Sir Michael Atiyah
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Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
specialising in
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
. His contributions include the
Atiyah–Singer index theorem In differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the space ...
and co-founding
topological K-theory In mathematics, topological -theory is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, by means of ideas now recognised as (general) K-theory that were introduced by Alexander Grothendieck. The early ...
. He was awarded the
Fields Medal The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of Mathematicians, International Congress of the International Mathematical Union (IMU), a meeting that takes place e ...
in 1966 and the Abel Prize in 2004.


Early life and education

Atiyah was born on 22 April 1929 in
Hampstead Hampstead () is an area in London, England, which lies northwest of Charing Cross, located mainly in the London Borough of Camden, with a small part in the London Borough of Barnet. It borders Highgate and Golders Green to the north, Belsiz ...
,
London London is the Capital city, capital and List of urban areas in the United Kingdom, largest city of both England and the United Kingdom, with a population of in . London metropolitan area, Its wider metropolitan area is the largest in Wester ...
, England, the son of Jean (née Levens) and Edward Atiyah. His mother was Scottish and his father was a Lebanese Orthodox Christian. He had two brothers, Patrick (deceased) and Joe, and a sister, Selma (deceased). Atiyah went to primary school at the Diocesan school in
Khartoum Khartoum or Khartum is the capital city of Sudan as well as Khartoum State. With an estimated population of 7.1 million people, Greater Khartoum is the largest urban area in Sudan. Khartoum is located at the confluence of the White Nile – flo ...
, Sudan (1934–1941), and to secondary school at Victoria College in
Cairo Cairo ( ; , ) is the Capital city, capital and largest city of Egypt and the Cairo Governorate, being home to more than 10 million people. It is also part of the List of urban agglomerations in Africa, largest urban agglomeration in Africa, L ...
and
Alexandria Alexandria ( ; ) is the List of cities and towns in Egypt#Largest cities, second largest city in Egypt and the List of coastal settlements of the Mediterranean Sea, largest city on the Mediterranean coast. It lies at the western edge of the Nile ...
(1941–1945); the school was also attended by
European nobility Nobility is a social class found in many societies that have an aristocracy. It is normally appointed by and ranked immediately below royalty. Nobility has often been an estate of the realm with many exclusive functions and characteristics. T ...
displaced by the
Second World War World War II or the Second World War (1 September 1939 – 2 September 1945) was a World war, global conflict between two coalitions: the Allies of World War II, Allies and the Axis powers. World War II by country, Nearly all of the wo ...
and some future leaders of Arab nations. He returned to England and
Manchester Grammar School The Manchester Grammar School (MGS) is a highly Selective school, selective Private_schools_in_the_United_Kingdom, private day school for boys aged 7-18 in Manchester, England, which was founded in 1515 by Hugh Oldham (then Bishop of Exeter). ...
for his HSC studies (1945–1947) and did his
national service National service is a system of compulsory or voluntary government service, usually military service. Conscription is mandatory national service. The term ''national service'' comes from the United Kingdom's National Service (Armed Forces) Act ...
with the
Royal Electrical and Mechanical Engineers The Corps of Royal Electrical and Mechanical Engineers (REME ) is the maintenance arm of the British Army that maintains the equipment that the Army uses. The corps is described as the "British Army's professional engineers". History Prior t ...
(1947–1949). His
undergraduate Undergraduate education is education conducted after secondary education and before postgraduate education, usually in a college or university. It typically includes all postsecondary programs up to the level of a bachelor's degree. For example, ...
and
postgraduate Postgraduate education, graduate education, or graduate school consists of academic or professional degrees, certificates, diplomas, or other qualifications usually pursued by post-secondary students who have earned an undergraduate (bachelor' ...
studies took place at
Trinity College, Cambridge Trinity College is a Colleges of the University of Cambridge, constituent college of the University of Cambridge. Founded in 1546 by King Henry VIII, Trinity is one of the largest Cambridge colleges, with the largest financial endowment of any ...
(1949–1955). He was a
doctoral A doctorate (from Latin ''doctor'', meaning "teacher") or doctoral degree is a postgraduate academic degree awarded by universities and some other educational institutions, derived from the ancient formalism '' licentia docendi'' ("licence to teach ...
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 was a member of the
British Humanist Association Humanists UK, known from 1967 until May 2017 as the British Humanist Association (BHA), is a charitable organisation which promotes secular humanism and aims to represent non-religious people in the UK through a mixture of charitable servic ...
. During his time at Cambridge, he was president of The Archimedeans.


