Geometry and Analysis of Manifolds

DMS-0804095摘要本提案涉及几何和物理中出现的几何和解析问题。PI将继续研究Einstein方程和Yang-Mills方程的存在性问题和奇异性形成。对于爱因斯坦方程，PI将主要集中在4维或卡勒情况下。他还将研究相关的里奇流及其奇异性发展及其与代数几何中的投影流形分类的相互作用。相应的问题，自对偶度量在4维也将进行研究。对于Yang-Mills方程，PI将专注于如何紧致其自对偶解的空间及其在构造新不变量方面的应用。PI之前发现杨-米尔斯场沿着沿着经典极小曲面（如肥皂泡或子曲面）形成奇点。他喜欢进一步探索这一点，特别是自对偶杨米尔斯场和校准几何之间的相互作用。研究者还打算继续研究辛几何中的问题，包括变形4-流形中的辛曲面和构造新的变形不变量。爱因斯坦方程和杨-米尔斯方程在过去几十年中在我们的物理学、几何学和拓扑学研究中发挥了基础作用。佩雷尔曼对庞加莱猜想的解答就是一个很好的例子。重要的和中心的问题包括研究什么时候可以解决这些方程，这些解决方案的性质，他们如何发展奇异行为。 理解这些解与其他数学分支之间的联系也很重要，例如代数几何和微分拓扑。这些问题的解决将为我们提供对潜在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研究的启发。通过这个研究项目，PI还打算开发研究弯曲空间，辛几何的新工具，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.