Editing Michael Atiyah (section)

===Gauge theory (1977–1985)===
{{Main|Gauge theory (mathematics)}}
[[File:Camposcargas.svg|thumb|right|On the left, two nearby monopoles of the same polarity repel each other, and on the right two nearby monopoles of opposite polarity form a [[dipole]]. These are abelian monopoles; the non-abelian ones studied by Atiyah are more complicated.]]
Many of his papers on gauge theory and related topics are reprinted in volume 5 of his collected works.{{sfn|Atiyah|1988e}} A common theme of these papers is the study of moduli spaces of solutions to certain [[non-linear partial differential equation]]s, in particular the equations for instantons and monopoles. This often involves finding a subtle correspondence between solutions of two seemingly quite different equations. An early example of this which Atiyah used repeatedly is the [[Penrose transform]], which can sometimes convert solutions of a non-linear equation over some real manifold into solutions of some linear holomorphic equations over a different complex manifold.

In a series of papers with several authors, Atiyah classified all instantons on 4-dimensional Euclidean space. It is more convenient to classify instantons on a sphere as this is compact, and this is essentially equivalent to classifying instantons on Euclidean space as this is conformally equivalent to a sphere and the equations for instantons are conformally invariant. With Hitchin and Singer<ref>{{harvnb|Atiyah|1988e|loc=papers 94, 97}}</ref> he calculated the dimension of the moduli space of irreducible self-dual connections (instantons) for any principal bundle over a compact 4-dimensional Riemannian manifold (the [[Atiyah–Hitchin–Singer theorem]]). For example, the dimension of the space of SU<sub>2</sub> instantons of rank ''k''>0 is 8''k''−3. To do this they used the Atiyah–Singer index theorem to calculate the dimension of the tangent space of the moduli space at a point; the tangent space is essentially the space of solutions of an elliptic differential operator, given by the linearization of the non-linear Yang–Mills equations. These moduli spaces were later used by Donaldson to construct his [[Donaldson invariant|invariants of 4-manifolds]].
Atiyah and Ward used the Penrose correspondence to reduce the classification of all instantons on the 4-sphere to a problem in algebraic geometry.<ref>{{harvnb|Atiyah|1988e|loc=paper 95}}</ref> With Hitchin he used ideas of Horrocks to solve this problem, giving the [[ADHM construction]] of all instantons on a sphere; Manin and Drinfeld found the same construction at the same time, leading to a joint paper by all four authors.<ref>{{harvnb|Atiyah|1988e|loc=paper 96}}</ref> Atiyah reformulated this construction using [[quaternion]]s and wrote up a leisurely account of this classification of instantons on Euclidean space as a book.<ref>{{harvnb|Atiyah|1988e|loc=paper 99}}</ref>

{{quote box
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|quote=The mathematical problems that have been solved or techniques that have arisen out of physics in the past have been the lifeblood of mathematics.
|source=Michael Atiyah<ref>{{harvnb|Atiyah|1988a|loc = paper 19, p. 13}}</ref>
}}
Atiyah's work on instanton moduli spaces was used in Donaldson's work on [[Donaldson theory]]. Donaldson showed that the moduli space of (degree 1) instantons over a compact simply connected [[4-manifold]] with positive definite intersection form can be compactified to give a cobordism between the manifold and a sum of copies of complex projective space. He deduced from this that the intersection form must be a sum of one-dimensional ones, which led to several spectacular applications to smooth 4-manifolds, such as the existence of non-equivalent [[smooth structure]]s on 4-dimensional Euclidean space. Donaldson went on to use the other moduli spaces studied by Atiyah to define [[Donaldson invariant]]s, which revolutionized the study of smooth 4-manifolds, and showed that they were more subtle than smooth manifolds in any other dimension, and also quite different from topological 4-manifolds. Atiyah described some of these results in a survey talk.<ref>{{harvnb|Atiyah|1988e|loc=paper 112}}</ref>

[[Green's function]]s for linear partial differential equations can often be found by using the [[Fourier transform]] to convert this into an algebraic problem. Atiyah used a non-linear version of this idea.<ref>{{harvnb|Atiyah|1988e|loc=paper 101}}</ref> He used the Penrose transform to convert the Green's function for the conformally invariant Laplacian into a complex analytic object, which turned out to be essentially the diagonal embedding of the Penrose twistor space into its square. This allowed him to find an explicit formula for the conformally invariant Green's function on a 4-manifold.

