Michael Atiyah
Encyclopedia
Sir Michael Francis Atiyah, OM, FRS, FRSE (born 22 April 1929) is a British mathematician
working in geometry
.
Atiyah grew up in Sudan
and Egypt
but spent most of his academic life in the United Kingdom at Oxford
and Cambridge
, and in the United States at the Institute for Advanced Study
. He has been president of the Royal Society
(1990–1995), master of Trinity College, Cambridge
(1990–1997), chancellor of the University of Leicester
(1995–2005), and president of the Royal Society of Edinburgh
(2005–2008). He is currently retired, and is an honorary professor at the University of Edinburgh
.
Atiyah's mathematical collaborators include Raoul Bott
, Friedrich Hirzebruch
and Isadore Singer
, and his students include Graeme Segal
, Nigel Hitchin
and Simon Donaldson
. 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 widely 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
s, which are responsible for some subtle corrections in quantum field theory
. He was awarded the Fields Medal
in 1966, the Copley Medal
in 1988, and the Abel Prize
in 2004.
Atiyah was born in Hampstead
, London
, to Lebanese
writer Edward Atiyah
and Scot
Jean Atiyah (née Levens). Patrick Atiyah
, professor of law, is his brother; he has one other brother, Joe, and a sister, Selma. He went to primary school at the Diocesan school in Khartoum
, Sudan (1934–1941) and to secondary school at Victoria College
in Cairo
and Alexandria
(1941–1945); the school was also attended by European nobility displaced by the Second World War and some future leaders of Arab nations. He returned to England
and Manchester Grammar School
for his HSC
studies (1945–1947) and did his national service with the Royal Electrical and Mechanical Engineers
(1947–1949). His undergraduate and postgraduate studies took place at Trinity College, Cambridge
(1949–1955). He was a doctoral student of William V. D. Hodge and was awarded a doctorate in 1955 for a thesis entitled Some Applications of Topological Methods in Algebraic Geometry.
Atiyah married Lily Brown on 30 July 1955, with whom he has three sons. He 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
(1958–1961). In 1961, he moved to the University of Oxford
, where he was a reader
and professor
ial fellow at St Catherine's College
(1961–1963). He became Savilian Professor of Geometry
and a professorial fellow of New College, Oxford
from 1963 to 1969. He then took up a three year professorship at the Institute for Advanced Study in 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.
Atiyah has been active on the international scene, for instance as president of the Pugwash Conferences on Science and World Affairs
from 1997 to 2002. 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).
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), Chancellor
of the University of Leicester
(1995–2005), and president of the Royal Society of Edinburgh
(2005–2008). He is now retired and is an honorary professor at the University of Edinburgh
.
Atiyah has collaborated with many other 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, all of whom he met at the Institute for Advanced Study
in Princeton in 1955. His other collaborators include J. Frank Adams (Hopf invariant
problem), Jürgen Berndt (projective planes), Roger Bielawski (Berry–Robbins problem), Howard Donnelly (L-function
s), Vladimir G. Drinfeld (instantons), Johan L. Dupont (singularities of vector field
s), Lars Gårding
(hyperbolic differential equation
s), Nigel J. Hitchin
(monopoles), William V. D. Hodge (Integrals of the second kind), Michael Hopkins (K-theory), Lisa Jeffrey (topological Lagrangians), John D. S. Jones (Yang–Mills theory), Juan Maldacena (M-theory), Yuri I. Manin
(instantons), Nick S. Manton
(Skyrmions), Vijay K. Patodi
(Spectral asymmetry), A. N. Pressley (convexity), Elmer Rees
(vector bundles), Wilfried Schmid
(discrete series representations), Graeme Segal
(equivariant K-theory), Alexander Shapiro
(Clifford algebras), L. Smith (homotopy groups of spheres), Paul Sutcliffe
(polyhedra), David O. Tall
(lambda rings), John A. Todd (Stiefel manifold
s), Cumrun Vafa
(M-theory), Richard S. Ward
(instantons) and Edward Witten
(M-theory, topological quantum field theories).
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
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 1985,
Brian Sanderson 1963,
Rolph Schwarzenberger 1960,
Graeme Segal 1967,
David Tall 1966,
and Graham White 1982.
Other contemporary mathematicians who influenced Atiyah include Roger Penrose
, Lars Hörmander
, Alain Connes
and Jean-Michel Bismut
. Atiyah said that the mathematician he most admired was Hermann Weyl
, and that his favorite mathematicians from before the 20th century were Bernhard Riemann
and William Rowan Hamilton
.
Atiyah's early papers on algebraic geometry (and some general papers) are reprinted in the first volume of his collected works.
As an undergraduate Atiyah was interested in classical projective geometry, and wrote his first paper: a short note on twisted cubics. He started research under W. V. D. Hodge
and won the Smith's prize
for 1954 for a sheaf-theoretic
approach to ruled surface
s, which encouraged Atiyah to continue in mathematics, rather than switch to his other interests—architecture and archaeology.
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. While in Princeton he classified vector bundle
s 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, and then showing that the space of indecomposable vector bundles of given degree and positive dimension can be identified with the elliptic curve. He also studied double points on surfaces, giving the first example of a flop, a special birational transformation of 3-fold
s that was later heavily used in Mori
's work on minimal model
s for 3-folds. Atiyah's flop can also be used to show that the universal marked family of K3 surface
s is non-Hausdorff.
Atiyah's works on K-theory, including his book on K-theory are reprinted in volume 2 of his collected works.
The simplest 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.
Topological K-theory
was discovered by Atiyah and Friedrich Hirzebruch
who were 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-group; they shortly after extended it to K-groups of all degrees, 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 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 used K-theory to explain some relations between Steenrod operations 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 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).
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
:
The same year they proved the result 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, that extension was complicated. However 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,
, which fibred over BG with fibre X:
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 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.
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, and in a related paper 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, and gave another version of it in his book. With Bott and Shapiro he analysed the relation of Bott periodicity to the periodicity of Clifford algebras; 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 he found a proof of several generalizations using elliptic operator
s; this new proof used an idea that he used to give a particularly short and easy proof of Bott's original periodicity theorem.
Atiyah's work on index theory is reprinted in volumes 3 and 4 of his collected works.
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.
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.
The index problem for elliptic differential operators was posed in 1959 by Gel'fand. 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 their 1963 paper. 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. Their first published proof 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. 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.
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. 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.
Atiyah and Segal combined this fixed point theorem with the index theorem as follows.
If there is a compact 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 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.
Atiyah solved a problem asked independently by Hörmander
and Gel'fand, about whether complex powers of analytic functions define distributions
. Atiyah used 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.
As an application of the equivariant index theorem, Atiyah and Hirzeburch showed that manifolds with effective circle actions have vanishing Â-genus. (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. 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.
Atiyah, Bott and Vijay K. Patodi
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. This resulted in a series of papers on spectral asymmetry, which were later unexpectedly used in theoretical physics, in particular in Witten's work on anomalies.
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.
Atiyah 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 and Schmid to give a geometric construction, using square integrable harmonic spinors, of Harish-Chandra's discrete series representation
s 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.
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.
Many of his papers on gauge theory and related topics are reprinted in volume 5 of his collected works. A common theme of these papers is the study of moduli spaces of solutions to certain non-linear partial differential equations, 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 classifing instantons on Euclidean space as this is conformally equivalent to a sphere and the equations for instantons are conformally invariant. With Hitchin and Singer he calculated the dimension of the moduli space of irreducible self-dual connections (instantons) for any principle bundle over a compact 4-dimensional Riemannian manifold. For example, the dimension of the space of SU2 instantons of rank k>0 is 8k−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 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. 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. Atiyah reformulated this construction using quaternion
s and wrote up a leisurely account of this classification of instantons on Euclidean space as a book.
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 invariants, 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.
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. 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, 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 groups 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.
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.
Atiyah and R. Bott used Morse theory
and the Yang–Mills equations over a Riemann surface
to reproduce and extending the results of Harder and Narasimhan.
An old result due to 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, and with Pressley gave a related generalization to infinite dimensional loop groups.
