Editing Michael Atiyah (section)

==Career and research==
[[File:Fuld Hall, Institute for Advanced Study, Princeton, NJ.jpg|thumb|right|The [[Institute for Advanced Study]] in Princeton, where Atiyah was professor from 1969 to 1972]]
Atiyah spent the academic year 1955–1956 at the [[Institute for Advanced Study, Princeton]], then returned to [[Cambridge University]], where he was a research fellow and assistant [[lecturer]] (1957–1958), then a university [[lecturer]] and tutorial [[fellow]] at [[Pembroke College, Cambridge]] (1958–1961). In 1961, he moved to the [[University of Oxford]], where he was a [[reader (academic rank)|reader]] and [[professor]]ial fellow at [[St Catherine's College, Oxford|St Catherine's College]] (1961–1963).<ref name="cv1"/> He became [[Savilian Professor of Geometry]] and a professorial fellow of [[New College, Oxford]], from 1963 to 1969. He took up a three-year professorship at the Institute for Advanced Study in [[Princeton, New Jersey|Princeton]] after which he returned to Oxford as a [[Royal Society]] Research Professor and professorial fellow of St Catherine's College. He was president of the [[London Mathematical Society]] from 1974 to 1976.<ref name="cv1"/>

{{quote box
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|quote= I started out by changing local currency into foreign currency everywhere I travelled as a child and ended up making money. That's when my father realised that I would be a mathematician some day.
|source=Michael Atiyah<ref>{{citation |first=Amba |last=Batra |date=8 November 2003 |url=http://cities.expressindia.com/fullstory.php?newsid=67555 |title=Maths guru with Einstein's dream prefers chalk to mouse. (Interview with Atiyah.) |access-date=2008-08-14 |publisher=Delhi newsline |archive-url=https://web.archive.org/web/20090208121300/http://cities.expressindia.com/fullstory.php?newsid=67555 |archive-date=8 February 2009 }}</ref>
}}
Atiyah was president of the [[Pugwash Conferences on Science and World Affairs]] from 1997 to 2002.<ref name="cv2">{{harvnb|Atiyah|2004|p=ix}}</ref> He also contributed to the foundation of the [[InterAcademy Panel on International Issues]], the Association of European Academies (ALLEA), and the [[European Mathematical Society]] (EMS).<ref>{{citation|url=https://www.ams.org/notices/200406/comm-abel.pdf|title=Atiyah and Singer receive 2004 Abel prize|journal=[[Notices of the American Mathematical Society]]|year=2006|issue=6|pages=650–651|volume=51|access-date=14 August 2008|archive-url=https://web.archive.org/web/20080910170610/http://www.ams.org/notices/200406/comm-abel.pdf|archive-date=10 September 2008|url-status=live}}</ref>

Within the United Kingdom, he was involved in the creation of the [[Isaac Newton Institute for Mathematical Sciences]] in Cambridge and was its first director (1990–1996). He was [[President of the Royal Society]] (1990–1995), [[Master of Trinity College, Cambridge]] (1990–1997),<ref name="cv2"/> [[Chancellor (education)|Chancellor]] of the [[University of Leicester]] (1995–2005),<ref name="cv2"/> and president of the [[Royal Society of Edinburgh]] (2005–2008).<ref>{{citation|url=http://www.royalsoced.org.uk/international/other_academies/norway.htm|title=Royal Society of Edinburgh announcement|access-date=14 August 2008|archive-url=https://web.archive.org/web/20081120025242/http://www.royalsoced.org.uk/international/other_academies/norway.htm|archive-date=20 November 2008|url-status=live}}</ref> From 1997 until his death in 2019 he was an honorary professor in the [[University of Edinburgh]].  He was a Trustee of the [[James Clerk Maxwell Foundation]].<ref>{{Cite web|date=2019|title=James Clerk Maxwell Foundation Annual Report and Summary Accounts|url=https://clerkmaxwellfoundation.org/Trustees_Report_2019.pdf}}</ref>

