Stochastic Currents

My thesis is based on a generalization of electrical current to higher dimensions. Specifically, I’m interested in quantization results for the current generated by sub-objects under stochastic motion. This breaks into two cases: discrete (CW complexes with Markov processes) or continuous (smooth manifolds with stochastic vector fields), and the tools involved vary for each. One such tool in the discrete case are higher spanning trees, and we have enumerated them using Reidemeister torsion, an invariant in Algebraic K-theory. All of this work generalizes the 1-dimensional graph results of my advisors, Vladimir Chernyak and John Klein. The interplay between these two settings also provides a nice framework to compare invariants in one to those in the other, e.g. Reidemeister to Ray-Singer torsion, using Witten-style deformations.