–32 in Symposium on anomalies, geometry, topology (University of Chicago, 28–30 March 1985). Edited by W. A. Bardeen and A. R. White. World Scientific (Singapore), 1985. MR 850843 Zbl 0651.58035 incollection

Alan R. White	Related
William A. Bardeen	Related

M. Atiyah: “Commentary on the article of Manin,” pp. 103–109 in Arbeitstagung Bonn 1984 (Max-Planck-Institut für Mathematik, Bonn, 15–22 June 1984). Edited by F. Hirzebruch, J. Schwermer, and S. Suter. Lecture Notes in Mathematics 1111. Springer (Berlin), 1985. The article is Yu. I. Manin, “New dimensions in geometry,” from the same volume. MR 797417 Zbl 0595.53071 incollection

F. Hirzebruch	Related	Volume
Silke Suter	Related
Yuri Ivanovich Manin	Related
Joachim Schwermer	Related

M. Atiyah, N. J. Hitchin, J. T. Stuart, and M. Tabor: “Low-energy scattering of nonabelian magnetic monopoles [and discussion],” Philos. Trans. Roy. Soc. London Ser. A 315 : 1533 (1985), pp. 459–469. MR 836746 article

Michael Tabor	Related
Nigel J. Hitchin	Related
John Trevor Stuart	Related

M. F. Atiyah: “Identifying progress in mathematics,” pp. 24–41 in The identification of progress in learning (Colmar, France, 26–28 March 1983). Edited by T. Hägerstrand. Cambridge University Press, 1985. MR 807796 incollection

M. Atiyah: “Mathematics and the computer revolution,” pp. 43–51 in The influence of computers and informatics on mathematics and its teaching (Strasbourg, March, 1985). Edited by A. G. Howson and J.-P. Kahane. Cambridge University Press, 1986. Reprinted in Atiyah’s Collected works, vol. 1, pp. 327–348.

Albert Geoffrey Howson	Related
Jean-Pierre Kahane	Related

M. F. Atiyah: “Mathematics and the computer revolution,” pp. 43–51 in The influence of computers and informatics on mathematics and its teaching (Strasbourg, March 1985). Edited by A. G. Howson and J.-P. Kahane. Cambridge University Press, 1986. Lecture given at the Locarno Conference “1984: Comincia il futuro” (May 1984). Republished in Atiyah’s Collected works, vol. 1. Italian translation published in Nuova Civiltà della Macchine 2:3 (1984). incollection

Albert Geoffrey Howson	Related
Jean-Pierre Kahane	Related

M. F. Atiyah: “Magnetic monopoles in hyperbolic spaces,” pp. 1–33 in Vector bundles on algebraic varieties (Bombay, 9–16 January 1984). Studies in Mathematics 11. Tata Institute for Fundamental Research (Bombay), 1987. MR 893593 Zbl 0722.53063 incollection

M. Atiyah: “On the work of Simon Donaldson,” pp. 3–6 in Proceedings of the International Congress of Mathematicians (Berkeley, CA, 3–11 August 1986), vol. 2. Edited by A. Gleason. American Mathematical Society (Providence, RI), 1987. MR 934209 Zbl 0666.01010 incollection

Andrew Mattei Gleason	Related	Volume
S. K. Donaldson	Related

M. Atiyah: “The logarithm of the Dedekind \( \eta \)-function,” Math. Ann. 278 : 1–4 (1987), pp. 335–380. MR 909232 Zbl 0648.58035 article

M. Atiyah: “Topology and differential equations,” pp. 45–52 in ICIAM ’87: Proceedings of the first International Conference on Industrial and Applied Mathematics (Paris, 29 June–3 July 1987). Edited by J. McKenna and R. Temam. SIAM (Philadelphia, PA), 1988. MR 976850 Zbl 0687.35001 incollection

In many areas of differential equations topological methods can provide important qualitative information. This particularly applies to complicated problems where the number of variables is large. Examples will be given illustrating the scope and power of topological ideas.

For constant coefficient linear hyperbolic systems the fundamental solution is supported in the forward “light-cone”. For general systems this cone has many compartments and, in some of these (called lacunas) the fundamental solution may vanish. The determination of lacunas is a topological problem (solved originally by Petrovsky) involving the topology of multi-dimensional contour integrals in the complex domain.

For elliptic boundary value problems of Cauchy–Riemann or Dirac type the index, measuring the difference between the number of solutions of the problem and its adjoint, is a topological invariant which can be computed from the geometrical data. This has important applications to models in elementary particle physics.

For a periodic family of self-adjoint elliptic operators the spectral flow, counting the net number of eigenvalues which change sign over a period, is a topological invariant. This can be computed from the geometrical data and can be used to derive sharp bounds on gaps in the spectrum.

In non-linear PDE, “solitons” may have a topological origin, and soliton-interaction can be related to underlying topological features. This is illustrated by models in 2 and 3 spatial dimensions, following original ideas of Skyrme.

James McKenna	Related
Roger Temam	Related

M. F. Atiyah: “Characters of semi-simple Lie groups,” pp. 489–558 in Collected works, vol. 4: Index theory 2. Oxford Science Publications. Clarendon Press and Oxford University Press (Oxford, New York), 1988. Lectures given at Mathematical Institute, Oxford, 1976. incollection

M. Atiyah: “New invariants of 3- and 4-dimensional manifolds,” pp. 285–299 in The mathematical heritage of Hermann Weyl (Durham, NC, 12–16 May 1987). Edited by R. O. Wells, Jr. Proceedings of Symposia in Pure Mathematics 48. American Mathematical Society (Providence, RI), 1988. Russian translation published in Uspekhi Mat. Nauk 45:4(274) (1990). MR 974342 Zbl 0667.57018 incollection

Hermann Weyl	Related
Raymond O'Neil Wells, Jr	Related

M. F. Atiyah: “Geometry and analysis in the nineteen eighties,” pp. 317–326 in Collected works, vol. 2: \( K \)-theory. Oxford Science Publications. Clarendon Press and Oxford University Press (Oxford, New York), 1988. incollection

M. Atiyah: “Topological quantum field theories,” Inst. Hautes Études Sci. Publ. Math. 68 : 1 (January 1988), pp. 175–186. MR 1001453 Zbl 0692.53053 article

M. Atiyah and N. Hitchin: The geometry and dynamics of magnetic monopoles. M. B. Porter Lectures. Princeton University Press, 1988. Russian translation published as Geometriya i dinamika magnitnykh monopolei (1991). MR 934202 Zbl 0671.53001 book

M. F. Atiyah: “Speech on conferment of Feltrinelli Prize,” pp. 309–316 in Collected works, vol. 2: \( K \)-theory. Oxford Science Publications. Clarendon Press and Oxford University Press (Oxford, New York), 1988. Atiyah received the Antonio Feltrinelli Prize from the Accademia Nazionale dei Lincei in 1981. incollection

M. F. Atiyah: “Geometry, topology and physics,” Quart. Journ. Royal Astrophysics Soc. 29 : 3 (September 1988), pp. 287–299. Delivered as the 11th Arthur Milne Lecture at Oxford University. Republished in Atiyah’s Collected works, vol. 6. article

The relation between Geometry and Physics is a very ancient one going back to the earliest times, but in the modern period it is best exemplified by Einstein’s Theory of General Relativity. As is well known, Einstein interpreted gravitational force in terms of the curvature of space-time and Maxwell’s Theory of Electro-Magnetism was subsequently interpreted in a somewhat similar manner. At the presetnt time these geometrical ideas have been extended to incorporate the other fundamental forces of nature, the “weak” and “strong” forces that are encountered at the nuclear level. As a result we are witnessing now a remarkable interaction between mathematicians and physicists which has been extremely fruitful.

