A DEFORMATION-GROUPOID APPROACH TO THE BAUM-CONNES ASSEMBLY MAP

In a recent preprint, together with Bai-Ling Wang and Hang Wang, following the ideas of Connes and Moscovici, we give an explicit formula (in terms of a pairing of forms and currents) for the pairing of the left-hand side of the BC map of a discrete group and the periodic cyclic cohomology of the group algebra. In this talk, I want to explain the main source leading to such results, that is the use of deformation groupoids. I will also try to explain how this approach extends to other settings (more general groups or groupoids) and how it also allows us to consider more "involved" geometric cycles in a very natural way. For example, one can naturally consider an orbifold version of the left-hand side of the BC assembly map. The first part of the talk will be a quick but basic introduction to the main ideas around deformation groupoids and their use in wrong-way functoriality.