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Wolfe Lecture: Fixed-energy harmonic functions

4:00 pm Thursday, March 24, 2015
Richard Kenyon (Brown University)

Abstract: The discrete Laplacian on graphs is one of the most fundamental and useful operators in graph theory, discrete probability, and network science. One naturally assigns positive weights, or conductances, to edges. The Dirichlet problem in this context is to find a harmonic function with specified boundary values. Here we change the problem slightly: we solve the Dirichlet problem for finite networks, fixing ``edge energies" rather than fixing conductances. More precisely, we show that for any given choice of edge energies there is a choice of conductances for which the resulting harmonic function realizes those energies. In fact the set of solutions is the number of compatible acyclic orientations of the graph. For rational data, the Galois group of the totally real algebraic numbers acts naturally on this set of solutions. We also consider scaling limits on large graphs which lead to novel PDEs. As a discrete geometry application we study fixed-area rectangulations of planar domains. This is joint work with Aaron Abrams.

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