Geometric flows and four-dimensional geometry

这个项目位于几何学和偏微分方程组的交界处。PI将进一步研究自然几何演化方程，旨在理解四维几何和拓扑的各个方面。其中一个项目是了解几乎厄米特流形上几何流的奇点形成和长时间行为，这是由PI与G.Tian联合提出的。这其中的一个方面是充分了解多闭合流的长时间行为如何与复杂曲面的分类有关。另一个项目是了解黎曼度规的杨-米尔斯能量梯度流的亚临界行为。这里的一个主要目标是了解在初始能量较小的四维流形上可能出现的奇点。利用几何和曲率来理解空间是数学和科学中的一个基本思想。通常，相关的曲率性质与与自然定律有关的方程密切相关。这里提出的这种几何演化方程经常出现在各种物理环境中，并在包括图像分析和材料科学在内的各个领域得到应用。拟议的研究将推进基本的几何和解析思想，这是此类应用的基础。

This project lies at the interface of geometry and partial differential equations. The PI will further the study of natural geometric evolution equations designed to understand aspects of four dimensional geometry and topology. One project is to understand the singularity formation and long time behavior of geometric flows on almost Hermitian manifolds, previously introduced by the PI in joint work with G. Tian. One aspect of this is to fully understand how the long time behavior of the pluriclosed flow relates to the classification of complex surfaces. Another project is to understand the subcritical behavior of the gradient flow of the Yang-Mills energy of a Riemannian metric. One major goal here is to understand the possible singularities which can develop on four-manifolds with small initial energy.The use of geometry and curvature in understanding spaces is a fundamental idea in mathematics and science. Frequently the relevant curvature properties are closely related to equations relating to the laws of nature. Geometric evolution equations of the kind proposed here arise frequently in various physical contexts, and see application to various areas including image analysis and material science. The proposed research will advance fundamental geometric and analytic ideas which are the foundation of such applications.