Geometry and spectrum of random hyperbolic surfaces

https://hu-berlin.zoom.us/j/61686623112

The aim of this talk is to describe typical compact hyperbolic surfaces: results will be stated for most surfaces rather than every single one of them. In order to motivate this idea, I will first present examples introduced in literature as limiting cases of famous theorems, and argue that they might be seen as "atypical". This will allow us to appreciate the contrast with a fast-growing family of new results in both geometry and spectral theory, which are established with probability close to one, while being false for these atypical surfaces. I will in particular present different probabilistic models that are used to study typical surfaces, and discuss results on the distribution of eigenvalues and the geometry of long geodesics.