One of the main problems in complex geometric analysis is to detect the existence of "canonical" Kahler metrics in a given class on a compact complex (Kahler) manifold. In particular one seeks necessary and sufficient conditions for the existence of a Kahler Einstein metric on a Fano manifold. The presence of positive curvature makes this extremely difficult to answer and has led to a striking series of conjectures which relate the existence of these special metrics (solutions to the complex Monge-Ampere equation, a fully non-linear p.d.e ) to the algebraic geometry of the pluri-anticanonical images of the manifold. This geometry was suspected to be related to Mumford's deep "Geometric Invariant Theory". Recently this speculation has been completely justified by the PI (building upon work of Gang Tian) and it is the aim of this proposal to finish the proof of the "standard conjectures" in the Fano case and to develop and extend the entire Theory in the context of representations of algebraic groups. Here the PI hopes to make contact with Mikio Sato's beautiful theory of prehomogeneous vector spaces.
Broadly speaking, there are two ways to mathematically approach, or model, a given problem: continuously, or discretely. These approaches are traditionally mutually exclusive. Analysis (differential equations in particular) is the time honored subject in the continuous domain, combinatorics and algebra (the study of enumerating a finite amount of data) is the hallmark of the discrete approach. In this proposal these two methods come together-the PI will explore the question of how the solution to an equation from analysis might be obtained by an infinite sequence of a purely finite (but large) collection of data. The equation arose in Einsteins' theory of Gravitation, whereas the finite set of data (hyperdiscriminants and resultants) arose in the work of Arthur Cayley a great Victorian era English mathematician.
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 was 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 the proposal was 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. The results of the proposal (when combined with results of other authors) confirm that combinatorial polyhedral methods from Algebraic geometry actually give necessary and sufficient conditions for the solvability of the Complex Monge Ampere equation. The main task now is to develop these tools and check this combinatorial condition in meaningful examples.
