Links between Algebraic Geometry and Complex Analysis

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An old idea, going back at least as far as Newton and probably much further, shows how it is possible to start with an polynomial equation that you wish to solve and end up with a certain polygon that captures important information of the original equation. Newton exploited this idea in his work on finding numerical solutions to polynomial equations, thereby allowing him to perform computations many centuries before any computers had been invented.One of the pieces of research in this proposal concerns a modern incarnation of this idea in the framework of algebraic geometry. Whereas Newton was considering a single polynomial equation, we now know how this works for several such equations simultaneously. An idea of Okounkov in the early 1980s showed how one can construct a certain solid in Euclidean space that similar to the Newton polygon but this time to associated an algebraic variety, and discovered that this shape captures some of the geometry of the original variety. One of the aims here is to study the geometry of this Okounkov body and to develop it as a tool connection algebraic and complex analysis.A second area of research in this proposal concerns a study of what is known as the Kahler-Einstein equations. These are some important differential equations whose solution should be thought of as giving the "best" shape of a space under consideration. These equations are analogous to the Einstein equations in general relativity, and have applications in various parts of pure mathematics and mathematical physics. One problem, however, is that the Kahler-Einstein equations are too complicated to be solved directly. In fact in many cases even knowing if there is a solution is beyond our current knowledge. However a deep and fascinating idea due to Yau-Tian-Donaldson states that it should be possible to detect the whether such a solution exists within algebraic geometry. In this proposal we aim to explore this circle of ideas, and to extend it to other frameworks and other kinds of differential equations.