Lecture 1

Julia Heller: Coxeter groups and buildings

Groups generated by reflections appear in various different contexts, for instance as Weyl groups in Lie theory. Coxeter groups are the underlying abstractions of reflection groups. The present rich geometric structure also carries over to their simplicial realizations, the Coxeter complexes.
In turn, Coxeter complexes are the building blocks of buildings, which themselves are highly symmetric simplicial complexes: Jacques Tits introduced buildings to understand semisimple algebraic groups.

In the lecture, we introduce the above concepts and aim to understand the relation between the mentioned groups and spaces, illustrated by various examples.

You can find lecture notes and further material in our RTG-sharepoint.

Lecture 2
Zachary Greenberg: Cluster Algebras for Geometers

To introduce cluster algebras we begin with a classic geometric problem, the study of Teichmüller space on bordered surfaces. First we describe the developing pair description of a hyperbolic structure. Using this we construct two coordinate systems on Teichmüller space that depend on an ideal triangulation of the surface. Finally we explore how these coordinates “mutate” when we change triangulation.