Non-commutative index theorem

Maxim Braverman

Abstract:  One of the most exciting achievements of mathematics in the second half of the 20'th century is the Atiyah-Singer index theorem, asserting that the analytical and topological index of an elliptic operator on a compact manifold coincide. This theorem allows, in particular, to prove that certain differential equations have solutions without solving the equation. Many important generalizations of this theorem have been obtained in the last 40 years.

In the talk I will review the classical Atiyah-Singer index theorem. In particular, I'll give a short introduction to the topological K-theory. Then I will discuss "non-commutative" generalizations: K-theory of operator algebras and non-commutative index theorems. If the time permits, I'll present some important applications of these non-commutative generalizations to the usual (commutative) geometry.