# Lectures

## RTG Lecture

Sommer Semester 2024

Plese see RTG Lecture Abstracts.

### RTG Lecture

Winter Semester 2023/24

Please see RTG Lecture Abstracts.

**RTG Lecture**

Sommer Semester 2023

Please see RTG Lecture Abstracts.

**RTG Lecture**

Winter Semester 2022/23

Please see RTG Lecture Abstracts.

### Sommer Semester 2022

Lecture 1

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.*

#### Lecture 2

#### 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

Lecture 2: The Goldman symplectic structure

Lecture 3: The mapping class group action

#### Lecture 3

#### 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.*

**Past RTG Lectures**

Winter Semester 2021/22

RTG Lecture "Asymptotic Invariants and Limits of Groups and Spaces" Some TOP Topology (Tam Nguyen Phan), Billiards (Lucas Dahinden)

Summer Semester 2021

RTG Lecture "Asymptotic Invariants and Limits of Groups and Spaces" Aspects of Teichmüller-Thurston Theory (James Farre), Cobordism theory (Georg Frenck)

Winter Semester 2020/21

RTG Lecture "Asymptotic Invariants and Limits of Groups and Spaces" Kähler groups and Geometric Group Theory (Claudio Llosa), Surgery theory and the Farrell-Jones conjecture (Carmen Rovi)

Summer Semester 2020

RTG Lecture "Asymptotic Invariants and Limits of Groups and Spaces" A short introduction to amenable groups (Alessandro Carderi), The top Lyapunov exponent (Thi Ndang)

Winter Semester 2019/20

RTG Lecture "Asymptotic Invariants and Limits of Groups and Spaces" Filling arithmetic groups (Enrico Leuzinger), Controlled topology applied to study aspherical manifolds (Tom Farrell)

Summer Semester 2019

RTG Lecture "Asymptotic Invariants and Limits of Groups and Spaces"

Gromov's nonsqueezing theorem (Urs Fuchs), CAT(0) cube complexes and applications to group theory and low-dimensional topology (Jonas Beyrer)

Winter Semester 2018/19

RTG Lecture "Asymptotic Invariants and Limits of Groups and Spaces"

Intoduction to arithmetic groups and their cohomology (Steffen Kionke), An introduction to geometric representation theory (Andrew Sanders)

Summer Semester 2018

RTG Lecture "Asymptotic Invariants and Limits of Groups and Spaces"

Geometric Convergence and Finiteness Theorems and their Applications (Wilderich Tuschmann), Coxeter Groups and Geometry (Gye-Seon Lee)

Winter Semester 2017/18

RTG Lecture "Asymptotic Invariants and Limits of Groups and Spaces"

The Willmore functional (Tobias Lamm), Thurston's theory of surfaces (Daniele Alessandrini)

Summer Semester 2017

RTG Lecture "Asymptotic Invariants and Limits of Groups and Spaces"

Introduction to parabolic geometries (Karin Melnick), Incidence structures on flag varieties and rigidity (Beatrice Pozzetti), CAT(0) groups and geometry (Petra Schwer)

Winter Semester 2016/17

RTG Lecture "Asymptotic Invariants and Limits of Groups and Spaces"

Torsion Invariants (Roman Sauer), Harmonic Maps (Andy Sanders)

#### Research Seminars

Topology (KA)

Differential Geometry (KA)

Geometry (HD)

Geometric Analysis (KA)

Symplectic Geometry (HD)