Atiyah's mathematical collaborators included [[Raoul Bott]], [[Friedrich Hirzebruch]]<ref>{{Cite journal |last1=Atiyah |first1=Michael |author-link1=Michael Atiyah |year=2014 |title=Friedrich Ernst Peter Hirzebruch 17 October 1927 – 27 May 2012 |journal=[[Biographical Memoirs of Fellows of the Royal Society]] |volume=60 |pages=229–247 |doi=10.1098/rsbm.2014.0010 |doi-access=free |title-link=Friedrich Hirzebruch}}</ref> and [[Isadore Singer]], and his students included [[Graeme Segal]], [[Nigel Hitchin]], [[Simon Donaldson]], and [[Edward Witten]].<ref name="Adv">{{cite web |title=Edward Witten – Adventures in physics and math (Kyoto Prize lecture 2014) |url=http://www.sns.ias.edu/ckfinder/userfiles/files/ComemorativeLecturePopular%281%29.pdf |access-date=30 October 2016 |archive-date=23 August 2016 |archive-url=https://web.archive.org/web/20160823223743/http://www.sns.ias.edu/ckfinder/userfiles/files/ComemorativeLecturePopular(1).pdf }}</ref> Together with Hirzebruch, he laid the foundations for [[topological K-theory]], an important tool in [[algebraic topology]], which, informally speaking, describes ways in which spaces can be twisted. His best known result, the [[Atiyah–Singer index theorem]], was proved with Singer in 1963 and is used in counting the number of independent solutions to [[differential equation]]s. Some of his more recent work was inspired by [[theoretical physics]], in particular [[instanton]]s and [[monopole (mathematics)|monopole]]s, which are responsible for some corrections in [[quantum field theory]]. He was awarded the [[Fields Medal]] in 1966 and the [[Abel Prize]] in 2004.

===Collaborations===
[[File:Mathematical Institute, University of Oxford.jpg|thumb|right|The old [[The Mathematical Institute, University of Oxford|Mathematical Institute]] (now the Department of Statistics) in [[Oxford]], where Atiyah supervised many of his students]]
Atiyah collaborated with many mathematicians. His three main collaborations were with [[Raoul Bott]] on the [[Atiyah–Bott fixed-point theorem]] and many other topics, with [[Isadore M. Singer]] on the [[Atiyah–Singer index theorem]], and with [[Friedrich Hirzebruch]] on topological K-theory,<ref>{{harvnb|Atiyah|2004|p=9}}</ref> all of whom he met at the [[Institute for Advanced Study]] in Princeton in 1955.<ref>{{harvnb|Atiyah|1988a|p=2}}</ref> His other collaborators included; [[J. Frank Adams]] ([[Hopf invariant]] problem), Jürgen Berndt (projective planes), Roger Bielawski (Berry–Robbins problem), Howard Donnelly ([[L-function]]s), [[Vladimir Drinfeld|Vladimir G. Drinfeld]] (instantons), Johan L. Dupont (singularities of [[vector field]]s), [[Lars Gårding]] ([[Hyperbolic partial differential equation|hyperbolic differential equation]]s), [[Nigel Hitchin|Nigel J. Hitchin]] (monopoles), [[William V. D. Hodge]] (Integrals of the second kind), [[Michael Hopkins (mathematician)|Michael Hopkins]] ([[K-theory]]), [[Lisa Jeffrey]] (topological Lagrangians), John D. S. Jones (Yang–Mills theory), [[Juan Maldacena]] (M-theory), [[Yuri I. Manin]] (instantons), [[Nick Manton|Nick S. Manton]] (Skyrmions), [[Vijay Kumar Patodi|Vijay K. Patodi]] (spectral asymmetry), A. N. Pressley (convexity), [[Elmer Rees]] (vector bundles), [[Wilfried Schmid]] (discrete series representations), [[Graeme Segal]] ([[K-theory|equivariant K-theory]]), Alexander Shapiro<ref>{{MathGenealogy|name=Alexander Shapiro|id=41807}}</ref> (Clifford algebras), L. Smith (homotopy groups of spheres), [[Paul Sutcliffe]] (polyhedra), [[David O. Tall]] (lambda rings), [[J. A. Todd|John A. Todd]] ([[Stiefel manifold]]s), [[Cumrun Vafa]] (M-theory), [[Richard S. Ward]] (instantons) and [[Edward Witten]] (M-theory, topological quantum field theories).<ref>{{harvnb|Atiyah|2004|pp=xi-xxv}}</ref>

