Many problems in applied mathematics require the solution of the heat equation in unbounded domains. Integral equation methods are particularly appropriate for the solution of such problems for several reasons: they are unconditionally stable, they are insensitive to the complexity of the geometry, and they do not require the artificial truncation of the computational domain as do finite difference and finite element techniques. Methods of this type, however, have not become widespread due to the high cost of evaluating heat potentials. I will discuss an optimal time algorithm which is based on a novel spectral representation of the heat kernel.