Geometry and topology
Discrete hyperbolic geometry
Abstract: I will talk about combinatorial maps, which are discrete surfaces built by gluing polygons together. They have been given a lot of attention in the last 60 years, and here we will focus on the geometric properties of large random maps, in a rather new regime where the genus of the underlying surface goes to infinity.
In this regime, the objects exhibit some natural hyperbolic properties, which turn out to be surprinsigly close to another (continuous) model of random hyperbolic surfaces : the Weil—Petersson probability measure when the genus goes to infinity.
I will give an overview of the existing results and open problems, guaranteed without technical details.