Algebraic curves, or closed Riemann surfaces, play an important role in several areas of mathematics. In this talk, I will discuss so-called moduli spaces that parametrize these objects, together with some extra structure. We will see that their cohomology groups are related to a profinite property of the mapping class group, whose definition I will recall. Then, I will explain some work from my thesis, which uses tools from p-adic geometry to obtain vanishing statements for these cohomologies at infinite level in high degrees. Speaker(s): Emanuel Reinecke (Max Planck Institute for Mathematics)