Problems in Analysis at the Interface with Geometry and Physics

Pi：Duong H.Phong，哥伦比亚大学DMS-0245371 ABSTRACT建议解决几何和物理分析界面上的几个问题。第一个问题是稳定分析方法的发展。对于奇点和振荡积分的研究，以及在微分几何中寻找正则度量的基本问题，稳定方法已经成为必不可少的方法。正则度量被期望等价于几何不变量理论意义下的稳定性概念，并且这一整体概念与稳定估计之间的关系有待研究。该提案中的第二个问题是关于弦散射幅度的费曼规则的发展。费曼规则由于在双环或更高的情况下遇到几何困难，即“超模”而无法使用。但现在的情况要好得多，这要归功于Eric D‘Hoker和PI最近在最简单的两环情况下克服了这一困难。建议中的第三个问题是超对称规范理论和弦理论中可积结构的构造/识别。一个基本的工具是由PI与Igor Krichever合作开发的求解孤子方程的新的哈密顿方法。这些都是理论物理和应用数学中的核心问题。对自然规律最基本层面的探索越来越多地依赖于几何原理和对称性。本文所讨论的问题--稳定性和正则度规、弦理论的费曼规则、孤子方程--对于理解过去二十年来引领数学和理论物理研究的两个理论，即弦理论和超对称性及其基本的几何结构是至关重要的。

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.