Differential Equations in Geometry

建议：DMS 9803347首席研究员游星东我们建议应用非线性偏微分方程组的技巧来解决几何中的几个问题：一般时空中重心和角动量的确定问题，以及它们在SchwarzChild解的非线性稳定性中的应用。研究耦合爱因斯坦-狄拉克方程的问题及其物理意义。将非线性分析应用于简单复数的问题，包括可用于组合问题和离散群论的图形和建筑物。通过使用哈密顿方程来变形黎曼度量的问题，并了解它们的长时间行为，这可能导致对三个流形的拓扑的理解。我们还想研究Calabi-Yau流形中的拉格朗日三个流形的空间，它的模空间将有助于理解弦理论中的对偶问题。这一提议的成功将带来数学与现代和经典物理学之间的强烈互动。在这里，经典物理学包括广义相对论，强大的几何和分析技术在其中发挥着非常重要的作用。现代物理学包括弦理论，它对数学的影响是巨大的。非线性方程组的研究应该在理解低维几何和拓扑学方面取得突破。对这些方程的透彻理解将为解决应用数学中的难题带来新的技术。希望最终一些流体方程可以用类似的技术来处理。

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.