M. Atiyah: “The impact of physics on geometry,” pp. 1–9 in Differential geometrical methods in theoretical physics (Como, Italy, 1987). Edited by K. Bleuler and M. Werner. NATO ASI Series C: Mathematical and Physical Sciences 250. Kluwer Academic (Dordrecht), 1988. MR 981370 Zbl 0657.53054 incollection

Konrad Bleuler	Related
M. Werner	Related

M. Atiyah: Collected works, vol. 5: Gauge theories. Oxford Science Publications. The Clarendon Press and Oxford University Press (Oxford, New York), 1988. MR 951896 Zbl 0691.53003 book

M. Atiyah: Collected works, vol. 4: Index theory 2. Oxford Science Publications. The Clarendon Press and Oxford University Press (Oxford and New York), 1988. MR 951895 book

M. Atiyah: Collected works, vol. 3: Index theory 1. Oxford Science Publications. The Clarendon Press and Oxford University Press (Oxford and New York), 1988. MR 951894 Zbl 0724.53001 book

M. Atiyah: Collected works, vol. 2: \( K \)-theory. Oxford Science Publications. The Clarendon Press and Oxford University Press (Oxford and New York), 1988. MR 951893 Zbl 0724.55001 book

M. Atiyah: Collected works, vol. 1: Early papers; general papers. Oxford Science Publications. The Clarendon Press and Oxford University Press (Oxford and New York), 1988. MR 951892 Zbl 0935.01034 book

M. F. Atiyah: “The index of elliptic operators,” pp. 475–498 in Collected works, vol. 3: Index theory 1. Oxford Science Publications. Oxford University Press, 1988. Colloquium Lectures (Dallas, 1973), American Mathematical Society.

M. F. Atiyah: “The frontier between geometry and physics,” Jahresber. Deutsch. Math.-Verein. 91 : 4 (1989), pp. 149–158. Bulgarian translation published in Fiz.-Mat. Spis. 33(66) (1991). MR 1027061 Zbl 0679.53074 article

M. Atiyah and G. Segal: “On equivariant Euler characteristics,” J. Geom. Phys. 6 : 4 (1989), pp. 671–677. MR 1076708 Zbl 0708.19004 article

M. F. Atiyah and N. S. Manton: “Skyrmions from instantons,” Phys. Lett. B 222 : 3–4 (1989), pp. 438–442. MR 1001325 article

M. F. Atiyah: “The geometry and physics of knots,” pp. 1–17 in Miniconference on geometry and physics (Canberra, 20–23 February 1989). Edited by M. N. Barber and M. K. Murray. Proceedings of the Centre for Mathematical Analysis 22. Australian National University (Canberra), 1989. See also 1990 book of the same title. MR 1027859 Zbl 0696.57002 incollection

Michael N. Barber	Related
Michael K. Murray	Related

M. F. Atiyah and D. W. Anderson: \( K \)-theory, 2nd edition. Advanced Book Classics. Addison-Wesley (Redwood City, CA), 1989. Original edition published by W. A. Benjamin in 1967. MR 1043170 book

M. Atiyah: “The Jones–Witten invariants of knots,” pp. 7–16 in Séminaire Bourbaki 1989/90. Astérisque 189–190. Société Mathématique de France (Paris), 1990. Exposé no. 715. MR 1099870 Zbl 0739.57001 incollection

M. Atiyah: “New invariants of 3- and 4-dimensional manifolds,” Uspekhi Mat. Nauk 45 : 4(274) (1990), pp. 3–16, 192. Russian translation of article from The mathematical heritage of Hermann Weyl (1988). MR 1075385 Zbl 0709.57018 article

M. Atiyah: “Quantum field theory and low-dimensional geometry,” pp. 1–13 in Common trends in mathematics and quantum field theories (Kyoto University, Japan, 17–19 May 1990). Edited by T. Eguchi, T. Inami, and T. Miwa. Progress of Theoretical Physics Supplements 102. 1990. MR 1182158 Zbl 0804.53098 incollection

Takeo Inami	Related
T. Miwa	Related
T. Eguchi	Related

M. Atiyah: “On framings of 3-manifolds,” Topology 29 : 1 (1990), pp. 1–7. MR 1046621 Zbl 0716.57011 article

M. Atiyah: “The icosahedron,” Math. Medley 18 : 1 (1990), pp. 1–12. MR 1106158 Zbl 0723.51001 article

M. Atiyah: The geometry and physics of knots. Lezioni Lincee. Cambridge University Press, 1990. These notes arise from lectures presented in Florence under the auspices of the Accademia dei Lincei. Russian translation published as Geometriya i fizika uzlov (1995). See also Miniconference on geometry and physics (1989). MR 1078014 Zbl 0729.57002 book

M. F. Atiyah and L. Jeffrey: “Topological Lagrangians and cohomology,” J. Geom. Phys. 7 : 1 (1990), pp. 119–136. MR 1094734 Zbl 0721.58056 article

M. Atiyah: “Hyper-Kähler manifolds,” pp. 1–13 in Complex geometry and analysis (Pisa, 23–27 May 1988). Edited by V. Villani. Lecture Notes in Mathematics 1422. Springer (Berlin), 1990. MR 1055838 incollection

M. F. Atiyah: “Representations of braid groups,” pp. 115–122 in Geometry of low-dimensional manifolds (Durham University, UK, July 1989), vol. 2: Symplectic manifolds and Jones–Witten theory. Edited by S. K. Donaldson and C. B. Thomas. London Mathematical Society Lecture Note Series 151. Cambridge University Press, 1990. Notes by S. K. Donaldson. MR 1171912 Zbl 0735.57001 incollection

Charles B. Thomas	Related
S. K. Donaldson	Related

M. Atiyah: “Magnetic monopoles and the Yang–Baxter equations,” pp. 2761–2774 in Topological methods in quantum field theory (Trieste, 11–15 June 1990), published as Internat. J. Modern Phys. A 6 : 16 (1991). MR 1117746 Zbl 0757.53038 incollection

M. Atiyah: “The European Mathematical Society,” pp. 1–5 in Miscellanea mathematica. Edited by P. J. Hilton, F. Hirzebruch, and R. Remmert. Springer (Berlin), 1991. MR 1131114 incollection

