DescriptionIn this thesis, we will investigate the convergence of discrete conformal metrics to the classical uniformization metric on Riemannian surfaces. We prove that for a reasonable geodesic triangle mesh on a smooth closed orientable surface, there exists a discrete conformal factor for the induced piecewise linear metric. And the difference between this discrete conformal factor and the classical uniformization factor is controlled by the maximal edge length of the triangulation. The estimates rely on collections of discrete elliptic estimates and isoperimetric inequalities for triangle meshes. The case for genus h >= 1 is a joint work with Tianqi Wu, the case for genus h = 0 is a joint work with Tianqi Wu and Yanwen Luo.