# Trimester Seminar

**Venue:** HIM lecture hall, Poppelsdorfer Allee 45

Organizers: Alan L. Carey, Victor Gayral, Matthias Lesch, Walter van Suijlekom, Raimar Wulkenhaar

## Tuesday, September 2

15:00 - 16:00 Welcome Meeting

Abstract: Welcome meeting with the participants and organizers of the Trimester Program.

## Friday, October 17

10:30 - 11:30 Thomas Schick: Large scale index theory and applications to positive scalar curvature

Abstract: The Dirac operator is a fundamental object of differential geometry and analysis on a manifold M. A fundamental property is that its index (which is of topological nature) has to vanish if the manifold admits a metric of positive scalar curvature - thus providing obstructions to the existence of such a metric.

Large scale (or coarse) index theory allows to systematically use of K-theory of operator algebras and the associated index theory to generalize these methods from compact to non-compact manifolds.

In the talk

- (after a short introduction into the general aspects of index theory of the Dirac operator)
- we will introduce into large scale index theory,
- explain how the Dirac operator fits into the picture
- and derive one particular application which actually also gives new information about closed manifolds: a codimension one or two index obstruction.

## Tuesday, November 4

15:00 - 16:00 Partha Chakraborty: On Dirac type DGA

Abstract: Dirac DGA was constructed by Connes as a shadow of the a univarsal DGA. We will study modifications of that concept. This is joint work with Satyajit Guin.

## Wednesday, November 5

14:00 - 15:00 Joachim Cuntz: Index theorems in the framework of bivariant K-Theory

Abstract: We sketch a very short proof for the index theorem of Baum-Douglas-Taylor and explain how this implies the index theorems of Kasparov and Atiyah-Singer.

15:00 - 16:00 Partha Chakraborty: An invariant for certain dynamical systems

Abstract: We propose an invariant for an ergodic C*dynamical system where dynamics is governed by a compact quantum group. In particular this makes sense for compact quantum groups. We will compute this for some examples. This is joint work with Arup Pal.

## Monday, November 10

11:00 - 12:00 Paul Baum: Expanders and Morita-compatible exact crossed-products

Abstract: An expander or expander family is a sequence of finite graphs X_{1}, X_{2}, X_{3}, ... which is efficiently connected. A discrete group G which contains an expander in its Cayley graph is a counter-example to the Baum-Connes (BC) conjecture with coefficients. Such groups are known to exist and are referred to as Gromov groups or Gromov monsters. These are the only known examples of non-exact groups. The left side of BC with coefficients "sees" any group as if the group were exact. This talk will indicate how to make a change in the right side of BC with coefficients so that the right side also "sees" any group as if the group were exact. This corrected form of BC with coefficients uses the unique minimal exact and Morita compatible intermediate crossed-product. For exact groups (i.e. all groups except the Gromov groups) there is no change in BC with coefficients. In the corrected form of BC with coefficients any Gromov group acting on the coefficient algebra obtained from its expander is not a counter-example.

Thus at the present time (November, 2014) there is no known counter-example to the corrected form of BC with coefficients. The above is joint work with E. Guentner and R. Willett.

## Wednesday, November 12

15:00 - 16:00 Joachim Cuntz: Witt vectors and cyclic homology