F. Hirzebruch	Related	Volume
Reinhold Remmert	Related
Peter John Hilton	Related

M. Atiyah and N. Khitchin: Geometriya i dinamika magnitnykh monopolei [The geometry and dynamics of magnetic monopoles]. Mir (Moscow), 1991. Russian translation of The geometry and dynamics of magnetic monopoles (1988). MR 1137269 Zbl 0754.53003 book

M. Atiyah: “On the work of Edward Witten,” pp. 31–35 in Proceedings of the International Congress of Mathematicians (Kyoto, 21–29 August 1990), vol. 1. Edited by I. Satake. Mathematical Society of Japan (Tokyo), 1991. MR 1159202 Zbl 0742.01013 incollection

Ichirō Satake	Related
Edward Witten	Related

M. F. Atiyah: “The frontier between geometry and physics,” Fiz.-Mat. Spis. 33(66) : 1–2 (1991), pp. 41–49. Bulgarian translation from Jahresber. Deutsch. Math.-Verein. 91:4 (1989). MR 1142704 article

M. F. Atiyah: “A new knot invariant, I,” pp. 1–9 in Topological quantum field theories and geometry of loop spaces (Budapest, 25 June–1 July 1989). Edited by L. Fehér, A. Stipsicz, and J. Szenthe. World Scientific (River Edge, NJ), 1992. MR 1260745 incollection

János Szenthe	Related
András Stipsicz	Related
László Fehér	Related

M. F. Atiyah: “A new knot invariant, II: Topological quantum field theories and the Jones polynomial,” pp. 10–15 in Topological quantum field theories and geometry of loop spaces (Budapest, 25 June–1 July 1989). Edited by L. Fehér, A. Stipsicz, and J. Szenthe. World Scientific (River Edge, NJ), 1992. MR 1260746 incollection

János Szenthe	Related
András Stipsicz	Related
László Fehér	Related

M. F. Atiyah: The mysteries of space, 1992. Selected Lectures in Mathematics. One 60-minute video tape. American Mathematical Society (Providence, RI). The 1991 Josiah Willard Gibbs Lecture presented in San Francisco, CA, January 1991. MR 1192780 Zbl 0803.53001 misc

M. F. Atiyah: “Mathematics and the physical world,” pp. 186–210 in Symbolien metsässä [In the forest of symbols]. Edited by O. Pekonen. Art House Osakeyhtiö (Helsinki), 1992. MR 1444861 incollection

M. Atiyah: “Address of the president, Sir Michael Atiyah, given at the anniversary meeting on 29 November 1991,” Notes and Records Roy. Soc. London 46 : 1 (1992), pp. 155–169. MR 1148537 Zbl 0978.01519 article

M. F. Atiyah: “Mathematics: Queen and servant of the sciences,” Proc. Am. Phil. Soc. 137 : 4 (December 1993), pp. 527–531. 250th anniversary issue. Republished in Asian J. Math. 3:1 (1999) and The founders of index theory (2009). article

M. F. Atiyah and N. S. Manton: “Geometry and kinematics of two skyrmions,” Comm. Math. Phys. 153 : 2 (1993), pp. 391–422. MR 1218307 Zbl 0778.53051 article

In Skyrme’s soliton model of baryons, a single Skyrmion has six degrees of freedom, so it is expected that two-Skyrmion dynamics at modest energies can be modelled by motion on a 12-dimensional space of Skyrme fields. A candidate for this space is generated by the gradient flow of the potential energy function, descending from the unstable, baryon number two, hedgehog solutions of the Skyrme field equation. An apparently very similar space is obtained by restricting the gradient flow to the Skyrme fields derived from \( \mathit{SU}(2) \) Yang–Mills instantons of charge two. On both of these spaces, one may quotient out by the group of translations and isospin rotations. Hartshome’s geometrical description of charge two instantons leads us to a conjecture for the global structure of the 6-dimensional quotient space. The conjectured structure is that of complex projective 3-space, with complex conjugate points on one projective plane identified and the real points in this plane removed.

M. Atiyah: “Mathematics as a basic science,” Current Sci. 65 : 12 (1993), pp. 912–917. MR 1263343 article

M. F. Atiyah: “American Philosophical Society dinner address, 30 April 1993,” Proc. Am. Phil. Soc. 137 : 4 (1993), pp. 704–707. Republished in Atiyah’s Collected works, vol. 6. article

M. F. Atiyah: “Anniversary address by the President,” Royal Society News supplement (December 1994), pp. i–iv, 535–540. Republished in Atiyah’s Collected works, vol. 6. See also version in Notes and Records Roy. Soc. London 49:1 (1995). article

M. Atiyah, A. Borel, G. J. Chaitin, D. Friedan, J. Glimm, J. J. Gray, M. W. Hirsch, S. MacLane, B. B. Mandelbrot, D. Ruelle, A. Schwarz, K. Uhlenbeck, R. Thom, E. Witten, and C. Zeeman: “Responses to ‘Theoretical mathematics: Toward a cultural synthesis of mathematics and theoretical physics’, by A. Jaffe and F. Quinn,” Bull. Am. Math. Soc., New Ser. 30 : 2 (April 1994), pp. 178–207. Zbl 0803.01014 ArXiv math/9404229 article

Benoit B. Mandelbrot	Related
Edward Witten	Related
David Ruelle	Related
René Thom	Related
Daniel Harry Friedan	Related
E. Christopher Zeeman	Related
Arthur Jaffe	Related
Karen Uhlenbeck	Related	Volume
Armand Borel	Related
James G. Glimm	Related
Jeremy J. Gray	Related
Albert S. Schwarz	Related	Volume
Frank Stringfellow Quinn, III	Related
S. Mac Lane	Related	Volume
Gregory John Chaitin	Related
Morris W. Hirsch	Related

@article {key0803.01014z,
	AUTHOR = {Atiyah, Michael and Borel, Armand and
		 Chaitin, G. J. and Friedan, Daniel and
		 Glimm, James and Gray, Jeremy J. and
		 Hirsch, Morris W. and MacLane, Saunders
		 and Mandelbrot, Benoit B. and Ruelle,
		 David and Schwarz, Albert and Uhlenbeck,
		 Karen and Thom, Ren\'e and Witten, Edward
		 and Zeeman, Christopher},
	TITLE = {Responses to ``Theoretical mathematics:
		 {T}oward a cultural synthesis of mathematics
		 and theoretical physics'', by {A}.~{J}affe
		 and {F}.~{Q}uinn},
	JOURNAL = {Bull. Am. Math. Soc., New Ser.},
	FJOURNAL = {Bulletin of the American Mathematical
		 Society. New Series},
	VOLUME = {30},
	NUMBER = {2},
	MONTH = {April},
	YEAR = {1994},
	PAGES = {178--207},
	DOI = {10.1090/S0273-0979-1994-00503-8},
	NOTE = {ArXiv:math/9404229. Zbl:0803.01014.},
	ISSN = {0273-0979},
}

