The analysis of non-Markovian, self-interacting diffusions, which has motivations from probability, physics, biology, etc., is intimately connected with that of an associated stochastic interface. In this talk, we will look at dynamical fluctuations of this interface, and derive a KPZ-type stochastic PDE as a scaling limit. Unlike the usual KPZ equation, the geometry of the underlying manifold plays an important role in the analysis of this SPDE. Deriving the SPDE from the diffusion model is based on a novel "local-to-global" stochastic homogenization principle. Studying the SPDE itself amounts to relatively modern ideas in stochastic analysis coupled with analysis of pseudo-differential operators on manifolds. Based on joint work with Amir Dembo.