Introduction to the Galois Theory of Linear Differential Equations

This paper is an expanded version of the 10 lectures I gave as the 2006 London Mathematical Society Invited Lecture Series at the Heriot-Watt University, 31 July 4 August 2006†. My goal was to give the audience an introduction to the algebraic, analytic and algorithmic aspects of the Galois theory of linear differential equations by focusing on some of the main ideas and philosophies and on examples. There are several texts ([Beu92, Kap76, Kol76, Mag94, dPS03] to name a few) that give detailed expositions and I hope that the taste offered here will encourage the reader to dine more fully with one of these. The rest of the paper is organized as follows. In Section 1.2, What is a Linear Differential Equation?, I discuss three ways to think about linear differential equations: scalar equations, linear systems and differential modules. Just as it is useful to think of linear maps in terms of linear equations, matrices and associated modules, it will be helpful in future sections to go back and forth between the different ways of presenting linear differential equations. In Section 1.3, Basic Galois Theory and Applications, I will give the basic definitions and describe the Galois correspondence. In addition I will describe the notion of monodromy and its relation to the Galois