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.