Editing Michael Atiyah (section)

===Index theory (1963–1984)===
[[File:Isadore Singer 1977.jpeg|thumb|right|[[Isadore Singer]] (in 1977), who worked with Atiyah on index theory]]
{{Main|Atiyah–Singer index theorem}}
Atiyah's work on index theory is reprinted in volumes 3 and 4 of his collected works.<ref>{{harvnb|Atiyah|1988c}}</ref><ref>{{harvnb|Atiyah|1988d}}</ref>

The index of a differential operator is closely related to the number of independent solutions (more precisely, it is the differences of the numbers of independent solutions of the differential operator and its adjoint). There are many hard and fundamental problems in mathematics that can easily be reduced to the problem of finding the number of independent solutions of some differential operator, so if one has some means of finding the index of a differential operator these problems can often be solved. This is what the Atiyah–Singer index theorem does: it gives a formula for the index of certain differential operators, in terms of topological invariants that look quite complicated but are in practice usually straightforward to calculate.{{Citation needed|date=November 2010}}

Several deep theorems, such as the [[Hirzebruch–Riemann–Roch theorem]], are special cases of the Atiyah–Singer index theorem. In fact the index theorem gave a more powerful result, because its proof applied to all compact complex manifolds, while Hirzebruch's proof only worked for projective manifolds. There were also many new applications: a typical one is calculating the dimensions of the moduli spaces of instantons. The index theorem can also be run "in reverse": the index is obviously an integer, so the formula for it must also give an integer, which sometimes gives subtle integrality conditions on invariants of manifolds. A typical example of this is [[Rochlin's theorem]], which follows from the index theorem.{{Citation needed|date=November 2010}}

{{quote box
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|quote=The most useful piece of advice I would give to a mathematics student is always to suspect an impressive sounding Theorem if it does not have a special case which is ''both'' simple ''and'' non-trivial.
|source=Michael Atiyah<ref>{{harvnb|Atiyah|1988a|loc = paper 17, p. 76}}</ref>
}}
The index problem for [[elliptic differential operator]]s was posed in 1959 by [[Israil Gelfand|Gel'fand]].<ref>{{harvnb|Gel'fand|1960}}</ref> He noticed the homotopy invariance of the index, and asked for a formula for it by means of [[topological invariant]]s. Some of the motivating examples included the [[Riemann–Roch theorem]] and its generalization the [[Hirzebruch–Riemann–Roch theorem]], and the [[Hirzebruch signature theorem]]. [[Friedrich Hirzebruch|Hirzebruch]] and [[Armand Borel|Borel]] had proved the integrality of the [[Â genus]] of a spin manifold, and Atiyah suggested that this integrality could be explained if it were the index of the [[Dirac operator]] (which was rediscovered by Atiyah and Singer in 1961).

The first announcement of the Atiyah–Singer theorem was their 1963 paper.<ref>{{harvnb|Atiyah|Singer|1963}}</ref> The proof sketched in this announcement was inspired by Hirzebruch's proof of the [[Hirzebruch–Riemann–Roch theorem]] and was never published by them, though it is described in the book by Palais.<ref>{{harvnb|Palais|1965}}</ref> Their first published proof<ref>{{harvnb|Atiyah|Singer|1968a}}</ref> was more similar to Grothendieck's proof of the [[Grothendieck–Riemann–Roch theorem]], replacing the [[cobordism]] theory of the first proof with [[K-theory]], and they used this approach to give proofs of various generalizations in a sequence of papers from 1968 to 1971.

Instead of just one elliptic operator, one can consider a family of elliptic operators parameterized by some space ''Y''. In this case the index is an element of the K theory of ''Y'', rather than an integer.<ref>{{harvnb|Atiyah|1988c|loc=paper 67}}</ref> If the operators in the family are real, then the index lies in the real K theory of ''Y''. This gives a little extra information, as the map from the real K theory of ''Y'' to the [[complex K-theory]] is not always injective.<ref>{{harvnb|Atiyah|1988c|loc=paper 68}}</ref>

[[File:Graeme Segal.jpeg|thumb|right|Atiyah's former student [[Graeme Segal]] (in 1982), who worked with Atiyah on [[K-theory|equivariant K-theory]]]]
With Bott, Atiyah found an analogue of the [[Lefschetz fixed-point formula]] for elliptic operators, giving the Lefschetz number of an endomorphism of an [[elliptic complex]] in terms of a sum over the fixed points of the endomorphism.<ref>{{harvnb|Atiyah|1988c|loc=papers 61, 62, 63}}</ref> As special cases their formula included the [[Weyl character formula]], and several new results about elliptic curves with complex multiplication, some of which were initially disbelieved by experts.<ref>{{harvnb|Atiyah|1988c|p=3}}</ref>
Atiyah and Segal combined this fixed point theorem with the index theorem as follows.
If there is a compact [[Group action (mathematics)|group action]] of a group ''G'' on the compact manifold ''X'', commuting with the elliptic operator, then one can replace ordinary K-theory in the index theorem with [[Equivariant algebraic K-theory|equivariant K-theory]].
For trivial groups ''G'' this gives the index theorem, and for a finite group ''G'' acting with isolated fixed points it gives the Atiyah–Bott fixed point theorem. In general it gives the index as a sum over fixed point submanifolds of the group ''G''.<ref>{{harvnb|Atiyah|1988c|loc=paper 65}}</ref>

