On the irregular Riemann-Hilbert correspondence

TL;DR

This paper extends the Riemann-Hilbert correspondence to irregular singularities, solving a longstanding open problem by developing an enhanced category of perverse sheaves, with implications for complex differential equations.

Contribution

It provides a solution to the irregular Riemann-Hilbert correspondence, generalizing Kashiwara's work to irregular cases and introducing an enhanced category of perverse sheaves.

Findings

Solved the irregular Riemann-Hilbert problem

Developed an enhanced category of perverse sheaves

Provided examples illustrating the main ideas

Abstract

The original Riemann-Hilbert problem asks to find a Fuchsian ordinary differential equation with prescribed singularities and monodromy in the complex line. In the early 1980's Kashiwara solved a generalized version of the problem, valid on complex manifolds of any dimension. He presented it as a correspondence between regular holonomic D-modules and perverse sheaves. The analogous problem where one drops the regularity condition remained open for about thirty years. We solved it in the paper that received a 2024 Frontiers of Science Award. Our construction requires in particular an enhancement of the category of perverse sheaves. Here, using some examples in dimension one, we wish to convey the gist of the main ingredients used in our work. This is a written account of a talk given by the first named author at the International Congress of Basic Sciences on July 2024 in Beijing.