His later research on [[gauge field theories]], particularly [[Yang–Mills]] theory, stimulated important interactions between [[geometry]] and [[theoretical physics|physics]], most notably in the work of Edward Witten.<ref>{{Cite web |url=http://www.sns.ias.edu/ckfinder/userfiles/files/ComemorativeLecturePopular%281%29.pdf |title=Edward Witten – Adventures in physics and math |access-date=30 October 2016 |archive-url=https://web.archive.org/web/20160823223743/http://www.sns.ias.edu/ckfinder/userfiles/files/ComemorativeLecturePopular(1).pdf |archive-date=23 August 2016 |url-status=live  }}</ref>

{{quote box
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|quote=If you attack a mathematical problem directly, very often you come to a dead end, nothing you do seems to work and you feel that if only you could peer round the corner there might be an easy solution. There is nothing like having somebody else beside you, because he can usually peer round the corner.
|source=Michael Atiyah<ref>{{harvnb|Atiyah|1988a|loc = paper 12, p. 233}}</ref>
}}
Atiyah's students included 
Peter Braam 1987,
[[Simon Donaldson]] 1983,
[[K. David Elworthy]] 1967,
Howard Fegan 1977,
Eric Grunwald 1977,
[[Nigel Hitchin]] 1972,
Lisa Jeffrey 1991,
[[Frances Kirwan]] 1984,
[[Peter Kronheimer]] 1986,
[[Ruth Lawrence]] 1989,
[[George Lusztig]] 1971,
[[Jack Morava]] 1968,
Michael Murray 1983,
Peter Newstead 1966,
[[Ian R. Porteous]] 1961,
[[John Roe (mathematician)|John Roe]] 1985,
Brian Sanderson	1963,
[[Rolph Ludwig Edward Schwarzenberger|Rolph Schwarzenberger]] 1960,
Graeme Segal 1967,
David Tall 1966,
and Graham White 1982.<ref name="genealogy"/>

Other contemporary mathematicians who influenced Atiyah include [[Roger Penrose]], [[Lars Hörmander]], [[Alain Connes]] and [[Jean-Michel Bismut]].<ref>{{harvnb|Atiyah|2004|p=10}}</ref> Atiyah said that the mathematician he most admired was [[Hermann Weyl]],<ref>{{harvnb|Atiyah|1988a|p=307}}</ref> and that his favourite mathematicians from before the 20th century were [[Bernhard Riemann]] and [[William Rowan Hamilton]].<ref>{{citation |url= http://www.superstringtheory.com/people/atiyah.html |title= Interview with Michael Atiyah |publisher= superstringtheory.com |access-date= 14 August 2008 |archive-url= https://web.archive.org/web/20080914135040/http://www.superstringtheory.com/people/atiyah.html |archive-date= 14 September 2008 |url-status= live }}</ref>

The seven volumes of Atiyah's collected papers include most of his work, except for his commutative algebra textbook;<ref>{{harvnb|Atiyah|Macdonald|1969}}</ref> the first five volumes are divided thematically and the sixth and seventh arranged by date.