In his paper with Jones,<ref>{{harvnb|Atiyah|1988e|loc=paper 102}}</ref> he studied the topology of the moduli space of SU(2) instantons over a 4-sphere. They showed that the natural map from this moduli space to the space of all connections induces epimorphisms of [[homology group]]s in a certain range of dimensions, and suggested that it might induce isomorphisms of homology groups in the same range of dimensions. This became known as the [[Atiyah–Jones conjecture]], and was later proved by several mathematicians.<ref>{{harvnb|Boyer|Hurtubise|Mann|Milgram|1993}}</ref>

Harder and [[M. S. Narasimhan]] described the cohomology of the [[moduli space]]s of [[stable vector bundle]]s over [[Riemann surface]]s by counting the number of points of the moduli spaces over finite fields, and then using the Weil conjectures to recover the cohomology over the complex numbers.<ref>{{harvnb|Harder|Narasimhan|1975}}</ref>
Atiyah and [[R. Bott]] used [[Morse theory]] and the [[Yang–Mills equation]]s over a [[Riemann surface]] to reproduce and extending the results of Harder and Narasimhan.<ref>{{harvnb|Atiyah|1988e|loc=papers 104–105}}</ref>

An old result due to [[Issai Schur|Schur]] and Horn states that the set of possible diagonal vectors of an Hermitian matrix with given eigenvalues is the convex hull of all the permutations of the eigenvalues. Atiyah proved a generalization of this that applies to all compact [[symplectic manifold]]s acted on by a torus, showing that the image of the manifold under the moment map is a convex polyhedron,<ref>{{harvnb|Atiyah|1988e|loc=paper 106}}</ref> and with Pressley gave a related generalization to infinite-dimensional loop groups.<ref>{{harvnb|Atiyah|1988e|loc=paper 108}}</ref>

Duistermaat and Heckman found a striking formula, saying that the push-forward of the [[Liouville measure]] of a [[moment map]] for a torus action is given exactly by the stationary phase approximation (which is in general just an asymptotic expansion rather than exact). Atiyah and Bott<ref>{{harvnb|Atiyah|1988e|loc=paper 109}}</ref> showed that this could be deduced from a more general formula in [[equivariant cohomology]], which was a consequence of well-known [[localization formula for equivariant cohomology|localization theorem]]s. Atiyah showed<ref>{{harvnb|Atiyah|1988e|loc=paper 110}}</ref> that the moment map was closely related to [[geometric invariant theory]], and this idea was later developed much further by his student [[F. Kirwan]]. Witten shortly after applied the [[Duistermaat–Heckman formula]] to loop spaces and showed that this formally gave the Atiyah–Singer index theorem for the Dirac operator; this idea was lectured on by Atiyah.<ref>{{harvnb|Atiyah|1988e|loc=paper 124}}</ref>

With Hitchin he worked on [[magnetic monopole]]s, and studied their scattering using an idea of [[Nick Manton]].<ref>{{harvnb|Atiyah|1988e|loc=papers 115, 116}}</ref> His book<ref>{{harvnb|Atiyah|Hitchin|1988}}</ref> with Hitchin gives a detailed description of their work on [[magnetic monopoles]]. The main theme of the book is a study of a moduli space of [[magnetic monopoles]]; this has a natural Riemannian metric, and a key point is that this metric is complete and [[hyperkähler]]. The metric is then used to study the scattering of two monopoles, using a suggestion of N. Manton that the geodesic flow on the moduli space is the low energy approximation to the scattering. For example, they show that a head-on collision between two monopoles results in 90-degree scattering, with the direction of scattering depending on the relative phases of the two monopoles. He also studied monopoles on hyperbolic space.<ref>{{harvnb|Atiyah|1988e|loc=paper 118}}</ref>

Atiyah showed<ref>{{harvnb|Atiyah|1988e|loc=paper 117}}</ref> that instantons in 4 dimensions can be identified with instantons in 2 dimensions, which are much easier to handle. There is of course a catch: in going from 4 to 2 dimensions the structure group of the gauge theory changes from a finite-dimensional group to an infinite-dimensional loop group. This gives another example where the moduli spaces of solutions of two apparently unrelated nonlinear partial differential equations turn out to be essentially the same.

Atiyah and Singer found that anomalies in quantum field theory could be interpreted in terms of index theory of the Dirac operator;<ref>{{harvnb|Atiyah|1988e|loc=papers 119, 120, 121}}</ref> this idea later became widely used by physicists.