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 showed that this could be deduced from a more general formula in equivariant cohomology
, which was a consequence of well-known localization theorems. Atiyah showed 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.
With Hitchin he worked on magnetic monopole
s, and studied their scattering using an idea of Nick Manton
. His book 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 hyperkahler. 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.
Atiyah showed 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; this idea later became widely used by physicists.
Many of the papers in the 6th volume of his collected works are surveys, obituaries, and general talks. Since its publication, Atiyah has continued to publish, including several surveys, a popular book, and another paper with Segal
on twisted K-theory.
One paper 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, 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. His book 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 explains Witten's Lagrangian giving the Donaldson invariants.
He studied skyrmion
s with Nick Manton, 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 were 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 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.
With Juan Maldacena and Cumrun Vafa
, and E. Witten he described the dynamics of M-theory
on manifolds with G2 holonomy. These papers seem to be the first time that Atiyah has worked on exceptional Lie groups.
In his papers with M. Hopkins
and G. Segal he returned to his earlier interest of K-theory, describing some twisted forms of K-theory with applications in theoretical physics.
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
and the Atiyah–Singer theorem, for which he also won the Abel Prize
jointly with Isadore Singer
in 2004.
Among other prizes he has received are the Royal Medal
of the Royal Society
in 1968, the De Morgan Medal
of the London Mathematical Society
in 1980, the Antonio Feltrinelli Prize
from the Accademia Nazionale dei Lincei in 1981, the King Faisal International Prize for Science in 1987, the Copley Medal
of the Royal Society in 1988, the Benjamin Franklin Medal for Distinguished Achievement in the Sciences of the American Philosophical Society
in 1993, the Jawaharlal Nehru Birth Centenary Medal
of the Indian National Science Academy
in 1993, the President's Medal from the Institute of Physics
in 2008, the Grande Médaille
of the French Academy of Sciences
in 2010 and the Grand Officier of the French Légion d'honneur in 2011.
He was elected a foreign member of the National Academy of Sciences
, the American Academy of Arts and Sciences
(1969), the Academie des Sciences, the Akademie Leopoldina, the Royal Swedish Academy, the Royal Irish Academy
, the Royal Society of Edinburgh
, the American Philosophical Society
, the Indian National Science Academy
, the Chinese Academy of Science, the Australian Academy of Science
, the Russian Academy of Science, the Ukrainian Academy of Science, the Georgian Academy of Science, the Venezuela Academy of Science, the Norwegian Academy of Science and Letters
, the Royal Spanish Academy of Science, the Accademia dei Lincei
and the Moscow Mathematical Society
.
Atiyah has been awarded honorary degrees by the universities of Bonn, Warwick, Durham, St. Andrews, Dublin, Chicago, Cambridge, Edinburgh, Essex, London, Sussex, Ghent, Reading, Helsinki, Salamanca, Montreal, Wales, Lebanon, Queen's (Canada), Keele, Birmingham, UMIST, Brown, Heriot–Watt, Mexico, Oxford, Hong Kong (Chinese University), The Open University, American University of Beirut, the Technical University of Catalonia and Leicester.
Atiyah was made a Knight Bachelor
in 1983 and made a member of the Order of Merit in 1992.
The Michael Atiyah building at the University of Leicester
and the Michael Atiyah Chair in Mathematical Sciences at the American University of Beirut
were named after him.
on the rationality of the L2-Betti numbers.. An announcement of the index theorem. Reprinted in .. This gives a proof using K theory instead of cohomology. Reprinted in .. This reformulates the result as a sort of Lefschetz fixed point theorem, using equivariant K theory. Reprinted in .. This paper shows how to convert from the K-theory version to a version using cohomology. Reprinted in . This paper studies families of elliptic operators, where the index is now an element of the K-theory of the space parametrizing the family. Reprinted in .. This studies families of real (rather than complex) elliptic operators, when one can sometimes squeeze out a little extra information. Reprinted in .. This states a theorem calculating the Lefschetz number of an endomorphism of an elliptic complex. Reprinted in . (reprinted in )and . Reprinted in . These give the proofs and some applications of the results announced in the previous paper.; Reprinted in .; . Reprinted in .
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
working in geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
.
Atiyah grew up in Sudan
Sudan
Sudan , officially the Republic of the Sudan , is a country in North Africa, sometimes considered part of the Middle East politically. It is bordered by Egypt to the north, the Red Sea to the northeast, Eritrea and Ethiopia to the east, South Sudan to the south, the Central African Republic to the...
and Egypt
Egypt
Egypt , officially the Arab Republic of Egypt, Arabic: , is a country mainly in North Africa, with the Sinai Peninsula forming a land bridge in Southwest Asia. Egypt is thus a transcontinental country, and a major power in Africa, the Mediterranean Basin, the Middle East and the Muslim world...
but spent most of his academic life in the United Kingdom at Oxford
Oxford
The city of Oxford is the county town of Oxfordshire, England. The city, made prominent by its medieval university, has a population of just under 165,000, with 153,900 living within the district boundary. It lies about 50 miles north-west of London. The rivers Cherwell and Thames run through...
and Cambridge
Cambridge
The city of Cambridge is a university town and the administrative centre of the county of Cambridgeshire, England. It lies in East Anglia about north of London. Cambridge is at the heart of the high-technology centre known as Silicon Fen – a play on Silicon Valley and the fens surrounding the...
, and in the United States at the Institute for Advanced Study
Institute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...
. He has been president of the Royal Society
Royal Society
The Royal Society of London for Improving Natural Knowledge, known simply as the Royal Society, is a learned society for science, and is possibly the oldest such society in existence. Founded in November 1660, it was granted a Royal Charter by King Charles II as the "Royal Society of London"...
(1990–1995), master of Trinity College, Cambridge
Trinity College, Cambridge
Trinity College is a constituent college of the University of Cambridge. Trinity has more members than any other college in Cambridge or Oxford, with around 700 undergraduates, 430 graduates, and over 170 Fellows...
(1990–1997), chancellor of the University of Leicester
University of Leicester
The University of Leicester is a research-led university based in Leicester, England. The main campus is a mile south of the city centre, adjacent to Victoria Park and Wyggeston and Queen Elizabeth I College....
(1995–2005), and president of the Royal Society of Edinburgh
Royal Society of Edinburgh
The Royal Society of Edinburgh is Scotland's national academy of science and letters. It is a registered charity, operating on a wholly independent and non-party-political basis and providing public benefit throughout Scotland...
(2005–2008). He is currently retired, and is an honorary professor at the University of Edinburgh
University of Edinburgh
The University of Edinburgh, founded in 1583, is a public research university located in Edinburgh, the capital of Scotland, and a UNESCO World Heritage Site. The university is deeply embedded in the fabric of the city, with many of the buildings in the historic Old Town belonging to the university...
.
Atiyah's mathematical collaborators include Raoul Bott
Raoul Bott
Raoul Bott, FRS was a Hungarian mathematician known for numerous basic contributions to geometry in its broad sense...
, Friedrich Hirzebruch
Friedrich Hirzebruch
Friedrich Ernst Peter Hirzebruch is a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation.-Life:He was born in Hamm, Westphalia...
and Isadore Singer
Isadore Singer
Isadore Manuel Singer is an Institute Professor in the Department of Mathematics at the Massachusetts Institute of Technology...
, and his students include Graeme Segal
Graeme Segal
Graeme Bryce Segal is a British mathematician, and professor at the University of Oxford.Segal was educated at the University of Sydney, where he received his BSc degree in 1961. He went on to receive his D.Phil...
, Nigel Hitchin
Nigel Hitchin
Nigel Hitchin is a British mathematician working in the fields of differential geometry, algebraic geometry, and mathematical physics.-Academic career:...
and Simon Donaldson
Simon Donaldson
Simon Kirwan Donaldson FRS , is an English mathematician known for his work on the topology of smooth four-dimensional manifolds. He is now Royal Society research professor in Pure Mathematics and President of the Institute for Mathematical Science at Imperial College London...