===Algebraic geometry (1952–1958)===
{{Main|Algebraic geometry}}
[[File:Twisted cubic curve.png|thumb|right|250px|A [[twisted cubic curve]], the subject of Atiyah's first paper]]
Atiyah's early papers on algebraic geometry (and some general papers) are reprinted in the first volume of his collected works.<ref>{{harvnb|Atiyah|1988a}}</ref>

As an undergraduate Atiyah was interested in classical projective geometry, and wrote his first paper: a short note on [[twisted cubics]].<ref>{{harvnb|Atiyah|1988a|loc=paper 1}}</ref> He started research under [[W. V. D. Hodge]] and won the [[Smith's prize]] for 1954 for a [[Sheaf (mathematics)|sheaf-theoretic]] approach to [[ruled surface]]s,<ref>{{harvnb|Atiyah|1988a|loc=paper 2}}</ref> which encouraged Atiyah to continue in mathematics, rather than switch to his other interests—architecture and archaeology.<ref>{{harvnb|Atiyah|1988a|p= 1}}</ref>
His PhD thesis with Hodge was on a sheaf-theoretic approach to [[Solomon Lefschetz]]'s theory of integrals of the second kind on algebraic varieties, and resulted in an invitation to visit the Institute for Advanced Study in Princeton for a year.<ref>{{harvnb|Atiyah|1988a|loc=papers 3, 4}}</ref> While in Princeton he classified [[vector bundle]]s on an [[elliptic curve]] (extending [[Alexander Grothendieck]]'s classification of vector bundles on a genus 0 curve), by showing that any vector bundle is a sum of (essentially unique) indecomposable vector bundles,<ref>{{harvnb|Atiyah|1988a|loc=paper 5}}</ref> and then showing that the space of indecomposable vector bundles of given degree and positive dimension can be identified with the elliptic curve.<ref>{{harvnb|Atiyah|1988a|loc=paper 7}}</ref> He also studied double points on surfaces,<ref>{{harvnb|Atiyah|1988a|loc=paper 8}}</ref> giving the first example of a [[flop (algebraic geometry)|flop]], a special birational transformation of [[3-fold]]s that was later heavily used in [[Shigefumi Mori]]'s work on [[minimal model (birational geometry)|minimal model]]s for 3-folds.<ref>{{harvnb|Matsuki|2002}}.</ref> Atiyah's flop can also be used to show that the universal marked family of [[K3 surface]]s is not [[Hausdorff space|Hausdorff]].<ref>{{harvnb|Barth|Hulek|Peters|Van de Ven|2004}}</ref>

===K-theory (1959–1974)===
{{Main|K-theory}}
[[File:Möbius strip.jpg|thumb|right|250px|A [[Möbius band]] is the simplest non-trivial example of a [[vector bundle]].]]
Atiyah's works on [[K-theory]], including his book on K-theory<ref>{{harvnb|Atiyah|1989}}</ref> are reprinted in volume 2 of his collected works.<ref>{{harvnb|Atiyah|1988b}}</ref>

The simplest nontrivial example of a vector bundle is the [[Möbius band]] (pictured on the right): a strip of paper with a twist in it, which represents a rank 1 vector bundle over a circle (the circle in question being the centerline of the Möbius band). K-theory is a tool for working with higher-dimensional analogues of this example, or in other words for describing higher-dimensional twistings: elements of the K-group of a space are represented by vector bundles over it, so the Möbius band represents an element of the K-group of a circle.<ref>{{cite arXiv |last1=Atiyah |first1=Michael |author1-link=Michael Atiyah |year=2000 |title=K-Theory Past and Present |eprint=math/0012213}}</ref>

Topological K-theory was discovered by Atiyah and [[Friedrich Hirzebruch]]<ref>{{harvnb|Atiyah|1988b|loc=paper 24}}</ref> who were inspired by Grothendieck's proof of the [[Grothendieck–Riemann–Roch theorem]] and Bott's work on the [[Bott periodicity theorem|periodicity theorem]]. This paper only discussed the zeroth K-group; they shortly after extended it to K-groups of all degrees,<ref name="paper28">{{harvnb|Atiyah|1988b|loc=paper 28}}</ref> giving the first (nontrivial) example of a [[generalized cohomology theory]].

