Every Gromov hyperbolic metric space has a boundary at infinity; this a topological space endowed with extra structure (a "quasisymmetric gauge" in Sullivan's terminology). One can ask when the boundary structure is equivalent, in a natural sense, to the standard quasisymmetric structure on the 2-sphere. In spirit, this is an attempt to generalize the classical uniformization theorem to a metric space setting. The motivation for such a generalization comes from work of Semmes, and from Thurston's hyperbolization conjecture for 3-manifolds.
In the lecture, I will explain the background and motivation, and then discuss joint work with Mario Bonk. This uses recently developed analytical tools (e.g. Poincare inequalities on metric spaces) that have appeared in work by Heinonen-Koskela, Semmes, and Cheeger, to prove uniformization theorems and answer a question of Semmes.