Singular foliations and the Baum-Connes conjecture

In earlier work of ours with G. Skandalis (Paris 7) we introduced a longitudinal pseudodifferential calculus for very "bad" singular foliations, as well as the associated analytic index. Calculating the K-theory of the foliation C*-algebra is the crucial next step. In this lecture we will discuss such a calculation for at least one case, namely the action of SO(3) on $\mathbb{R}^3$. What we learn from this calculation is that there are different kinds of "badness", which we can express explicitly in terms of a "singularity height". Using this notion, we will discuss how to formulate the Baum-Connes conjecture (left hand side and assembly map) for quite a large class of singular foliations. This is joint work with G. Skandalis.