Uniformization of metric surfaces

The classical uniformization theorem states that every simply connected Riemann surface is conformally equivalent to the unit disc, the complex plane, or the Riemann sphere. In this talk we are interested in non-smooth generalizations of this statement, where conformality is replaced by quasisymmetry or (weak) quasiconformality. Our goal is to demonstrate that every metric surface of locally finite  area admits a suitable parametrization. If time admits, we will see how weakly quasiconformal uniformization can be applied to obtain 2-dimensional Lipschitz-volume rigidity results.