Sir Michael Francis Atiyah OM FRS FRSE FMedSci FREng[5]
(/əˈtiːə/; born 22 April 1929)[1] is an English mathematician
specialising in geometry.[6]
Atiyah grew up in
Contents 1 Education and early life 2 Career and research 2.1 Collaborations
2.2
3 Personal life 4 References 4.1 Sources 4.2 External links Education and early life[edit] Great Court of Trinity College, Cambridge, where Atiyah was a student and later Master Atiyah was born in Hampstead, London, England, to a Lebanese father,
the academic, Eastern Orthodox,
The
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).[12] He became Savilian
I started out by changing local currency into foreign currency everywhere I travelled as a child and ended up making money. That’s when my father realised that I would be a mathematician some day. Michael Atiyah[13] Atiyah has been active on the international scene, for instance as
president of the
The old Mathematical Institute (now the Department of Statistics) in Oxford, where Atiyah supervised many of his students Atiyah has collaborated with many other mathematicians. His three main
collaborations were with
If you attack a mathematical problem directly, very often you come to a dead end, nothing you do seems to work and you feel that if only you could peer round the corner there might be an easy solution. There is nothing like having somebody else beside you, because he can usually peer round the corner. Michael Atiyah[22] Atiyah's many students include Peter Braam 1987,
A twisted cubic curve, the subject of Atiyah's first paper Atiyah's early papers on algebraic geometry (and some general papers)
are reprinted in the first volume of his collected works.[27]
As an undergraduate Atiyah was interested in classical projective
geometry, and wrote his first paper: a short note on twisted
cubics.[28] He started research under
A
Atiyah's works on K-theory, including his book on K-theory[37] are
reprinted in volume 2 of his collected works.[38]
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).
The
K ( B G ) ≅ R ( G ) ∧ . displaystyle K(BG)cong R(G)^ wedge . The same year[45] 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,[46]
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.[47]
It was shown that under suitable conditions the completion of the
equivariant
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
Algebra is the offer made by the devil to the mathematician. The devil says: `I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine.' Michael Atiyah[49] He introduced[50] the
Main article: Atiyah–Singer index theorem Atiyah's work on index theory is reprinted in volumes 3 and 4 of his collected works.[58][59] The index of a differential operator is closely related to the number of independent solutions (more precisely, it is the differences of the numbers of independent solutions of the differential operator and its adjoint). There are many hard and fundamental problems in mathematics that can easily be reduced to the problem of finding the number of independent solutions of some differential operator, so if one has some means of finding the index of a differential operator these problems can often be solved. This is what the Atiyah–Singer index theorem does: it gives a formula for the index of certain differential operators, in terms of topological invariants that look quite complicated but are in practice usually straightforward to calculate.[citation needed] Several deep theorems, such as the Hirzebruch–Riemann–Roch theorem, are special cases of the Atiyah–Singer index theorem. In fact the index theorem gave a more powerful result, because its proof applied to all compact complex manifolds, while Hirzebruch's proof only worked for projective manifolds. There were also many new applications: a typical one is calculating the dimensions of the moduli spaces of instantons. The index theorem can also be run "in reverse": the index is obviously an integer, so the formula for it must also give an integer, which sometimes gives subtle integrality conditions on invariants of manifolds. A typical example of this is Rochlin's theorem, which follows from the index theorem.[citation needed] The most useful piece of advice I would give to a mathematics student is always to suspect an impressive sounding Theorem if it does not have a special case which is both simple and non-trivial. Michael Atiyah[60] The index problem for elliptic differential operators was posed in
1959 by Gel'fand.[61] 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
Atiyah's former student
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.[67] 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.[68] 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.[69]
Atiyah[70] 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.[71]
As an application of the equivariant index theorem, Atiyah and
Hirzebruch showed that manifolds with effective circle actions have
vanishing Â-genus.[72] (Lichnerowicz showed that if a manifold has a
metric of positive scalar curvature then the
Raoul Bott, who worked with Atiyah on fixed point formulas and several other topics Atiyah, Bott and Vijay K. Patodi[74] 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,[75] which were later unexpectedly used in theoretical physics, in particular in Witten's work on anomalies. The lacunas discussed by Petrovsky, Atiyah, Bott and Gårding are similar to the spaces between shockwaves of a supersonic object. The fundamental solutions of linear hyperbolic partial differential
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.[76]
Atiyah[77] 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[78] 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.[79]
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.[80]
On the left, two nearby monopoles of the same polarity repel each other, and on the right two nearby monopoles of opposite polarity form a dipole. These are abelian monopoles; the non-abelian ones studied by Atiyah are more complicated. Many of his papers on gauge theory and related topics are reprinted in
volume 5 of his collected works.[81] 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[82] 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
The mathematical problems that have been solved or techniques that have arisen out of physics in the past have been the lifeblood of mathematics. Michael Atiyah[86] 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
Edward Witten, whose work on invariants of manifolds and topological quantum field theories was influenced by Atiyah Many of the papers in the 6th volume[103] of his collected works are
surveys, obituaries, and general talks. Since its publication, Atiyah
has continued to publish, including several surveys, a popular
book,[104] and another paper with Segal on twisted K-theory.
