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Abstract

We will introduce the concept of combinatorial curvature defined on planar graphs, followed by several versions of the combinatorial Gauss-Bonnet theorem. The Gauss-Bonnet theorem involving boundary turns (geodesic curvature) is particularly useful because the combinatorial interpretation of boundary turns can help us prove certain problems. These include a generalized version of the He-Schramm conjecture for circle packings—an open question posed by Lyons and Peres—and another open question by Angel, Benjamini, and Horesh. Additionally, we will explain some recent results related to combinatorial curvature during the talk.