Michael Atiyah

Sir Michael Francis Atiyah, OM ( born April 22, 1929 in London) is an English mathematician.

Life and work

He was born the son of a Scottish woman and a Lebanese. He went to Cairo and Manchester to school, did his military service and began to study at Cambridge. After his doctorate, he was there 1954 " Fellow" of the Trinity College. He was also in 1955 at the Institute for Advanced Study at Princeton. In 1958 he joined the University of Oxford, where he received the 1963 " Savilian " Professor of Geometry. He remained in Oxford (up to a professorship at the Institute for Advanced Study from 1969 to 1972 ) until 1990, when he was the newly founded Isaac Newton Institute for Mathematics in Cambridge and director at the same time the "master" of Trinity College.

He was in the 1960s, along with Friedrich Hirzebruch a founder of K-theory, one defined with the help of vector bundles cohomology theory, and proved in 1963 along with Isadore M. Singer the Atiyah-Singer index theorem, that of one of the most important mathematical propositions 20th century is considered. The sentence expresses the analytical index of an elliptic linear differential operator ( for example, Laplace operator, Dirac operator ) on a compact manifold M by topological invariants of M from (varies by type of operator, in the simplest case by the Euler -Poincare characteristic ). It therefore combines analysis with topology and thus also has applications in modern physics. The " index " of the differential operator is the difference of the dimensions of his solution vector space and its adjoint operator. He also participated in the new simpler proofs of this theorem using the heat equation (diffusion equation) in the 1970s ( with Raoul Bott, Patodi 1973). With Graeme Segal and Raoul Bott, he also gave a proof of the theorem formulated as a variant of the Lefschetz fixed-point theorem in equivariant K- theory. The index set also has applications in algebraic geometry - it contains the theorem of Riemann -Roch as a special case. In connection with the index theorem "discovered" Atiyah and Singer also the Dirac operator for mathematics.

The motivation for the famous phrase came from Israel Gelfand (On elliptic equations, Russ. Mathematical Surveys, 1960).

Atiyah has rendered outstanding services in recent decades as a tireless mediator between mathematics and physics. In particular, he was interested in the non- Abelian gauge theories (Yang -Mills theories ) that are non-linear differential equations on manifolds, and he examined together with Raoul Bott ( fixed-point theorem of Atiyah - Bott ). But also imposed particularly by Edward Witten in the mathematics of quantum field theory methods (of new knot invariants on topological quantum field theories to supersymmetry and string theory ), he resorted to. Many ideas Atiyah were reversed from the 1980s taken by physicists (including new evidence of its index set ) and further developed.

His students include Simon Donaldson, Nigel Hitchin, Peter Kronheimer, Graeme Segal, Frances Kirwan and Ruth Lawrence.

Prizes and awards

  • Fields Medal (1966 ), with a plenary lecture at the ICM (Global aspects of the theory of elliptic differential operators )
  • Royal Medal of the Royal Society (1968 )
  • Antonio Feltrinelli Prize of the Accademia Nazionale dei Lincei (1981 )
  • King Faisal International Prize for Science (1987 )
  • Appointed Knight (1983 )
  • Copleymedaille (1988)
  • Member of the British Order of Merit (1992 )
  • Abel Prize (2004) jointly with Singer
  • He was invited speaker at the International Mathematical Congresses in Stockholm 1962 ( The Grothendieck ring in geometry and topology ), 1970 in Nice (Elliptic operators and singularities of vector fields ) and 1978 in Helsinki ( Geometrical aspects of gauge theories ).
  • He is a Fellow of the American Mathematical Society.

Writings

  • Collected works. 5 volumes, 1988
  • K -theory. Benjamin 1967 1989
  • With Iagolnitzer (ed. and co-author): Fields medalist lectures. World Scientific 1997, 2003
  • The geometry and physics of knots. Cambridge 1990
  • (Ed. and co-author) The representation theory of Lie groups. 1977
  • Introduction to commutative algebra. Addison-Wesley 1969
  • With Isadore M. Singer: The index of elliptic operators on compact manifolds. Bull AMS 1963 proof then. In a series of articles in the Annals of Mathematics, reprinted in Volume 3 of the Collected Works
  • With Raoul Bott: A Lefschetz fixed point formula for elliptic operators. Bull AMS 1966
  • Advice to a young mathematician, Timothy Gowers June Barrow- in Green, Imre Leader ( Editor): The Princeton Companion to Mathematics, Princeton University Press, 2008, pp. 1006-1010
Princeton (New Jersey) Isadore Singer Atiyah–Singer index theorem Fredholm-Operator Laplace-Operator Dirac-Operator Euler characteristic Atiyah–Bott fixed-point theorem Jesse Douglas Laurent Schwartz Atle Selberg Kunihiko Kodaira Jean-Pierre Serre Klaus Roth Paul Cohen (mathematician) Alexander Grothendieck Stephen Smale Alan Baker (mathematician) Heisuke Hironaka Sergei Novikov (mathematician) David Mumford Charles Fefferman Alain Connes Gerd Faltings Michael Freedman Vaughan Jones Jean Bourgain Pierre-Louis Lions Jean-Christophe Yoccoz Curtis T. McMullen Laurent Lafforgue Andrei Okounkov Terence Tao Wendelin Werner Elon Lindenstrauss Lennart Carleson Jacques Tits Mikhail Leonidovich Gromov John Tate Yakov Sinai Integrated Authority File
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