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Dive into the research topics where Michael Atiyah is active.

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Featured researches published by Michael Atiyah.


Mathematical Proceedings of the Cambridge Philosophical Society | 1975

Spectral asymmetry and Riemannian Geometry. I

Michael Atiyah; V. K. Patodi; I. M. Singer

1. Introduction . The main purpose of this paper is to present a generalization of Hirzebruchs signature theorem for the case of manifolds with boundary. Our result is in the framework of Riemannian geometry and can be viewed as analogous to the Gauss–Bonnet theorem for manifolds with boundary, although there is a very significant difference between the two cases which is, in a sense, the central topic of the paper. To explain this difference let us begin by recalling that the classical Gauss–Bonnet theorem for a surface X with boundary Y asserts that the Euler characteristic E(X) is given by a formula: where K is the Gauss curvature of X and σ is the geodesic curvature of Y in X . In particular if, near the boundary, X is isometric to the product Y x R + , the boundary integral in (1.1) vanishes and the formula is the same as for closed surfaces. Similar remarks hold in higher dimensions. Now if X is a closed oriented Riemannian manifold of dimension 4, there is another formula relating cohomological invariants with curvature in addition to the Gauss–Bonnet formula. This expresses the signature of the quadratic form on H 2 ( X , R) by an integral formula where p 1 is the differential 4-form representing the first Pontrjagin class and is given in terms of the curvature matrix R by p 1 = (2π) −2 Tr R 2 . It is natural to ask if (1.2) continues to hold for manifolds with boundary, provided the metric is a product near the boundary. Simple examples show that this is false, so that in general


Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences | 1978

Selfduality in Four-Dimensional Riemannian Geometry

Michael Atiyah; I.M. Singer; Nigel Hitchin

We present a self-contained account of the ideas of R. Penrose connecting four-dimensional Riemannian geometry with three-dimensional complex analysis. In particular we apply this to the self-dual Yang-Mills equations in Euclidean 4-space and compute the number of moduli for any compact gauge group. Results previously announced are treated with full detail and extended in a number of directions.


Physics Letters A | 1978

Construction of instantons

Michael Atiyah; Nigel Hitchin; V.G. Drinfeld; Yu. I. Manin

A complete construction, involving only linear algebra, is given for all self-dual euclidean Yang-Mills fields.


Topology | 1984

The moment map and equivariant cohomology

Michael Atiyah; Raoul Bott


Publications Mathématiques de l'IHÉS | 1988

Topological quantum field theories

Michael Atiyah


Transactions of the American Mathematical Society | 1957

Complex analytic connections in fibre bundles

Michael Atiyah

© Publications mathematiques de l’I.H.E.S., 1988, tous droits reserves. L’acces aux archives de la revue « Publications mathematiques de l’I.H.E.S. » (http://www. ihes.fr/IHES/Publications/Publications.html), implique l’accord avec les conditions generales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systematique est constitutive d’une infraction penale. Toute copie ou impression de ce fichier doit contenir la presente mention de copyright.


American Mathematical Monthly | 1968

Seminar on the Atiyah-Singer Index Theorem.

Richard S. Palais; Michael Atiyah

Introduction. 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. The ideas outlined above are developed in considerable detail, and they are applied in particular to a problem first studied by Weil [17], namely the problem of characterizing those fibre bundles which arise from a representation of the fundamental group. We show how Weils main result fits into the general picture, and we discuss various aspects of the problem. As no complete exposition of the theory of complex analytic fibre bundles has as yet been published, this paper should start with a basic exposition of this nature. However this would be a major undertaking in itself, and instead we shall simply summarize in ?1 the terminology and results on vector bundles which we require, and for the rest we refer to Grothendieck [8], Serre [12], and Hirzebruch [9].


Bulletin of the American Mathematical Society | 1963

The index of elliptic operators on compact manifolds

Michael Atiyah; I. M. Singer

The description for this book, Seminar on Atiyah-Singer Index Theorem. (AM-57), will be forthcoming.


Communications in Mathematical Physics | 1977

Instantons and algebraic geometry

Michael Atiyah; R. S. Ward

© Association des collaborateurs de Nicolas Bourbaki, 1964, tous droits réservés. L’accès aux archives du séminaire Bourbaki (http://www.bourbaki. ens.fr/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.


Publications Mathématiques de l'IHÉS | 1961

Characters and cohomology of finite groups

Michael Atiyah

Minimum action solutions for SU(2) Yang-Mills fields in Euclidean 4-space correspond, via the Penrose twistor transform, to algebraic bundles on the complex projective 3-space. These bundles in turn correspond to algebraic curves. The implication of these results for the Yang-Mills fields is described. In particular all solutions are rational and can be constructed from a series of AnsätzeAl forl≧1.

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I. M. Singer

Massachusetts Institute of Technology

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V. K. Patodi

Massachusetts Institute of Technology

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Armand Borel

Institute for Advanced Study

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Edward Witten

Institute for Advanced Study

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