The systole of a random surface

Bram Petri (Fribourg/Brown)

Abstract :

There are multiple notions of random surfaces. In this talk a random surface will be a surface constructed by randomly gluing together an even number of triangles that carry a fixed metric. If one chooses a specific hyperbolic metric then the set of all possible surfaces obtained by performing this procedure will be dense in every moduli space of compact surfaces. This means that using this construction, one can ask questions about the geometry of a typical hyperbolic surface. The model lends itself particularly well to studying high genus surfaces. For example it turns out that the expected value of the length of the shortest non-contractible curve, the systole, of such a surface converges to a constant. In this talk I will explain what goes into the proof of this fact and how this relates to the theory of random regular graphs.