. Together with Hirzebruch, he laid the foundations for topological K-theory
Topological K-theory
In mathematics, topological K-theory is a branch of algebraic topology. It was founded to study vector bundles on general topological spaces, by means of ideas now recognised as K-theory that were introduced by Alexander Grothendieck...
, an important tool in algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
, which, informally speaking, describes ways in which spaces can be twisted. His best known result, the Atiyah–Singer index theorem
Atiyah–Singer index theorem
In differential geometry, the Atiyah–Singer index theorem, proved by , states that for an elliptic differential operator on a compact manifold, the analytical index is equal to the topological index...
, was proved with Singer in 1963 and is widely used in counting the number of independent solutions to differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s. Some of his more recent work was inspired by theoretical physics, in particular instanton
Instanton
An instanton is a notion appearing in theoretical and mathematical physics. Mathematically, a Yang–Mills instanton is a self-dual or anti-self-dual connection in a principal bundle over a four-dimensional Riemannian manifold that plays the role of physical space-time in non-abelian gauge theory...
s and monopole
Monopole (mathematics)
In mathematics, a monopole is a connection over a principal bundle G with a section of the associated adjoint bundle. The connection and Higgs field should satisfy the Bogomolnyi equation and be of finite action....
s, which are responsible for some subtle corrections in quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
. He was awarded the Fields Medal
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...
in 1966, the Copley Medal
Copley Medal
The Copley Medal is an award given by the Royal Society of London for "outstanding achievements in research in any branch of science, and alternates between the physical sciences and the biological sciences"...
in 1988, and the Abel Prize
Abel Prize
The Abel Prize is an international prize presented annually by the King of Norway to one or more outstanding mathematicians. The prize is named after Norwegian mathematician Niels Henrik Abel . It has often been described as the "mathematician's Nobel prize" and is among the most prestigious...
in 2004.
Biography
Hampstead
Hampstead is an area of London, England, north-west of Charing Cross. Part of the London Borough of Camden in Inner London, it is known for its intellectual, liberal, artistic, musical and literary associations and for Hampstead Heath, a large, hilly expanse of parkland...
, London
London
London is the capital city of :England and the :United Kingdom, the largest metropolitan area in the United Kingdom, and the largest urban zone in the European Union by most measures. Located on the River Thames, London has been a major settlement for two millennia, its history going back to its...
, to Lebanese
Lebanon
Lebanon , officially the Republic of LebanonRepublic of Lebanon is the most common term used by Lebanese government agencies. The term Lebanese Republic, a literal translation of the official Arabic and French names that is not used in today's world. Arabic is the most common language spoken among...
writer Edward Atiyah
Edward Atiyah
Edward Selim Atiyah was a Lebanese author and political activist. He was born in 1903 and died in 1964. He is best known for his 1946 autobiography An Arab Tells His Story, and his 1955 book The Arabs....
and Scot
Scot
A Scot is a member of an ethnic group indigenous to Scotland, derived from the Latin name of Irish raiders, the Scoti.Scot may also refer to:People with the given name Scot:* Scot Brantley , American football linebacker...
Jean Atiyah (née Levens). Patrick Atiyah
Patrick Atiyah
Patrick S. Atiyah QC FBA is an English lawyer and academic. He is best known for his work as a common lawyer, particularly in the law of contract and for advocating reformation or abolition of the law of tort. He was made a Fellow of the British Academy in 1979.-Biography:Atiyah is a son of the...
, professor of law, is his brother; he has one other brother, Joe, and a sister, Selma. He went to primary school at the Diocesan school in Khartoum
Khartoum
Khartoum is the capital and largest city of Sudan and of Khartoum State. It is located at the confluence of the White Nile flowing north from Lake Victoria, and the Blue Nile flowing west from Ethiopia. The location where the two Niles meet is known as "al-Mogran"...
, Sudan (1934–1941) and to secondary school at Victoria College
Victoria College, Alexandria
Victoria College, Alexandria, was founded in 1902 under the impetus of the recently ennobled Evelyn Baring, 1st Earl of Cromer of the Barings Bank, that was heavily invested in Egyptian stability. For years the British Consul-General was ex officio on the board of Victoria College...
in Cairo
Cairo
Cairo , is the capital of Egypt and the largest city in the Arab world and Africa, and the 16th largest metropolitan area in the world. Nicknamed "The City of a Thousand Minarets" for its preponderance of Islamic architecture, Cairo has long been a centre of the region's political and cultural life...
and Alexandria
Alexandria
Alexandria is the second-largest city of Egypt, with a population of 4.1 million, extending about along the coast of the Mediterranean Sea in the north central part of the country; it is also the largest city lying directly on the Mediterranean coast. It is Egypt's largest seaport, serving...
(1941–1945); the school was also attended by European nobility displaced by the Second World War and some future leaders of Arab nations. He returned to England
England
England is a country that is part of the United Kingdom. It shares land borders with Scotland to the north and Wales to the west; the Irish Sea is to the north west, the Celtic Sea to the south west, with the North Sea to the east and the English Channel to the south separating it from continental...
and Manchester Grammar School
Manchester Grammar School
The Manchester Grammar School is the largest independent day school for boys in the UK . It is based in Manchester, England...
for his HSC
Higher School Certificate (UK)
The Higher School Certificate was a United Kingdom educational attainment standard qualification, established in 1918 by the Secondary Schools Examination Council . The Higher School Certificate Examination was usually taken at age 18, or two years after the School Certificate. It was abolished...
studies (1945–1947) and did his national service with the Royal Electrical and Mechanical Engineers
Royal Electrical and Mechanical Engineers
The Corps of Royal Electrical and Mechanical Engineers is a corps of the British Army that has responsibility for the maintenance, servicing and inspection of almost every electrical and mechanical piece of equipment within the British Army from Challenger II main battle tanks and WAH64 Apache...
(1947–1949). His undergraduate and postgraduate studies took place at Trinity College, Cambridge
Trinity College, Cambridge
Trinity College is a constituent college of the University of Cambridge. Trinity has more members than any other college in Cambridge or Oxford, with around 700 undergraduates, 430 graduates, and over 170 Fellows...
(1949–1955). He was a doctoral student of William V. D. Hodge and was awarded a doctorate in 1955 for a thesis entitled Some Applications of Topological Methods in Algebraic Geometry.
Lecturer
Lecturer is an academic rank. In the United Kingdom, lecturer is a position at a university or similar institution, often held by academics in their early career stages, who lead research groups and supervise research students, as well as teach...
(1957–1958), then a university lecturer and tutorial fellow at Pembroke College
Pembroke College, Cambridge
Pembroke College is a constituent college of the University of Cambridge, England.The college has over seven hundred students and fellows, and is the third oldest college of the university. Physically, it is one of the university's larger colleges, with buildings from almost every century since its...
(1958–1961). In 1961, he moved to the University of Oxford
University of Oxford
The University of Oxford is a university located in Oxford, United Kingdom. It is the second-oldest surviving university in the world and the oldest in the English-speaking world. Although its exact date of foundation is unclear, there is evidence of teaching as far back as 1096...
, where he was a reader
Reader (academic rank)
The title of Reader in the United Kingdom and some universities in the Commonwealth nations like Australia and New Zealand denotes an appointment for a senior academic with a distinguished international reputation in research or scholarship...
and professor
Professor
A professor is a scholarly teacher; the precise meaning of the term varies by country. Literally, professor derives from Latin as a "person who professes" being usually an expert in arts or sciences; a teacher of high rank...
ial fellow at St Catherine's College
St Catherine's College, Oxford
St Catherine's College, often called Catz, is one of the constituent colleges of the University of Oxford in England. Its motto is Nova et Vetera...
