Linear and nonlinear geometric evolution equations

微分几何研究空间的拓扑、几何和解析性质。由于所有的物理事件都发生在一个空间中，根据爱因斯坦的相对论，底层空间的空间度量特性对空间中发生的物理事件有根本性的影响，包括我们生活的空间。这种关系使微分几何这门学科处于数学和一些科学基本问题的中心位置。现代微分几何诉诸于线性和非线性偏微分方程的理论/方法。本文提出用演化偏微分方程的方法研究微分几何，重点从热力学和统计力学的角度研究熵、单调性和相关的尖锐微分估计动机。高斯曲率流的研究对图像处理、仿射微分几何和凸几何等学科具有重要的意义。本文对里奇流的研究促进了对空间微分拓扑结构的认识，是与高能物理直接相关的理论物理。这一建议所涉及的问题的解决，推动了上述相关学科的发展。该提案的外展部分向公众传播新的结果，并为圣地亚哥及其他地区社区的数学教育做出贡献。它还有助于代表性不足群体的培训和早期职业发展。该建议的技术方面涉及发展的熵的尖锐单调性和相关的尖锐点明智的估计。微分几何问题的研究依赖于求解演化方程，如里奇流方程、高斯曲率流方程或其他非线性演化方程，以及/或研究所得到的解的微妙性质。现代偏微分方程理论将各种线性和非线性方程的解的研究简化为某些熵量的单调性和相关的逐点估计。本提案中提出的去正则化估计将过去几十年来发展起来的已经影响深远的梯度估计、Hessian估计和曲率估计技术的范围扩大到更一般的设置，通过减少所涉及的解决方案的规则性，同时编码更多的几何信息。它们已经并将产生许多强大的几何结果。所开发的技术反过来又可以促进相关偏微分方程的研究，这通常在偏微分方程理论中占有中心地位。

Differential geometry studies the topological, geometric and analytic properties of the spaces. Since all physical events take places in a space, by Einstein's relativity the spatial metric properties of the underlying space have fundamental impacts on the physical events happening in the space, including the very one we live in. This relation places the subject of the differential geometry in a central position of the mathematics and some fundamental issues of sciences. Modern differential geometry appeals to the theory/methods of linear and nonlinear partial differential equations. The proposal involves the study of differential geometry via the method of the evolutional partial differential equations with the focus on the study of the entropy, the monotonicity, and related sharp differential estimates motives by thermodynamics and statistical mechanics. The study of Gauss curvature flow has important consequences on the subjects of the image processing, affine differential geometry and convex geometry. The study of the Ricci flow in the proposal advances the understanding of the differential topological structure of the spaces and is related to the theoretic physics with direct bearing on the high energy physics. The resolving of the problems involved in the proposal advances the above mentioned related subjects in sciences. The out-reach components of the proposal disseminate the new results to general public and contributes towards the mathematical education in the community in the greater area of San Diego and beyond. It also contributes to the training and the early career development of under-represented groups. The technical aspects of the proposal involve the development of the sharp monotonicity of the entropy and related sharp point wise estimates. The study of differential geometric problems relies on solving evolution equations such as the Ricci flow equation, Gauss curvature flow equation or other nonlinear evolution equations, and/or the study of delicate properties of the solutions obtained. The theory of modern partial differential equations reduces the study of the solutions of various linear and nonlinear equations to the monotonicity of certain entropic quantities and related point wise estimates. The de-regularizing estimates proposed in this proposal broadens the scope of the already far-reaching gradient estimates, Hessian estimates and curvature estimates techniques developed in the last several decades to more general settings, by requiring less regularity of the solutions involved, while encoding far more geometric information. They have produced and will produce much powerful geometric consequences. The techniques developed can in turn advances the study of the related partial differential equations, which usually occupies a central role in the theory of the partial differential equations.