One paper[105] is a detailed study of the
But for most practical purposes, you just use the classical groups. The exceptional Lie groups are just there to show you that the theory is a bit bigger; it is pretty rare that they ever turn up. Michael Atiyah[111] With
Books[edit] This subsection lists all books written by Atiyah; it omits a few books that he edited. Atiyah, Michael F.; Macdonald, Ian G. (1969), Introduction to
commutative algebra,
Selected papers[edit] Atiyah, Michael F. (1961), "Characters and cohomology of finite
groups", Inst. Hautes Études Sci. Publ. Math., 9: 23–64,
doi:10.1007/BF02698718 . Reprinted in (Atiyah 1988b, paper 29).
Atiyah, Michael F.; Hirzebruch, Friedrich (1961), "Vector bundles and
homogeneous spaces", Proc. Sympos. Pure Math. AMS, 3: 7–38 .
Reprinted in (Atiyah 1988b, paper 28).
Atiyah, Michael F.; Segal, Graeme B. (1969), "Equivariant K-Theory and
Completion", Journal of Differential Geometry, 3: 1–18 .
Reprinted in (Atiyah 1988b, paper 49).
Atiyah, Michael F. (1976), "Elliptic operators, discrete groups and
von Neumann algebras", Colloque "Analyse et Topologie" en l'Honneur de
Awards and honours[edit] The premises of the Royal Society, where Atiyah was president from 1990 to 1995 In 1966, when he was thirty-seven years old, he was awarded the Fields
Medal,[116] for his work in developing K-theory, a generalized
So I don't think it makes much difference to mathematics to know that there are different kinds of simple groups or not. It is a nice intellectual endpoint, but I don't think it has any fundamental importance. Michael Atiyah, commenting on the classification of finite simple groups[111] He was elected a foreign member of the National Academy of Sciences,
the
I had to wear a sort of bulletproof vest after that! Michael Atiyah, commenting on the reaction to the previous quote[129] Atiyah was made a
^ a b ATIYAH, Sir Michael (Francis). ukwhoswho.com. Who's Who. 2014
(online edition via
Sources[edit] Boyer, Charles P.; Hurtubise, J. C.; Mann, B. M.; Milgram, R. J.
(1993), "The topology of instanton moduli spaces. I. The
Atiyah–Jones conjecture", Annals of Mathematics, Second Series, The
Annals of Mathematics, Vol. 137, No. 3, 137 (3): 561–609,
doi:10.2307/2946532, ISSN 0003-486X, JSTOR 2946532,
MR 1217348
Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius
(2004), Compact Complex Surfaces, Berlin: Springer, p. 334,
ISBN 978-3-540-00832-3
Gel'fand, Israel M. (1960), "On elliptic equations", Russ. Math.
