Michael Atiyah

Michael Atiyah
Born	(1929-04-22) April 22, 1929 Hampstead, London, England
Nationality	United Kingdom
Alma mater	Trinity College, Cambridge
Awards	Fields Medal (1966) Copley Medal (1988) Abel Prize (2004)
Scientific career
Fields	Mathematics
Institutions	University of Cambridge University of Oxford Institute for Advanced Study University of Leicester University of Edinburgh

Sir Michael Francis Atiyah, OM, FRS, FRSE (b. April 22, 1929) is a British mathematician, widely considered one of the greatest geometers of the 20th century. His path-breaking work with Isadore Singer led to the proof of the Atiyah-Singer index theorem in the 1960s, a result that has helped pave the way for the development of several branches of mathematics since then.

He had also founded, earlier and together with Friedrich Hirzebruch, the study of another major tool in algebraic topology: topological K-theory. It was inspired by Alexander Grothendieck's work on generalising the Riemann-Roch theorem, and has since generated algebraic K-theory and many applications to mathematical physics.

Biography

Atiyah was born in Hampstead, London to a Scottish mother Jean and Lebanese writer Edward Atiyah. Patrick Atiyah, professor of law, is his brother. He grew up mostly in Cairo, Egypt, and Sudan. He later went to Manchester Grammar School and then Trinity College, Cambridge. He was a student of W. V. D. Hodge at Cambridge, where he was awarded a doctorate in 1955 for a thesis entitled Some Applications of Topological Methods in Algebraic Geometry. He was one of the founders, with Friedrich Hirzebruch, of topological K-theory, a branch of algebraic topology. He has collaborated with many other mathematicians, for example with Raoul Bott and Isadore Singer on the Atiyah–Bott fixed-point theorem and related developments leading to the Atiyah-Singer index theorem. This led to work in representation theory, and on the heat equation on manifolds. His later research on gauge field theories, particularly Yang-Mills theory, stimulated important interactions between geometry and physics, most notably in the work of Edward Witten.

Atiyah's many students include Simon Donaldson, Nigel Hitchin, Peter Kronheimer, Graeme Segal, George Lusztig, Jack Morava, Frances Kirwan, Lisa Jeffrey, Ruth Lawrence and John Roe.

Career

Atiyah rejuvenated British mathematics during his years at University of Oxford and Cambridge. He was also one of the driving forces behind the creation of the Isaac Newton Institute for Mathematical Sciences in Cambridge and became its first director. He received the Royal Medal of the Royal Society in 1968 and its Copley Medal in 1988. He served as president of the London Mathematical Society (1974 - 1976). In the 1990s, he has been president of the Royal Society, and master of Trinity College, Cambridge.

He was appointed Savilian Professor of Geometry at the University of Oxford in 1963, and successed by Ioan MacKenzie James in 1969.

Atiyah has also been active on the international scene. He has served as president of the Pugwash Conferences on Science and World Affairs. He was responsible for the founding of the InterAcademy Panel on International Issues, a global network of the world's scientific academies which aims to help its member academies to shape public policy in areas related to science. He also instigated the formation of the Association of European Academies (ALLEA), and has played an important role in the shaping of today’s European Mathematical Society (EMS).

Atiyah is now retired and an honorary professor at the University of Edinburgh. He served as Chancellor of the University of Leicester between 1995 and 2005, from where he received a Distinguished Honorary Fellowship in 2007. He has also been professor of mathematics at the Institute for Advanced Study in Princeton, New Jersey. Atiyah has been the president of the Royal Society of Edinburgh since 2005.

Awards and honours

In 1966, when he was thirty-seven years old, he was awarded the Fields Medal, for his work in developing K-theory, a generalized Lefschetz fixed-point theorem (jointly with Raoul Bott) and the Atiyah-Singer theorem, for which he also won, in 2004, the Abel Prize jointly with Isadore Singer.

Among other prizes he has received are the Feltrinelli Prize from the Accademia Nazionale dei Lincei (1981) and the King Faisal International Prize for Science (1987). He is a foreign member of the Russian Academy of Sciences.

Atiyah was knighted in 1983 and made a member of the Order of Merit in 1992.

He is listed as a Distinguished Supporter of the British Humanist Association.