Several results showed that the newly introduced K-theory was in some ways more powerful than ordinary cohomology theory. Atiyah and Todd<ref>{{harvnb|Atiyah|1988b|loc=paper 26}}</ref> used K-theory to improve the lower bounds found using ordinary cohomology by Borel and Serre for the [[James number]], describing when a map from a complex [[Stiefel manifold]] to a sphere has a cross section. ([[J. Frank Adams|Adams]] and Grant-Walker later showed that the bound found by Atiyah and Todd was best possible.) Atiyah and Hirzebruch<ref>{{harvnb|Atiyah|1988a|loc=papers 30,31}}</ref> used K-theory to explain some relations between [[Steenrod operation]]s and [[Todd class]]es that Hirzebruch had noticed a few years before. The original solution of the [[Hopf invariant one problem]] operations by J. F. Adams was very long and complicated, using secondary cohomology operations. Atiyah showed how primary operations in K-theory could be used to give a short solution taking only a few lines, and in joint work with Adams<ref>{{harvnb|Atiyah|1988b|loc=paper 42}}</ref> also proved analogues of the result at odd primes.

[[File:Atiyah-Hirzebruch.jpeg|thumb|right|250px|Michael Atiyah and [[Friedrich Hirzebruch]] (right), the creators of K-theory]]
The [[Atiyah–Hirzebruch spectral sequence]] relates the ordinary cohomology of a space to its generalized cohomology theory.<ref name="paper28" /> (Atiyah and Hirzebruch used the case of K-theory, but their method works for all cohomology theories).

Atiyah showed<ref>{{harvnb|Atiyah|1961}}</ref> that for a finite group ''G'', the K theory of its [[classifying space]], ''BG'', is isomorphic to the [[completion (ring theory)|completion]] of its [[representation ring|character ring]]:
:<math> K(BG) \cong R(G)^{\wedge}.</math>
The same year<ref>{{harvnb|Atiyah|Hirzebruch|1961}}</ref> they proved the result for ''G'' any [[Compact group|compact]] [[Connected space|connected]] [[Lie group]]. Although soon the result could be extended to ''all'' compact Lie groups by incorporating results from [[Graeme Segal]]'s thesis,<ref>{{harvnb|Segal|1968}}</ref> that extension was complicated. However a simpler and more general proof was produced by introducing [[Equivariant algebraic K-theory|equivariant K-theory]], ''i.e.'' equivalence classes of ''G''-vector bundles over a compact ''G''-space ''X''.<ref>{{harvnb|Atiyah|Segal|1969}}</ref> It was shown that under suitable conditions the completion of the equivariant K theory of ''X'' is [[isomorphic]] to the ordinary K-theory of a space, <math>X_G</math>, which fibred over ''BG'' with fibre ''X'':
:<math>K_G(X)^{\wedge} \cong K(X_G). </math>

The original result then followed as a corollary by taking ''X'' to be a point: the left hand side reduced to the completion of ''R(G)'' and the right to ''K(BG)''. See [[Atiyah–Segal completion theorem]] for more details.

He defined new generalized homology and cohomology theories called bordism and [[cobordism]], and pointed out that many of the deep results on cobordism of manifolds found by [[René Thom]], [[C. T. C. Wall]], and others could be naturally reinterpreted as statements about these cohomology theories.<ref>{{harvnb|Atiyah|1988b|loc=paper 34}}</ref> Some of these cohomology theories, in particular complex cobordism, turned out to be some of the most powerful cohomology theories known.

{{quote box
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| quote = "Algebra is the offer made by the devil to the mathematician. The devil says: `I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine."
| source = Michael Atiyah<ref>{{harvnb|Atiyah|2004|loc = paper 160, p. 7}}</ref>
}}
He introduced<ref name="paper37">{{harvnb|Atiyah|1988b|loc=paper 37}}</ref> the [[J-group]] ''J''(''X'') of a finite complex ''X'', defined as the group of stable fiber homotopy equivalence classes of [[sphere bundle]]s; this was later studied in detail by [[J. F. Adams]] in a series of papers, leading to the [[Adams conjecture]].