M. Atiyah: “Contribution to the collected works of Raoul Bott,” pp. xxix–xxx in Raoul Bott: Collected papers, vol. 2: Differential operators. Edited by R. D. MacPherson. Contemporary Mathematicians. Birkhäuser (Boston), 1994. MR 1290365 incollection

Raoul H. Bott	Related	Volume
Robert Duncan MacPherson	Related

M. Atiyah: Geometriya i fizika uzlov [The geometry and physics of knots]. Mir (Moscow), 1995. Russian translation of The geometry and physics of knots (1990). See also Miniconference on geometry and physics (1989). MR 1397266 book

M. Atiyah: “Quantum theory and geometry,” J. Math. Phys. 36 : 11 (1995), pp. 6069–6072. MR 1355898 Zbl 0860.58001 article

M. Atiyah: “Quantum physics and the topology of knots,” pp. 5–14 in XI-th International Congress of Mathematical Physics (Paris, 18–23 July 1994). Edited by D. Iagolnitzer. International Press (Cambridge, MA), 1995. MR 1370664 Zbl 1052.57500 incollection

M. F. Atiyah: “Hyperbolic differential equations and algebraic geometry (after Petrowsky),” pp. 87–99 in Séminaire Bourbaki 10: Années 1966/67–1967/68. Société Mathématique de France (Paris), 1995. Exposé no. 319. Republication of 1968 original. MR 1610456 incollection

M. F. Atiyah: “Book review: ‘Conversations on mind, matter and mathematics’,” Times High. Educ. Suppl. (29 September 1995). Book by Jean-Pierre Changeux and Alain Connes (Princeton University Press, 1995). Republished in Atiyah’s Collected works, vol. 6. article

Jean-Pierre Changeux	Related
Alain Connes	Related

M. Atiyah: “Address of the President, Sir Michael Atiyah, O.M., given at the anniversary meeting on 30 November 1994,” Notes and Records Roy. Soc. London 49 : 1 (1995), pp. 141–151. See also version in Royal Society News (December 1994). MR 1325201 Zbl 0978.01520 article

J. Lebowitz, M. Atiyah, E. Brézin, A. Connes, J. Fröhlich, D. Gross, A. Jaffe, L. Kadanoff, and D. Ruelle: “Round table: Physics and mathematics,” pp. 691–705 in XI-th International Congress of Mathematical Physics (Paris, 18–23 July 1994). Edited by D. Iagolnitzer. International Press (Cambridge, MA), 1995. MR 1370725 Zbl 1052.00520 incollection

Alain Connes	Related
David Ruelle	Related
Josef Fröhlich	Related
Arthur Jaffe	Related
L. Kadanoff	Related
Édouard Brézin	Related
D. Gross	Related
Joel Louis Lebowitz	Related
Daniel Iagolnitzer	Related

M. Atiyah: “Reflections on geometry and physics,” pp. 1–6 in Surveys in differential geometry: Proceedings of the conference on geometry and topology (Cambridge, MA, 23–25 April 1993). Edited by C.-C. Hsiung and S.-T. Yau. Surveys in differential geometry (Journal of Differential Geometry supplements) 2. International Press (Cambridge, MA), 1995. MR 1375254 Zbl 0867.57031 incollection

Chuan-Chih Hsiung	Related
Shing-Tung Yau	Related

M. Atiyah: “The geometry of classical particles,” pp. 1–15 in Proceedings of the conference on geometry and topology held at Harvard University (Cambridge, MA, 23–25 April 1993). Edited by C.-C. Hsiung and S.-T. Yau. Surveys in Differential Geometry 2. International Press (Somerville, MA), 1995. MR 1919420 Zbl 1050.55502 incollection

Chuan-Chih Hsiung	Related
Shing-Tung Yau	Related

M. Atiyah: “Floer homology,” pp. 105–108 in The Floer memorial volume. Edited by H. Hofer, C. H. Taubes, A. Weinstein, and E. Zehnder. Progress in Mathematics 133. Birkhäuser (Basel), 1995. MR 1362825 Zbl 0837.58011 incollection

Helmut Hofer	Related
Eduard J. Zehnder	Related
Clifford Henry Taubes	Related
Andreas Floer	Related
Alan Weinstein	Related

M. Atiyah: “Reflections on geometry and physics,” pp. 423–428 in Geometry, topology, & physics: Lectures of a conference in honor of Raoul Bott’s 70th birthday (Harvard University, April 1993). Edited by S.-T. Yau. Conference proceedings and lecture notes in geometry and topology 4. International Press (Cambridge, MA), 1995. MR 1358626 incollection

Raoul H. Bott	Related	Volume
Shing-Tung Yau	Related

M. Atiyah: “Geometry and physics,” Math. Gaz. 80 : 487 (1996), pp. 78–82. Zbl 0864.57003 article

M. Atiyah: “Friedrich Hirzebruch: An appreciation,” pp. 1–5 in Proceedings of the Hirzebruch 65 conference on algebraic geometry (Ramat Gan, Israel, 2–7 May 1993). Edited by M. Teicher. Israel Mathematical Conference Proceedings 9. Bar-Ilan University (Ramat Gan, Israel), 1996. Zbl 0834.01012 incollection

F. Hirzebruch	Related	Volume
Mina Teicher	Related

M. Atiyah, A. Borel, D. Friedan, J. J. Gray, and M. W. Hirsch: “Proofs, physics and things around them,” Pokroky Mat. Fyz. Astron. 41 : 2 (1996), pp. 73–81. In Czech. Zbl 0863.01010 article

Daniel Harry Friedan	Related
Armand Borel	Related
Jeremy J. Gray	Related
Morris W. Hirsch	Related

M. F. Atiyah: “The index of elliptic operators,” pp. 115–127 in Fields Medallists’ lectures. Edited by M. F. Atiyah and D. Iagolnitzer. World Scientific Series in 20th Century Mathematics 5. World Scientific (River Edge, NJ), 1997. Republication of notes distributed at AMS conference (1973). MR 1622942 incollection

M. F. Atiyah: “The work of Serge Novikov,” pp. 195–197 in Fields Medallists’ lectures. Edited by M. F. Atiyah and D. Iagolnitzer. World Scientific Series in 20th Century Mathematics 5. World Scientific (River Edge, NJ), 1997. MR 1622933 incollection

Daniel Iagolnitzer	Related
Sergey Petrovich Novikov	Related

M. F. Atiyah: “The work of Edward Witten,” pp. 514–518 in Fields Medallists’ lectures. Edited by M. F. Atiyah and D. Iagolnitzer. World Scientific Series in 20th Century Mathematics 5. World Scientific (River Edge, NJ), 1997. MR 1622921 incollection

