Problems in Analysis at the Interface with Geometry and Physics

几何与物理交叉点的分析问题

• 批准号: 0245371
• 负责人: Duong Phong
• 金额: $ 58.87万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0245371&HistoricalAwards=false
• 关键词: Problems; Analysis; Interface; Geometry; Physics

PI: Duong H. Phong, Columbia UniversityDMS-0245371ABSTRACTIt is proposed to address several problems at the interface of analysis with geometry and physics. A first problem is the development of stable analytic methods. Stable methods have emerged as essential for the study of singularities and oscillatory integrals, and for the basic problem of finding canonical metrics in differential geometry. Canonical metrics are expected to be equivalent to the notion of stability in the sense of geometric invariant theory, and the relation between this global notion and stable estimates is to be investigated. The second problem in the proposal is the development of Feynman rules for string scattering amplitudes. Feynman rules had been unavailable due to a geometric difficulty encountered for two-loops or higher, namely ``supermoduli". But the situation is now much more promising, thanks to the recent success of Eric D'Hoker and the PI in overcoming this difficulty in the simplest case of two-loops. The third problem in the proposal is the construction/identification of integrable structures in supersymmetric gauge and string theories. A basic tool is a new Hamiltonian approach to soliton equations developed by the PI in collaboration with Igor Krichever.These are core problems in theoretical physics and applied mathematics. The search of natural laws at their most fundamental level relies more and more on geometric principles and symmetry.The problems addressed in the present proposal - stability and canonical metrics, Feynman rules for string theory, soliton equations - are essential to the understanding of the two theories which have led research in both mathematics and theoretical physics for the last two decades, namely string theory and supersymmetry, together with their underlying geometric structures.

PI：Duong H. Phong，哥伦比亚大学DMS-0245371摘要提出了在几何和物理分析的界面上解决几个问题。第一个问题是稳定分析方法的发展。稳定的方法已经成为研究奇点和振荡积分，以及在微分几何中寻找规范度量的基本问题的关键。在几何不变理论的意义下，正则度量被期望等价于稳定性的概念，并且这个整体概念和稳定估计之间的关系有待于研究。建议中的第二个问题是发展弦散射振幅的费曼规则。费曼规则已经无法使用，因为遇到了几何困难的两个循环或更高，即“supermoduli”。但现在的情况是更有希望，由于最近成功的埃里克D 'Hoker和PI在克服这个困难，在最简单的情况下，两个循环。第三个问题是在超对称规范和弦理论中的可积结构的构造/识别。一个基本的工具是由PI与Igor Krichever合作开发的孤子方程的新Hamilton方法。这些是理论物理和应用数学的核心问题。在最基本的层次上寻找自然定律越来越依赖于几何原理和对称性。本提案中所解决的问题--稳定性和正则度规、弦论的费曼规则、孤子方程--对于理解在过去二十年中引领数学和理论物理研究的两个理论，即弦论和超对称性，是必不可少的。以及它们的基本几何结构。