abstract: The two lectures in Week 3 described the basic features of Ricci flow. The schedule for Week 5 is as follows:
Talk 1: Perelman's differential Harnack estimate. This talk is a continuation of the introduction, but will be essentially self-contained. I will first review the differential Harnack estimates of Li-Yau (for the scalar heat equation) and Hamilton for the Ricci flow, and then discuss how Perelman's differential Harnack estimate for the conjugate backward heat equation associated to the Ricci flow could be derived from similar method of Li-Yau-Hamilton. One can then relate the L-function of Perelman to the optimal path integral of Li-Yau type.
Talk 2: Gaussian densities and stability of Ricci solitons. In this talk I will present the recent joint work with R. Hamilton and T. Ilmanen on the second variation of Perelman's \lambda and \nu functionals for the Ricci flow, and the linear stability of examples. We also define the "central density" of a shrinking Ricci soliton and compute its values for certain examples in dimension 4. Using these tools, one can sometimes predict or limit the formation of singularities in the Ricci flow. In particular, we can show that certain Einstein manifolds are unstable for the Ricci flow in the sense that generic perturbations acquire higher entropy and thus can never return near the original metric.
Talk 3: The Ricci flow on Kaehler manifolds. I will survey the developments on the Ricci flow on Kaehler manifolds and present some open problems.