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Fixed-energy harmonic functions

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If you have a question about this talk, please contact Perla Sousi.

This is joint work with Aaron Abrams. We study the map from conductances to edge energies for harmonic functions on graphs with Dirichlet boundary conditions. We prove that for any compatible acyclic orientation and choice of energies there is a unique choice of conductances such that the associated harmonic function realizes those orientations and energies.

For rational energies and boundary data the Galois group of $\Q^{tr}$ (the totally real algebraic numbers) over $\Q$ permutes the enharmonic functions, acting on the set of compatible acyclic orientations.

Connections with square ice and SLE _{12} (based on work with Angel, Miller, Sheffield, Wilson) will be briefly discussed.

This talk is part of the Probability series.

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