Links between Algebraic Geometry and Complex Analysis

一个古老的想法，至少可以追溯到牛顿，甚至更远，展示了如何从一个你想要解决的多项式方程开始，并最终得到一个捕捉原始方程重要信息的多边形。牛顿利用这一思想在他的工作中寻找数值解多项式方程，从而使他能够执行计算许多世纪前，任何计算机已被发明。在这一建议的研究件涉及现代化身的这一想法在代数几何的框架。牛顿只考虑一个多项式方程，现在我们知道如何同时处理多个这样的方程。一个想法Okounkov在20世纪80年代初表明，如何可以构建一个特定的固体在欧几里德空间，类似于牛顿多边形，但这一次相关联的代数品种，并发现这种形状捕捉一些几何的原始品种。这里的目的之一是研究几何的Okounkov体，并开发它作为一个工具连接代数和复杂的分析。第二个领域的研究在这个建议涉及的研究是什么被称为卡勒-爱因斯坦方程。这些是一些重要的微分方程，其解应该被认为是给出了所考虑的空间的“最佳”形状。这些方程类似于广义相对论中的爱因斯坦方程，并且在纯数学和数学物理的各个部分都有应用。然而，一个问题是，Kahler-Einstein方程太复杂，无法直接求解。事实上，在许多情况下，即使知道是否有解决方案也超出了我们目前的知识范围。然而，一个深刻的和迷人的想法，由于姚田-唐纳森指出，它应该是可能的检测是否存在这样的解决方案内代数几何。在这个提议中，我们的目标是探索这个思想圈，并将其扩展到其他框架和其他类型的微分方程。

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.