We present a construction of multiresolution analyses and wavelets on manifolds and graphs, that efficiently represent a diffusion process,
and its Green's function, at all scales, in a multiresolution compressed form. We discuss motivations, applications, examples,
including: clustering and regularized function approximation on data sets and manifolds, atomic decompositions, nonlinear dimensionality
reduction, homogenization of diffusion equations, Markov chains. Collaborators include JC Bremer, RR Coifman, W Goetzman, PW Jones, S Lafon, AD Szlam, J Walden.