Geometric flows on Riemannian and Kaehler manifolds

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The first part of the project is about the geometric and analytic properties of the Ricci flow equation and their applications to the study of geometry. Namely to study the classifications of self-similar solutions, e.g gradient solitons and ancient solutions, as well as the convexity type estimates in general. Such results have far-reaching consequences in the singularity analysis, and applications of Ricci flow in the study of geometric-topological structure of the manifolds. The second theme of the project is on the sharp gradient estimates of Li-Yau-Hamilton type, related monotonicity formulae and applications in geometric nonlinear PDEs. Their relations to physics, statistical mechanics will be studied too. The aim is to discover a fundamental physical/geometric principle to unify various sharp estimates and monotonicity formulae. It will also provide the guideline for further discovery of the new monotonicity formulae in other geometric PDEs.Since all physical event takes place in a space, the subject of differential geometry which studies the geometric properties of the space has important consequence in every physical event. This project mainly involves the study of partial differential equations of parabolic type which arise from differential geometry, and their applications to the understanding of various geometric/topological properties of manifolds. This area lies in the center of the current mathematics. It naturally connects various area of mathematics, such as topology, Riemannian geometry, partial differential geometry, Lie groups, as well as mathematical physics. The techniques developed can be useful in understanding problems in economics, material sciences and bio-sciences.