Sir Michael Francis Atiyah (; born 22 April
1929) is a British-Lebanese mathematician
specialising in geometry.Atiyah grew up in
Sudan and Egypt but spent most of his academic
life in the United Kingdom at University of
Oxford and University of 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). Since 1997, he has been 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 used in counting
the number of independent solutions to differential
equations. Some of his more recent work was
inspired by theoretical physics, in particular
instantons and monopoles, which are responsible
for some subtle corrections in quantum field
theory. He was awarded the Fields Medal in
1966 and the Abel Prize in 2004.
== Education and early life ==
Atiyah was born in Hampstead, London, England,
the son of Jean (née Levens) and Edward Atiyah.
His mother was Scottish and his father was
a Lebanese Orthodox Christian. He has two
brothers, Patrick and Joe, and a sister, Selma
(deceased). Atiyah 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.
== Career and research ==
Atiyah spent the academic year 1955–1956
at the Institute for Advanced Study, Princeton,
then returned to Cambridge University, where
he was a research fellow and assistant lecturer
(1957–1958), then a university lecturer
and tutorial fellow at Pembroke College, Cambridge
(1958–1961). In 1961, he moved to the University
of Oxford, where he was a reader and professorial
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 was 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). Since
1997, he has been an honorary professor in
the University of Edinburgh. He is a Trustee
of the James Clerk Maxwell Foundation.
=== Collaborations ===
Atiyah has collaborated with many mathematicians.
His three main collaborations were with Raoul
Bott on the Atiyah–Bott fixed-point theorem
and many other topics, with Isadore M. Singer
on the Atiyah–Singer index theorem, and
with Friedrich Hirzebruch on topological K-theory,
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-functions), Vladimir G. Drinfeld
(instantons), Johan L. Dupont (singularities
of vector fields), Lars Gårding (hyperbolic
differential equations), 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 manifolds),
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 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.The seven volumes of Atiyah's
collected papers include most of his work,
except for his commutative algebra textbook;
the first five volumes are divided thematically
and the sixth and seventh arranged by date.
=== Algebraic geometry (1952–1958) ===
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 surfaces, 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 bundles on an elliptic curve
(extending Alexander Grothendieck's classification
of vector bundles on a genus 0 curve), by
showing that any vector bundle is a sum of
(essentially unique) indecomposable vector
bundles, 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-folds that was later heavily used in
Shigefumi Mori's work on minimal models for
3-folds. Atiyah's flop can also be used to
show that the universal marked family of K3
surfaces is non-Hausdorff.
=== K theory (1959–1974) ===
Atiyah's works on K-theory, including his
book on K-theory are reprinted in volume 2
of his collected works.The simplest nontrivial
example of a vector bundle is the Möbius
band (pictured on the right): a strip of paper
with a twist in it, which represents a rank
1 vector bundle over a circle (the circle
in question being the centerline of the Möbius
band). K-theory is a tool for working with
higher-dimensional analogues of this example,
or in other words for describing higher-dimensional
twistings: elements of the K-group of a space
are represented by vector bundles over it,
so the Möbius band represents an element
of the K-group of a circle.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 classes that
Hirzebruch had noticed a few years before.
The original solution of the Hopf invariant
one problem operations by J. F. Adams was
very long and complicated, using secondary
cohomology operations. Atiyah showed how primary
operations in K-theory could be used to 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:
K
(
B
G
)
≅
R
(
G
)
∧
.
{\displaystyle K(BG)\cong R(G)^{\wedge }.}
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,
X
G
{\displaystyle X_{G}}
, which fibred over BG with fibre X:
K
G
(
X
)
∧
≅
K
(
X
G
)
.
{\displaystyle K_{G}(X)^{\wedge }\cong K(X_{G}).}
The original result then followed as a corollary
by taking X to be a point: the left hand side
reduced to the completion of R(G) and the
right to K(BG). See Atiyah–Segal completion
theorem for more details.
He defined new generalized homology and cohomology
theories called bordism and cobordism, and
pointed out that many of the deep results
on cobordism of manifolds found by René Thom,
C. T. C. Wall, and others could be naturally
reinterpreted as statements about these cohomology
theories. 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. He found a proof
of several generalizations using elliptic
operators; 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
and its generalization the Hirzebruch–Riemann–Roch
theorem, and the Hirzebruch signature theorem.
Hirzebruch and Borel had proved the integrality
of the Â genus of a spin manifold, and Atiyah
suggested that this integrality could be explained
if it were the index of the Dirac operator
(which was rediscovered by Atiyah and Singer
in 1961).
