# Groupoids and Algebras of Singular Foliations

## Meeting Details

Abstract: The holonomy groupoid of a foliated manifold $M$ is, roughly speaking, the thing you get if you try to make the leaf equivalence relation $R \subseteq M \times M$ into a smooth manifold. Androulidakis-Skandalis have generalized the constructions of this groupoid and its associated algebras to make them applicable to a very general class of singular foliations considered by Stefan-Sussmann. To get an idea just how wild these singular foliations are allowed to be, note that any Lie group action whatsoever determines a singular foliation, whereas one only expects to get a regular foliation when the action is more or less free. An interesting feature (bug?) of their approach is that singular foliations depend on more detailed data than just the partition into leaves; the extra data is stored in the collection of leafwise vector fields. This infinitesimal information can make a big difference to the groupoids and algebras. I'll try to draw this point out by discussing what happens for probably the simplest "singular foliation" you could imagine, namely the partition of the circle into two "leaves": a point and an arc.