With Hirzebruch he extended the [[Grothendieck–Riemann–Roch theorem]] to complex analytic embeddings,<ref name="paper37" /> and in a related paper<ref>{{harvnb|Atiyah|1988b|loc=paper 36}}</ref> they showed that the [[Hodge conjecture]] for integral cohomology is false. The Hodge conjecture for rational cohomology is, as of 2008, a major unsolved problem.<ref>{{citation|url=http://www.claymath.org/millennium/Hodge_Conjecture/Official_Problem_Description.pdf|publisher=The Clay Math Institute|title=The Hodge conjecture|first=Pierre|last=Deligne|access-date=14 August 2008|archive-url=https://web.archive.org/web/20080827172255/http://www.claymath.org/millennium/Hodge_Conjecture/Official_Problem_Description.pdf|archive-date=27 August 2008}}</ref>

The [[Bott periodicity theorem]] was a central theme in Atiyah's work on K-theory, and he repeatedly returned to it, reworking the proof several times to understand it better. With Bott he worked out an elementary proof,<ref>{{harvnb|Atiyah|1988b|loc=paper 40}}</ref> and gave another version of it in his book.<ref>{{harvnb|Atiyah|1988b|loc=paper 45}}</ref> With Bott and [[Alexander Shapiro|Shapiro]] he analysed the relation of Bott periodicity to the periodicity of [[Clifford algebras]];<ref>{{harvnb|Atiyah|1988b|loc=paper 39}}</ref> although this paper did not have a proof of the periodicity theorem, a proof along similar lines was shortly afterwards found by R. Wood.  He found a proof of several generalizations using [[elliptic operator]]s;<ref>{{harvnb|Atiyah|1988b|loc=paper 46}}</ref> this new proof used an idea that he used to give a particularly short and easy proof of Bott's original periodicity theorem.<ref>{{harvnb|Atiyah|1988b|loc=paper 48}}</ref>

===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>

===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
|align=right
|width=33%
|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.

===Later work (1986–2019)===
[[File:Edward Witten at Harvard.jpg|thumb|right|[[Edward Witten]], whose work on invariants of manifolds and [[topological quantum field theories]] was influenced by Atiyah]]
Many of the papers in the 6th volume<ref>{{harvs|nb|first=Michael|last=Atiyah|year1=2004}}</ref> of his collected works are surveys, obituaries, and general talks. Atiyah continued to publish subsequently, including several surveys, a popular book,<ref>{{harvnb|Atiyah|2007}}</ref> and another paper with [[Graeme Segal|Segal]] on [[twisted K-theory]].

One paper<ref>{{harvnb|Atiyah|2004|loc=paper 127}}</ref> is a detailed study of the [[Dedekind eta function]] from the point of view of topology and the index theorem.

Several of his papers from around this time study the connections between [[quantum field theory]], [[knot theory|knots]], and [[Donaldson theory]]. He introduced the concept of a [[topological quantum field theory]], inspired by Witten's work and Segal's definition of a conformal field theory.<ref>{{harvnb|Atiyah|2004|loc=paper 132}}</ref> His book "The Geometry and Physics of Knots"<ref>{{harvnb|Atiyah|1990}}</ref> describes the new [[knot invariant]]s found by [[Vaughan Jones]] and [[Edward Witten]] in terms of topological quantum field theories, and his paper with L. Jeffrey<ref>{{harvnb|Atiyah|2004|loc=paper 139}}</ref> explains Witten's Lagrangian giving the [[Donaldson invariant]]s.

He studied [[skyrmion]]s with Nick Manton,<ref>{{harvnb|Atiyah|2004|loc=papers 141, 142}}</ref> finding a relation with [[magnetic monopoles]] and [[instanton]]s, and giving a conjecture for the structure of the [[moduli space]] of two [[skyrmions]] as a certain [[subquotient]] of complex [[projective 3-space]].

