Differential Equations in Geometry

摘要 提案：DMS 9803347主要研究者：Shing-Tung Yau 我们建议应用非线性偏微分方程的技术 解决几何学中的几个问题： 一般时空中的引力、角动量及其在 Schwarzchild解的非线性稳定性 学习问题 耦合Einstein-Dirac方程及其物理意义 的 将非线性分析应用于单纯形的问题，包括 图形和建筑物，可用于组合问题， 离散群理论 利用黎曼度量变形的问题 汉密尔顿方程和理解他们的长期行为， 从而理解三个流形拓扑。 我们也想 研究了Calabi-Yau流形中的Lagrange三流形空间， 模空间将有助于理解弦论中的对偶问题。 该提案的成功将带来以下方面的强有力互动： 数学、现代物理和古典物理。 在这里经典物理学 包括广义相对论，其中强几何和分析 技术起着非常重要作用。 现代物理学包括弦 这个理论对数学的影响是巨大的。 研究 非线性方程组应该在理解上有所突破 低维几何和拓扑。 深入了解这样的 方程组应引入新技术来解决难题， 应用数学 希望最终能找到一些流体方程 用类似的技术处理。

Abstract Proposal: DMS 9803347 Principal Investigator: Shing-Tung Yau We propose to apply techniques of nonlinear partial differential equations to solve several problems in geometry: the problem of defining center of gravity, angular momentum in general space time and their applications to the nonlinear stability of Schwarzchild solution. The problem of studying coupled Einstein-Dirac equation and their physical significance. The problem of applying nonlinear analysis to simplicial complices, including graphs and buildings that can be used in combinatorial questions and discrete group theory. The problem of deforming Riemannian metric by using Hamilton's equation and understanding their long time behavior that may lead to understanding of topology of three manifolds. We also want to study the space of Lagrangian three manifolds in Calabi-Yau manifolds whose moduli space will lead to understanding duality questions in string theory. The success of this proposal will bring in strong interaction between mathematics and modern and classical physics. Here classical physics include general relativity, where strong geometrical and analytical technique play a very important role. Modern physics includes String theory whose impact on mathematics has been tremendous. The study of nonlinear system of equations should give a breakthrough in understanding low dimensional geometry and topology . A thorough understanding of such equations should bring in new technique to solve difficult problems in applied mathematics. Hopefully, eventually some fluid equations can be treated by similar technique.