Lecturer: Claude Mitschi (Université de Strasbourg, France).

Title: Introduction to the classical and parameterized Picard-Vessiot theories and related monodromy problems

Abstract: The first part of these lectures aims at getting the audience familiar with modern forms of the classical differential Galois theory initiated by Picard and Vessiot . We will in particular present the generalization, due to Cassidy and Singer, of this theory to parameterized linear differential equations. In the second part we will refresh basics facts in analytic continuation such as the classical monodromy theorem and show how monodromy is closely related to analytic differential Galois theory.

In the third part we will prove some density theorems in analytic differential Galois theory and their analogues in parameterized Picard-Vessiot theory. We will end the course by proving the inverse problem of parameterized Picard-Vessiot theory for analytic families of differential equations with regular singularities, using a parameterized version of the weak Riemann-Hilbert problem.