Abstract: The starting point for the heat equation proof of the Atiyah-Singer index theorem is the celebrated McKean-Singer formula and the asymptotic expansion of the heat kernel. In this first lecture the following topics will be covered: basic Sobolev space theory, including Garding's inequality; finiteness and regularity results for elliptic PDE's on compact manifolds, Hodge decomposition theorem for elliptic complexes, existence of the heat kernel and its asymptotic expansion. (Talk 2 of 3)