Daniel Iagolnitzer	Related
Edward Witten	Related

M. F. Atiyah: “The work of Simon Donaldson,” pp. 377–380 in Fields Medallists’ lectures. Edited by M. F. Atiyah and D. Iagolnitzer. World Scientific Series in 20th Century Mathematics 5. World Scientific (River Edge, NJ), 1997. MR 1622912 incollection

S. K. Donaldson	Related
Daniel Iagolnitzer	Related

M. F. Atiyah: “An introduction to topological quantum field theories,” pp. 1–7 in Proceedings of Gökova geometry-topology conference 1996 (Gökova, Turkey, 27–31 May 1996), published as Turkish J. Math. 21 : 1. Issue edited by S. Akbulut, T. Önder, and R. J. Stern. Scientific and Technological Research Council of Turkey (Ankara), 1997. Republished in Atiyah’s Collected works, vol. 6. MR 1456155 Zbl 0890.57019 incollection

Turgut Önder	Related
Selman Akbulut	Related
Ronald John Stern	Related

M. Atiyah: “Topology and quantum physics: From knots to quarks,” Bol. Acad. Cienc. Fis. Mat. Nat. 56 : 181–186 (1997), pp. 15–20. Zbl 1072.81504 article

M. Atiyah: “Geometry and physics: Where are we going?,” pp. 1–7 in Geometry and physics (Aarhus, Denmark, 1995). Edited by J. E. Andersen, J. Dupont, H. Pederson, and A. Swann. Lecture Notes in Pure and Applied Mathematics 184. Marcel Dekker (New York), 1997. MR 1423153 Zbl 0865.53063 incollection

Johan L. Dupont	Related
Jørgen Ellegaard Andersen	Related
Henrik Pederson	Related
Andrew Swann	Related

M. F. Atiyah: “The Dirac equation and geometry,” pp. 108–124 in A. Pais, M. Jacob, D. I. Olive, and M. F. Atiyah: Paul Dirac: The man and his work. Edited by P. Goddard. Cambridge University Press, 1998. MR 1606723 incollection

Paul Dirac	Related
Maurice Jacob	Related
David Ian Olive	Related
Abraham Pais	Related
Peter Goddard	Related

M. Atiyah: “Duality and quantum field theory,” pp. 1–7 in Topics in symplectic 4-manifolds (Irvine, CA, 28–30 March 1996). Edited by R. J. Stern. First International Press Lecture Series 1. International Press (Cambridge, MA), 1998. MR 1635693 Zbl 0927.57031 incollection

A. Pais, M. Jacob, D. I. Olive, and M. F. Atiyah: Paul Dirac: The man and his work. Edited by P. Goddard. Cambridge University Press, 1998. MR 1611403 Zbl 0917.01032 book

Paul Dirac	Related
Maurice Jacob	Related
David Ian Olive	Related
Abraham Pais	Related
Peter Goddard	Related

M. Atiyah: “Roger Penrose: A personal appreciation,” pp. 3–7 in The geometric universe (Oxford, June 1996). Edited by S. A. Huggett, L. J. Mason, K. P. Tod, S. Tsou, and N. M. J. Woodhouse. Oxford University Press (New York), 1998. MR 1634501 Zbl 0904.01010 incollection

Nicholas Michael John Woodhouse	Related
Lionel J. Mason	Related
S. A. Huggett	Related
Kenneth Paul Tod	Related
Sheung Tsou	Related
Roger Penrose	Related

M. F. Atiyah: “Obituary: John Arthur Todd,” Bull. London Math. Soc. 30 : 3 (1998), pp. 305–316. MR 1608134 Zbl 0927.01042 article

M. Atiyah: “Mathematics and the real world,” pp. 807–812 in Current and future challenges in the applications of mathematics, published as Quart. Appl. Math. 56 : 4 (1998). MR 1668728 Zbl 1159.00312 incollection

M. Atiyah: “Mathematics: Queen and servant of the sciences,” pp. xxiii–xxvi in Sir Michael Atiyah: A great mathematician of the twentieth century, published as Asian J. Math. 3 : 1 (1999). Republished from Proc. Am. Phil. Soc. 137:4 (1993). See also The founders of index theory (2009). Zbl 0957.01038 incollection

M. F. Atiyah: “Obituary: Kunihiko Kodaira,” Bull. London Math. Soc. 31 : 4 (1999), pp. 489–493. MR 1687532 Zbl 0928.01020 article

M. Atiyah: “The conscience of science,” pp. xxvii–xxxviii in Sir Michael Atiyah: A great mathematician of the twentieth century, published as Asian J. Math. 3 : 1. International Press (Somerville, MA), 1999. Schrödinger lecture, given at Imperial College, March 18th, 1997. Zbl 0957.01039 incollection

M. Atiyah: “Physics and geometry: A look at the last twenty years,” pp. 1–8 in Algebraic geometry: Hirzebruch 70 (Warsaw, 11–16 May 1998). Edited by P. Pragacz, M. Szurek, and J. Wiśniewski. Contemporary Mathematics 241. American Mathematical Society (Providence, RI), 1999. MR 1718133 Zbl 0945.14026 incollection

F. Hirzebruch	Related	Volume
Michał Szurek	Related
Piotr Pragacz	Related
Jarosław Wiśniewski	Related

M. Atiyah: “Geometry and physics in the 20th century,” pp. 1–9 in The mathematical sciences after the year 2000 (Beirut, 11–15 January 1999). Edited by K. Bitar, A. Chamseddine, and W. Sabra. World Scientific (River Edge, NJ), 2000. MR 1799435 Zbl 0994.01011 incollection

Ali H. Chamseddine	Related
Wafic Sabra	Related
Khalil Bitar	Related

M. Atiyah: “Some personal reminiscences,” pp. 257–267 in Oxford figures. Edited by J. Fauvel, R. Flood, and R. J. Wilson. Oxford University Press (New York), 2000. MR 1750206 incollection

Robin James Wilson	Related
John Fauvel	Related
Raymond Flood	Related

M. F. Atiyah: “Preface” in Mathematics: Frontiers and perspectives. Edited by V. Arnold, M. Atiyah, P. Lax, and B. Mazur. American Mathematical Society (Providence, RI), 2000. Republished in Atiyah’s Collected works, vol. 6. incollection

Peter D. Lax	Related
Vladimir Igorevich Arnold	Related
Barry C. Mazur	Related

M. Atiyah: “100 years of mathematics,” Normat 48 : 3 (2000), pp. 123–126, 144. MR 1811714 Zbl 0969.01015 article

M. Atiyah: Equivariant cohomology and representations of the symmetric group. Preprint, 2000. ArXiv math/0012215 techreport

M. Atiyah: “Mathematics in the 20th century,” Amer. Math. Mon. 108 : 7 (2001), pp. 654–666. See also versions in Math. Today 37:2 (2001), Contemporary trends in algebraic geometry and algebraic topology (2002), N.T.M. 10:1 (2002), Bull. Lond. Math. Soc. 34:1 (2002), Wiadom. Mat. 39 (2003) and Adv. Math. (China) 33:1 (2004). MR 1862105 Zbl 1081.01502 article