Career and research

Atiyah spent the academic year 1955–1956 at the Institute for Advanced Study, Princeton, then returned to
Cambridge University The University of Cambridge is a Public university, public collegiate university, collegiate research university in Cambridge, England. Founded in 1209, the University of Cambridge is the List of oldest universities in continuous operation, wo ...
, where he was a research fellow and assistant
lecturer Lecturer is an academic rank within many universities, though the meaning of the term varies somewhat from country to country. It generally denotes an academic expert who is hired to teach on a full- or part-time basis. They may also conduct re ...
(1957–1958), then a university
lecturer Lecturer is an academic rank within many universities, though the meaning of the term varies somewhat from country to country. It generally denotes an academic expert who is hired to teach on a full- or part-time basis. They may also conduct re ...
and tutorial
fellow A fellow is a title and form of address for distinguished, learned, or skilled individuals in academia, medicine, research, and industry. The exact meaning of the term differs in each field. In learned society, learned or professional society, p ...
at
Pembroke College, Cambridge Pembroke College is a constituent college of the University of Cambridge, England. The college is the third-oldest college of the university and has over 700 students and fellows. It is one of the university's larger colleges, with buildings from ...
(1958–1961). In 1961, he moved to the
University of Oxford The University of Oxford is a collegiate university, collegiate research university in Oxford, England. There is evidence of teaching as early as 1096, making it the oldest university in the English-speaking world and the List of oldest un ...
, where he was a reader and
professor Professor (commonly abbreviated as Prof.) is an Academy, academic rank at university, universities and other tertiary education, post-secondary education and research institutions in most countries. Literally, ''professor'' derives from Latin ...
ial fellow at St Catherine's College (1961–1963). He became
Savilian Professor of Geometry The position of Savilian Professor of Geometry was established at the University of Oxford in 1619. It was founded (at the same time as the Savilian Professor of Astronomy, Savilian Professorship of Astronomy) by Henry Savile (Bible translator), ...
and a professorial fellow of
New College, Oxford New College is a constituent college of the University of Oxford in the United Kingdom. Founded in 1379 by Bishop William of Wykeham in conjunction with Winchester College as New College's feeder school, New College was one of the first col ...
, from 1963 to 1969. He took up a three-year professorship at the Institute for Advanced Study in
Princeton Princeton University is a private Ivy League research university in Princeton, New Jersey, United States. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the Unit ...
after which he returned to Oxford as a
Royal Society The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...
Research Professor and professorial fellow of St Catherine's College. He was president of the
London Mathematical Society The London Mathematical Society (LMS) is one of the United Kingdom's Learned society, learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh ...
from 1974 to 1976. Atiyah was president of the
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. It was fo ...
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 The European Mathematical Society (EMS) 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. The curren ...
(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 The president of the Royal Society (PRS), also known as the Royal Society of London, is the elected Head of the Royal Society who presides over meetings of the society's council. After an informal meeting (a lecture) by Christopher Wren at Gres ...
(1990–1995), Master of Trinity College, Cambridge (1990–1997),
Chancellor Chancellor () is a title of various official positions in the governments of many countries. The original chancellors were the of Roman courts of justice—ushers, who sat at the (lattice work screens) of a basilica (court hall), which separa ...
of the
University of Leicester The University of Leicester ( ) is a public university, public research university based in Leicester, England. The main campus is south of the city centre, adjacent to Victoria Park, Leicester, Victoria Park. The university's predecessor, Univ ...
(1995–2005), and president of the
Royal Society of Edinburgh The Royal Society of Edinburgh (RSE) is Scotland's national academy of science and letters. It is a registered charity that operates on a wholly independent and non-partisan basis and provides public benefit throughout Scotland. It was establis ...
(2005–2008). From 1997 until his death in 2019 he was an honorary professor in the
University of Edinburgh The University of Edinburgh (, ; abbreviated as ''Edin.'' in Post-nominal letters, post-nominals) is a Public university, public research university based in Edinburgh, Scotland. Founded by the City of Edinburgh Council, town council under th ...
. He was a Trustee of the
James Clerk Maxwell Foundation The James Clerk Maxwell Foundation is a Office of the Scottish Charity Regulator, registered Scottish charity set up in 1977. By supporting physics and mathematics, it honors one of the greatest physicists, James Clerk Maxwell (1831–1879), an ...
. Atiyah's mathematical collaborators included
Raoul Bott Raoul Bott (September 24, 1923 – December 20, 2005) was a Hungarian-American mathematician known for numerous foundational contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem, the Morse–Bott function ...
,
Friedrich Hirzebruch Friedrich Ernst Peter Hirzebruch ForMemRS (17 October 1927 – 27 May 2012) was a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation. He has been described as ...
and
Isadore Singer Isadore Manuel Singer (May 3, 1924 – February 11, 2021) was an American mathematician. He was an Emeritus Institute Professor in the Department of Mathematics at the Massachusetts Institute of Technology and a Professor Emeritus of Mathemat ...
, and his students included Graeme Segal, Nigel Hitchin,
Simon Donaldson Sir Simon Kirwan Donaldson (born 20 August 1957) is an English mathematician known for his work on the topology of smooth function, smooth (differentiable) four-dimensional manifolds, Donaldson–Thomas theory, and his contributions to Kähl ...
, and
Edward Witten Edward Witten (born August 26, 1951) is an American theoretical physics, theoretical physicist known for his contributions to string theory, topological quantum field theory, and various areas of mathematics. He is a professor emeritus in the sc ...
. Together with Hirzebruch, he laid the foundations for
topological K-theory In mathematics, topological -theory is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, by means of ideas now recognised as (general) K-theory that were introduced by Alexander Grothendieck. The early ...
, an important tool in
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, which, informally speaking, describes ways in which spaces can be twisted. His best known result, the
Atiyah–Singer index theorem In differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the space ...
, 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 Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
, in particular
instanton An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. M ...
s and monopoles, which are responsible for some corrections in
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
. He was awarded the
Fields Medal The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of Mathematicians, International Congress of the International Mathematical Union (IMU), a meeting that takes place e ...
in 1966 and the Abel Prize in 2004.


