An invitation to singular foliations

Abstract

These lecture notes attempt to invite the reader towards the theory of singular foliations, both smooth and holomorphic. In addition to a systematic review of the foundations, and an attempt to put in order examples and several elementary constructions, we detail several recent tools developed for non-commutative geometry, in particular the holonomy groupoid of Androulidakis and Skandalis and various methods for resolving singularities. We also introduce various homotopic notions, and end with a series of open questions.