Electrical networks, hyperplane arrangements and matroids

An electrical network is a graph with Dirichlet boundary conditions. In this talk, I will show how to associate a hyperplane arrangement and matroid to any electrical network. These objects allow us to view electrical networks in a new light, by generalizing the elegant ways in which graphic arrangements capture information about graphs. Time permitting, I will present results on harmonic functions with prescribed edge energies; on the relationship between supersolvable arrangements and chordal graphs; and on matroid quotients formed by geometric duals of electrical networks embedded in surfaces.