Lecture 1

Lecturer
Roman Sauer

Title
The systolic inequality for aspherical manifolds

Abstract
Gromov’s systolic inequality bounds the the minimal length of a non-contractible loop on a closed aspherical Riemannian manifold in terms of its volume. The proofs of this fundamental inequality in Riemannian geometry inspired many developments around isoperimetric inequalities, large-scale invariants and even scalar curvature.

We start this lecture series by presenting Guth's proof of this inequality for the torus. We then present quite recent developments around Uryson width which lead to a shorter proof of the systolic inequality in general.

Lecture 2

Lecturer
Tommaso Scognami

Title
Local systems on Riemann surfaces and character varieties

Abstract
Local systems on a Riemann surface X are geometric objects which arise naturally in the study of holomorphic differential equations on the surface. Character varieties for  the Riemann surface X are the moduli spaces of local systems on it, i.e. geometric objects parametrizing isomorphism classes of local systems, or, equivalently, representations of the fundamental group of X. The aim of these lectures is to give an introduction to the study of local systems on X and of its associated character varieties.

In particular, the main topics of the lectures would be:
1) The Riemann-Hilbert correspondence for X, which explains how to relate local systems on X to vector bundles with connections on X. This will cover the first three lectures approximately.
2) A result of Hausel and Rodriguez-Villegas, who found interesting topological invariants of character varieties using the surprising technique of counting points over finite fields. This will cover the last two lectures approximately.

Outlines of the lectures:

1st: Linear differential equations on Riemann surfaces, monodromy and regularity condition.
2nd: Local systems, representations of the fundamental groups and vector bundles with connections.
3rd: Riemann-Hilbert correspondence for Riemann surfaces.
4th: Character varieties for compact Riemann surfaces and non-abelian Hodge correspondence
5th: Cohomology of character varieties. Hausel and Rodriguez-Villegas result.