K-Energy Maps and the Stability of Pairs

K 能量图和对的稳定性

• 批准号: 1405972
• 负责人: Sean Paul
• 金额: $ 17.65万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-15 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1405972&HistoricalAwards=false
• 关键词: Energy; Maps; Stability; Pairs

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.

在科学和工程中的大多数数学应用中，所考虑的现象的数学模型采用微分方程的形式。通常这个方程涉及许多空间和时间变量。更重要的是，建模方程的形式通常是高度非线性的。尽管在过去的200年里，在开发解微分方程的方法（以及理解解的定性性质）方面取得了巨大的进步，但我们仍然远远没有一个完整的理论来处理它们，我们接受对特定类别的微分方程的研究。这个建议是关于理解所谓的复蒙日-安培方程。这个方程的起源既有几何的，也有物理的。一个著名的密切相关的方程（仍未完全理解）是爱因斯坦广义相对论中的场方程。这个提议的目的是从几何的角度来理解这个方程，并从一个叫做代数几何的非常不同的数学分支中开发高效的工具，以帮助在具体情况下证明解确实存在。该项目汇集了数学的几个领域：经典射影代数几何（Cayley形式和射影对偶变体），环型变体和凸多面体，偏微分方程和作用泛函的强制估计，以及pi发现的一种新的几何不变理论-即半稳定对的概念。正如前一段所建议的那样，目标是将Mabuchi泛函的强制估计（或下界）转换为多面体组合问题，然后在尽可能多的情况下解决这个问题。所有这一切都是出于在代数变量上构造规范度量的愿望——通常是光滑的或具有相对温和的奇点。