Differential Equations in Geometry

AbstractAward：DMS-0306600首席研究员：姚成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学中的许多问题，而且要在以前互不相关的数学学科中建立一个连贯的图景。 对黑洞形成的理解已经引起了对极小曲面、逆平均流和各种全局非线性微分方程的积极研究。 它们还应该指导数值相对性，使计算更精确。 我们最近关于准定域能的工作，为理解强引力场下爱因斯坦方程的结构提供了一条途径。 爱因斯坦方程与汉密尔顿开创的里奇流有很大关系。 Perelman从物理学的熵出发，提出了一些新的观点，通过对下面方程的深入理解，我们可以更好地理解三维流形的拓扑。 对镜像几何学和瞬子计算的渴望导致了代数几何的重要发现。 我们期望镜像对称的解析方面将使几何学家研究Calabi-Yau流形中的特殊拉格朗日曲面。 这对微分几何和数学物理都将是重要的贡献。 特殊Lagrange流形的存在性定理和特殊Lagrange流形族的详细结构将有助于对流形几何结构的深入理解。 数学和几何的最终大统一令人兴奋.

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.