Collaborations

Atiyah collaborated with many mathematicians. His three main collaborations were with
Raoul Bott Raoul Bott (September 24, 1923 – December 20, 2005) was a Hungarian-American mathematician known for numerous foundational contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem, the Morse–Bott function ...
on the
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''. This is a sys ...
and many other topics, with Isadore M. Singer on the
Atiyah–Singer index theorem In differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the space ...
, and with
Friedrich Hirzebruch Friedrich Ernst Peter Hirzebruch ForMemRS (17 October 1927 – 27 May 2012) was a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation. He has been described as ...
on topological K-theory, all of whom he met at the
Institute for Advanced Study The Institute for Advanced Study (IAS) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally preeminent scholars, including Albert Ein ...
in Princeton in 1955. His other collaborators included; J. Frank Adams (
Hopf invariant In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between ''n''-spheres. __TOC__ Motivation In 1931 Heinz Hopf used Clifford parallels to construct the '' Hopf map'' :\eta\colon S^ ...
problem), Jürgen Berndt (projective planes), Roger Bielawski (Berry–Robbins problem), Howard Donnelly (
L-function In mathematics, an ''L''-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An ''L''-series is a Dirichlet series, usually convergent on a half-plane, that may gi ...
s), Vladimir G. Drinfeld (instantons), Johan L. Dupont (singularities of
vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...
s),
Lars Gårding Lars Gårding (7 March 1919 – 7 July 2014) was a Swedish mathematician. He made notable contributions to the study of partial differential equations and partial differential operators. He was a professor of mathematics at Lund University in S ...
( hyperbolic differential equations), Nigel J. Hitchin (monopoles), William V. D. Hodge (Integrals of the second kind), Michael Hopkins (
K-theory In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometr ...
), 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 Wilfried Schmid (born May 28, 1943) is a German-American mathematician who works in Hodge theory, representation theory, and automorphic forms. After graduating as valedictorian of Princeton University's class of 1964, Schmid earned his Ph.D. at ...
(discrete series representations), Graeme Segal ( equivariant K-theory), Alexander Shapiro (Clifford algebras), L. Smith (homotopy groups of spheres), Paul Sutcliffe (polyhedra),
David O. Tall David Orme Tall (1941-2024) was an Emeritus Professor in Mathematical Thinking at the University of Warwick. One of his early influential works is the joint paper with Vinner " Concept image and concept definition in mathematics with particular ...
(lambda rings), John A. Todd (
Stiefel manifold In mathematics, the Stiefel manifold V_k(\R^n) is the set of all orthonormal ''k''-frames in \R^n. That is, it is the set of ordered orthonormal ''k''-tuples of vectors in \R^n. It is named after Swiss mathematician Eduard Stiefel. Likewise one ...
s),
Cumrun Vafa Cumrun Vafa (, ; born 1 August 1960) is an Iranian-American theoretical physicist and the Hollis Professor of Mathematicks and Natural Philosophy at Harvard University. Early life and education Cumrun Vafa was born in Tehran, Iran on 1 August 1 ...
(M-theory), Richard S. Ward (instantons) and
Edward Witten Edward Witten (born August 26, 1951) is an American theoretical physics, theoretical physicist known for his contributions to string theory, topological quantum field theory, and various areas of mathematics. He is a professor emeritus in the sc ...
(M-theory, topological quantum field theories). His later research on gauge field theories, particularly Yang–Mills theory, stimulated important interactions between
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
and
physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
, most notably in the work of Edward Witten. Atiyah's students included Peter Braam 1987,
Simon Donaldson Sir Simon Kirwan Donaldson (born 20 August 1957) is an English mathematician known for his work on the topology of smooth function, smooth (differentiable) four-dimensional manifolds, Donaldson–Thomas theory, and his contributions to Kähl ...
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 George Lusztig (born ''Gheorghe Lusztig''; May 20, 1946) is a Romanian-born American mathematician and Abdun Nur Professor at the Massachusetts Institute of Technology (MIT). He was a Norbert Wiener Professor in the Department of Mathematics f ...
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 Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, Philosophy of science, philosopher of science and Nobel Prize in Physics, Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics i ...
, Lars Hörmander,
Alain Connes Alain Connes (; born 1 April 1947) is a French mathematician, known for his contributions to the study of operator algebras and noncommutative geometry. He was a professor at the , , Ohio State University and Vanderbilt University. He was awar ...
and
Jean-Michel Bismut Jean-Michel Bismut (born 26 February 1948) is a French mathematician who has been a professor at the Université Paris-Sud since 1981. His mathematical career covers two apparently different branches of mathematics: probability theory and diff ...
. Atiyah said that the mathematician he most admired was
Hermann Weyl Hermann Klaus Hugo Weyl (; ; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist, logician and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, ...
, and that his favourite mathematicians from before the 20th century were
Bernhard Riemann Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the f ...
and
William Rowan Hamilton Sir William Rowan Hamilton (4 August 1805 – 2 September 1865) was an Irish astronomer, mathematician, and physicist who made numerous major contributions to abstract algebra, classical mechanics, and optics. His theoretical works and mathema ...
. 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 Sir William Vallance Douglas Hodge (; 17 June 1903 – 7 July 1975) was a British mathematician, specifically a geometer. His discovery of far-reaching topological relations between algebraic geometry and differential geometry—an area no ...
and won the
Smith's prize Smith's Prize was the name of each of two prizes awarded annually to two research students in mathematics and theoretical physics at the University of Cambridge from 1769. Following the reorganization in 1998, they are now awarded under the names ...
for 1954 for a sheaf-theoretic approach to
ruled surface In geometry, a Differential geometry of surfaces, surface in 3-dimensional Euclidean space is ruled (also called a scroll) if through every Point (geometry), point of , there is a straight line that lies on . Examples include the plane (mathemat ...
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 (; 3 September 1884 – 5 October 1972) was a Russian-born American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equatio ...
'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 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 (for example X could be a topological space, a manifold, or an algebraic variety): to eve ...
s on an
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the ...
(extending
Alexander Grothendieck Alexander Grothendieck, later Alexandre Grothendieck in French (; ; ; 28 March 1928 â€“ 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research ext ...
'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 Floating point operations per second (FLOPS, flops or flop/s) is a measure of computer performance in computing, useful in fields of scientific computations that require floating-point calculations. For such cases, it is a more accurate measur ...
, a special birational transformation of 3-folds that was later heavily used in
Shigefumi Mori is a Japanese mathematician, known for his work in algebraic geometry, particularly in relation to the classification of three-folds. He won the Fields Medal in 1990. Career Mori completed his Ph.D. titled "The Endomorphism Rings of Some Abelian ...
's work on minimal models for 3-folds. Atiyah's flop can also be used to show that the universal marked family of
K3 surface In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with а trivial canonical bundle and irregularity of a surface, irregularity zero. An (algebraic) K3 surface over any field (mathematics), field ...
s is not Hausdorff.


