Topological Mating of CRTs & Geometry of Brownian Excursion

In this talk, I will introduce the framework to glue together a pair of independent continuum random trees (CRTs). This "mating" of the two trees draws an important connection between the space-filling Schramm–Loewner evolution (SLE) and the Liouville quantum gravity (LQG). We will go through the definition of these random trees as well as their associated random process—Brownian excursion. Using the techniques from Stochastic Calculus and Moore's theorem, we will show that the gluing process almost surely produces a topological 2-sphere.