Groupoids and singular manifolds

TL;DR

This paper reviews how Lie groupoids are applied in singular analysis, index theory, and non-commutative geometry, highlighting their role in generalizing classical theorems and addressing complex geometric problems.

Contribution

It provides an overview of Lie groupoid applications in index theory, including proofs of the Atiyah-Singer theorem and progress on the Baum-Connes conjecture for Lie groupoids.

Findings

Groupoid proofs of the Atiyah-Singer index theorem

Progress on index problems for non-compact manifolds

Generalizations of pseudodifferential calculus for boundary problems

Abstract

We describe how Lie groupoids are used in singular analysis, index theory and non-commutative geometry and give a brief overview of the theory. We also expose groupoid proofs of the Atiyah-Singer index theorem and discuss the Baum-Connes conjecture for Lie groupoids. With the help of the general framework of Lie groupoids and related structures we survey recent progress on problems which were outside the scope of the original work of Atiyah and Singer. This includes the Atiyah-Singer type index problem for many classes of non-compact manifolds (e.g. manifolds with a Lie structure at infinity). We also consider generalizations of the pseudodifferential calculus on Lie groupoids, e.g. for boundary value problems.