K-theory (1959–1974)

Atiyah's works on
K-theory In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometr ...
, 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 Friedrich Ernst Peter Hirzebruch ForMemRS (17 October 1927 – 27 May 2012) was a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation. He has been described as ...
who were inspired by Grothendieck's proof of the
Grothendieck–Riemann–Roch theorem In mathematics, specifically in algebraic geometry, the Grothendieck–Riemann–Roch theorem is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is ...
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 In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
. 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 In mathematics, the Stiefel manifold V_k(\R^n) is the set of all orthonormal ''k''-frames in \R^n. That is, it is the set of ordered orthonormal ''k''-tuples of vectors in \R^n. It is named after Swiss mathematician Eduard Stiefel. Likewise one ...
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 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 encounte ...
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 In mathematics, the Atiyah–Hirzebruch spectral sequence is a spectral sequence for calculating generalized cohomology, introduced by in the special case of topological K-theory. For a CW complex X and a generalized cohomology theory E^\bullet, ...
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 In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e., a topological space all of whose homotopy groups are trivial) by a proper free ...
, ''BG'', is isomorphic to the completion of its
character ring In mathematics, especially in the area of algebra known as representation theory, the representation ring (or Green ring after J. A. Green) of a group is a ring formed from all the (isomorphism classes of the) finite-dimensional linear representat ...
: : K(BG) \cong R(G)^. The same year they proved the result for ''G'' any
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
. 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 In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to the ordinary K-theory of a space, X_G, which fibred over ''BG'' with fibre ''X'': :K_G(X)^ \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 In mathematics, cobordism is a fundamental equivalence relation on the class of compact space, compact manifolds of the same dimension, set up using the concept of the boundary (topology), boundary (French ''wikt:bord#French, bord'', giving ''cob ...
, and pointed out that many of the deep results on cobordism of manifolds found by
René Thom René Frédéric Thom (; 2 September 1923 – 25 October 2002) was a French mathematician, who received the Fields Medal in 1958. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became ...
, 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 bundle In the mathematical field of topology, a sphere bundle is a fiber bundle in which the fibers are spheres S^n of some dimension ''n''. Similarly, in a disk bundle, the fibers are disks D^n. From a topological perspective, there is no difference betw ...
s; this was later studied in detail by J. F. Adams in a series of papers, leading to the Adams conjecture. With Hirzebruch he extended the
Grothendieck–Riemann–Roch theorem In mathematics, specifically in algebraic geometry, the Grothendieck–Riemann–Roch theorem is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is ...
to complex analytic embeddings, and in a related paper they showed that the
Hodge conjecture In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties. In simple terms, the Hodge conjectur ...
for integral cohomology is false. The Hodge conjecture for rational cohomology is, as of 2008, a major unsolved problem. The
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 comple ...
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 Shapiro, and its variations such as Shapira, Schapiro, Schapira, Sapir, Sapira, Spira, Spiro, Sapiro, Szapiro/Szpiro in Polish and Chapiro in French (more at "See also"), is a Jewish Ashkenazi surname. Etymology The surname is derived from ...
he analysed the relation of Bott periodicity to the periodicity of
Clifford algebras In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure of a distinguished subspace. As -algebras, they generalize the real numbers ...
; although this paper did not have a proof of the periodicity theorem, a proof along similar lines was shortly afterwards found by R. Wood. He found a proof of several generalizations using
elliptic operator 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 im ...
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 In mathematics, the Hirzebruch–Riemann–Roch theorem, named after Friedrich Hirzebruch, Bernhard Riemann, and Gustav Roch, is Hirzebruch's 1954 result generalizing the classical Riemann–Roch theorem on Riemann surfaces to all complex algeb ...
, 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 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 im ...
s 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 invariant In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological space ...
s. Some of the motivating examples included the
Riemann–Roch theorem The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It re ...
and its generalization the
Hirzebruch–Riemann–Roch theorem In mathematics, the Hirzebruch–Riemann–Roch theorem, named after Friedrich Hirzebruch, Bernhard Riemann, and Gustav Roch, is Hirzebruch's 1954 result generalizing the classical Riemann–Roch theorem on Riemann surfaces to all complex algeb ...
, and the Hirzebruch signature theorem.
Hirzebruch Friedrich Ernst Peter Hirzebruch ForMemRS (17 October 1927 – 27 May 2012) was a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation. He has been described as ...
and
Borel Borel may refer to: People * Antoine Borel (1840–1915), a Swiss-born American businessman * Armand Borel (1923–2003), a Swiss mathematician * Borel (author), 18th-century French playwright * Borel (1906–1967), pseudonym of the French actor ...
had proved the integrality of the
 genus Â, â ( a- circumflex) is a letter of the Inari Sami, Skolt Sami, Romanian, Vietnamese and Mizo alphabets. This letter also appears in French, Friulian, Frisian, Portuguese, Turkish, Walloon, and Welsh languages as a variant of the l ...
of a spin manifold, and Atiyah suggested that this integrality could be explained if it were the index of the
Dirac operator In mathematics and in quantum mechanics, a Dirac operator is a first-order differential operator that is a formal square root, or half-iterate, of a second-order differential operator such as a Laplacian. It was introduced in 1847 by William Ham ...
(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 In mathematics, the Hirzebruch–Riemann–Roch theorem, named after Friedrich Hirzebruch, Bernhard Riemann, and Gustav Roch, is Hirzebruch's 1954 result generalizing the classical Riemann–Roch theorem on Riemann surfaces to all complex algeb ...
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 In mathematics, specifically in algebraic geometry, the Grothendieck–Riemann–Roch theorem is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is ...
, replacing the
