Differential Equations and Monodromy

TL;DR

This paper reviews classical and recent results on differential equations, focusing on hypergeometric equations, monodromy representations, and their algebraic properties, highlighting explicit descriptions and recent advances in the field.

Contribution

It provides an overview of historical and recent results on hypergeometric differential equations and their monodromy groups, including explicit descriptions and properties like thin-ness and arithmeticity.

Findings

Levelt's theorem explicitly determines monodromy representations.

Recent results on thin-ness and arithmeticity of hypergeometric monodromy groups.

Overview of Zariski closure of monodromy groups without proofs.

Abstract

In these expository notes, we describe results of Cauchy, Fuchs and Pochhammer on differential equations. We then apply these results to hypergeometric differential equation of type $_nF_{n-1}$ and describe Levelt's theorem determining the monodromy representation explicitly in terms of the hypergeometric equation. We also give a brief overview, without proofs, of results of Beukers and Heckman, on the Zariski closure of the monodromy group of the hypergeometric equation. In the last section, we recall some recent results on thin-ness and arithmeticity of hypergeometric monodromy groups.