Geometry and Analysis of Manifolds

DMS-0804095摘要本建议涉及几何和物理中的几何和解析问题。PI将继续研究爱因斯坦方程和杨-米尔斯方程的存在问题和奇点形成。对于爱因斯坦方程，圆周率将主要集中在4维或在卡勒的情况下。他还将研究相关的Ricci流及其奇点发展，以及它与代数几何中分类射影流形的相互作用。还将研究4维自对偶度量的相应问题。对于Yang-Mills方程，PI将集中于如何紧化其自对偶解的空间以及它们在构造新不变量方面的应用。PI之前发现，杨-米尔斯场沿着经典的极小表面形成奇点，如肥皂泡或亚种。他喜欢更深入地探讨这一点，特别是自对偶杨-米尔场和校准几何之间的相互作用。他还打算继续他在辛几何问题上的研究，包括变形4-流形中的辛曲面和构造新的变形不变量。在过去的几十年里，爱因斯坦方程和杨-米尔斯方程在我们的物理、几何和拓扑学的研究中扮演了重要的角色。佩雷尔曼对庞加莱猜想的解答就是一个很好的例子。重要和核心的问题包括研究什么时候可以解这些方程，这些解的性质是什么，它们是如何形成奇异行为的。了解这些解与其他数学分支(如代数几何和微分拓扑学)之间的联系也很重要。这些问题的解决将为我们提供对底层空间几何的新的深刻理解。本研究项目所涉及的问题也受到了物理学中弦理论的研究的启发。通过这一研究项目，PI还打算为研究曲线空间、辛几何开发新的工具，并为一些物理理论提供新的数学基础。

Abstract for DMS-0804095This proposal concerns geometric and analytic problems which arise from geometry and physics. The PI will continue his study on existence problem and singularity formation of the Einstein equation and the Yang-Mills equation. For the Einstein equation, the PI will focus mainly in dimension 4 or in the Kahler case. He will also study related Ricci flow and its singularity development and its interaction with classifying projective manifolds in algebraic geometry. Corresponding problems for self-dual metrics in dimension 4 will be also studied. For the Yang-Mills equation, the PI will focus on how to compactify spaces of its self-dual solutions and their applications to constructing new invariants. The PI found before that the Yang-Mills fields forms singularity along classical minimal surfaces like soap bubbles or subvarieties. He likes to explore this further and particularly, the interaction between self-dual Yang-Mills fields and calibrated geometry. The PI also intends to continue his study on problems in symplectic geometry, including deforming symplectic surfaces in 4-manifolds and constructing new deformation invariants.The Einstein and the Yang-Mills equations have played a fundamental role in our study of physics and geometry and topology in last few decades. Perelman's solution for the Poincare conjecture is an excellent example. Important and central problems include studying when one can solve those equations, what properties of those solutions found, how they develop singular behaviors. It is also important to understand the connection between these solutions and other branches of mathematics, such as, algebraic geometry and differential topology. The resolution of these problems will provide new profound understanding geometry of underlying spaces.The problems involved in this research project were also inspired by the study of the string theory in physics. Through this research project, the PI also intends to develop new tools for studying curved spaces, symplectic geometry and provide new mathematical foundation for some physical theories.