Differential Equations and Monodromy

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.