Title: Algebraic, analytic and topological aspects of linear (ordinary) differential equations

Lecturer: Claude Mitschi, Strasbourg University

Date: April 8 - 27, 2011

Time: Every Friday, 15:30 to 18:30 (April 8), 14:00 to 17:00 (other dates)

Place: Room 1218 at BICMR, Resource Plaza, Peking University
Prof.Mitschi's lectures this week will be held on Thursday （4.21）at 15:30 to 17:30 in Room 1213 and also on Friday（4.22） 14:00-17:00 in Room 1218.

Abstract:
We will address various problems on linear differential equations in one complex variable: analytic questions arising from analytic continuation of the solutions, algebraic questions in the language of differential Galois theory and topological questions related to the Riemann-Hilbert problem.

Plan:
1. Algebraic aspects: differential Galois theory (over general fields)

• - differential algebras, Picard-Vessiot algebras
• - linear algebraic groups
• - the differential Galois group of a linear differential equation
• - existence and uniqueness of Picard-Vessiot field extensions, Galois correspondence, solvability

2. Analytic aspects

• - analytic continuation, monodromy theorem
• - singularities of linear equations over the Riemann sphere, analytic continuation of solutions

3. Analytic differential Galois theory for equations over the Riemann sphere

• - local and global Galois groups
• - Schlesinger's theorem relating the monodromy group and the Galois group
• - the inverse problem

4.Introduction to the Riemann-Hilbert problem

• - Levelt's theory for regular singularities
• - vector bundles and connections
• - the Riemann-Hilbert problem