Problems in Analysis at the Interface with Geometry and Physics

摘要提出了一种解决几何和物理分析界面上的几个问题的方法。第一个问题是稳定分析方法的发展。稳定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标准度量的基本问题所必需的。期望正则度量等价于几何不变理论意义上的稳定性概念，并研究这一整体概念与稳定估计之间的关系。该提案中的第二个问题是弦散射振幅的费曼规则的发展。由于在双环或更高的情况下遇到几何困难，即“超模”，费曼规则无法使用。但是，由于Eric D'Hoker和PI最近在最简单的双环情况下成功地克服了这一困难，现在的情况更有希望了。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/识别问题。一个基本工具是由PI与Igor krichhever合作开发的新的哈密顿方法。这些都是理论物理和应用数学的核心问题。对最基本的自然规律的探索越来越依赖于几何原理和对称。本提案中讨论的问题——稳定性和规范度量、弦理论的费曼规则、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——即弦理论和超对称理论，以及它们潜在的几何结构——至关重要。

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.