cobordism In mathematics, cobordism is a fundamental equivalence relation on the class of compact space, compact manifolds of the same dimension, set up using the concept of the boundary (topology), boundary (French ''wikt:bord#French, bord'', giving ''cob ...
theory of the first proof with
K-theory In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometr ...
, 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 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 . There is a closely related formula for the ch ...
, 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 In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under ...
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 In mathematics and physics (more specifically thermodynamics), the heat equation is a parabolic partial differential equation. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quanti ...
. If the
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
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 Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
, in particular in Witten's work on anomalies. The fundamental solutions of linear
hyperbolic partial differential equation In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n - 1 derivatives. More precisely, the Cauchy problem can ...
s 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 Lars Gårding (7 March 1919 – 7 July 2014) was a Swedish mathematician. He made notable contributions to the study of partial differential equations and partial differential operators. He was a professor of mathematics at Lund University in S ...
, 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 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. It is a special type of C*-algebra. Von Neumann al ...
; 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 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(''G''). In the Plancherel measur ...
s of
semisimple Lie group In mathematics, a simple Lie group is a connected space, connected nonabelian group, non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple ...
s. In the course of this work they found a more elementary proof of Harish-Chandra's fundamental theorem on the local integrability of characters of Lie groups. 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 8''k''−3. To do this they used the Atiyah–Singer index theorem to calculate the dimension of the tangent space of the moduli space at a point; the tangent space is essentially the space of solutions of an elliptic differential operator, given by the linearization of the non-linear Yang–Mills equations. These moduli spaces were later used by Donaldson to construct his 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 In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...
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 In mathematics, and especially gauge theory, Donaldson theory is the study of the topology of smooth 4-manifolds using moduli spaces of anti-self-dual instantons. It was started by Simon Donaldson (1983) who proved Donaldson's theorem restricting ...
. Donaldson showed that the moduli space of (degree 1) instantons over a compact simply connected
4-manifold In mathematics, a 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. T ...
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 In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows mathematical analysis to be performed on the manifold. Definition A smooth structure on a manifold M ...
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 In mathematics, a Green's function (or Green function) is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if L is a linear dif ...
s for linear partial differential equations can often be found by using the
Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
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 group In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely-related usages. The most direct usage of the term is to take the ''homology of a chain complex'', resulting in a sequence of abelian grou ...
s in a certain range of dimensions, and suggested that it might induce isomorphisms of homology groups in the same range of dimensions. This became known as the Atiyah–Jones conjecture, and was later proved by several mathematicians. Harder and M. S. Narasimhan described the cohomology of the
moduli space In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme (mathematics), scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of suc ...
s of
stable vector bundle In mathematics, a stable vector bundle is a (holomorphic or algebraic) vector bundle that is stable in the sense of geometric invariant theory. Any holomorphic vector bundle may be built from stable ones using Harder–Narasimhan filtration. Stable ...
s over
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
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 In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differenti ...
and the Yang–Mills equations over a
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
to reproduce and extending the results of Harder and Narasimhan. An old result due to
Schur Schur is a German or Jewish surname. Notable people with the surname include: * Alexander Schur (born 1971), German footballer * Dina Feitelson-Schur (1926–1992), Israeli educator * Friedrich Schur (1856-1932), German mathematician * Fritz 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 In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sy ...
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 In mathematics, specifically in symplectic geometry, the momentum map (or, by false etymology, moment 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. ...
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 In mathematics, equivariant cohomology (or ''Borel cohomology'') is a cohomology theory from algebraic topology which applies to topological spaces with a ''group action''. It can be viewed as a common generalization of group cohomology and an ord ...
, which was a consequence of well-known
localization theorem In mathematics, particularly in integral calculus, the localization theorem allows, under certain conditions, to infer the nullity of a function (mathematics), function given only information about its continuity (mathematics), continuity and the v ...
s. Atiyah showed that the moment map was closely related to
geometric invariant theory In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas from the paper in class ...
, 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 In particle physics, a magnetic monopole is a hypothetical particle that is an isolated magnet with only one magnetic pole (a north pole without a south pole or vice versa). A magnetic monopole would have a net north or south "magnetic charge". ...
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 In particle physics, a magnetic monopole is a hypothetical particle that is an isolated magnet with only one magnetic pole (a north pole without a south pole or vice versa). A magnetic monopole would have a net north or south "magnetic charge". ...
. The main theme of the book is a study of a moduli space of
magnetic monopoles In particle physics, a magnetic monopole is a hypothetical particle that is an isolated magnet with only one magnetic pole (a north pole without a south pole or vice versa). A magnetic monopole would have a net north or south "magnetic charge". ...
; 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 (1986–2019)

