Tuesday 14 May 2019, 16:00-17:00 in HG00.310
Shantanu Dave (Vienna)
Geometry of the Baum-Connes conjecture
The Baum-Connes Conjecture (BCC) is a statement that relates the geometry of a locally compact group to its analysis. It is a super-conjecture, that is it implies several conjectures such as Novikov's higher signature conjecture, the Gromov-Larson conjecture etc. However the BCC is not understood for many groups such as \(3\times 3\) integer matrices with determinant one \(\mathrm{SL}(3,\mathbb Z)\). After a brief introduction to the BCC the talk will describe a program to attack the problem for discrete subgroups of Lie groups. First we present the geometry "on different parts of infinity" for the corresponding symmetric spaces. Secondly we will talk about how this geometry provides the analysis on the groups. The structures appearing at infinity are filtered manifolds and they include foliations, contact manifolds with additional structures. The analysis creates hypoelliptic operators. We shall finish with the challenges in realising this approach to the BCC.