Mathematics Colloquia and Seminars

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Discrete differential geometry is of great interest because it 1) allows computable versions of differential geometry so as to allow numerical calculations and modeling, and 2) provides an alternative to geometric/complex analysis that replaces the differential geometry with a combination of classical geometry (Euclidean/hyperbolic) and combinatorics. We will pursue some of these themes in regard to conformal geometry of surfaces. Discrete conformal structures are a particular class of piecewise Euclidean or hyperbolic geometry on a triangulation, and give rise to versions of the Riemann Mapping Theorem and the Uniformization of Surfaces. The main tool in analysis of these structures is the ability to compute variational formulas for curvature in terms of the Laplacian of the variation; this has been well-studied in the case of Riemannian geometry, and the discrete form relates certain angle variations to graph Laplacians with specific geometric choices of weights. We will explain how such discrete conformal structures can be classified. Some additional topics for discussion include progress on discrete conformal variations in three dimensions, applications of discrete conformal structures (e.g., to sensor network routing), and the role of geometric flows (combinatorial Ricci flows, etc.).