(1961–1963). He became Savilian Professor of Geometry
Savilian Professor of Geometry
The position of Savilian Professor of Geometry was established at the University of Oxford in 1619. It was founded by Sir Henry Savile, a mathematician and classical scholar who was Warden of Merton College, Oxford and Provost of Eton College, reacting to what has been described as "the wretched...
and a professorial fellow of New College, Oxford
New College, Oxford
New College is one of the constituent colleges of the University of Oxford in the United Kingdom.- Overview :The College's official name, College of St Mary, is the same as that of the older Oriel College; hence, it has been referred to as the "New College of St Mary", and is now almost always...
from 1963 to 1969. He then took up a three year professorship at the Institute for Advanced Study in Princeton
Princeton, New Jersey
Princeton is a community located in Mercer County, New Jersey, United States. It is best known as the location of Princeton University, which has been sited in the community since 1756...
after which he returned to Oxford as a Royal Society
Royal Society
The Royal Society of London for Improving Natural Knowledge, known simply as the Royal Society, is a learned society for science, and is possibly the oldest such society in existence. Founded in November 1660, it was granted a Royal Charter by King Charles II as the "Royal Society of London"...
Research Professor and professorial fellow of St Catherine's College. He was president of the London Mathematical Society
London Mathematical Society
-See also:* American Mathematical Society* Edinburgh Mathematical Society* European Mathematical Society* List of Mathematical Societies* Council for the Mathematical Sciences* BCS-FACS Specialist Group-External links:* * *...
from 1974 to 1976.
Atiyah has been active on the international scene, for instance as president of the Pugwash Conferences on Science and World Affairs
Pugwash Conferences on Science and World Affairs
The Pugwash Conferences on Science and World Affairs is an international organization that brings together scholars and public figures to work toward reducing the danger of armed conflict and to seek solutions to global security threats...
from 1997 to 2002. He also contributed to the foundation of the InterAcademy Panel on International Issues
InterAcademy Panel on International Issues
The InterAcademy Panel on International Issues is a global network consisting of over 100 national science academies. Founded in 1993, its stated goal is to help member academies advise the public on the scientific aspects of critical global issues...
, the Association of European Academies (ALLEA), and the European Mathematical Society
European Mathematical Society
The European Mathematical Society is a European organization dedicated to the development of mathematics in Europe. Its members are different mathematical societies in Europe, academic institutions and individual mathematicians...
(EMS).
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
President of the Royal Society
The president of the Royal Society is the elected director of the Royal Society of London. After informal meetings at Gresham College, the Royal Society was founded officially on 15 July 1662 for the encouragement of ‘philosophical studies’, by a royal charter which nominated William Brouncker as...
(1990–1995), Master of Trinity College, Cambridge (1990–1997), Chancellor
Chancellor (education)
A chancellor or vice-chancellor is the chief executive of a university. Other titles are sometimes used, such as president or rector....
of the University of Leicester
University of Leicester
The University of Leicester is a research-led university based in Leicester, England. The main campus is a mile south of the city centre, adjacent to Victoria Park and Wyggeston and Queen Elizabeth I College....
(1995–2005), and president of the Royal Society of Edinburgh
Royal Society of Edinburgh
The Royal Society of Edinburgh is Scotland's national academy of science and letters. It is a registered charity, operating on a wholly independent and non-party-political basis and providing public benefit throughout Scotland...
(2005–2008). He is now retired and is an honorary professor at the University of Edinburgh
University of Edinburgh
The University of Edinburgh, founded in 1583, is a public research university located in Edinburgh, the capital of Scotland, and a UNESCO World Heritage Site. The university is deeply embedded in the fabric of the city, with many of the buildings in the historic Old Town belonging to the university...
.
Collaborations
Raoul Bott
Raoul Bott, FRS was a Hungarian mathematician known for numerous basic contributions to geometry in its broad sense...
on the Atiyah–Bott fixed-point theorem
Atiyah–Bott fixed-point theorem
In mathematics, the Atiyah–Bott fixed-point theorem, proven by Michael Atiyah and Raoul Bott in the 1960s, is a general form of the Lefschetz fixed-point theorem for smooth manifolds M , which uses an elliptic complex on M...
and many other topics, with Isadore M. Singer on the Atiyah–Singer index theorem
Atiyah–Singer index theorem
In differential geometry, the Atiyah–Singer index theorem, proved by , states that for an elliptic differential operator on a compact manifold, the analytical index is equal to the topological index...
, and with Friedrich Hirzebruch
Friedrich Hirzebruch
Friedrich Ernst Peter Hirzebruch is a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation.-Life:He was born in Hamm, Westphalia...
on topological K-theory, all of whom he met at the Institute for Advanced Study
Institute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...
in Princeton in 1955. His other collaborators include J. Frank Adams (Hopf invariant
Hopf invariant
In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between spheres.- Motivation :In 1931 Heinz Hopf used Clifford parallels to construct the Hopf map\eta\colon S^3 \to S^2,...
problem), Jürgen Berndt (projective planes), Roger Bielawski (Berry–Robbins problem), Howard Donnelly (L-function
L-function
The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases...
s), Vladimir G. Drinfeld (instantons), Johan L. Dupont (singularities of vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
s), Lars Gårding
Lars Gårding
Lars Gårding is a Swedish mathematician. He has made notable contributions to the study of partial differential operators. He is a professor emeritus of mathematics at Lund University in Sweden...
(hyperbolic differential equation
Hyperbolic partial differential equation
In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation that, roughly speaking, has a well-posed initial value problem for the first n−1 derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data along...
s), Nigel J. Hitchin
Nigel Hitchin
Nigel Hitchin is a British mathematician working in the fields of differential geometry, algebraic geometry, and mathematical physics.-Academic career:...
(monopoles), William V. D. Hodge (Integrals of the second kind), Michael Hopkins (K-theory), Lisa Jeffrey (topological Lagrangians), John D. S. Jones (Yang–Mills theory), Juan Maldacena (M-theory), Yuri I. Manin
Yuri I. Manin
Yuri Ivanovitch Manin is a Soviet/Russian/German mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical logic to theoretical physics.-Biography:...
(instantons), Nick S. Manton
Nick Manton
Nicholas Stephen Manton is a mathematician at the University of Cambridge whose work has mostly concerned solitons in particle physics. He is perhaps best known for his paper on sphalerons and for his work on the interaction of BPS monopoles. He has also worked on skyrmions and, with Michael...
(Skyrmions), Vijay K. Patodi
Vijay Kumar Patodi
Vijay Kumar Patodi was an Indian mathematician who made fundamental contributions to differential geometry and topology. He was the first mathematician to apply heat equation methods to the proof of the Index Theorem for elliptic operators...
(Spectral asymmetry), A. N. Pressley (convexity), Elmer Rees
Elmer Rees
Elmer Gethin Rees, CBE, FRSE is a mathematician with publications in area ranging from topology, differential geometry, algebraic geometry, linear algebra and Morse theory to robotics...
(vector bundles), Wilfried Schmid
Wilfried Schmid
Wilfried Schmid is a German-American mathematician who works in Hodge theory, representation theory, and automorphic forms. He earned his Ph.D. at University of California, Berkeley in 1967 under the direction of Phillip Griffiths, and then taught at Berkeley and Columbia University, becoming a...
(discrete series representations), Graeme Segal
Graeme Segal
Graeme Bryce Segal is a British mathematician, and professor at the University of Oxford.Segal was educated at the University of Sydney, where he received his BSc degree in 1961. He went on to receive his D.Phil...
(equivariant K-theory), Alexander Shapiro
Sascha Schapiro
Alexander "Sascha" Schapiro , also known by the noms de guerre Alexander Tanarov, Sascha Piotr, and Sergei, was a Ukrainian anarchist revolutionary and father of eminent 20th century mathematician Alexandre Grothendieck....
(Clifford algebras), L. Smith (homotopy groups of spheres), Paul Sutcliffe
Paul Sutcliffe
Paul Sutcliffe is professor of mathematics, formerly at the University of Kent, now at Durham University. Using supercomputers he has modelled topological solitons. In a famous result he showed that skyrmions look like buckminsterfullerene. He was awarded the LMS Whitehead Prize in 2006.-External...
(polyhedra), David O. Tall
David O. Tall
David Orme Tall is a mathematics education theorist at the University of Warwick. One of his most influential works is the joint paper with Vinner Concept image and concept definition.... The "concept image" is a notion in cognitive theory. It consists of all the cognitive structure in the...
