Laura Monk

The Probabilistic Method in Hyperbolic Geometry

The aim of this talk is to illustrate how a probabilistic viewpoint can be incredibly fruitful in all fields of mathematics, even "fundamental" fields such as hyperbolic geometry. The probabilistic method consists in establishing properties that are true "for most X" rather than "for all X", using a random model to sample the object X we wish to study. It allows to prove results in spite of some pathological counter-examples, and also to benefit from powerful additional tools. I will narrate the story of this idea in hyperbolic geometry, from the first combinatorial model by Brooks and Makover, to more recent models, based on random covers and Weil-Petersson geometry. The main character we will follow during this journey is the diameter, a very simple quantity that cannot be bounded a priori, but is now very well-known thanks to this new approach.