A translation surface is a connected surface with a translation structure. For translation surfaces of finite type, there exists a rich theory, for example on moduli spaces of finite translation surfaces.
The goal of this project is to approach the definition of a good topology on the space of all
translation surfaces by a detour. Instead of studying the space itself, we study four types of invariants
that are living in spaces which are better understood:
- geometric invariants such as Cheeger constants or systoles,
- Siegel-Veech constants,
- Veech groups, and
- saddle connection complexes.