(lambda rings), John A. Todd (Stiefel manifold
Stiefel manifold
In mathematics, the Stiefel manifold Vk is the set of all orthonormal k-frames in Rn. That is, it is the set of ordered k-tuples of orthonormal vectors in Rn. It is named after Swiss mathematician Eduard Stiefel...
s), Cumrun Vafa
Cumrun Vafa
Cumrun Vafa is an Iranian-American leading string theorist from Harvard University where he started as a Harvard Junior Fellow. He is a recipient of the 2008 Dirac Medal.-Birth and education:...
(M-theory), Richard S. Ward
Richard S. Ward
Richard Samuel Ward FRS is a professor of mathematics at Durham University. He is most famous for his extension of Roger Penrose's twistor theory to nonlinear cases...
(instantons) and Edward Witten
Edward Witten
Edward Witten is an American theoretical physicist with a focus on mathematical physics who is currently a professor of Mathematical Physics at the Institute for Advanced Study....
(M-theory, topological quantum field theories).
His later research on gauge field theories, particularly Yang–Mills
Yang–Mills
Yang–Mills theory is a gauge theory based on the SU group. Wolfgang Pauli formulated in 1953 the first consistent generalization of the five-dimensional theory of Kaluza, Klein, Fock and others to a higher dimensional internal space...
theory, stimulated important interactions between geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
and physics
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, most notably in the work of Edward Witten.
Atiyah's many students include
Peter Braam 1987,
Simon Donaldson
Simon Donaldson
Simon Kirwan Donaldson FRS , is an English mathematician known for his work on the topology of smooth four-dimensional manifolds. He is now Royal Society research professor in Pure Mathematics and President of the Institute for Mathematical Science at Imperial College London...
1983,
K. David Elworthy
K. David Elworthy
Kenneth David Elworthy is a Professor Emeritus of Mathematics at the University of Warwick.He works on stochastic analysis, stochastic differential equations and geometric analysis....
1967,
Howard Fegan 1977,
Eric Grunwald 1977,
Nigel Hitchin
Nigel Hitchin
Nigel Hitchin is a British mathematician working in the fields of differential geometry, algebraic geometry, and mathematical physics.-Academic career:...
1972,
Lisa Jeffrey 1991,
Frances Kirwan
Frances Kirwan
Frances Clare Kirwan, FRS is a British mathematician, currently a Professor of Mathematics at the University of Oxford.Educated at Oxford High School, she studied at the University of Cambridge. She took a D.Phil at Oxford in 1984, supervised by Michael Atiyah...
1984,
Peter Kronheimer 1986,
Ruth Lawrence
Ruth Lawrence
Ruth Elke Lawrence-Naimark is an Associate Professor of mathematics at the Einstein Institute of Mathematics, Hebrew University of Jerusalem, and a researcher in knot theory and algebraic topology. Outside academia, she is best known for being a child prodigy in mathematics.- Youth :Ruth Lawrence...
1989,
George Lusztig 1971,
Jack Morava
Jack Morava
Jack Johnson Morava is an American topologist.Of Czech and Appalachian descent, he was raised in Mercedes, Texas ; an early interest in topology was strongly encouraged by his parents...
1968,
Michael Murray 1983,
Peter Newstead 1966,
Ian R. Porteous
Ian R. Porteous
Ian Robertson Porteous was a Scottish mathematician at the University of Liverpool and an educator on Merseyside. He is best known for three books on geometry and modern algebra...
1961,
John Roe 1985,
Brian Sanderson 1963,
Rolph Schwarzenberger 1960,
Graeme Segal 1967,
David Tall 1966,
and Graham White 1982.
Other contemporary mathematicians who influenced Atiyah include Roger Penrose
Roger Penrose
Sir Roger Penrose OM FRS is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College...
, Lars Hörmander
Lars Hörmander
Lars Valter Hörmander is a Swedish mathematician who has been called "the foremost contributor to the modern theory of linear partial differential equations". He was awarded the Fields Medal in 1962, and the Wolf Prize in 1988...
, Alain Connes
Alain Connes
Alain Connes is a French mathematician, currently Professor at the Collège de France, IHÉS, The Ohio State University and Vanderbilt University.-Work:...
and Jean-Michel Bismut
Jean-Michel Bismut
Jean-Michel Bismut is a French mathematician who has been a Professor at the Université Paris-Sud since 1981.He found a heat equation proof for the Atiyah–Singer index theorem. In 1990 he was awarded the Prix Ampere of the Academy of Sciences. He was elected as a member of the French Academy of...
. Atiyah said that the mathematician he most admired was Hermann Weyl
Hermann Weyl
Hermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...
, and that his favorite mathematicians from before the 20th century were Bernhard Riemann
Bernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....
and William Rowan Hamilton
William Rowan Hamilton
Sir William Rowan Hamilton was an Irish physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and techniques...
.
Mathematical work
The six volumes of Atiyah's collected papers include most of his work, except for his commutative algebra textbook and a few works written since 2004.Algebraic geometry (1952–1958)
As an undergraduate Atiyah was interested in classical projective geometry, and wrote his first paper: a short note on twisted cubics. He started research under W. V. D. Hodge
W. V. D. Hodge
William Vallance Douglas Hodge FRS was a Scottish mathematician, specifically a geometer.His discovery of far-reaching topological relations between algebraic geometry and differential geometry—an area now called Hodge theory and pertaining more generally to Kähler manifolds—has been a major...
and won the Smith's prize
Smith's Prize
The Smith's Prize was the name of each of two prizes awarded annually to two research students in theoretical Physics, mathematics and applied mathematics at the University of Cambridge, Cambridge, England.- History :...
for 1954 for a sheaf-theoretic
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
approach to ruled surface
Ruled surface
In geometry, a surface S is ruled if through every point of S there is a straight line that lies on S. The most familiar examples are the plane and the curved surface of a cylinder or cone...
s, which encouraged Atiyah to continue in mathematics, rather than switch to his other interests—architecture and archaeology.
His PhD thesis with Hodge was on a sheaf-theoretic approach to Solomon Lefschetz
Solomon Lefschetz
Solomon Lefschetz was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations.-Life:...
'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. While in Princeton he classified vector bundle
Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
s on an elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
(extending Grothendieck
Alexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
'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, and then showing that the space of indecomposable vector bundles of given degree and positive dimension can be identified with the elliptic curve. He also studied double points on surfaces, giving the first example of a flop, a special birational transformation of 3-fold
3-fold
In algebraic geometry, a 3-fold or threefold is a 3-dimensional algebraic variety.The Mori program showed that 3-folds have minimal models....
s that was later heavily used in Mori
Shigefumi Mori
-References:*Heisuke Hironaka, Fields Medallists Lectures, Michael F. Atiyah , Daniel Iagolnitzer ; World Scientific Publishing, 2007. ISBN 9810231172...
's work on minimal model
Minimal model (birational geometry)
In algebraic geometry, the minimal model program is part of the birational classification of algebraic varieties. Its goal is to construct a birational model of any complex projective variety which is as simple as possible...
s for 3-folds. Atiyah's flop can also be used to show that the universal marked family of K3 surface
K3 surface
In mathematics, a K3 surface is a complex or algebraic smooth minimal complete surface that is regular and has trivial canonical bundle.In the Enriques-Kodaira classification of surfaces they form one of the 5 classes of surfaces of Kodaira dimension 0....
s is non-Hausdorff.
K theory (1959–1974)
The simplest example of a vector bundle is the Möbius band
Mobius Band
Mobius Band is an electronic rock trio from Brooklyn, New York consisting of Noam Schatz , Peter Sax , and Ben Sterling .-History:...
(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.
Topological K-theory
K-theory
In mathematics, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It...
was discovered by Atiyah and Friedrich Hirzebruch
Friedrich Hirzebruch
Friedrich Ernst Peter Hirzebruch is a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation.-Life:He was born in Hamm, Westphalia...
who were 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-group; they shortly after extended it to K-groups of all degrees, 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 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
Stiefel manifold
In mathematics, the Stiefel manifold Vk is the set of all orthonormal k-frames in Rn. That is, it is the set of ordered k-tuples of orthonormal vectors in Rn. It is named after Swiss mathematician Eduard Stiefel...
