The Baum-Connes conjecture has been an outstanding focal point of noncommutative geometry for over twenty years. The first part of this lecture course will show how to apply new ideas from homological algebra and homotopy theory to understand the conjecture more conceptually and to widen its scope towards locally compact quantum groups. The second part of the course will be devoted to applications of the Baum-Connes conjecture and its relations to classical topology.

The following topics will be covered:

1. Categorical aspects of Kasparov's KK-theory
2. Duality in Kasparov theory
3. Homological algebra in triangulated categories
4. The Baum-Connes assembly map for locally compact groups
5. Dirac-dual-Dirac method
6. The Baum-Connes conjecture for discrete quantum groups that are duals of compact groups
7. Towards the Baum-Connes conjecture for locally compact quantum groups
8. Corollaries of the Baum-Connes conjecture
9. Baum-Douglas model for K-homology
10. Equivariant-bivariant Chern character
11. Geometric KK-theory

Prerequisites: Basic knowledge of K-theory of C*-algebras and elementary category theory.