The first announcement of the Atiyah–Singer
theorem was their 1963 paper. 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 Hirzebruch 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,
and also introduced the Atiyah–Patodi–Singer
eta invariant. 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 equations often have
Petrovsky lacunas: 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 representations
of semisimple Lie groups. In the course of
this work they found a more elementary proof
of Harish-Chandra's fundamental theorem on
the local integrability of characters of Lie
groups.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) ===
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 classifying instantons on Euclidean space
as this is conformally equivalent to a sphere
and the equations for instantons are conformally
invariant. With Hitchin and Singer he calculated
the dimension of the moduli space of irreducible
self-dual connections (instantons) for any
principal bundle over a compact 4-dimensional
Riemannian manifold (the Atiyah–Hitchin–Singer
theorem). For example, the dimension of the
space of 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
quaternions 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 structures 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
functions 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 spaces of stable vector bundles
over Riemann surfaces 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 manifolds 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
monopoles, 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 hyperkähler. The metric is then used
to study the scattering of two monopoles,
using a suggestion of N. Manton that the geodesic
flow on the moduli space is the low energy
approximation to the scattering. For example,
they show that a head-on collision between
two monopoles results in 90-degree scattering,
with the direction of scattering depending
on the relative phases of the two monopoles.
He also studied monopoles on hyperbolic space.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) ===
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 invariants 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 skyrmions with Nick Manton, finding
a relation with magnetic monopoles and instantons,
and giving a conjecture for the structure
of the moduli space of two skyrmions as a
certain subquotient of complex projective
3-space.
Several papers were inspired by a question
of Jonathan Robbins (called the Berry–Robbins
problem), who asked if there is a map from
the configuration space of n points in 3-space
to the flag manifold of the unitary group.
Atiyah gave an affirmative answer to this
question, but felt his solution was too computational
and studied a conjecture that would give a
more natural solution. He also related the
question to Nahm's equation, and introduced
the Atiyah conjecture on configurations.
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 October 2016, he claimed a short proof
of the non-existence of complex structures
on the 6-sphere. His proof, like many predecessors,
is considered flawed by the mathematical community,
even after the proof was rewritten in a revised
form.In September 2018, at the 2018 Heidelberg
Laureate Forum, he claimed a simple proof
of the Riemann hypothesis, one of the most
important and challenging problems in mathematics.
His claim was met with skepticism from the
mathematical community.
== Bibliography ==
=== Books ===
This subsection lists all books written by
Atiyah; it omits a few books that he edited.
=== Selected papers ===
=== Awards and honours ===
In 1966, when he was thirty-seven years old,
he was awarded the Fields Medal, for his work
in developing K-theory, a generalized Lefschetz
fixed-point theorem 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 Académie
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.
In 2012, he became a fellow of the American
Mathematical Society. He was also appointed
as a Honorary Fellow of the Royal Academy
of Engineering in 1993.
Atiyah has been awarded honorary degrees by
the universities of Birmingham, Bonn, Chicago,
Cambridge, Dublin, Durham, Edinburgh, Essex,
Ghent, Helsinki, Lebanon, Leicester, London,
Mexico, Montreal, Oxford, Reading, Salamanca,
St. Andrews, Sussex, Wales, Warwick, the American
University of Beirut, Brown University, Charles
University in Prague, Harvard University,
Heriot–Watt University, Hong Kong (Chinese
University), Keele University, Queen's University
(Canada), The Open University, Technical University
of Catalonia, and UMIST.
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.
== Personal life ==
Atiyah married Lily Brown on 30 July 1955,
with whom he has three sons, John, David and
Robin. Atiyah's eldest son John died on 24
June 2002 while on a walking holiday in the
Pyrenees with his wife Maj-Lis. Lily Atiyah
died on 13 March 2018 at the age of 90.
== References ==
=== Sources ===
== External links ==
Michael Atiyah tells his life story at Web
of Stories
The celebrations of Michael Atiyah's 80th
birthday in Edinburgh, 20-24 April 2009
Mathematical descendants of Michael Atiyah
"Sir Michael Atiyah on math, physics and fun",
superstringtheory.com, Official Superstring
theory web site], retrieved 2008-08-14
Atiyah, Michael, Beauty in Mathematics (video,
3m14s), retrieved 2008-08-14
Atiyah, Michael, The nature of space (Online
lecture), retrieved 2008-08-14
Batra, Amba (8 November 2003), Maths guru
with Einstein's dream prefers chalk to mouse.
(Interview with Atiyah.), Delhi newsline,
retrieved 2008-08-14
Michael Atiyah at the Mathematics Genealogy
Project
Halim, Hala (1998), "Michael Atiyah:Euclid
and Victoria", Al-Ahram Weekly On-line (391),
archived from the original on 2004-08-16,
retrieved 2008-08-26
Meek, James (21 April 2004), "Interview with
Michael Atiyah", The Guardian, London, retrieved
2008-08-14
Sir Michael Atiyah FRS, Isaac Newton Institute,
retrieved 2008-08-14
"Atiyah and Singer receive 2004 Abel prize"
(PDF), Notices of the American Mathematical
Society, 51 (6): 650–651, 2006, retrieved
2008-08-14
Raussen, Martin; Skau, Christian (24 May 2004),
Interview with Michael Atiyah and Isadore
Singer, retrieved 2008-08-14
Photos of Michael Francis Atiyah, Oberwolfach
photo collection, retrieved 2008-08-14
Wade, Mike (21 April 2009), Maths and the
bomb: Sir Michael Atiyah at 80, London: Timesonline,
retrieved 2010-05-12
List of works of Michael Atiyah from Celebratio
Mathematica
