Title: The metric geometry of surfaces

Abstract: A well-known part of classical mathematics is the differential geometry of smooth surfaces, developed by historical mathematicians such as Euler and Gauss. A fundamental result on the topic is the uniformization theorem, proved by Poincaré and Koebe in 1907, which states that any smooth Riemannian surface can be mapped conformally onto a surface of constant curvature. Since then, the geometry of surfaces has been investigated in increasing generality by several research communities. Non-smooth surfaces were perhaps first systematically studied as part of the field of Alexandrov geometry beginning in the 1930s, leading to a well-developed theory of surfaces of bounded integral curvature. Surfaces also form part of the modern field of analysis on metric spaces, which has led to several striking uniformization theorems for general classes of metric surfaces. In recent work with P. Creutz and D. Ntalampekos, we show how a geometric theory of surfaces in the same spirit as Alexandrov geometry can be developed under the single minimal geometric assumption of locally finite area. In particular, we use no assumption on curvature, or any of the previous assumptions from analysis on metric spaces. As an application, we obtain a new uniformization theorem for surfaces that is essentially the strongest possible for the non-fractal setting.