Many of the papers in the 6th volume of his collected works are surveys, obituaries, and general talks. Atiyah continued to publish subsequently, including several surveys, a popular book, and another paper with
Segal Segal, and its variants including Sagal, Segel, Sigal or Siegel, is a family name which is primarily Ashkenazi Jewish. The name is said to be derived from Hebrew ''segan leviyyah'' (assistant to the Levites) although a minority of sources claim ...
on
twisted K-theory In mathematics, twisted K-theory (also called K-theory with local coefficients) is a variation on K-theory, a mathematical theory from the 1950s that spans algebraic topology, abstract algebra and operator theory. More specifically, twisted K-theo ...
. One paper is a detailed study of the
Dedekind eta function In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string ...
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 In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
,
knots A knot is a fastening in rope or interwoven lines. Knot or knots may also refer to: Other common meanings * Knot (unit), of speed * Knot (wood), a timber imperfection Arts, entertainment, and media Films * ''Knots'' (film), a 2004 film * ''Kn ...
, and
Donaldson theory In mathematics, and especially gauge theory, Donaldson theory is the study of the topology of smooth 4-manifolds using moduli spaces of anti-self-dual instantons. It was started by Simon Donaldson (1983) who proved Donaldson's theorem restricting ...
. He introduced the concept of a
topological quantum field theory In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory that computes topological invariants. While TQFTs were invented by physicists, they are also of mathemati ...
, inspired by Witten's work and Segal's definition of a conformal field theory. His book "The Geometry and Physics of Knots" describes the new
knot invariant In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) 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 i ...
s found by
Vaughan Jones Sir Vaughan Frederick Randal Jones (31 December 19526 September 2020) was a New Zealand mathematician known for his work on von Neumann algebras and knot polynomials. He was awarded a Fields Medal in 1990. Early life Jones was born in Gisbo ...
and
Edward Witten Edward Witten (born August 26, 1951) is an American theoretical physics, theoretical physicist known for his contributions to string theory, topological quantum field theory, and various areas of mathematics. He is a professor emeritus in the sc ...
in terms of topological quantum field theories, and his paper with L. Jeffrey explains Witten's Lagrangian giving the Donaldson invariants. He studied
skyrmion In particle theory, the skyrmion () is a topologically stable field configuration of a certain class of non-linear sigma models. It was originally proposed as a model of the nucleon by (and named after) Tony Skyrme in 1961. As a topological solito ...
s with Nick Manton, finding a relation with
magnetic monopoles In particle physics, a magnetic monopole is a hypothetical particle that is an isolated magnet with only one magnetic pole (a north pole without a south pole or vice versa). A magnetic monopole would have a net north or south "magnetic charge". ...
and
instanton An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. M ...
s, and giving a conjecture for the structure of the
moduli space In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme (mathematics), scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of suc ...
of two skyrmions as a certain
subquotient In the mathematical fields of category theory and abstract algebra, a subquotient is a quotient object of a subobject. Subquotients are particularly important in abelian categories, and in group theory, where they are also known as sections, thou ...
of complex projective 3-space. Several papers were inspired by a question o
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 In mathematics, the Atiyah conjecture on configurations is a conjecture introduced by stating that a certain ''n'' by ''n'' matrix depending on ''n'' points in R3 is always non-singular Singular may refer to: * Singular, the grammatical numbe ...
. With Juan Maldacena and
Cumrun Vafa Cumrun Vafa (, ; born 1 August 1960) is an Iranian-American theoretical physicist and the Hollis Professor of Mathematicks and Natural Philosophy at Harvard University. Early life and education Cumrun Vafa was born in Tehran, Iran on 1 August 1 ...
, and E. Witten he described the dynamics of
M-theory In physics, M-theory is a theory that unifies all Consistency, consistent versions of superstring theory. Edward Witten first conjectured the existence of such a theory at a string theory conference at the University of Southern California in 1 ...
on manifolds with G2 holonomy. These papers seem to be the first time that Atiyah 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 Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
. 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. At the 2018 Heidelberg Laureate Forum, he claimed to have solved the
Riemann hypothesis In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in pure ...
, Hilbert's eighth problem, by contradiction using the
fine-structure constant In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Alpha, Greek letter ''alpha''), is a Dimensionless physical constant, fundamental physical constant that quantifies the strength of the el ...
. Again, the proof did not hold up and the hypothesis remains one of the six unsolved
Millennium Prize Problems The Millennium Prize Problems are seven well-known complex mathematics, mathematical problems selected by the Clay Mathematics Institute in 2000. The Clay Institute has pledged a US $1 million prize for the first correct solution to each problem ...
in mathematics, as of 2025.


