This is an introductory course on discrete conformal geometry on polyhedral surfaces.

We will start with hyperbolic geometry and then introduce two notions of discrete conformality for polyhedral surfaces. The first comes from Thurston’s work on circle packing and the related discrete Ricci flow on surfaces.  The second comes from discretization of conformal factors and hyperbolic convex hulls.

We will prove several fundamental theorems on circle packing. These include Andreev-Koebe-Thurston theorem, colin de Verdiere’s variational principle and its generalization by Chow-Luo, Rodin-Sullivan’s work on convergence of circle packing maps and He-Schramm’s work on rigidity of circle packing.

On the second topics of discrete conformality coming from vertex scaling, we will introduce briefly the related work of Penner on decorated Teichmuller spaces, hyperbolic convex hull in 3-dimensions and discrete uniformization theorem for both compact and non-compact polyhedral surfaces.

The students should know some basic differential geometry,  complex analysis and basic topology.