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)

In 2001, Connes and Landi proved that certain classes of Riemannian manifolds admits an isospetral deformation defined by the isometric toric action. This construction is a vast generalization of NC tori and does include the NC tori. In my talk I will outline the idea and discuss possible consequences.

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 3 of 3)