In a noncompact Riemannian manifold like $M$ above, heat tends to zero as time tends to infinity. In this talk, we will investigate the question, ``How fast does heat decay to zero?'' For the type of manifolds considered here, the answer depends upon a function known as the harmonic profile of the manifold. We will compute the profile and the heat decay in simple cases before discussing cases where computing the profile is a hard open problem. The talk will contain explanations as to why the heat equation is my favorite PDE and plentiful pictures.