Geometric flows on Riemannian and Kaehler manifolds

该项目的第一部分是关于Ricci流动方程的几何和解析性质及其在几何研究中的应用。即研究自相似解的分类，例如梯度孤子解和古解，以及一般的凸型估计。这些结果对奇性分析以及Ricci流在流形几何拓扑结构研究中的应用都有深远的影响。该项目的第二个主题是Li-Yau-Hamilton类型的尖锐梯度估计，相关的单调性公式以及在几何非线性偏微分方程组中的应用。它们与物理学、统计力学的关系也将被研究。其目的是发现一个基本的物理/几何原理来统一各种精确的估计和单调性公式。由于所有的物理事件都发生在空间中，所以研究空间几何性质的微分几何学科在每一个物理事件中都有重要的意义。这个项目主要涉及从微分几何中产生的抛物型偏微分方程组的研究，以及它们在理解流形的各种几何/拓扑性质方面的应用。这一领域处于当前数学的中心。它自然地连接了数学的各个领域，如拓扑学、黎曼几何、偏微分几何、李群以及数学物理。所开发的技术可能有助于理解经济学、材料科学和生物科学中的问题。

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.