His Erdős number is 3, via a chain of collaborations involving Laurel A. Smith and Persi Diaconis.

Mathematical work

The six volumes (Atiyah 2004b) of Atiyah's collected papers include most of his work, except for his commutative algebra textbook (Atiyah & Macdonald 1969) and a few works written since 2004.

Early work

Atiyah's early work (and general papers) are collected in (Michael Atiyah 1988a). His first few papers are all on algebraic geometry.

HIs first paper was a short note, written as an undergraduate, on twisted cubics (Atiyah 1988a, papers 1). His second paper (Atiyah 1988a, papers 2), giving a sheaf-theoretic approach to ruled surfaces, won the Smith's prize for 1954, which encouraged Atiyah to continue in mathematics, rather than switch to his other interests—architecture and archeology. His early work includes his papers with W. V. D. Hodge on a sheaf-theoretic approach to S. Lefschetz's theory of integrals of the second kind on algebraic varieties; this work was essentially his PhD thesis (Atiyah 1988a, papers 3, 4). He also classified vector bundles on an elliptic curve (extending 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 (Atiyah 1988a, paper 5), and then showing that the space of indecomposable vector bundles of given degree and positive dimension can be identified with the elliptic curve (Atiyah 1988a, papers 7). In a 1958 paper (Atiyah 1988a, papers 8) he studied double points on surfaces. This paper gives the first example of a special birational transformation of 3-folds, that was later called a "flop" and heavily used in Mori's work on 3-folds. Atiyah's flop can be used to show that the universal marked family of K3 surfaces is non-Hausdorff.

K theory

Atiyah's workss on K-theory, including his book (Michael Atiyah 1989) are collected in (Michael Atiyah 1988b).

Topological K-theory was discovered by Atiyah (1988b, paper24) (with some assistance from Hirzebruch), inspired by Grothendieck's proof of the Grothendieck-Riemann-Roch theorem and Bott's work on the periodicity theorem. This paper only discussed the zeroth K-theory group; it was shortly after extended to K-groups of all degrees in Atiyah (1988b, paper 28), giving was 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. In Atiyah (1988b, paper 26) he and Todd 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. (Adams and Grant-Walker later showed that the bound found by Atiyah and Todd was best possible.) Atiyah and Hirzebruch (Atiyah 1988a, papers 30,31) used K-theory to explain some relations between Steenrod operations and Todd classes 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 gives a short solution taking only a few lines, and in joint work with Adams (Atiyah 1988b, paper 42) also proved analogues of the result at odd primes.

The Atiyah-Hirzebruch spectral sequence relates the ordinary cohomology of a space to its generalized cohomology theory (Atiyah and Hirzebruch used the case of K-theory, but their method works for all cohomology theories).

In (Atiyah 1961), Atiyah showed that for a finite group G, the K-theory of its classifying space, BG, is isomorphic to the completion of its character ring:

$K(BG)\cong R(G)^{\wedge }.$

The same year, in (Atiyah & Hirzebruch 1961), the result was proved for G any compact connected Lie group. Although soon the result could be extended to all compact Lie groups by incorporating results from Graeme Segal's thesis (Segal 1968), that extension was complicated. However, in (Atiyah & Segal 1969), a simpler and more general proof was produced by introducing equivariant K-theory, i.e. equivalence classes of G-vector bundles over a compact G-space X. 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, $X_{G}$, which fibred over BG with fibre X:

$K_{G}(X)^{\wedge }\cong K(X_{G}).$

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.

In the paper (Atiyah 1988b, paper 34) 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 R. Thom, C. T. C. Wall, and others could be naturally reinterpreted as statements about these cohomology theories. Some of these cohomology theories, in particular complex cobordism, turned out to be some of the most powerful cohomology theories known.

In (Atiyah 1988b, paper 37) he introduced the J-group J(X) of a finite complex X, defined as the group of stable fiber homotopy equivalence classes of sphere bundles; 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 (Atiyah 1988b, paper 37), and in a related paper (Atiyah 1988b, paper 36) 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.)

