Unni Namboodiri Lectures in Geometry and Topology

Dimensions and C*-algebras

Monday April 7, 4:00 pm, E 206

The study of the possible dimensions of projective modules over C*- algebras has been a perennial topic in operator algebras since the earliest work of von Neumann. But the subject took on new life in the 1980's when it emerged that encoded within these dimensions is information of real geometric interest, and when K-theoretic techniques were developed to extract this information. I shall present a survey of these topics, ending with an introductory account of the Baum-Connes conjecture, which is the main organizing principle governing this meeting of geometry, C*-algebras and K-theory.


How to prove the Baum-Connes conjecture

Tuesday April 8, 4:30 pm, E 206

As a by-product of his work with Singer on the index theorem, Atiyah discovered a remarkably simple (or at least conceptually simple) proof of the Bott periodicity theorem using elliptic operators. As I shall explain, most of the progress on the Baum-Connes conjecture has been achieved by framing the conjecture as a generalized, equivariant form of Bott periodicity, and then borrowing from Atiyah's argument.
On the positive side, this approach has led to quite a bit of progress on the conjecture and its geometric corollaries. Moreover some of the geometric progress so achieved appears to be very difficult to recast in purely geometric terms. On the negative side, obstacles arise that have more to do with unitary representation theory than geometry.


C*-algebras and Lie group representations

Thursday April 10, 4:30pm, E 206

I shall try to give an account of the connection between the Baum- Connes conjecture and the representation theory of semisimple groups. Even though this is, in some sense, one of the oldest aspects of the conjecture, the situation is still very unclear. However there are clear links to the problem of realizing discrete series representations, and to a proposal of Mackey to parametrize the duals of semisimple groups.