## 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.