Surv., 15 (3): 113–123, Bibcode:1960RuMaS..15..113G,
doi:10.1070/rm1960v015n03ABEH004094 . Reprinted in volume 1 of
his collected works, p. 65–75, ISBN 0-387-13619-3. On page
120 Gel'fand suggests that the index of an elliptic operator should be
expressible in terms of topological data.
Harder, G.; Narasimhan, M. S. (1975), "On the cohomology groups of
moduli spaces of vector bundles on curves", Mathematische Annalen, 212
(3): 215–248, doi:10.1007/BF01357141, ISSN 0025-5831,
MR 0364254
Matsuki, Kenji (2002), Introduction to the Mori program, Universitext,
Berlin, New York: Springer-Verlag, ISBN 978-0-387-98465-0,
MR 1875410
Palais, Richard S. (1965), Seminar on the Atiyah–Singer Index
Theorem,
External links[edit] Find more aboutMichael Atiyahat's sister projects Quotations from Wikiquote
Academic offices Preceded by George Porter President of the Royal Society 1990–1995 Succeeded by Sir Aaron Klug Preceded by Sir Andrew Huxley Master of Trinity College, Cambridge 1990–1997 Succeeded by Amartya Sen Preceded by The Lord Porter of Luddenham Chancellor of the University of Leicester 1995–2005 Succeeded by Sir Peter Williams Preceded by
Lord Sutherland of Houndwood
President of the
Awards and achievements Preceded by Robin Hill Copley Medal 1988 Succeeded by César Milstein v t e
v t e Fields Medalists 1936 Ahlfors Douglas 1950 Schwartz Selberg 1954 Kodaira Serre 1958 Roth Thom 1962 Hörmander Milnor 1966 Atiyah Cohen Grothendieck Smale 1970 Baker Hironaka Novikov Thompson 1974 Bombieri Mumford 1978 Deligne Fefferman Margulis Quillen 1982 Connes Thurston Yau 1986 Donaldson Faltings Freedman 1990 Drinfeld Jones Mori Witten 1994 Bourgain Lions Yoccoz Zelmanov 1998 Borcherds Gowers Kontsevich McMullen 2002 Lafforgue Voevodsky 2006 Okounkov Perelman (declined) Tao Werner 2010 Lindenstrauss Ngô Smirnov Villani 2014 Avila Bhargava Hairer Mirzakhani Book Category Mathematics portal v t e Savilian Professors Chairs established by Sir Henry Savile Savilian Professors of Astronomy John Bainbridge (1620) John Greaves (1642) Seth Ward (1649) Christopher Wren (1661) Edward Bernard (1673) David Gregory (1691) John Caswell (1709) John Keill (1712) James Bradley (1721) Thomas Hornsby (1763) Abraham Robertson (1810) Stephen Rigaud (1827) George Johnson (1839) William Donkin (1842) Charles Pritchard (1870) Herbert Turner (1893) Harry Plaskett (1932) Donald Blackwell (1960) George Efstathiou (1994) Joseph Silk (1999) Steven Balbus (2012) Savilian Professors of Geometry Henry Briggs (1619) Peter Turner (1631) John Wallis (1649) Edmond Halley (1704) Nathaniel Bliss (1742) Joseph Betts (1765) John Smith (1766) Abraham Robertson (1797) Stephen Rigaud (1810) Baden Powell (1827) Henry John Stephen Smith (1861) James Joseph Sylvester (1883) William Esson (1897) Godfrey Harold Hardy (1919) Edward Charles Titchmarsh (1931) Michael Atiyah (1963) Ioan James (1969) Richard Taylor (1995) Nigel Hitchin (1997)
v t e Presidents of the Royal Society 17th century Viscount Brouncker (1662)
Joseph Williamson (1677)
18th century
19th century
20th century
21st century Robert May (2000)
Martin Rees (2005)
Sir
v t e Copley Medallists (1951–2000)
Authority control WorldCat Identities VIAF: 44275873 LCCN: n50030722 ISNI: 0000 0001 1025 7457 GND: 118848453 SELIBR: 234572 SUDOC: 067638252 BNF: cb12339740x (data) MGP: 30949 NLA: 35009787 NDL: 01034751 NKC: jn20000600 |