Alain Connes

fl. 1947, Draguignan, France

AnalysisGeometryMathematical physicsFields Medal 1982Crafoord Prize in Mathematics 2001

French mathematician; classified injective von Neumann algebra factors and founded noncommutative geometry, with applications ranging from foliations and index theory to the Standard Model.

Alain Connes was born on 1 April 1947 in Draguignan in the south of France. He studied at the École Normale Supérieure (1966–1970) and completed his doctorate at the CNRS under Jacques Dixmier in 1973, with a thesis that introduced the modular theory now named after him and Tomita. He became director of research at the CNRS, then a permanent professor at the IHÉS (1979), and held the chair of Analysis and Geometry at the Collège de France from 1984 to 2017.

His work on von Neumann algebras in the 1970s — the subdivision of type III factors, the automorphism-group machinery, and the uniqueness theorem for the injective II_1 factor — largely completed the classification program initiated by Murray and von Neumann. He then turned to the geometry encoded by noncommutative algebras: the cyclic cohomology he invented, the noncommutative Chern character, and the spectral triple formalism laid out in his 1994 book 'Noncommutative Geometry' rebuilt differential geometry, index theory and Riemannian geometry on C*-algebras. The Connes embedding problem, posed in 1976 about approximating II_1 factors by finite-dimensional matrices, was shown in 2020 to be equivalent to the disproved form of Tsirelson's problem, with consequences for quantum information.

With collaborators he applied noncommutative geometry to the Standard Model of particle physics, derived the Higgs mechanism geometrically, and pursued a program to attack the Riemann hypothesis through the noncommutative space of adèle classes. He received the Fields Medal in 1982 and the Crafoord Prize in 2001, and is a member of the French Academy of Sciences.

Workplaces: Collège de France, Institut des Hautes Études Scientifiques (IHÉS)

France

Award-winning work

Fields Medal 1982For contributions to operator algebras, in particular the theory of factors of von Neumann algebras. Connes solved the classification problem for injective factors of type II and type III: in his thesis he had already divided type III factors into a finer classification by a real parameter, and in 1976 he proved that an injective type II_1 factor is uniquely determined (isomorphic to the hyperfinite factor). The machinery he developed — the modular operator theory applied to von Neumann algebras — became the basis of noncommutative geometry, where 'noncommutative spaces' replace manifolds and classical geometric notions are rephrased in algebraic terms.

Contributions, linked to the library

Library topicOperator theoryCompleted the classification of injective von Neumann algebra factors and founded noncommutative geometry.Great problemKadison–Singer problemThe Connes embedding problem on II_1 factors was long known to be equivalent to a positive answer to Kadison–Singer; its 2020 negative resolution (via Tsirelson's problem) came after Marcus–Spielman–Srivastava proved the Kadison–Singer conjecture in 2013.