# Geometry and Physics

Hausdorff Trimester Program

May - August 2008

**Organizers:** Alain Connes, Matthias Lesch, Matilde Marcolli

The interactions between mathematics and quantum physics played an important role in recent developments in mathematics. Areas of mathematics which have been inspired very much from physics are Conformal Field Theory, Mirror Symmetry, and Noncommutative Geometry.

The aim of this Hausdorff Trimester Program was to bring together leading experts from these areas. The program was divided into two main parts. The first, May and June, concentrated on quantum field theories, whereas the second, June and August, focussed around non-commutative geometry. The modern geometric quantum field theory involves mathematical areas like topology, differential geometry and functional analysis, whereas non-commutative geometry has its idea in the fact that a (nice) space is completely characterized by the C*-algebra of continuous functions, which allows a generalization, mainly propagated by and based on work of A. Connes, by considering non-commutative C*-algebras. Whereas in the first topic one of the big problems is to construct topological or geometric quantum field theories and to study their properties, a guiding principle for the second topic is to generalize theorems known for manifolds like index theorems to the non-commutative world. In both cases the interaction with theoretical physics is a central theme.