Abstract for the first lecture (Thursday, July 9th, 10:15 am): The aim of this talk and the next is to explain two important aspects of equivariant Kasparov theory, especially for smooth manifolds: duality, and the topological description of equivariant KK-groups using equivariant correspondences. In the first talk we will review the basic definitions of KK-theory, including the Thom isomorphism. We then explain duality, which gives a way of reducing KK groups to K-theory groups with support conditions. This will be used in the second talk to prove that equivariant KK-groups for smooth manifolds can be described in topological terms.

Robin Deeley and Elkaioum Moutuou provided us with lecturer notes for this lecture, thank you!

Abstract for the second lecture (Tuesday, July 14th, 10:15 am): Building on the first talk, we describe equivariant Kasparov theory for smooth manifolds in purely topological terms, using an appropriate theory of equivariant correspondences. There is always a map from the topological theory to the analytic theory, and it is an isomorphism under some hypotheses related to the existence of certain equivariant vector bundles. We conclude by listing various geometric situations under which the hypotheses hold, and by showing how and why these considerations are important for equivariant index theory.