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 used K-theory to explain some relations between Steenrod operations and Todd class
Todd class
In mathematics, the Todd class is a certain construction now considered a part of the theory in algebraic topology of characteristic classes. The Todd class of a vector bundle can be defined by means of the theory of Chern classes, and is encountered where Chern classes exist — most notably...
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 also proved analogues of the result at odd primes.
Atiyah showed that for a finite group G, the K-theory
K-theory
In mathematics, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It...
of its classifying space
Classifying space
In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG by a free action of G...
, BG, is isomorphic to the completion
Completion (ring theory)
In abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules. Completion is similar to localization, and together they are among the most basic tools in analysing commutative rings. Complete commutative rings have...
of its character ring
Representation ring
In mathematics, especially in the area of algebra known as representation theory, the representation ring of a group is a ring formed from all the linear representations of the group. For a given group, the ring will depend on the base field of the representations...
:
The same year they proved the result for G any compact
Compact group
In mathematics, a compact group is a topological group whose topology is compact. Compact groups are a natural generalisation of finite groups with the discrete topology and have properties that carry over in significant fashion...
connected
Connection (mathematics)
In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. There are a variety of kinds of connections in modern geometry, depending on what sort of data one wants to transport...
Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
. Although soon the result could be extended to all compact Lie groups by incorporating results from Graeme Segal
Graeme Segal
Graeme Bryce Segal is a British mathematician, and professor at the University of Oxford.Segal was educated at the University of Sydney, where he received his BSc degree in 1961. He went on to receive his D.Phil...
's thesis, that extension was complicated. However 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,
, which fibred over BG with fibre X: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
Cobordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary of a manifold. Two manifolds are cobordant if their disjoint union is the boundary of a manifold one dimension higher. The name comes...
, and pointed out that many of the deep results on cobordism of manifolds found by R. Thom, C. T. C. Wall
C. T. C. Wall
Charles Terence Clegg Wall is a leading British mathematician, educated at Marlborough and Trinity College, Cambridge. He is an emeritus professor of the University of Liverpool, where he was first appointed Professor in 1965...
, 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.
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, and in a related paper they showed that the Hodge conjecture
Hodge conjecture
The Hodge conjecture is a major unsolved problem in algebraic geometry which relates the algebraic topology of a non-singular complex algebraic variety and the subvarieties of that variety. More specifically, the conjecture says that certain de Rham cohomology classes are algebraic, that is, they...
for integral cohomology is false. The Hodge conjecture for rational cohomology is, as of 2008, a major unsolved problem.
The Bott periodicity theorem
Bott periodicity theorem
In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by , which proved to be of foundational significance for much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy...
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, and gave another version of it in his book. With Bott and Shapiro he analysed the relation of Bott periodicity to the periodicity of Clifford algebras; 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 he found a proof of several generalizations using elliptic operator
Elliptic operator
In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which implies the key property that the principal symbol is...
s; this new proof used an idea that he used to give a particularly short and easy proof of Bott's original periodicity theorem.
Index theory (1963–1984)
Atiyah's work on index theory is reprinted in volumes 3 and 4 of his collected works.
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.
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.
The index problem for elliptic differential operators was posed in 1959 by Gel'fand. 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
Riemann–Roch theorem
The Riemann–Roch theorem is an important tool in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles...
and its generalization the Hirzebruch–Riemann–Roch theorem, and the Hirzebruch signature theorem. Hirzebruch
Friedrich Hirzebruch
Friedrich Ernst Peter Hirzebruch is a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation.-Life:He was born in Hamm, Westphalia...
and Borel
Armand Borel
Armand Borel was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993...
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
Dirac operator
In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian...
(which was rediscovered by Atiyah and Singer in 1961).
The first announcement of the Atiyah–Singer theorem was their 1963 paper. 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. Their first published proof was more similar to Grothendieck's proof of the Grothendieck–Riemann–Roch theorem, replacing the cobordism
Cobordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary of a manifold. Two manifolds are cobordant if their disjoint union is the boundary of a manifold one dimension higher. The name comes...
theory of the first proof with K-theory
K-theory
In mathematics, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It...
, 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. 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.
Elliptic complex
In mathematics, in particular in partial differential equations and differential geometry, an elliptic complex generalizes the notion of an elliptic operator to sequences. Elliptic complexes isolate those features common to the de Rham complex and the Dolbeault complex which are essential for...
in terms of a sum over the fixed points of the endomorphism. As special cases their formula included the Weyl character formula
Weyl character formula
In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by ....
, and several new results about elliptic curves with complex multiplication, some of which were initially disbelieved by experts.
Atiyah and Segal combined this fixed point theorem with the index theorem as follows.
If there is a compact group action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
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 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.
Atiyah solved a problem asked independently by Hörmander
Lars Hörmander
Lars Valter Hörmander is a Swedish mathematician who has been called "the foremost contributor to the modern theory of linear partial differential equations". He was awarded the Fields Medal in 1962, and the Wolf Prize in 1988...
and Gel'fand, about whether complex powers of analytic functions define distributions
Distribution (mathematics)
In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...
. Atiyah used Hironaka
Heisuke Hironaka
is a Japanese mathematician. After completing his undergraduate studies at Kyoto University, he received his Ph.D. from Harvard while under the direction of Oscar Zariski. He won the Fields Medal in 1970....
'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.
As an application of the equivariant index theorem, Atiyah and Hirzeburch showed that manifolds with effective circle actions have vanishing Â-genus. (Lichnerowicz showed that if a manifold has a metric of positive scalar curvature then the Â-genus vanishes.)
With Elmer Rees
Elmer Rees
Elmer Gethin Rees, CBE, FRSE is a mathematician with publications in area ranging from topology, differential geometry, algebraic geometry, linear algebra and Morse theory to robotics...
, 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. Horrocks
Geoffrey Horrocks
Geoffrey Horrocks is a British mathematician whose work on vector bundles has been important for the ADHM construction.He was a professor at Newcastle University until his retirement in 1998.-References:...
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.
Vijay Kumar Patodi
Vijay Kumar Patodi was an Indian mathematician who made fundamental contributions to differential geometry and topology. He was the first mathematician to apply heat equation methods to the proof of the Index Theorem for elliptic operators...
gave a new proof of the index theorem using the heat equation
Heat equation
The heat equation is an important partial differential equation which describes the distribution of heat in a given region over time...
.
If the manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of 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
Signature operator
In mathematics, the signature operator is an elliptic differential operator defined on a certain subspace of the space of differential forms on an even-dimensional compact Riemannian manifold, whose analytic index is the same as the topological signature of the manifold if the dimension of the...
) 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 asymmetry, which were later unexpectedly used in theoretical physics, in particular in Witten's work on anomalies.
Hyperbolic partial differential equation
In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation that, roughly speaking, has a well-posed initial value problem for the first n−1 derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data along...
s often have Petrovsky lacuna
Petrovsky lacuna
In mathematics, a Petrovsky lacuna, named for the Russian mathematician I. G. Petrovsky, is a region where the fundamental solution of a linear hyperbolic partial differential equation vanishes....
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
Lars Gårding
Lars Gårding is a Swedish mathematician. He has made notable contributions to the study of partial differential operators. He is a professor emeritus of mathematics at Lund University in Sweden...
, Atiyah wrote three papers updating and generalizing Petrovsky's work.
Atiyah 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
Von Neumann algebra
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. They were originally introduced by John von Neumann, motivated by his study of single operators, group...
; this index is in general real rather than integer valued. This version is called the L2 index theorem, and was used by Atiyah and Schmid to give a geometric construction, using square integrable harmonic spinors, of Harish-Chandra's discrete series representation
Discrete series representation
In mathematics, a discrete series representation is an irreducible unitary representation of a locally compact topological group G that is a subrepresentation of the left regular representation of G on L²...
s 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.
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.
Gauge theory (1977–1985)
Penrose transform
In mathematical physics, the Penrose transform, introduced by , is a complex analogue of the Radon transform that relates massless fields on spacetime to cohomology of sheaves on complex projective space...
