The assembly map and the Baum-Connes conjecture.

Nigel Higson (Penn State University)
Thursday January 28th, 2010 RLM 9.166

These talks will be about the C*-algebra approach to index theory and K-theory that was proposed by Atiyah and worked out in detail by Kasparov.

In the last lecture I'll discuss the most famous application of Kasparov's work - to the Novikov higher signature conjecture and the Baum-Connes conjecture.  I'll sketch the proof of both conjectures for Gromov's a-T-menable groups (these are groups that act properly on an infinite-dimensional Euclidean space, and include amenable groups, free groups, Coxeter groups and others).  The argument uses an interesting noncommutative C*-algebra that serves as a proxy for the commutative algebra of continuous functions on a Euclidean space (which isn't itself very useful when the space is infinite-dimensional).  This algebra, together with a closely related Bott-Dirac operator on the Euclidean space,  may have other applications.