Analysis on singular spaces: Lie manifolds and operator algebras

TL;DR

This paper surveys the connections between analysis on singular spaces, specifically Lie manifolds, and operator algebras, highlighting new insights and results, with a focus on index theory and applications to PDEs.

Contribution

It introduces a new presentation linking analysis on Lie manifolds with operator algebras, including some original results and a comprehensive overview of the field.

Findings

Analysis on Lie manifolds provides a bridge to operator algebras.

Groupoids associated with Lie manifolds facilitate understanding of operator algebra structures.

Applications to index theory and PDEs are significant outcomes of this framework.

Abstract

We discuss and develop some connections between analysis on singular spaces and operator algebras, as presented in my sequence of four lectures at the conference "Noncommutative geometry and applications," Frascati, Italy, June 16-21, 2014. Therefore this paper is mostly a survey paper, but the presentation is new, and there are included some new results as well. In particular, Sections 3 and 4 provide a complete short introduction to analysis on noncompact manifolds that is geared towards a class of manifolds--called "Lie manifolds"--that often appears in practice. Our interest in Lie manifolds is due to the fact that they provide the link between analysis on singular spaces and operator algebras. The groupoids integrating Lie manifolds play an important background role in establishing this link because they provide operator algebras whose structure is often well understood. The initial motivation for the work surveyed here--work that spans over close to two decades--was to develop the index theory of stratified singular spaces. Meanwhile, several other applications have emerged as well, including applications to Partial Differential Equations and Numerical Methods. These will be mentioned only briefly, however, due to the lack of space. Instead, we shall concentrate on the applications to Index theory.