Title: Differential Harnack inequalities and the Ricci flow
Abstract: In the classical theory of parabolic PDEs one can use the maximum principle to prove Harnack's inequality, which provides local estimates for solutions depending upon the geometry of the underlying manifold. Seeking to clarify this geometric dependence, Li and Yau (1986) discovered "differential Harnack inequalities" that can be integrated along paths to produce classical Harnack-type estimates. This in turn allowed them to obtain estimates on the heat kernel, eigenvalues, and Betti numbers among other things.
In this talk I will explain the Li-Yau differential Harnack inequality, its geometric meaning, and some of the aforementioned estimates. As time permits, I will discuss the further advancements made by Hamilton in this area: analogous Harnack-type estimates for the Ricci flow which play a key role in analyzing the singularities that arise when evolving a Riemannian metric under said flow.
