In most applications of mathematics to science and engineering the mathematical model of the phenomena under consideration takes the form of a differential equation. Usually this equation involves many spatial and temporal variables. Even more, the modeling equation is usually highly nonlinear in form. Although great progress has been made in the past 200 years in developing methods to solve differential equations (as well as in understanding the qualitative nature of the solutions) we are still far from having a complete theory on how to deal with them, and we accept instead a study of particular classes of differential equations. This proposal is concerned with understanding what is known as the complex Monge-Ampere equation. The origin of this equation is both geometric and physical. A famous closely related equation (still not understood completely) is Einstein's field equation from General relativity. The aim of this proposal is to understand this equation from the point of view of geometry and to develop efficient and effective tools from a very different branch of mathematics called algebraic geometry to aid in showing-in concrete cases-that solutions actually exist.
This project brings together several areas of mathematics: classical projective algebraic geometry (Cayley forms and projectively dual varieties), toric varieties and convex polytopes, partial differential equations and coercive estimates for action functionals and a new kind of Geometric Invariant theory discovered by the PI-namely the notion of a semistable pair. The goal is, as suggested in the previous paragraph, to convert the coercive estimate (or lower bound) for the Mabuchi functional into a problem in polyhedral combinatorics and then solve this problem in as many cases as possible. All of this is motivated by the desire to construct canonical metrics on algebraic varieties-usually smooth or with relatively mild singularities.