M. Atiyah: “Mathematics in the 20th century,” Math. Today 37 : 2 (2001), pp. 46–53. See also versions in Amer. Math. Mon. 108:7 (2001), Contemporary trends in algebraic geometry and algebraic topology (2002), N.T.M. 10:1 (2002), Bull. Lond. Math. Soc. 34:1 (2002), Wiadom. Mat. 39 (2003) and Adv. Math. (China) 33:1 (2004). MR 1840757 article

M. Atiyah, J. Maldacena, and C. Vafa: “An M-theory flop as a large \( N \) duality,” J. Math. Phys. 42 : 7 (2001), pp. 3209–3220. MR 1840340 Zbl 1061.81056 article

Juan Maldacena	Related
Cumrun Vafa	Related

M. Atiyah: “Configurations of points,” pp. 1375–1387 in Topological methods in the physical sciences (London, 2000), published as R. Soc. Lond. Philos. Trans. Ser. A 359 : 1784 (July 2001). MR 1853626 Zbl 1037.58007 incollection

Berry & Robbins, in their discussion of the spin-statistics theorem in quantum mechanics, were led to ask the following question. Can one construct a continuous map from the configuration space of \( n \) distinct particles in 3-space to the flag manifold of the unitary group \( U(n) \)? I shall discuss this problem and various generalizations of it. In particular, there is a version in which \( U(n) \) is replaced by an arbitrary compact Lie group. It turns out that this can be treated using Nahm’s equations, which are an integrable system of ordinary differential equations arising from the self-dual Yang–Mills equations. Our topological problem is therefore connected with physics in two quite different ways, once at its origin and once at its solution.

M. Atiyah: “Equivariant cohomology and representations of the symmetric group,” Chinese Ann. Math. Ser. B 22 : 1 (2001), pp. 23–30. MR 1823127 Zbl 1057.20007 article

M. Atiyah: “\( K \)-theory past and present,” pp. 411–417 in Sitzungsberichte der Berliner Mathematischen Gesellschaft, 1997–2000. Berliner Mathematischen Gesellschaft, 2001. MR 2091892 Zbl 1061.19500 ArXiv math/0012213 incollection

M. Atiyah: “Mathematics in the 20th century,” pp. 1–21 in Contemporary trends in algebraic geometry and algebraic topology (Tianjin, China, 9–13 October 2000). Edited by S.-S. Chern, L. Fu, and R. M. Hain. Nankai Tracts in Mathematics 5. World Scientific (River Edge, NJ), 2002. See also versions in Amer. Math. Mon. 108:7 (2001), Math. Today 37:2 (2001), N.T.M. 10:1 (2002), Bull. Lond. Math. Soc. 34:1 (2002), Wiadom. Mat. 39 (2003) and Adv. Math. (China) 33:1 (2004). MR 1945353 Zbl 1038.01014 incollection

S. S. Chern	Related	Volume
Richard M. Hain	Related
Lei Fu	Related

@incollection {key1945353m,
	AUTHOR = {Atiyah, Michael},
	TITLE = {Mathematics in the 20th century},
	BOOKTITLE = {Contemporary trends in algebraic geometry
		 and algebraic topology},
	EDITOR = {Shiing-Shen Chern and Lei Fu and Richard
		 Martin Hain},
	SERIES = {Nankai Tracts in Mathematics},
	NUMBER = {5},
	PUBLISHER = {World Scientific},
	ADDRESS = {River Edge, NJ},
	YEAR = {2002},
	PAGES = {1--21},
	DOI = {10.1142/9789812777416_0001},
	NOTE = {(Tianjin, China, 9--13 October 2000).
		 See also versions in \textit{Amer. Math.
		 Mon.} \textbf{108}:7 (2001), \textit{Math.
		 Today} \textbf{37}:2 (2001), \textit{N.T.M.}
		 \textbf{10}:1 (2002), \textit{Bull.
		 Lond. Math. Soc.} \textbf{34}:1 (2002),
		 \textit{Wiadom. Mat.} \textbf{39} (2003)
		 and \textit{Adv. Math. (China)} \textbf{33}:1
		 (2004). MR:1945353. Zbl:1038.01014.},
	ISBN = {9789810249540},
}

M. Atiyah: “On the unreasonable effectiveness of physics in mathematics,” pp. 25–38 in Highlights of mathematical physics (London, 17–22 July 2000). Edited by A. S. Fokas, J. Halliwell, T. Kibble, and B. Zegarlinski. American Mathematical Society (Providence, RI), 2002. MR 2001571 Zbl 1150.00002 incollection

Boguslaw Zegarlinski	Related
Athanassios S. Fokas	Related
T. Kibble	Related
Jonathan J. Halliwell	Related

M. Atiyah: “Mathematics in the 20th century,” Bull. Lond. Math. Soc. 34 : 1 (2002), pp. 1–15. See also versions in Amer. Math. Mon. 108:7 (2001), Math. Today 37:2 (2001), Contemporary trends in algebraic geometry and algebraic topology (2002), N.T.M. 10:1 (2002), Wiadom. Mat. 39 (2003), Adv. Math. (China) 33:1 (2004). MR 1866422 Zbl 1022.01007 article

M. Atiyah and P. Sutcliffe: “The geometry of point particles,” R. Soc. Lond. Proc. Ser. A 458 : 2021 (2002), pp. 1089–1115. MR 1902577 Zbl 1010.58015 article

There is a very natural map from the configuration space of \( n \) distinct points in Euclidean 3-space into the flag manifold \( U(n)/U(1)^n \), which is compatible with the action of the symmetric group. The map is well defined for all configurations of points provided a certain conjecture holds, for which we provide numerical evidence. We propose some additional conjectures, which imply the first, and test these numerically. Motivated by the above map, we define a geometrical multi-particle energy function and compute the energy-minimizing configurations for up to 32 particles. These configurations comprise the vertices of polyhedral structures that are dual to those found in a number of complicated physical theories, such as Skyrmions and fullerenes. Comparisons with 2- and 3-particle energy functions are made. The planar restriction and the generalization to hyperbolic 3-space are also investigated.

