15:00 Igor Rivin (St.Andrews): Experiments with graphs
We describe experimental results on (mostly spectral) properties of various classes of random graphs (including random regular graphs, random planar triangulations, random Delaunay triangulations).
16:30 Richard Kenyon (Brown University): Fixed Energy Harmonic Functions
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 planar graphs one can construct associated tilings of planar regions with rectangles of prescribed areas. 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.