Quasisymmetric Koebe uniformization

By the Riemann Mapping Theorem every simply connected proper domain $D$ in the complex plane $\mathbb{C}$, is conformally equivalent to the unit disk $\mathbb{D}$. More generally, Koebe proved in 1920's that every finitely connected domain $D$ is conformal to a circle domain, or a domain whose complementary components are round disks or points. In 1909 Koebe conjectured that every planar domain is conformal to a circle domain. In 1993, He and Schramm proved Koebe's conjecture in the countable case. The general case is still open and is one of the hardest and most important open problems in complex analysis.

During the last 30 years, numerous modern versions of uniformization results have been proven, often motivated by problems in geometric group theory and dynamics. Instead of planar domains and conformal mappings, one now considers possibly fractal metric spaces and quasiconformal/quasisymmetric mappings between them. I will describe some of these results and time permitting will state some recent theorems (joint with Jon Rehmert) characterizing countably connected metric surfaces which are quasisymmetric to planar circle domains.