, 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 classifing instantons on Euclidean space as this is conformally equivalent to a sphere and the equations for instantons are conformally invariant. With Hitchin and Singer he calculated the dimension of the moduli space of irreducible self-dual connections (instantons) for any principle bundle over a compact 4-dimensional Riemannian manifold. For example, the dimension of the space of SU2 instantons of rank k>0 is 8k−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 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. With Hitchin he used ideas of Horrocks to solve this problem, giving the ADHM construction
ADHM construction
The ADHM construction or monad construction is the construction of all instantons using method of linear algebra by Michael Atiyah, Vladimir G. Drinfel'd, Nigel. J. Hitchin, Yuri I...
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. Atiyah reformulated this construction using quaternion
Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
s and wrote up a leisurely account of this classification of instantons on Euclidean space as a book.
Atiyah's work on instanton moduli spaces was used in Donaldson's work on Donaldson theory
Donaldson theory
Donaldson theory is the study of smooth 4-manifolds using gauge theory. It was started by Simon Donaldson who proved Donaldson's theorem restricting the possible quadratic forms on the second cohomology group of a compact simply connected 4-manifold....
. Donaldson showed that the moduli space of (degree 1) instantons over a compact simply connected 4-manifold
4-manifold
In mathematics, 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different...
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
Smooth structure
In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold....
s on 4 dimensional Euclidean space. Donaldson went on to use the other moduli spaces studied by Atiyah to define Donaldson invariants, 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.
Green's function
Green's function
In mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to specific initial conditions or boundary conditions...
s for linear partial differential equations can often be found by using the Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
to convert this into an algebraic problem. Atiyah used a non-linear version of this idea. 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, 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 groups 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.
Harder and M. S. Narasimhan described the cohomology of the moduli space
Moduli space
In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...
s of stable vector bundle
Stable vector bundle
In mathematics, a stable vector bundle is a vector bundle that is stable in the sense of geometric invariant theory. They were defined by .-Stable vector bundles over curves:...
s over Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
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.
Atiyah and R. Bott used Morse theory
Morse theory
In differential topology, the techniques of Morse theory give a very direct way of analyzing the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a differentiable function on a manifold will, in a typical case, reflect...
and the Yang–Mills equations over a Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
to reproduce and extending the results of Harder and Narasimhan.
An old result due to Schur
Issai Schur
Issai Schur was a mathematician who worked in Germany for most of his life. He studied at Berlin...
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
Symplectic manifold
In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology...
s acted on by a torus, showing that the image of the manifold under the moment map is a convex polyhedron, and with Pressley gave a related generalization to infinite dimensional loop groups.
Duistermaat and Heckman found a striking formula, saying that the push-forward of the Liouville measure of a moment map
Moment map
In mathematics, specifically in symplectic geometry, the momentum map is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The moment map generalizes the classical notions of linear and angular momentum...
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 showed that this could be deduced from a more general formula in equivariant cohomology
Equivariant cohomology
In mathematics, equivariant cohomology is a theory from algebraic topology which applies to spaces with a group action. It can be viewed as a common generalization of group cohomology and an ordinary cohomology theory....
, which was a consequence of well-known localization theorems. Atiyah showed 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
Duistermaat–Heckman formula
In mathematics, the Duistermaat–Heckman formula, due to , states that thepushforward of the canonical measure on a symplectic manifold under the moment map is a piecewise polynomial measure...
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.
With Hitchin he worked on magnetic monopole
Magnetic monopole
A magnetic monopole is a hypothetical particle in particle physics that is a magnet with only one magnetic pole . In more technical terms, a magnetic monopole would have a net "magnetic charge". Modern interest in the concept stems from particle theories, notably the grand unified and superstring...
s, and studied their scattering using an idea of Nick Manton
Nick Manton
Nicholas Stephen Manton is a mathematician at the University of Cambridge whose work has mostly concerned solitons in particle physics. He is perhaps best known for his paper on sphalerons and for his work on the interaction of BPS monopoles. He has also worked on skyrmions and, with Michael...
. His book 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 hyperkahler. 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.
Atiyah showed 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; this idea later became widely used by physicists.
Later work (1986 onwards)
Graeme Segal
Graeme Bryce Segal is a British mathematician, and professor at the University of Oxford.Segal was educated at the University of Sydney, where he received his BSc degree in 1961. He went on to receive his D.Phil...
on twisted K-theory.
One paper is a detailed study of the Dedekind eta function
Dedekind eta function
The Dedekind eta function, named after Richard Dedekind, is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive...
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, knots, and Donaldson theory. He introduced the concept of a topological quantum field theory
Topological quantum field theory
A topological quantum field theory is a quantum field theory which computes topological invariants....
, inspired by Witten's work and Segal's definition of a conformal field theory. His book describes the new knot invariant
Knot invariant
In the mathematical field of knot theory, a knot invariant is a quantity defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some invariants are indeed numbers, but invariants can range from the...
s found by Vaughan Jones
Vaughan Jones
Sir Vaughan Frederick Randal Jones, KNZM, FRS, FRSNZ is a New Zealand mathematician, known for his work on von Neumann algebras, knot polynomials and conformal field theory. He was awarded a Fields Medal in 1990, and famously wore a New Zealand rugby jersey when he accepted the prize...
and Edward Witten
Edward Witten
Edward Witten is an American theoretical physicist with a focus on mathematical physics who is currently a professor of Mathematical Physics at the Institute for Advanced Study....
in terms of topological quantum field theories, and his paper with L. Jeffrey explains Witten's Lagrangian giving the Donaldson invariants.
He studied skyrmion
Skyrmion
In theoretical physics, a skyrmion is a mathematical model used to model baryons . It was conceived by Tony Skyrme.-Overview:...
s with Nick Manton, finding a relation with magnetic monopoles and instanton
Instanton
An instanton is a notion appearing in theoretical and mathematical physics. Mathematically, a Yang–Mills instanton is a self-dual or anti-self-dual connection in a principal bundle over a four-dimensional Riemannian manifold that plays the role of physical space-time in non-abelian gauge theory...
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 were 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 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.
With Juan Maldacena and Cumrun Vafa
Cumrun Vafa
Cumrun Vafa is an Iranian-American leading string theorist from Harvard University where he started as a Harvard Junior Fellow. He is a recipient of the 2008 Dirac Medal.-Birth and education:...
, and E. Witten he described the dynamics of M-theory
M-theory
In theoretical physics, M-theory is an extension of string theory in which 11 dimensions are identified. Because the dimensionality exceeds that of superstring theories in 10 dimensions, proponents believe that the 11-dimensional theory unites all five string theories...
on manifolds with G2 holonomy. These papers seem to be the first time that Atiyah has worked on exceptional Lie groups.
In his papers with M. Hopkins
Michael J. Hopkins
Michael Jerome Hopkins is an American mathematician known for work in algebraic topology.-Life:He received his Ph.D. from Northwestern University in 1984 under the direction of Mark Mahowald. In 1984 he also received his D.Phil...
and G. Segal he returned to his earlier interest of K-theory, describing some twisted forms of K-theory with applications in theoretical physics.
Awards and honours
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...
, for his work in developing K-theory, a generalized Lefschetz fixed-point theorem
Lefschetz fixed-point theorem
In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X...
and the Atiyah–Singer theorem, for which he also won the Abel Prize
Abel Prize
The Abel Prize is an international prize presented annually by the King of Norway to one or more outstanding mathematicians. The prize is named after Norwegian mathematician Niels Henrik Abel . It has often been described as the "mathematician's Nobel prize" and is among the most prestigious...
jointly with Isadore Singer
Isadore Singer
Isadore Manuel Singer is an Institute Professor in the Department of Mathematics at the Massachusetts Institute of Technology...
in 2004.