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 (Atiyah 1988b, paper 40), and gave another version of it in his book (Atiyah 1988b, paper 45). With Bott and Schapiro he analysed the relation of Bott periodicity to the periodicity of Clifford algebras (Atiyah 1988b, paper 39); although this paper did not have a proof of the periodicity theorem, a proof along similar lines was shortly afterwards found by R. Wood. In (Atiyah 1988b, paper 46) he found a proof of several generalizations using elliptic operators; this new proof used an idea that he used in (Atiyah 1988b, paper 48) to give a particularly short and easy proof of Bott's original periodicity theorem.

Index theory 1

Atiyah's work on index theory is collected in (Michael Atiyah 1988c, 1988d).

The index problem for elliptic differential operators was posed in 1959 by Israel Gel'fand (1960). He noticed the homotopy invariance of the index, and asked for a formula for it by means of topological invariants. Some of the motivating examples included the Riemann–Roch theorem and its generalization the Hirzebruch–Riemann–Roch theorem, and the Hirzebruch signature theorem. Hirzebruch and Borel had proved the integrality of the  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 the paper (Atiyah & Singer 1963). The proof sketched in this announcement was never published by them, though it appears in the book (Palais 1965). Their first published proof (Atiyah & Singer 1968a) replaced the cobordism theory of the first proof with K-theory, and they used this to give proofs of various generalizations in a sequence of papers from 1968 to 1971, listed in the references below.

Atiyah, Bott, and Patodi (1973) gave a new proof of the index theorem using the heat equation, described in (Melrose 1993), (Berline, Getzler & Vergne 2004) and (Roe 1998).

• 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. This resulted in a series of papers on spectral assymetry, which were later unexpectedly used in theoretical physics, in particular in Witten's work on anomalies.

• 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. 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.

• If there is a group action of a group G on the compact manifold X, commuting with the elliptic operator, then one replaces ordinary K theory with equivariant K-theory. This was studied by Atiyah and Segal. Moreover, one gets generalizations of the Lefschetz fixed point theorem, with terms coming from fixed point submanifolds of the group G.

(Atiyah 1988c, paper 73) solves a problem asked independently by Hormander and Gel'fand, about whether complex powers of analytic functions define distributions. Atiyah used Hironaka's resolution of singularities to answer this affirmatively. A ingenious and elementary solution was found at about the same time by J. Bernstein, and discussed by Atiyah (1988a, paper 15).

Index theory 2

• Atiyah (1976) 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 L² index theorem, and was used by Atiyah & Schmid (1977) to rederive properties of the discrete series representations of semisimple Lie groups. 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.

In collaboration with Bott and Lars Gårding, Atiyah wrote three papers modernizing and generalizing Petrovsky's work on lacunas for fundamental solutions of linear partial differential equations (Atiyah 1988d, papers 84, 85, 86).

Gauge theory

Most of his papers on gauge theory are collected in (Michael Atiyah 1988e).

With Hitchin, Manin, and Drinfeld he introduced the ADHM construction of instantons on a sphere. He later used this and the Ward correspondence to classify all instantons on the 4-sphere. This later led to Donaldson's work on Donaldson theory.

With R. Bott he studied Yang-Mills equations over a Riemann surface.

With Hitchin he worked on magnetic monopoles, and studied their scattering using an idea of Nick Manton.

Recent work

Many of the papers in the 6th volume (Michael Atiyah 2004a) of his collected works are surveys, obituaries, and general talks. Since its publication, Atiyah has continued to publish, including several surveys, the popular book (Atiyah 2007), and another paper with Segal on twisted K-theory.

His book (Atiyah & Hitchin 1988) 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 hyperkahler. The metric is then used to study the scattering of two monopoles.

His paper (Atiyah 2004a, paper 127) 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 survey the connections between quantum field theory, knots, and Donaldson theory. In particular his book (Atiyah 1990) decribes the new knot invariants found by Vaughan Jones and Edward Witten in terms of topological quantum field theories, and his paper (Atiyah 2004a, paper 139) with L. Jeffrey explains Witten's Lagrangian giving the Donaldson invariants.

He studied skyrmions with Nick Manton (Atiyah 2004a, papers 141, 142), finding a relation with magnetic monopoles and instantons, and giving a conjecture for the structure of the moduli space of two skyrmions as a certain subquotient of complex projective 3-space.

A series of paper (Atiyah 2004a, papers 163, 164, 165, 166, 167, 168) was inspired by a question of M. Berry, 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 affirmitive 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.

