Problems in complex analysis, complex geometry, and mathematical physics

复分析、复几何和数学物理中的问题

• 批准号: 0757372
• 负责人: Duong Phong
• 金额: $ 71.36万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0757372&HistoricalAwards=false
• 关键词: Problems; complex; analysis; geometry; mathematical

This project focuses on a number of open problems about canonical metrics and stability in complex geometry, pluripotential theory and complex Monge-Ampere equations, Feynman rules in string theory, and integrable models related to gauge theories. These are fundamental problems in complex analysis and complex geometry that are also of great interest in algebraic geometry, partial differential equations, and mathematical physics. They are acknowledged to be difficult, but recent progress has revealed for them a very rich structure, as well as many unifying threads. The project builds upon previous research of the principal investigator, but it also branches out to explore relations with different concurrent lines of investigation. The principal investigator?s approaches to the various problems are tightly interwoven, so that progress on one could well lead to progress on others.Complex analysis and complex geometry are central fields in mathematics, whose role is essential in the very formulation and ultimate understanding of physical laws. Complex analytic methods are needed in every branch of both pure and applied mathematics. The geometric problems contemplated here either are rooted directly in attempts at understanding the laws of nature at their most fundamental level (as in the problems from string theory and gauge theories) or have strong analogies with basic equations from general relativity and other branches of science (as in the case of canonical metrics and Monge-Ampere equations). The proposed research will have an immediate beneficial effect on students and postdoctoral researchers at the principal investigator?s home institution. But it will also generate a lot of research problems and provide a fertile training ground for the many researchers in analysis and complex geometry nationwide. The principal investigator has actively encouraged junior people, irrespective of their affiliations, to participate in various components of this research. To this end, he plans to continue to disseminate the results of the research to a broad audience through lectures, survey papers, and graduate texts.

这个项目的重点是一些开放的问题，关于规范度量和稳定性在复杂的几何，多能理论和复杂的Monge-Ampere方程，费曼规则在弦理论，和可积模型相关的规范理论。这些是复分析和复几何中的基本问题，也是代数几何、偏微分方程和数学物理中的重要问题。它们被认为是困难的，但最近的进展为它们揭示了一个非常丰富的结构，以及许多统一的线程。该项目建立在主要研究者以前的研究基础上，但它也扩展到探索与不同并行调查线的关系。首席调查员？复分析和复几何是数学的中心领域，它们的作用对于物理定律的表述和最终理解是必不可少的。在纯数学和应用数学的每一个分支中都需要复分析方法。这里考虑的几何问题要么直接植根于试图在最基本的层次上理解自然定律（如弦论和规范理论的问题），要么与广义相对论和其他科学分支的基本方程有很强的相似性（如正则度规和蒙格-安培方程）。拟议的研究将有一个直接的有益影响，学生和博士后研究人员在主要研究者？s home institution.但它也将产生大量的研究问题，并为全国范围内的许多分析和复杂几何研究人员提供肥沃的训练场地。主要研究者积极鼓励年轻人，无论他们的从属关系，参与本研究的各个组成部分。为此，他计划继续通过讲座，调查论文和研究生教材向广大受众传播研究成果。