One of the aspects of the noncommutative index theory is the Baum-Connes conjecture, an outstanding open problem in the K-theory of the convolution C*-algebras of locally compact groups. It asserts that a certain index map whose receptacle are the K-groups of the convolution C*-algebras is an isomorphism. If true, it provides a mean of computing the aforementioned K-groups, and entails other consequences. Recently, the conjecture was proved for a large class of locally compact groups though counterexamples were found to a more general version of the conjecture (e.g., V.Lafforgue). A possible direction for further exploration is to extend the conjecture to cover locally compact quantum groups.

Syllabus: convolution C*-algebra of a locally compact group, proper actions of locally compact groups, universal spaces for proper actions (axiomatic and constructive approach), Kasparov KK-theory, assembly map, Peter-Weyl theory and proof of the conjecture for compact groups, generalized versions of the conjecture, categories of examples where the conjecture holds (e.g., amenable groupoids) and counterexamples to the generalized conjecture, important implications of the conjecture.