With J. Maldacena and C. Vafa (Atiyah 2004a, paper 169) and E. Witten (Atiyah 2004a, paper 170) he described the dynamics of M-theory on manifolds with G₂ holonomy.

In his papers with M. Hopkins (Atiyah 2004a, paper 172) and G. Segal (Atiyah 2004a, paper 173) he returned to his earlier interest of K-theory, describing some twisted forms of K-theory with applications in theoretical physics.

Books

• Atiyah, Michael Francis (1989), K-theory, Advanced Book Classics (2nd ed.), Addison-Wesley, ISBN 978-0-201-09394-0, MR1043170 Reprinted as (Atiyah 1988b, item 45)
• Atiyah, Michael Francis (1970), Vector fields on manifolds, Arbeitsgemeinschaft für Forschung des Landes Nordrhein-Westfalen, Heft 200, Cologne: Westdeutscher Verlag, MR0263102 Reprinted as (Atiyah 1988b, item 50)
• Atiyah, Michael Francis; Macdonald, I. G. (1969), Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., MR0242802 A classic textbook covering standard commuative algebra.
• Atiyah, Michael Francis (1974), Elliptic operators and compact groups, Lecture Notes in Mathematics, Vol. 401, Berlin, New York: Springer-Verlag, MR0482866 Reprinted as (Atiyah 1988c, item 78)
• Atiyah, Michael Francis (1979), Geometry of Yang-Mills fields, Scuola Normale Superiore Pisa, Pisa, MR554924 Reprinted as (Atiyah 1988e, item 99)
• Atiyah, Michael; Hitchin, Nigel (1988), The geometry and dynamics of magnetic monopoles, M. B. Porter Lectures, Princeton University Press, ISBN 978-0-691-08480-0, MR934202 Reprinted as (Atiyah 2004a, item 126)
• Atiyah, Michael (1990), The geometry and physics of knots, Lezioni Lincee. [Lincei Lectures], Cambridge University Press, ISBN 978-0-521-39521-2, MR1078014 Reprinted as (Atiyah 2004a, item 136)
• Atiyah, Michael (1988a), Collected works. Vol. 1 Early papers: general papers, Oxford Science Publications, The Clarendon Press Oxford University Press, ISBN 978-0-19-853275-0, MR951892
• Atiyah, Michael (1988b), Collected works. Vol. 2 K-theory, Oxford Science Publications, The Clarendon Press Oxford University Press, ISBN 978-0-19-853276-7, MR951892
• Atiyah, Michael (1988c), Collected works. Vol. 3 Index theory: 1, Oxford Science Publications, The Clarendon Press Oxford University Press, ISBN 978-0-19-853277-4, MR951892
• Atiyah, Michael (1988d), Collected works. Vol. 4 Index theory:2, Oxford Science Publications, The Clarendon Press Oxford University Press, ISBN 978-0-19-853278-1, MR951892
• Atiyah, Michael (1988e), Collected works. Vol. 5 Gauge theories, Oxford Science Publications, The Clarendon Press Oxford University Press, ISBN 978-0-19-853279-8, MR951892
• Atiyah, Michael (2004a), Collected works. Vol. 6, Oxford Science Publications, The Clarendon Press Oxford University Press, ISBN 978-0-19-853099-2, MR2160826
• Atiyah, Michael (2004b), Collected works. 6 volume set, Oxford Science Publications, The Clarendon Press Oxford University Press, ISBN 978-0-19-8520948
• Atiyah, Michael (2007), Siamo tutti matematici (Italian: We are all mathematicians), Roma: Di Renzo Editore, p. 96, ISBN 8883231570

Papers

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References

• O'Connor, John J.; Robertson, Edmund F., "Michael Atiyah", MacTutor History of Mathematics Archive, University of St Andrews
• Michael Atiyah at the Mathematics Genealogy Project
• Official biographical sketch, Isaac Newton Institute
• Official biographical sketch, Notices of the American Mathematical Society, following award of Abel prize
• Atiyah, Michael Francis; Iagolnitzer, Daniel (1997), Fields Medallists' Lectures, World Scientific, pp. 113–4, ISBN 9810231172, biographical sketch.