Differential Equations in Geometry

摘要奖：DMS-0306600主要研究人员：游星东我们建议将非线性偏微分方程、微分几何和代数几何的思想结合起来，研究广义相对论和弦理论中提出的问题。不仅要回答理论物理中的许多问题，而且要在以前无关的数学学科中构筑一幅连贯的图景。理解黑洞形成的动机已经导致了极小曲面、逆平均流和各种全局非线性微分方程组的活跃研究。他们还应该引导数值相对论来简化计算。我们最近关于准局域能量的工作应该为理解引力场较强时爱因斯坦方程的结构提供了一种途径。爱因斯坦方程与汉密尔顿首创的利玛奇流有很大关系。Perelman通过观察来自物理的熵得到了一些新的想法，通过更深入地理解基本方程，我们应该理解三维流形的拓扑。了解镜面对称性和瞬子计算的愿望促成了代数几何的重大发现。我们期望镜面对称性的解析方面将把几何学引入到研究Calabi-Yau流形中特殊的拉格朗日曲面。它将对微分几何和数学物理都有重要贡献。关于特殊拉格朗日的存在定理和这类流形的族的详细结构都将有助于更深入地理解流形上的几何结构。由几何学引领的数学和物理学的终极统一令人兴奋。

AbstractAward: DMS-0306600Principal Investigator: Shing-Tung YauWe propose to bring together ideas from the subject of nonlinearpartial differential equations, differential geometry andalgebraic geometry to study questions raised in generalrelativity and string theory. Not only many questions intheoretical physics should be answered, but also a coherentpicture in the previous unrelated subjects in mathematics shouldbe constructed. The motivation to understand black holeformation has already led active research in minimal surfaces,inverse mean flow and various global nonlinear differentialequations. They should also guide numerical relativities to doprecise calculations. Our recent work on quasilocal energyshould provide a way to understand the structure of Einsteinequation when the gravity field is strong. The Einstein equationis much related to the Ricci flow which was pioneered byHamilton. Some new ideas were obtained by Perelman by looking atthe entropy that come from physics.By deeper understanding the underlining equation, we shouldunderstand topology of three dimensional manifolds. The desireto understand mirror symmetrics and calculation of instantons hasled to important discovery in algebraic geometry. We expect thatanalytic aspects of mirror symmetry will bring geometers to studyspecial Lagrangian surfaces in Calabi-Yau manifolds. It will bean important contribution to both differential geometry andmathematical physics. Both existence theorems for specialLagrangian and the detail structure of family of such manifoldswill head to deeper understanding of geometric structures onmanifolds. The ultimate grand unification of mathematics andphysic led by geometry is exciting.