M. Atiyah: “Mathematics in the 20th century,” N.T.M. (N.S.) 10 : 1 (2002), pp. 25–39. See also versions in Amer. Math. Mon. 108:7 (2001), Math. Today 37:2 (2001), Contemporary trends in algebraic geometry and algebraic topology (2002), Bull. Lond. Math. Soc. 34:1 (2002), Wiadom. Mat. 39 (2003) and Adv. Math. (China) 33:1 (2004). MR 1894494 Zbl 0991.01015 article

M. F. Atiyah: “Hermann Weyl: 1885–1955,” Biog. Mem. Nat. Acad. Sci. 82 (2002), pp. 3–17. Republished in Atiyah’s Collected works, vol. 6. article

M. Atiyah: The Millennium prize problems, 2002. 60-minute videotape, Springer VideoMATH. From the CMI Millennium Meeting Collection (Collège de France, Paris, 24–25 May 2000). Atiyah gives a lecture on the Navier–Stokes equations. MR 2015194 Zbl 1093.00003 misc

M. Atiyah and E. Witten: “\( M \)-theory dynamics on a manifold of \( G_2 \) holonomy,” Adv. Theor. Math. Phys. 6 : 1 (2002), pp. 1–106. MR 1992874 Zbl 1033.81065 article

M. Atiyah and R. Bielawski: “Nahm’s equations, configuration spaces and flag manifolds,” Bull. Braz. Math. Soc. (N.S.) 33 : 2 (2002), pp. 157–176. MR 1940347 Zbl 1022.22007 ArXiv math/0110112 article

We give a positive answer to the Berry–Robbins problem for any compact Lie group \( G \), i.e. we show the existence of a smooth \( W \)-equivariant map from the space of regular triples in a Cartan subalgebra to the flag manifold \( G/T \). This map is constructed via solutions to Nahm’s equations and it is compatible with the \( \mathit{SO}(3) \) action, where \( \mathit{SO}(3) \) acts on \( G/T \) via a regular homomorphism from \( \mathit{SU}(2) \) to \( G \). We then generalize this picture to include an arbitrary homomorphism from \( \mathit{SU}(2) \) to \( G \). This leads to an interesting geometrical picture which appears to be related to the Springer representation of the Weyl group and the work of Kazhdan and Lusztig on representations of Hecke algebras.

M. Atiyah: “Mathematics in the 20th century,” Wiadom. Mat. 39 (2003), pp. 47–63. Polish translation of Bull. Lond. Math. Soc. 34:1 (2002). See also versions in Amer. Math. Mon. 108:7 (2001), Math. Today 37:2 (2001), Contemporary trends in algebraic geometry and algebraic topology (2002), N.T.M. 10:1 (2002) and Adv. Math. (China) 33:1 (2004). MR 2043772 article

M. Atiyah and J. Berndt: “Projective planes, Severi varieties and spheres,” pp. 1–27 in Surveys in differential geometry: Lectures on geometry and topology held in honor of Calabi, Lawson, Siu, and Uhlenbeck (Harvard University, 3–5 May 2002). Edited by S.-T. Yau. Surveys in Differential Geometry (Journal of Differential Geometry supplements) 8. International Press (Somerville, MA), 2003. MR 2039984 Zbl 1057.53040 ArXiv math/0206135 incollection

H. Blaine Lawson, Jr.	Related
Jürgen Berndt	Related
Yum-Tong Siu	Related
Karen Uhlenbeck	Related	Volume
Shing-Tung Yau	Related
Eugenio Calabi	Related

M. Atiyah and P. Sutcliffe: “Polyhedra in physics, chemistry and geometry,” Milan J. Math. 71 : 1 (2003), pp. 33–58. MR 2120915 Zbl 1050.52002 article

M. Atiyah: “The geometry behind some string theory dualities,” pp. 89–108 in UK-Japan Winter School 2004: Geometry and analysis towards quantum theory. Edited by M. A. Guest, Y. Maeda, K. G. Daigaku, and S. Kōgakuka. Keio University (Yokohama), 2004. MR 2131366 incollection

Yoshiaki Maeda	Related
Sūri Kōgakuka	Related
Martin A. Guest	Related
Kyōto Daigaku	Related

M. Atiyah: Collected works, vol. 6. Oxford Science Publications. The Clarendon Press and Oxford University Press (Oxford, New York), 2004. MR 2160826 Zbl 1099.01024 book

M. Atiyah and M. Hopkins: “A variant of \( K \)-theory: \( K_\pm \),” pp. 5–17 in Topology, geometry and quantum field theory (Oxford, 24–29 June 2002). Edited by U. Tillmann. Cambridge University Press, 2004. MR 2079369 Zbl 1090.19004 incollection

Topological \( K \)-theory has many variants which have been developed and exploited for geometric purposes. There are real or quaternionic versions, “real” \( K \)-theory in the sense of, equivariant \( K \)-theory and combinations of all these.

In recent years \( K \)-theory has found unexpected application in the physics of string theories and all variants of \( K \)-theory that had previously been developed appear to be needed. There are even variants, needed for the physics, which had previously escaped attention, and it is one such variant that is the subject of this paper.

This variant, denoted by \( K_{\pm}(X) \), was introduced by Witten in relation to “orientifolds”. The geometric situation concerns a manifold \( X \) with an involution \( \tau \) having a fixed sub-manifold \( Y \). On \( X \) one wants to study a pair of complex vector bundles \( (E^+, E^-) \) with the property that \( \tau \) interchanges them. If we think of the virtual vector bundle \( E^+ - E^- \), then \( \tau \) takes this into its negative, and \( K_{\pm}(X) \) is meant to be the appropriate K-theory of this situation.

In physics, \( X \) is a 10-dimensional Lorentzian manifold and maps \( \Sigma \to X \) of a surface \( \Sigma \) describe the world-sheet of strings. The symmetry requirements for the appropriate Feynman integral impose conditions that the putative \( K \)-theory \( K_{\pm}(X) \) has to satisfy.

The second author proposed a precise topological definition of \( K_{\pm}(X) \) which appears to meet the physics requirements, but it was not entirely clear how to link the physics with the geometry.

Michael Jerome Hopkins	Related
Ulrike Tillmann	Related

M. F. Atiyah: “Bakerian Lecture, 1975: Global geometry,” Amer. Math. Mon. 111 : 8 (2004), pp. 716–723. Republished from Proc. Roy. Soc. London Ser. A 347:1650 (1976). MR 2091548 Zbl 1187.58001 article

M. F. Atiyah: “The geometry and physics of knots,” pp. 289–304 in Collected works, vol. 6. Oxford Science Publications. Clarendon Press and Oxford University Press (Oxford, New York), 2004. See also Miniconference on geometry and physics (1989), the 1990 book of the same title and the Russian translation Geometriya i fizika uzlov (1995). incollection

M. Atiyah: “Mathematics in the 20th century,” Adv. Math. (China) 33 : 1 (2004), pp. 26–40. See also versions in Amer. Math. Mon. 108:7 (2001), Math. Today 37:2 (2001), Contemporary trends in algebraic geometry and algebraic topology (2002), N.T.M. 10:1 (2002), Bull. Lond. Math. Soc. 34:1 (2002) and Wiadom. Mat. 39 (2003). MR 2058440 article

M. Atiyah: “The impact of Thom’s cobordism theory,” Bull. Amer. Math. Soc. (N.S.) 41 : 3 (2004), pp. 337–340. MR 2058290 Zbl 1049.57500 article

M. Atiyah and G. Segal: “Twisted \( K \)-theory,” Ukr. Mat. Visn. 1 : 3 (2004), pp. 287–330. MR 2172633 Zbl 1151.55301 article

M. Atiyah: “Einstein and geometry,” Current Sci. 89 : 12 (2005), pp. 2041–2044. See also The legacy of Albert Einstein (2007). MR 2189856 article

A. Bundy, D. Mackenzie, M. Atiyah, and A. Macintyre: “Abstracts of additional presentations made at the Royal Society Discussion Meeting ‘The nature of mathematical proof’,” Philos. Trans. R. Soc. Lond., Ser. A 363 : 1835 (2005), pp. 2461.

