Three Lectures on Algebraic Microlocal Analysis

Abstract

This is a survey talk with some historical comments. I will first explain the notions of Sato’s hyperfunctions and microfunctions, at the origin of the story, and I will describe the Sato’s microlocalization functor which was first motivated by problems of analysis. Then I will briefly recall the main features of the microlocal theory of sheaves with emphasize on the functor \(\mu \)hom which will be an essential tool in the sequel. Then, I will construct the microlocal Euler class associated with trace kernels. This construction applies in particular to constructible sheaves on real manifolds and \(\mathscr {D}\)-modules (or more generally, elliptic pairs) on complex manifolds. Finally, I will first recall the construction of the sheaves of holomorphic functions with temperate growth or with exponential decay. These are not sheaves on the usual topology, but ind-sheaves, or else, sheaves on the subanalytic site. I will explain how these objects appear naturally in the study of irregular holonomic \(\mathscr {D}\)-modules.