Modern Mathematicians: Alain Connes

Alain Connes was born in Dpaca_mapraguignan, France in 1947. He entered École Normale Supérieure, one of the leading universities in Paris, in 1966 and graduated in 1970. American mathematician Robert Moore describes his thesis on the classification of factors of type III on operator algebras (in particular on von Neumann algebras), as:

“a major, stunning breakthrough in the classification problem.”

Connes has received many awards for his work, including the:

  • Prix Aimeé Berthé (1975)
  • Prix Pecot-Vimont (1976)
  • Gold Medal of the Centre National de la Recherche Scientifique (1977)
  • Prix Ampère from the Académie des Sciences in Paris (1980)
  • Prix de Electricité de France (1981)

However, Connes’ most notable achievement was being awarded the Fields Medal in 1982 (the ceremony was in 1983) due to his work on operator theory and in specific, as depicted by Japanese mathematician Huzihiro Araki, his:

(1) general classification and a structure theorem for factors of type III, obtained in his thesis;

(2) classification of automorphisms of the hyperfinite factor, which served as a preparation for the next contribution; 

(3) classification of injective factors;

(4) application of the theory of C*-algebras to foliations and differential geometry in general.

The study on von Neumann algebras began in the 1930s, when their factors were claconnes-alain_mp_771x900ssified. In the late 1960s there was a resurgence of interest on this topic.

Connes unified a number of ideas in the area that had been previously considered disparate. He also worked on some applications of operator algebras, for example their application to differential geometry. Additionally, his application of operator theory to noncommutative geometry produced new geometries. Furthermore, his later work has had meaningful impact in ergodic theory, which is the study of systems whose final state is independent of their initial state.

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