Donald Mackenzie	Related
Alan Richard Bundy	Related
Angus Macintyre	Related

M. Atiyah: “The interaction between geometry and physics,” pp. 1–15 in The unity of mathematics (Cambridge, MA, 31 August–4 September 2003). Edited by P. I. Ètingof, V. Retakh, and I. M. Singer. Progress in Mathematics 244. Birkhäuser (Boston, MA), 2006. MR 2181802 Zbl 1118.58002 incollection

Isadore M. Singer	Related	Volume
Vladimir Solomonovich Retakh	Related
Pavel Ilyich Ètingof	Related

M. Atiyah: “Mathematics: Art and science,” Bull. Amer. Math. Soc. (N.S.) 43 : 1 (2006), pp. 87–88. MR 2201551 Zbl 1121.00307 article

M. Atiyah and G. Segal: “Twisted \( K \)-theory and cohomology,” pp. 5–43 in Inspired by S. S. Chern. Edited by P. Griffiths. Nankai Tracts in Mathematics 11. World Scientific (Hackensack, NJ), 2006. MR 2307274 Zbl 1138.19003 ArXiv math/0510674 incollection

Phillip A. Griffiths	Related
S. S. Chern	Related	Volume
Graeme Bryce Segal	Related

M. Atiyah: “Einstein and geometry,” pp. 15–23 in The legacy of Albert Einstein. Edited by S. R. Wadia. World Scientific (Hackensack, NJ), 2007. See also Current Sci. 89:12 (2005). MR 2330810 Zbl 1136.01012 incollection

Spenta R. Wadia	Related
Albert Einstein	Related

M. F. Atiyah: Siamo tutti matematici [We are all mathematicians]. Di Renzo Editore (Rome), 2007. In Italian. book

M. Atiyah: “Preface,” pp. xvii in From probability to geometry, vol. I. Edited by X. Dai. Astérisque 327. Socíeté Mathématique de France (Paris), 2009. Volume in honor of the 60th birthday of Jean-Michel Bismut. MR 2642355 incollection

Xianzhe Dai	Related
Jean-Michel Bismut	Related

M. F. Atiyah: “A personal history,” pp. 5–15 in The founders of index theory: Reminiscences of and about Sir Michael Atiyah, Raoul Bott, Friedrich Hirzebruch, and I. M. Singer, 2nd edition. Edited by S.-T. Yau. International Press (Somerville, MA), 2009. Republished in Atiyah’s Collected works, vol. 6. incollection

F. Hirzebruch	Related	Volume
Raoul H. Bott	Related	Volume
Shing-Tung Yau	Related
Isadore M. Singer	Related	Volume

M. Atiyah: Edinburgh lectures on geometry, analysis and physics. Preprint, University of Edinburgh, 2010. ArXiv 1009.4827 techreport

M. Atiyah: “The art of mathematics,” Notices Amer. Math. Soc. 57 : 1 (2010), pp. 8. MR 2590111 article

M. Atiyah: “Working with Raoul Bott: From geometry to physics,” pp. 51–61 in A celebration of the mathematical legacy of Raoul Bott (Montréal, QC, 9–13 June 2008). Edited by P. R. Kotiuga. CRM Proceedings & Lecture Notes 50. American Mathematical Society (Providence, RI), 2010. MR 2648885 Zbl 1195.01022 incollection

Raoul H. Bott	Related	Volume
Peter Robert Kotiuga	Related

M. Atiyah: “Obituary: Raoul Harry Bott, FRS, 1923–2005,” Bull. Lond. Math. Soc. 42 : 1 (2010), pp. 170–180. Republication of an article in Biogr. Mems Fell. R. Soc. 53 (2007). MR 2586977 Zbl 1181.01039 article

M. Atiyah: “Mathematical work of Nigel Hitchin,” pp. 11–16 in The many facets of geometry. Edited by O. García-Prada, J. P. Bourguignon, and S. Salamon. Oxford University Press (New York), 2010. MR 2681683 Zbl 1206.01039 incollection

Nigel J. Hitchin	Related
Simon Salamon	Related
Oscar García-Prada	Related
Jean Pierre Bourguignon	Related

J.-P. Serre and M. Atiyah: “A tribute to Henri Cartan,” Notices Amer. Math. Soc. 57 : 8 (2010), pp. 946–950. MR 2667491 Zbl 1195.01081 article

Henri Cartan	Related
Jean-Pierre Serre	Related

M. Atiyah, R. Dijkgraaf, and N. Hitchin: “Geometry and physics,” Philos. Trans. R. Soc. Lond. Ser. A 368 : 1914 (2010), pp. 913–926. MR 2587923 Zbl pre05764159 article

Nigel J. Hitchin	Related
Robbert Dijkgraaf	Related

M. Atiyah and G. W. Moore: “A shifted view of fundamental physics,” pp. 1–15 in Surveys in differential geometry, XV: Perspectives in mathematics and physics. Edited by T. Mrowka and S.-T. Yau. Surv. Differ. Geom. 15. Int. Press (Somerville, MA), 2011. MR 2815723 Zbl 1238.83047 ArXiv 1009.3176 incollection

M. F. Atiyah, N. S. Manton, and B. J. Schroers: “Geometric models of matter,” Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 468 : 2141 (2012), pp. 1252–1279. MR 2910348 Zbl 1364.53046 ArXiv 1108.5151 article

Inspired by soliton models, we propose a description of static particles in terms of Riemannian 4-manifolds with self-dual Weyl tensor. For electrically charged particles, the 4-manifolds are non-compact and asymptotically fibred by circles over physical 3-space. This is akin to the Kaluza–Klein description of electromagnetism, except that we exchange the roles of magnetic and electric fields, and only assume the bundle structure asymptotically, away from the core of the particle in question. We identify the Chern class of the circle bundle at infinity with minus the electric charge and, at least provisionally, the signature of the 4-manifold with the baryon number. Electrically neutral particles are described by compact 4-manifolds. We illustrate our approach by studying the Taub–Newman, Unti, Tamburino (Taub–NUT) manifold as a model for the electron, the Atiyah–Hitchin manifold as a model for the proton, \( \mathbb{C}P^2 \) with the Fubini–Study metric as a model for the neutron and \( S^4 \) with its standard metric as a model for the neutrino.

Bernd J. Schroers	Related
Nicholas S. Manton	Related

Michael F. Atiyah

Works authored by Atiyah

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