Max Riestenberg: Discrete and faithful representations of surface groups | 6 Lectures

Significant interest in the study of representations of surface groups has developed over the recent decades. For instance, Hitchin famously proved that the character variety of surface group representations into PSL(n,R) has a distinguished component homeomorphic to a ball, and Labourie proved that all of the representations in this component are discrete and faithful.
 
This lecture series is intended to be a gentle introduction to this topic. We will start with geometric structures on surfaces, some reminders about Teichmüller theory, Goldman’s theorem on representations into PSL(2,R), and convex real projective structures. Later on we will discuss infinitesimal deformations of representations, actions on affine space, and maximal and positive representations.

Arnaud Maret: Character Varieties - a symplectic perspective | 3 Lectures  (26.4., 10.5. and 21.6.)

The goal of this minicourse is to introduce the notion of character variety. A character variety is, broadly speaking, a construction that associates a symplectic manifold to a surface and a Lie group. It comes with a natural action of the mapping class group of the surface that preserves its symplectic structure.

The minicourse will introduce all relevant notions and will, in particular, not assume any particular knowledge on symplectic geometry or mapping class groups. We will provide reminders on relevant notions of Lie theory, algebraic/analytic varieties and group cohomology.

Tentative plan:

Lecture 1: Representation and Character varieties - an introduction
Lecture 2: The Goldman symplectic structure
Lecture 3: The mapping class group action

Alexander Lytchak: CAT spaces | 3 Lectures (24.5, 5.7 and 19.7.)

In the lectures I will explain definitions, basic properties and some applications of metric spaces with curvature bounded above in the sense of Alexandrov.