Geometry and Analysis of Differentiable Manifolds

本课题所研究的问题涉及到几何和物理上的各种非线性微分方程组。这些方程是演化类型的，涉及到衡量空间如何弯曲的曲率。它们在通过数学手段认识自然世界方面起着基础性的作用，并与物理学场论的研究密切相关。它们还在几何和拓扑学中发现了许多深层次的应用。一个著名的例子是Perelman用Ricci流求解Poincare猜想。本项目中问题的解决将为一些物理理论提供数学基础，并将进一步深入应用于代数空间分类等长期存在的数学问题。这些方程最常见的现象是由于方程的非线性而产生的奇异行为。这种行为反映在演化过程中可能的分解中，并通过描述演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件具有挑战性的事情。这个项目将解决这些奇异解的一些基本问题，并探索它们在几何和拓扑学中的应用。国际和平研究所将就与该项目直接相关的主题进行讲座和教授研究生课程。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和拓宽他们的数学知识。这个项目涉及黎曼几何中的曲率流动和方程。对于Ricci流，PI将关注(1)其解在Kahler几何中的有限时间奇性形成；(2)奇性形成与底层空间几何之间的相互作用；(3)解的长期行为。对于厄米特曲率流，PI将开发新的分析工具来研究它是如何形成奇点的。其中最显著的厄米特曲率流是多闭流，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刻画流的最大存在性，以及它如何在4维上发展有限时间奇性。PI还打算将Einstein度量的紧性理论推广到更一般的Kahler度量和4维反自对偶度量。他还继续研究涉及某些规范方程的辛几何的基本问题。这些问题包括为允许哈密顿S1作用的辛流形构造新的形变不变量，以及为规范的线性Sigma模型提供数学理论。这些问题是辛几何中的重要问题，受到物理拓扑场理论的启发。

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.