Bob Lutz: Matroids arising from electrical networks

Abstract

In this talk I will introduce Dirichlet matroids, a generalization of graphic matroids defined by electrical networks. First, I will describe an interaction between duals of Dirichlet matroids and duals of circular electrical networks. Second, I will characterize the Bergman fans of Dirichlet matroids as explicit subcomplexes of graphic Bergman fans. For "complete" networks, we obtain a description in terms of phylogenetic trees. And third, time permitting, I will discuss a connection between electrical response matrices and the half-plane property of matroids. No prior knowledge of electrical networks is necessary!