Differential Equations in Geometry

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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.