Differential Equations in Geometry

Harvard University, Cambridge MA

Investigators

Abstract

Abstract Award: DMS-0306600 Principal Investigator: Shing-Tung Yau We propose to bring together ideas from the subject of nonlinear partial differential equations, differential geometry and algebraic geometry to study questions raised in general relativity and string theory. Not only many questions in theoretical physics should be answered, but also a coherent picture in the previous unrelated subjects in mathematics should be constructed. The motivation to understand black hole formation has already led active research in minimal surfaces, inverse mean flow and various global nonlinear differential equations. They should also guide numerical relativities to do precise calculations. Our recent work on quasilocal energy should provide a way to understand the structure of Einstein equation when the gravity field is strong. The Einstein equation is much related to the Ricci flow which was pioneered by Hamilton. Some new ideas were obtained by Perelman by looking at the entropy that come from physics. By deeper understanding the underlining equation, we should understand topology of three dimensional manifolds. The desire to understand mirror symmetrics and calculation of instantons has led to important discovery in algebraic geometry. We expect that analytic aspects of mirror symmetry will bring geometers to study special Lagrangian surfaces in Calabi-Yau manifolds. It will be an important contribution to both differential geometry and mathematical physics. Both existence theorems for special Lagrangian and the detail structure of family of such manifolds will head to deeper understanding of geometric structures on manifolds. The ultimate grand unification of mathematics and physic led by geometry is exciting.