An invitation to singular foliations

TL;DR

This paper provides an overview of singular foliations, covering foundational concepts, examples, recent non-commutative geometry tools, and open questions in the field.

Contribution

It systematically reviews singular foliation theory and introduces recent tools like the holonomy groupoid and singularity resolution methods.

Findings

Detailed review of singular foliation foundations

Introduction of holonomy groupoid in non-commutative geometry

Discussion of methods for resolving singularities

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.