Groupoids and singular foliations

Abstract: This doctoral thesis has two objectives. The first objective is to introduce a notion of equivalence for singular foliations that preserves their transverse geometry and is compatible with the notions of Morita equivalence of the holonomy groupoids and the transverse equivalence for regular foliations that appeared in the 1980's. The second one is to describe the structures behind quotients of singular foliations and to connect these results with their associated holonomy groupoids. It also wants to give an introduction to the notion of singular foliations as given by Androulidakis and Skandalis in arXiv:math/0612370, as well as to their relation with Lie groupoids and Lie algebroids.