Three lectures on Algebraic Microlocal Analysis

TL;DR

This paper provides an overview of fundamental concepts in algebraic microlocal analysis, including sheaf microsupport, Hochschild homology analogues, and applications to D-modules and irregular holonomic systems.

Contribution

It introduces a microlocal analogue of Hochschild homology for sheaves and explores applications to index theorems and irregular holonomic D-modules.

Findings

Construction of microlocal Hochschild homology for sheaves

Recovery of index theorems for D-modules

Development of ind-sheaves of holomorphic functions

Abstract

These three lectures present some fundamental and classical aspects of microlocal analysis. Starting with the Sato's microlocalization functor and the microsupport of sheaves, we then construct a microlocal analogue of the Hochschild homology for sheaves and apply it to recover index theorems for D-modules and elliptic pairs. In the third lecture, we construct the ind-sheaves of temperate and Whitney holomorphic functions and give some applications to the study of irregular holonomic D-modules.