Heat content asymptotics of bounded domains

For a bounded domain in a Riemannian manifold, we consider the solution of the heat equation with unit initial data and Dirichlet boundary conditions. Integrating the solution with respect to the space variable one obtains the function of time known in the literature as the "heat content" of the given domain. In this talk we show how the geometry of the boundary affects heat diffusion by examining the small time behavior of the heat content. In particular, we study a three term asymptotic expansion for polyhedral Euclidean domains, and give a recursive algorithm for the calculation of the entire asymptotic series when the boundary is smooth.