Noncommutative geometry

Abstract

Through algebraic geometry we became familiar with the correspondence between geometrical spaces and commutative algebra. The aim of this talk is to show an analogous correspondence, in the domain of real analysis, between geometrical spaces and algebras of functional analysis, going beyond the commutative case. This theory is based on three essential points:

• 1. The existence of many examples of spaces which arise naturally, such as Penrose's space of universes, the space of leaves of a foliation, the space of irreducible representations of a discrete group, for which the classical tools of analysis lose their pertinence, but which correspond in a very natural fashion to a noncommutative algebra.

• 2. The possibility of reformulating the classical tools of analysis such as measure, topology and calculus in algebraic and Hilbertian terms, so that their framework becomes noncommutative, the commutative case being neither isolated nor closed in the general theory.

• 3. The relationship with physics, the spaces used by physicists being noncommutative in many cases.

Fields Medal 1982 for his contribution to the theory of operator algebras, particularly the general classification and a structure theorem for factors of type III, classification of automorphisms of the hyperfinite factor, classification of injective factors, and applications of the theory of C*-algebras to foliations and differential geometry in general.

Transcribed from the videotape of the talk by Pere Ara and Carles Broto; revised by the author.