Atiyah<ref>{{harvnb|Atiyah|1988c|loc=paper 73}}</ref> solved a problem asked independently by [[Lars Hörmander|Hörmander]] and Gel'fand, about whether complex powers of analytic functions define [[Distribution (mathematics)|distributions]]. Atiyah used [[Heisuke Hironaka|Hironaka]]'s resolution of singularities to answer this affirmatively. An ingenious and elementary solution was found at about the same time by [[J. Bernstein]], and discussed by Atiyah.<ref>{{harvnb|Atiyah|1988a|loc=paper 15}}</ref>

As an application of the equivariant index theorem, Atiyah and Hirzebruch showed that manifolds with effective circle actions have vanishing [[Â-genus]].<ref>{{harvnb|Atiyah|1988c|loc=paper 74}}</ref> (Lichnerowicz showed that if a manifold has a metric of positive scalar curvature then the Â-genus vanishes.)

With [[Elmer Rees]], Atiyah studied the problem of the relation between topological and holomorphic vector bundles on projective space. They solved the simplest unknown case, by showing that all rank 2 vector bundles over projective 3-space have a holomorphic structure.<ref>{{harvnb|Atiyah|1988c|loc=paper 76}}</ref> [[Geoffrey Horrocks (mathematician)|Horrocks]] had previously found some non-trivial examples of such vector bundles, which were later used by Atiyah in his study of instantons on the 4-sphere.

[[File:Raoul Bott 1986.jpeg|thumb|right|[[Raoul Bott]], who worked with Atiyah on fixed point formulas and several other topics]]
Atiyah, Bott and [[Vijay Kumar Patodi|Vijay K. Patodi]]<ref>{{harvnb|Atiyah|Bott|Patodi|1973}}</ref> gave a new proof of the index theorem using the [[heat equation]].

If the [[manifold]] is allowed to have boundary, then some restrictions must be put on the domain of the elliptic operator in order to ensure a finite index. These conditions can be local (like demanding that the sections in the domain vanish at the boundary) or more complicated global conditions (like requiring that the sections in the domain solve some differential equation). The local case was worked out by Atiyah and Bott, but they showed that many interesting operators (e.g., the [[signature operator]]) do not admit local boundary conditions. To handle these operators, Atiyah, Patodi and Singer introduced global boundary conditions equivalent to attaching a cylinder to the manifold along the boundary and then restricting the domain to those sections that are square integrable along the cylinder, and also introduced the [[Atiyah–Patodi–Singer eta invariant]]. This resulted in a series of papers on spectral asymmetry,<ref>{{harvnb|Atiyah|1988d|loc=papers 80–83}}</ref> which were later unexpectedly used in [[theoretical physics]], in particular in Witten's work on anomalies.

[[File:Schlierenfoto Mach 1-2 Pfeilflügel - NASA.jpg|thumb|right|The lacunas discussed by Petrovsky, Atiyah, Bott and Gårding are similar to the spaces between shockwaves of a supersonic object.]]
The fundamental solutions of linear [[hyperbolic partial differential equation]]s often have [[Petrovsky lacuna]]s: regions where they vanish identically. These were studied in 1945 by [[I. G. Petrovsky]], who found topological conditions describing which regions were lacunas.
In collaboration with Bott and [[Lars Gårding]], Atiyah wrote three papers updating and generalizing Petrovsky's work.<ref>{{harvnb|Atiyah|1988d|loc=papers 84, 85, 86}}</ref>

Atiyah<ref>{{harvnb|Atiyah|1976}}</ref> showed how to extend the index theorem to some non-compact manifolds, acted on by a discrete group with compact quotient. The kernel of the elliptic operator is in general infinite-dimensional in this case, but it is possible to get a finite index using the dimension of a module over a [[von Neumann algebra]]; this index is in general real rather than integer valued. This version is called the ''L<sup>2</sup> index theorem,'' and was used by Atiyah and Schmid<ref>{{harvnb|Atiyah|Schmid|1977}}</ref> to give a geometric construction, using square integrable harmonic spinors, of Harish-Chandra's [[discrete series representation]]s of [[semisimple Lie group]]s. In the course of this work they found a more elementary proof of Harish-Chandra's fundamental theorem on the local integrability of characters of Lie groups.<ref>{{harvnb|Atiyah|1988d|loc=paper 91}}</ref>

With H. Donnelly and I. Singer, he extended Hirzebruch's formula (relating the signature defect at cusps of Hilbert modular surfaces to values of L-functions) from real quadratic fields to all totally real fields.<ref>{{harvnb|Atiyah|1988d|loc=papers 92, 93}}</ref>