Combinatorial geometric flows on triangulated surfaces

After proving the convergence of the Ricci flow on a two-dimensional closed surface for any initial data to the metric of constant curvature, the question of discretization of the flow naturally arose. Some conceptual difficulty is related to the fact that the metric on a triangulated surface is determined by the lengths of the edges, while the curvature is concentrated at the vertices. The naive version of the Ricci flow, as we will see, does not satisfy the desired property of convergence of the flow to the metric of constant curvature for any initial metric. A positive solution was found in a class of circle packing metrics, which itself is a very interesting combinatorial object. We will discuss the corresponding combinatorial Ricci flow, as well as some of its generalizations.