Geometry and Analysis of Differentiable Manifolds

本课题研究的问题涉及几何和物理中的各种非线性微分方程。这些方程是演化型的，涉及测量空间如何弯曲的曲率。它们在通过数学手段理解自然世界方面发挥着基础作用，并且与物理学中的场论研究密切相关。它们也在几何学和拓扑学中得到了广泛的应用。一个著名的例子是佩雷尔曼利用里奇流解庞加莱猜想。本课题问题的解决将为一些物理理论提供数学基础，并对代数空间的分类等长期存在的数学问题有更深远的应用。这些方程最常见的现象是由于方程的非线性所导致的奇异行为。这些行为反映在进化过程中可能出现的故障中，并以描述进化过程的这些方程的奇异解来描述。对这些奇异解有一个完整的数学理解仍然是一个挑战。本项目将探讨这些奇异解的一些基本问题，并探讨它们在几何和拓扑中的应用。PI将讲授与该项目直接相关的主题，并教授研究生课程。他还将举办一个几何工作研讨会，目的是帮助学生获得研究经验，拓宽他们的数学知识。这个项目涉及黎曼几何中的曲率流和方程。对于Ricci流，PI将关注(1)在Kahler几何中其解的有限时间奇点形成；(2)奇点的形成与底层空间几何的相互作用；(3)解的长期行为。对于厄米曲率流，PI将开发新的分析工具来研究它如何形成奇点。厄米曲率流中最突出的一种是多闭流，它与具有b场的非线性sigma模型的重整化群流有关。PI将进一步探索这种联系，一方面为弦理论中的对偶性提供新的数学见解，另一方面为有限时间奇点提供新的理解。对于辛曲率流，PI打算研究如何用上同调条件来表征该流的极大存在性，以及它如何在4维空间中产生有限时间奇点。PI还打算将爱因斯坦度量的紧性理论扩展到更一般的Kahler度量和四维反自对偶度量。他还继续研究辛几何中涉及某些规范方程的基本问题。这些问题包括构造具有哈密顿s1作用的辛流形的新的变形不变量，以及为规范线性σ模型提供数学理论。这些问题在辛几何中很重要，并受到物理学拓扑场论的启发。

The problems investigated in this project concern various nonlinear differential equations from geometry and physics. These equations are of the evolution type and involve the curvature which measures how a space is curved. They play a fundamental role in understanding the natural world through mathematical means and are closely related to the study of the field theories in physics. They have also found many deep applications in geometry and topology. A famous example is Perelman's solution of the Poincare conjecture by using Ricci flow. The resolution of the problems in this project will provide mathematical foundations for some physical theories and have further profound applications to long-standing mathematical problems such as the classification of algebraic spaces. The most common phenomena of these equations are their singular behaviors due to the nonlinearity of the equations. Such behaviors are reflected in the possible break-downs in the evolution process and described in terms of singular solutions to these equations which describe the evolution process. It is still challenging to have a complete mathematical understanding of these singular solutions. This project will address some basic problems on these singular solutions and explore their applications to geometry and topology. The PI will give lectures and teach graduate courses on topics directly related to this project. He will also run a geometry working seminar with a goal of helping students to gain research experiences and broaden their knowledge in mathematics.This project concerns curvature flows and equations in Riemannian geometry. 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 formation and geometry of the underlying spaces; (3) The long-time behavior of the solutions. For the Hermitian curvature flow, the PI will develop new analytic tools to study how it forms singularity. One of most prominent Hermitian curvature flow is the pluriclosed flow which is connected to the renormalization group flow of the nonlinear sigma model with B-field. The PI will further explore this connection and gives new mathematical insights for the duality in the string theory on one hand, new understanding of finite-time singularity on the other hand. For the symplectic curvature flow, the PI intends to study how to characterize the maximal existence of the flow by cohomological condition and how it develops finite-time singularity in dimension 4. The PI also intends to extend the compactness theory for Einstein metrics to a more general class of Kahler metrics and 4-dimensional anti-self-dual metrics. He also continues his study on fundamental problems in symplectic geometry which involve certain gauge equation. The problems include constructing new deformation invariants for symplectic manifolds which admit a Hamiltonian S1-action and providing a mathematical theory for the gauged linear sigma model. These problems are important in symplectic geometry and are inspired by the topological field theories in physics.