Abstract: 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.

For planar graphs one can associate tilings of planar regions with rectangles of prescribed areas.

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

Coffee will be served at 12:00 before the lecture
at Schreiber building 006