# Felix Klein Lectures

## On the Role of Symmetry in Noncommutative Geometry

### Henri Moscovici Ohio State University, Columbus, USA

**Date: **June 6-25, Tuesdays and Thursdays, 10-12 a.m.

**Venue: **Mathematics Center, Endenicher Allee 60, Lipschitzsaal

**Abstract:**

My lectures aim to illustrate a certain aspect of symmetry in noncommutative geometry, which could be aptly called symmetry bending. Such a feature is manifest in the construction of Hopf algebras associated to classical Lie pseudogroups, which emerged from the work of A. Connes and myself on the local index formula for transversely elliptic operators on foliations. While the transverse characteristic classes of foliations were well understood in the context of the (Gelfand-Fuks) Lie algebra cohomology, the appropriate tool in the dual K-homological context turns out to be the Hopf algebra version of cyclic cohomology. A spin-off of this approach reveals hidden geometric structures underlying classical constructs in the theory of modular forms. In the above constructions the symmetry bending is actually an after-effect of symmetry breaking, borne out by tracking and codifying the latter. A more common manifestation arises as deformation of classical symmetry. This aspect will be illustrated by a discussion of my recent joint work with A. Connes on the conformal geometry of noncommutative tori, intended to shed light on the notion of intrinsic curvature in the realm of noncommutative spaces.