# Lie groupoids, pseudodifferential calculus and index theory

## TL;DR

This paper reviews the role of Lie groupoids in noncommutative geometry, focusing on their C*-algebras and pseudodifferential calculus, and discusses recent advances and open questions in the field.

## Contribution

It provides a comprehensive overview of Lie groupoids' applications in index theory and noncommutative geometry, highlighting recent progress and future directions.

## Key findings

- Review of Lie groupoids and their C*-algebras
- Summary of recent advances in the field
- Open questions and potential developments

## Abstract

Alain Connes introduced the use of Lie groupoids in noncommutative geometry in his pioneering work on the index theory of foliations. In the present paper, we recall the basic notion involved: groupoids, their C*-algebras, their pseudodifferential calculus... We review several recent and older advances on the involvement of Lie groupoids in noncommutative geometry. We then propose some open questions and possible developments of the subject.