$C^*$-algebras and pseudodifferential calculus for singular foliations

We give an overview of joint work with Georges Skandalis over a number of years, which aims to set up the framework for Analysis on singular foliations. First we will discuss the definition of singular foliations, which needs to focus on the dynamical systems viewpoint rather than a partition to leaves. Then we will discuss the construction of the associated holonomy groupoid and the associated $C^*$-algebra(s). This algebra records the longitudinal pseudodifferential calculus of the foliation; its $K$-theory is the recipient of the analytic index for elliptic longitudinal pseudodifferential operators. Time permitting, we might also discuss the spectrum of the longitudinal Laplace operator.