Geometric equations and geometric applications

本课题研究黎曼几何中曲率方程的存在性和正则性问题。这些方程包括里奇流，爱因斯坦方程，杨-米尔斯方程。对于Ricci流，PI将重点关注(1)Kahler几何解的有限时间奇点形成；(2)奇点与底层空间几何的相互作用；(3)解的长期行为。PI将研究多闭流，这是一种由复杂几何结构产生的新型曲率流。他将发展它的解析理论和奇点形成。研究发现，该流与具有b场的非线性sigma模型的重整化群流之间存在着深刻的联系。PI将进一步探索这种联系，并为弦理论中的对偶性提供新的数学见解。对于爱因斯坦方程，PI将主要关注其解的存在性及其奇异解的行为。维4的情况特别有趣。我们还将研究四维的自对偶度量。它们可以用来研究底层空间的几何和拓扑结构。PI还将研究辛流如何在4维中发展有限时间奇点以及辛几何中的其他基本问题。问题包括辛曲面的同位素及其在辛4流形分类中的应用。PI也将继续他对辛流形的新不变量的构造，这些流形承认哈密顿S^1作用。这些不变量是通过研究Yang-Mills方程与Cauchy-Riemann方程的耦合来构造的。这些不变量及其扩展受到数学物理拓扑场论的启发，为物理理论提供了新的数学基础。这个项目中的问题自然来自于试图从几何和物理上理解非线性微分方程。这些方程涉及曲率，包括里奇流，静态爱因斯坦方程。它们在通过数学手段理解自然方面发挥了重要作用。他们还在几何和拓扑学中发现了许多深入的应用，例如低维空间的拓扑学。这些问题的解决将为某些物理理论提供数学基础，并将对长期存在的数学问题产生深远的应用。大多数自然现象是非线性的，具有奇异的行为。这些都反映在描述这些现象的曲率方程的可能的奇异解中。对这些奇异解有一个完整的数学理解仍然是一个挑战。这个项目将解决其中的一些基本问题。

This project concerns existence and regularity problems of curvature equations in Riemannian geometry. These equations include the Ricci flow, the Einstein equation, the Yang-Mills equation. For Ricci flow, the PI will focus on (1) finite-time singularity formation for its solutions in Kahler geometry; (2) The interaction between the singularity and geometry of the underlying spaces; (3) long-time behavior of the solutions. The PI will study the pluri-closed flow, a new curvature flow which arises from complex geometry. He will develop its analytic theory and singularity formation. It has been found that there is a deep connection between this flow and the renormalization group flow of the nonlinear sigma model with B-field. The PI will explore this connection further and give new mathematical insights for the duality in the string theory. For the Einstein equation, the PI will focus mainly on the existence of its solutions and behavior of its singular solutions. The case for dimension 4 is particularly interesting. Self-dual metrics in dimension 4 will be also studied. They can be used to study the geometry and topology of underlying spaces. The PI will also study how the symplectic flow develops finite-time singularity in dimension 4 as well as other fundamental problems in symplectic geometry. The problems include isotopy of symplectic surfaces and its applications to classifying symplectic 4-manifolds. The PI will also continue his construction of new invariants for symplectic manifolds which admit a Hamiltonian S^1-action. These invariants are constructed by studying the Yang-Mills equation coupled with Cauchy-Riemann equation. These invariants and their extensions are inspired by the topological field theories in mathematical physics and provide new mathematical foundations for physical theories. Problems in this project arose naturally from the attempts to understanding nonlinear differential equations from geometry and physics. These equations involve curvature and include Ricci flow, static Einstein equation. They played a fundamental role in understanding of nature through mathematical means. They also have found many deep applications in geometry and topology, such as the topology of low dimensional spaces. The resolution of these problems will provide mathematical foundations for some physical theories and will have profound applications to long-standing mathematical problems. Most natural phenomena are nonlinear and possess singular behaviors. These are reflected in possible singular solutions to the curvature equations which describe those phenomena. It is still challenging to have a complete mathematical understanding of these singular solutions. This project will address some of these basic problems.