This workshop is organized around a visit of Professor Jean-Michel Bismut to Toronto.  Professor Bismut is a leading expert in index theory, and he is renowned for his striking use of probability theory in this context.  More recently he has initiated the theory of the hypoelliptic Laplacian, with applications to geometry and orbital integrals. This offers the prospect of entirely new methods in the representation theory of real reductive Lie groups.  The workshop will report on recent progress on these topics, with a focus on the roles of index theory and the hypoelliptic Laplacian in representation theory.