Several papers<ref>{{harvnb|Atiyah|2004|loc=papers 163, 164, 165, 166, 167, 168}}</ref> were inspired by a question of [https://research-information.bristol.ac.uk/en/persons/jonathan-m-robbins(60101278-f877-4e14-946f-262f3b95d5ae).html Jonathan Robbins] (called the [[Berry–Robbins problem]]), who asked if there is a map from the configuration space of ''n'' points in 3-space to the flag manifold of the unitary group. Atiyah gave an affirmative answer to this question, but felt his solution was too computational and studied a conjecture that would give a more natural solution. He also related the question to [[Nahm's equation]], and introduced the [[Atiyah conjecture on configurations]].

{{quote box
|align=right
|width=33%
|quote=But for most practical purposes, you just use the classical groups. The exceptional Lie groups are just there to show you that the theory is a bit bigger; it is pretty rare that they ever turn up.
|source=Michael Atiyah<ref name="ReferenceA">{{harvnb|Atiyah|1988a|loc = paper 19, p. 19}}</ref>
}}
With [[Juan Maldacena]] and [[Cumrun Vafa]],<ref>{{harvnb|Atiyah|2004|loc=paper 169}}</ref> and [[E. Witten]]<ref>{{harvnb|Atiyah|2004|loc=paper 170}}</ref> he described the dynamics of [[M-theory]] on [[Joyce manifold|manifolds with G<sub>2</sub> holonomy]]. These papers seem to be the first time that Atiyah worked on exceptional Lie groups.

In his papers with [[Michael J. Hopkins|M. Hopkins]]<ref>{{harvnb|Atiyah|2004|loc=paper 172}}</ref> and G. Segal<ref>{{harvnb|Atiyah|2004|loc=paper 173}}</ref> he returned to his earlier interest of K-theory, describing some twisted forms of K-theory with applications in [[theoretical physics]].

In October 2016, he claimed<ref>{{cite arXiv |last=Atiyah |first=Michael |eprint=1610.09366 |title=The Non-Existent Complex 6-Sphere|class=math.DG |year=2016 }}</ref> a short proof of the non-existence of [[Complex manifold|complex structures]] on the 6-sphere. His proof, like many predecessors, is considered flawed by the mathematical community, even after the proof was rewritten in a revised form.<ref>{{citation|url=https://mathoverflow.net/q/263301|title=What is the current understanding regarding complex structures on the 6-sphere? (MathOverflow)|access-date=24 September 2018 }}</ref><ref>{{citation|url=https://mathoverflow.net/q/304071 |title=Atiyah's May 2018 paper on the 6-sphere (MathOverflow)|access-date=24 September 2018 }}</ref>

At the 2018 [[Klaus Tschira Foundation|Heidelberg Laureate Forum]], he claimed to have solved the [[Riemann hypothesis]], [[Hilbert's eighth problem]], [[proof by contradiction|by contradiction]] using the [[fine-structure constant]]. Again, the proof did not hold up and the hypothesis remains one of the six unsolved [[Millennium Prize Problems]] in mathematics, as of 2025.<ref>{{Cite news|url=https://www.science.org/content/article/skepticism-surrounds-renowned-mathematician-s-attempted-proof-160-year-old-hypothesis|title=Skepticism surrounds renowned mathematician's attempted proof of 160-year-old hypothesis|date=24 September 2018|work=Science {{!}} AAAS|access-date=26 September 2018|language=en|archive-url=https://web.archive.org/web/20180926115652/https://www.sciencemag.org/news/2018/09/skepticism-surrounds-renowned-mathematician-s-attempted-proof-160-year-old-hypothesis|archive-date=26 September 2018|url-status=live}}</ref><ref>{{cite news|url=https://www.newscientist.com/article/2180504-riemann-hypothesis-likely-remains-unsolved-despite-claimed-proof/|title=Riemann hypothesis likely remains unsolved despite claimed proof|access-date=24 September 2018|archive-url=https://web.archive.org/web/20180924175329/https://www.newscientist.com/article/2180504-riemann-hypothesis-likely-remains-unsolved-despite-claimed-proof/|archive-date=24 September 2018|url-status=live}}</ref>