ABSTRACT: I will explain how to construct geometric heat flows on manifolds through the use of stochastic parallel translation and give applications of this construction to the regularity of heat flows and spectral theory of
geometric Laplacians. Some applications are: stochastic characterizations of the Yang-Mills and Yang-Mills heat equation, a new proof of non-explosion for the Yang-Mills heat equation with small initial condition, calculations
of the mean of random holonomy and determination of the spectrum of the horizontal Laplacian on the Hopf fibration over complex and quaternionic projective space.