K-Stability of Algebraic Manifolds

代数流形的 K-稳定性

• 批准号: 0736269
• 负责人: Sean Paul
• 金额: $ 3.72万
• 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: 2006
• 资助国家: 美国
• 起止时间: 2006-12-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0736269&HistoricalAwards=false
• 关键词: Stability; Algebraic; Manifolds

AbstractAward: DMS-0505059Principal Investigator: Sean T. PaulThis proposal aims to relate the stability of a complexprojective variety to intrinsic geometric analysis on thevariety. Often, the PI is concerned with the small timeasymptotics of complicated rational energy integrals over thevariety under study. This idea was initiated by the PI in hisdissertation. The energies arise in connection with the problemof establishing the 0th order apriori estimate for solutions tothe Monge Ampere equation.Broadly speaking, there are two ways to mathematically approach,or model, a given problem: continuously, ordiscretely. Traditionally, these are quite seperateapproaches. Analysis (differential equations in particular) isthe time honoured subject in the continuous domain, combinatorics(the study of enumerating a finite amount of data) is thehallmark of the discrete approach. In this proposal these twomethods come together-the PI will explore the question of how thesolution to an equation from analysis might be obtained by apurely finite (but large) collection of data. The equationappeared in Einsteins' theory of Gravitation, whereas the finiteset of data arose in the mathematics of the 1800's, and isrelated to the theory of computer vision.

摘要奖项：DMS-0505059 首席研究员：Sean T. Paul 该提案旨在将复杂射影簇的稳定性与该簇的内在几何分析联系起来。通常，PI 与所研究的品种的复杂有理能量积分的小时间渐近有关。这个想法是由PI在他的论文中提出的。能量的产生与建立蒙日安培方程解的 0 阶先验估计的问题有关。一般来说，有两种方法可以对给定问题进行数学处理或建模：连续或离散。传统上，这些是完全独立的方法。分析（特别是微分方程）是连续领域中历史悠久的学科，组合学（枚举有限数据量的研究）是离散方法的标志。 在这个提案中，这两种方法结合在一起——PI 将探讨如何通过完全有限（但大量）的数据收集来获得分析方程的解的问题。该方程出现在爱因斯坦的万有引力理论中，而有限数据集则出现在1800年代的数学中，并且与计算机视觉理论相关。