Abstract: Many biological and physical systems are well-modeled by Brownian particles subject to gradient dynamics plus noise. Important for many applications is the net steady-state particle current or "flux" enabled by the noise and an additional driving force, but this flux is rarely computable analytically. Motivated by this, I will describe joint work with Yuliy Baryshnikov investigating the steady-state flux of nondegenerate diffusion processes on compact manifolds. Such a flux is associated to each one-dimensional real cohomology class and is equivalent to an asymptotic winding rate of trajectories. When the deterministic part of the dynamics is "gradient-like" in a certain sense, I will describe a graph-theoretic formula for the small-noise asymptotics of the flux (based on an extension of Freidlin-Wentzell large deviations theory). When additionally the deterministic part is locally gradient and close to a generic global gradient, there is a natural flux for which the graph-theoretic formula becomes Morse-theoretic and admits a description in terms of persistent homology. As an application, I will explain the paradoxical "negative resistance" phenomenon in Brownian transport discovered by Cecchi and Magnasco (1996).