Among other prizes he has received are the Royal Medal
Royal Medal
The Royal Medal, also known as The Queen's Medal, is a silver-gilt medal awarded each year by the Royal Society, two for "the most important contributions to the advancement of natural knowledge" and one for "distinguished contributions in the applied sciences" made within the Commonwealth of...
of the Royal Society
Royal Society
The Royal Society of London for Improving Natural Knowledge, known simply as the Royal Society, is a learned society for science, and is possibly the oldest such society in existence. Founded in November 1660, it was granted a Royal Charter by King Charles II as the "Royal Society of London"...
in 1968, the De Morgan Medal
De Morgan Medal
The De Morgan Medal is a prize for outstanding contribution to mathematics, awarded by the London Mathematical Society. The Society's most prestigious award, it is given in memory of Augustus De Morgan, who was the first President of the society....
of the London Mathematical Society
London Mathematical Society
-See also:* American Mathematical Society* Edinburgh Mathematical Society* European Mathematical Society* List of Mathematical Societies* Council for the Mathematical Sciences* BCS-FACS Specialist Group-External links:* * *...
in 1980, the Antonio Feltrinelli Prize
Antonio Feltrinelli Prize
The Antonio Feltrinelli Prize is a prestigious award for achievement in the arts, music, literature, history, philosophy, medicine, and physical and mathematical sciences. The award comes with a monetary grant, a certificate, and a gold medal. The prize is awarded once every five years in each...
from the Accademia Nazionale dei Lincei in 1981, the King Faisal International Prize for Science in 1987, the Copley Medal
Copley Medal
The Copley Medal is an award given by the Royal Society of London for "outstanding achievements in research in any branch of science, and alternates between the physical sciences and the biological sciences"...
of the Royal Society in 1988, the Benjamin Franklin Medal for Distinguished Achievement in the Sciences of the American Philosophical Society
American Philosophical Society
The American Philosophical Society, founded in 1743, and located in Philadelphia, Pa., is an eminent scholarly organization of international reputation, that promotes useful knowledge in the sciences and humanities through excellence in scholarly research, professional meetings, publications,...
in 1993, the Jawaharlal Nehru Birth Centenary Medal
of the Indian National Science Academy
Indian National Science Academy
The Indian National Science Academy , New Delhi is the apex body of Indian scientists representing all branches of science & technology.-History:...
in 1993, the President's Medal from the Institute of Physics
Institute of Physics
The Institute of Physics is a scientific charity devoted to increasing the practice, understanding and application of physics. It has a worldwide membership of around 40,000....
in 2008, the Grande Médaille
Grande Médaille
The Grande Médaille of the French Academy of Sciences, established in 1997, is awarded annually to a researcher who has contributed decisively to the development of science...
of the French Academy of Sciences
French Academy of Sciences
The French Academy of Sciences is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research...
in 2010 and the Grand Officier of the French Légion d'honneur in 2011.
He was elected a foreign member of the National Academy of Sciences
United States National Academy of Sciences
The National Academy of Sciences is a corporation in the United States whose members serve pro bono as "advisers to the nation on science, engineering, and medicine." As a national academy, new members of the organization are elected annually by current members, based on their distinguished and...
, the American Academy of Arts and Sciences
American Academy of Arts and Sciences
The American Academy of Arts and Sciences is an independent policy research center that conducts multidisciplinary studies of complex and emerging problems. The Academy’s elected members are leaders in the academic disciplines, the arts, business, and public affairs.James Bowdoin, John Adams, and...
(1969), the Academie des Sciences, the Akademie Leopoldina, the Royal Swedish Academy, the Royal Irish Academy
Royal Irish Academy
The Royal Irish Academy , based in Dublin, is an all-Ireland, independent, academic body that promotes study and excellence in the sciences, humanities and social sciences. It is one of Ireland's premier learned societies and cultural institutions and currently has around 420 Members, elected in...
, the Royal Society of Edinburgh
Royal Society of Edinburgh
The Royal Society of Edinburgh is Scotland's national academy of science and letters. It is a registered charity, operating on a wholly independent and non-party-political basis and providing public benefit throughout Scotland...
, the American Philosophical Society
American Philosophical Society
The American Philosophical Society, founded in 1743, and located in Philadelphia, Pa., is an eminent scholarly organization of international reputation, that promotes useful knowledge in the sciences and humanities through excellence in scholarly research, professional meetings, publications,...
, the Indian National Science Academy
Indian National Science Academy
The Indian National Science Academy , New Delhi is the apex body of Indian scientists representing all branches of science & technology.-History:...
, the Chinese Academy of Science, the Australian Academy of Science
Australian Academy of Science
The Australian Academy of Science was founded in 1954 by a group of distinguished Australians, including Australian Fellows of the Royal Society of London. The first president was Sir Mark Oliphant. The Academy is modelled after the Royal Society and operates under a Royal Charter; as such it is...
, the Russian Academy of Science, the Ukrainian Academy of Science, the Georgian Academy of Science, the Venezuela Academy of Science, the Norwegian Academy of Science and Letters
Norwegian Academy of Science and Letters
The Norwegian Academy of Science and Letters is a learned society based in Oslo, Norway.-History:The University of Oslo was established in 1811. The idea of a learned society in Christiania surfaced for the first time in 1841. The city of Throndhjem had no university, but had a learned...
, the Royal Spanish Academy of Science, the Accademia dei Lincei
Accademia dei Lincei
The Accademia dei Lincei, , is an Italian science academy, located at the Palazzo Corsini on the Via della Lungara in Rome, Italy....
and the Moscow Mathematical Society
Moscow Mathematical Society
The Moscow Mathematical Society is a society of Moscow mathematicians aimed at the development of mathematics in Russia.The first meeting of the society was . Nikolai Brashman was the first president of MMO. Victor Vassiliev is the current president of MMO....
.
Atiyah has been awarded honorary degrees by the universities of Bonn, Warwick, Durham, St. Andrews, Dublin, Chicago, Cambridge, Edinburgh, Essex, London, Sussex, Ghent, Reading, Helsinki, Salamanca, Montreal, Wales, Lebanon, Queen's (Canada), Keele, Birmingham, UMIST, Brown, Heriot–Watt, Mexico, Oxford, Hong Kong (Chinese University), The Open University, American University of Beirut, the Technical University of Catalonia and Leicester.
Atiyah was made a Knight Bachelor
Knight Bachelor
The rank of Knight Bachelor is a part of the British honours system. It is the most basic rank of a man who has been knighted by the monarch but not as a member of one of the organised Orders of Chivalry...
in 1983 and made a member of the Order of Merit in 1992.
The Michael Atiyah building at the University of Leicester
University of Leicester
The University of Leicester is a research-led university based in Leicester, England. The main campus is a mile south of the city centre, adjacent to Victoria Park and Wyggeston and Queen Elizabeth I College....
and the Michael Atiyah Chair in Mathematical Sciences at the American University of Beirut
American University of Beirut
The American University of Beirut is a private, independent university in Beirut, Lebanon. It was founded as the Syrian Protestant College by American missionaries in 1866...
were named after him.
Books by Atiyah
This subsection lists all books written by Atiyah; it omits a few books that he edited.. A classic textbook covering standard commutative algebra.. Reprinted as .. Reprinted as .. Reprinted as .. Reprinted as ....... First edition (1967) reprinted as .. Reprinted as ....Selected papers by Atiyah
. Reprinted in .. Reprinted in .. Reprinted in .. Reprinted in . Formulation of the Atiyah "Conjecture"Atiyah conjecture
In Mathematics, the Atiyah conjecture is a collective term for a number of statements about restrictions on possible values of l^2-Betti numbers.-History:...
on the rationality of the L2-Betti numbers.. An announcement of the index theorem. Reprinted in .. This gives a proof using K theory instead of cohomology. Reprinted in .. This reformulates the result as a sort of Lefschetz fixed point theorem, using equivariant K theory. Reprinted in .. This paper shows how to convert from the K-theory version to a version using cohomology. Reprinted in . This paper studies families of elliptic operators, where the index is now an element of the K-theory of the space parametrizing the family. Reprinted in .. This studies families of real (rather than complex) elliptic operators, when one can sometimes squeeze out a little extra information. Reprinted in .. This states a theorem calculating the Lefschetz number of an endomorphism of an elliptic complex. Reprinted in . (reprinted in )and . Reprinted in . These give the proofs and some applications of the results announced in the previous paper.; Reprinted in .; . Reprinted in .