The University of Michigan Combinatorics Seminar
The inverse (or Dirichlet-to-Neumann) problem in electrical networks asks one to recover the combinatorial structure of a network and its edge conductances from its response matrix. For planar networks embedded in a disk, the problem was studied and effectively solved by Curtis-Ingerman-Morrow, de Verdière-Gitler-Vertigan and Kenyon-Wilson. In the first part of the talk I will describe how the problem can be solved for a large class of networks embedded in a cylinder. Our approach uses an analog of the R-matrix for certain affine geometric crystals. It also makes use of Kenyon-Wilson's groves. There are a number of remarkable similarities between the theory of total positivity and that of electrical networks. In the second part of the talk I will offer one way to understand these similarities by introducing a family of Lie algebras/groups naturally associated with electrical networks. This is joint work with Thomas Lam.