Bibliography


Books

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

*. Reprinted in . *. Reprinted in . *. Reprinted in . *. Reprinted in . Formulation of the Atiyah "Conjecture" 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 In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
. 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 . *


Awards and honours

In 1966, when he was thirty-seven years old, he was awarded the
Fields Medal The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of Mathematicians, International Congress of the International Mathematical Union (IMU), a meeting that takes place e ...
, for his work in developing K-theory, a generalized
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. It is name ...
and the Atiyah–Singer theorem, for which he also won the Abel Prize jointly with
Isadore Singer Isadore Manuel Singer (May 3, 1924 – February 11, 2021) was an American mathematician. He was an Emeritus Institute Professor in the Department of Mathematics at the Massachusetts Institute of Technology and a Professor Emeritus of Mathemat ...
in 2004. Among other prizes he has received are the
Royal Medal The Royal Medal, also known as The Queen's Medal and The King's Medal (depending on the gender of the monarch at the time of the award), is a silver-gilt medal, of which three are awarded each year by the Royal Society. Two are given for "the mo ...
of the
Royal Society The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...
in 1968, the
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 The London Mathematical Society (LMS) is one of the United Kingdom's Learned society, learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh ...
in 1980, the Antonio Feltrinelli Prize from the
Accademia Nazionale dei Lincei The (; literally the "Academy of the Lynx-Eyed"), anglicised as the Lincean Academy, is one of the oldest and most prestigious European scientific institutions, located at the Palazzo Corsini on the Via della Lungara in Rome, Italy. Founded in ...
in 1981, the King Faisal International Prize for Science in 1987, the
Copley Medal The Copley Medal is the most prestigious award of the Royal Society of the United Kingdom, conferred "for sustained, outstanding achievements in any field of science". The award alternates between the physical sciences or mathematics and the bio ...
of the Royal Society in 1988, the Benjamin Franklin Medal for Distinguished Achievement in the Sciences of the
American Philosophical Society The American Philosophical Society (APS) is an American scholarly organization and learned society founded in 1743 in Philadelphia that promotes knowledge in the humanities and natural sciences through research, professional meetings, publicat ...
in 1993, the Jawaharlal Nehru Birth Centenary Medal of the
Indian National Science Academy The Indian National Science Academy (INSA) is a national academy in New Delhi New Delhi (; ) is the Capital city, capital of India and a part of the Delhi, National Capital Territory of Delhi (NCT). New Delhi is the seat of all three b ...
in 1993, the President's Medal from the
Institute of Physics The Institute of Physics (IOP) is a UK-based not-for-profit learned society and professional body that works to advance physics education, physics research, research and applied physics, application. It was founded in 1874 and has a worldwide ...
in 2008, the Grande Médaille of the
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 method, scientific research. It was at the forefron ...
in 2010 and the Grand Officier of the
French Légion d'honneur The National Order of the Legion of Honour ( ), formerly the Imperial Order of the Legion of Honour (), is the highest and most prestigious French national order of merit, both military and civil. Currently consisting of five classes, it was ...
in 2011. He was elected a foreign member of the
National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, NGO, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the ...
, the
American Academy of Arts and Sciences The American Academy of Arts and Sciences (The Academy) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, and other ...
(1969), the
Académie des 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 method, scientific research. It was at the forefron ...
, the Akademie Leopoldina, the
Royal Swedish Academy The Royal Swedish Academy of Sciences () is one of the Swedish Royal Academies, royal academies of Sweden. Founded on 2 June 1739, it is an independent, non-governmental scientific organization that takes special responsibility for promoting nat ...
, the
Royal Irish Academy The Royal Irish Academy (RIA; ), based in Dublin, is an academic body that promotes study in the natural sciences, arts, literature, and social sciences. It is Ireland's premier List of Irish learned societies, learned society and one of its le ...
, the
Royal Society of Edinburgh The Royal Society of Edinburgh (RSE) is Scotland's national academy of science and letters. It is a registered charity that operates on a wholly independent and non-partisan basis and provides public benefit throughout Scotland. It was establis ...
, the
American Philosophical Society The American Philosophical Society (APS) is an American scholarly organization and learned society founded in 1743 in Philadelphia that promotes knowledge in the humanities and natural sciences through research, professional meetings, publicat ...
, the
Indian National Science Academy The Indian National Science Academy (INSA) is a national academy in New Delhi New Delhi (; ) is the Capital city, capital of India and a part of the Delhi, National Capital Territory of Delhi (NCT). New Delhi is the seat of all three b ...
, the
Chinese Academy of Sciences The Chinese Academy of Sciences (CAS; ) is the national academy for natural sciences and the highest consultancy for science and technology of the People's Republic of China. It is the world's largest research organization, with 106 research i ...
, the
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 Soci ...
, the
Russian Academy of Science The Russian Academy of Sciences (RAS; ''Rossíyskaya akadémiya naúk'') consists of the national academy of Russia; a network of scientific research institutes from across the Russian Federation; and additional scientific and social units such ...
, the Ukrainian Academy of Science, the Georgian Academy of Science, the Venezuela Academy of Science, the
Norwegian Academy of Science and Letters The Norwegian Academy of Science and Letters (, DNVA) is a learned society based in Oslo, Norway. Its purpose is to support the advancement of science and scholarship in Norway. History The Royal Frederick University in Christiania was establis ...
, the Royal Spanish Academy of Science, the
Accademia dei Lincei The (; literally the "Academy of the Lynx-Eyed"), anglicised as the Lincean Academy, is one of the oldest and most prestigious European scientific institutions, located at the Palazzo Corsini on the Via della Lungara in Rome, Italy. Founded in ...
and the
Moscow Mathematical Society The Moscow Mathematical Society (MMS) is a society of Moscow mathematicians aimed at the development of mathematics in Russia. It was created in 1864, and Victor Vassiliev is the current president. History The first meeting of the society w ...
. In 2012, he became a fellow of the
American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
. He was also appointed as a Honorary
Fellow A fellow is a title and form of address for distinguished, learned, or skilled individuals in academia, medicine, research, and industry. The exact meaning of the term differs in each field. In learned society, learned or professional society, p ...
of the
Royal Academy of Engineering The Royal Academy of Engineering (RAEng) is the United Kingdom's national academy of engineering. The Academy was founded in June 1976 as the Fellowship of Engineering with support from Prince Philip, Duke of Edinburgh, who became the first senio ...
in 1993. Atiyah was 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, University of Waterloo, Wilfrid Laurier University, Technical University of Catalonia, and UMIST. Atiyah was made a
Knight Bachelor The title of Knight Bachelor is the basic rank granted to a man who has been knighted by the monarch but not inducted as a member of one of the organised Order of chivalry, orders of chivalry; it is a part of the Orders, decorations, and medals ...
in 1983 and made a member of the
Order of Merit The Order of Merit () is an order of merit for the Commonwealth realms, recognising distinguished service in the armed forces, science, art, literature, or the promotion of culture. Established in 1902 by Edward VII, admission into the order r ...
in 1992. The Michael Atiyah building at the
University of Leicester The University of Leicester ( ) is a public university, public research university based in Leicester, England. The main campus is south of the city centre, adjacent to Victoria Park, Leicester, Victoria Park. The university's predecessor, Univ ...
and the Michael Atiyah Chair in Mathematical Sciences at the
American University of Beirut The American University of Beirut (AUB; ) is a private, non-sectarian, and independent university chartered in New York with its main campus in Beirut, Lebanon. AUB is governed by a private, autonomous board of trustees and offers programs le ...
were named after him.


