Differential Equations in Geometry

点击翻译按钮获取中文摘要

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.