Personal life

Atiyah married Lily Brown on 30 July 1955, with whom he had three sons, John, David and Robin. Atiyah's eldest son John died on 24 June 2002 while on a walking holiday in the
Pyrenees The Pyrenees are a mountain range straddling the border of France and Spain. They extend nearly from their union with the Cantabrian Mountains to Cap de Creus on the Mediterranean coast, reaching a maximum elevation of at the peak of Aneto. ...
with his wife Maj-Lis. Lily Atiyah died on 13 March 2018 at the age of 90 while Sir Michael Atiyah died less than a year later on 11 January 2019, aged 89.


See also

*
List of presidents of the Royal Society The president of the Royal Society (PRS), also known as the Royal Society of London, is the elected Head of the Royal Society who presides over meetings of the society's council. After an informal meeting (a lecture) by Christopher Wren at Gresh ...


References


Sources

* * *. Reprinted in volume 1 of his collected works, p. 65–75, . On page 120 Gel'fand suggests that the index of an elliptic operator should be expressible in terms of topological data. * * *. This describes the original proof of the index theorem. (Atiyah and Singer never published their original proof themselves, but only improved versions of it.) *. *. *.


External links


Michael Atiyah tells his life story
at
Web of Stories Web of Stories is an online collection of thousands of autobiographical video-stories. Web of Stories, originally known as Science Archive, was set up to record the life stories of scientists. When it expanded to include the lives of authors, mov ...

The celebrations of Michael Atiyah's 80th birthday in Edinburgh, 20-24 April 2009Mathematical descendants of Michael Atiyah
* * * * * * * * * * * *
List of works of Michael Atiyah
from Celebratio Mathematica * * {{DEFAULTSORT:Atiyah, Michael 1929 births 2019 deaths 20th-century British mathematicians 21st-century British mathematicians Academics of the University of Edinburgh Abel Prize laureates Algebraic geometers Alumni of Trinity College, Cambridge British humanists Differential geometers English people of Lebanese descent English people of Scottish descent Fellows of the Academy of Medical Sciences (United Kingdom) Fellows of the American Academy of Arts and Sciences Fellows of the American Mathematical Society Fellows of the Australian Academy of Science Fellows of New College, Oxford Fellows of Pembroke College, Cambridge Fellows of St Catherine's College, Oxford Fellows of the Royal Society Fellows of the Royal Society of Edinburgh Fields Medalists Foreign associates of the National Academy of Sciences Foreign fellows of the Indian National Science Academy Foreign members of the Russian Academy of Sciences Institute for Advanced Study faculty International members of the American Philosophical Society Honorary Fellows of the Royal Academy of Engineering Knights Bachelor Masters of Trinity College, Cambridge Mathematicians from London Members of the French Academy of Sciences Members of the Norwegian Academy of Science and Letters Members of the Order of Merit Members of the Royal Swedish Academy of Sciences People associated with the University of Leicester People educated at Manchester Grammar School People from Hampstead Presidents of the Royal Society Recipients of the Copley Medal Royal Electrical and Mechanical Engineers soldiers Royal Medal winners Savilian Professors of Geometry Topologists Victoria College, Alexandria alumni