Problems at the Interface of Analysis with Geometry and Physics

建议：DMS-9800783首席研究员：Duong Phong摘要：在过去的几十年里，数学分析的影响有了很大的提高。然而，该领域正处于一个重要的新关头，分析方法必须与其他领域的技术和见解相结合，尤其是几何、概率和理论物理。这是提案中反复出现的特点。更具体地说，该方案解决了振荡积分的界和稳定性、Radon变换和简并的傅立叶积分算子的正则性、格林函数的渐近行为、超对称场和弦理论的精确解、哈密顿孤子理论和非线性WKB方法。这些可以说是当今数学和理论物理中最紧迫的问题之一。例如，振荡积分是任何物理散射过程的一个众所周知的特征，沿着这里建议的路线进行的进展可以极大地减少目前对其进行数值研究所需的大量工作。另一方面，提案中关于超对称理论的问题反映了最近的进展，表明三个看似不同的领域--即超对称规范理论、孤子方程和黎曼曲面--之间存在着深刻而令人惊讶的对应关系。这种对应的起源仍然是个谜，它很可能对正在进行的在最基本层面上理解自然规律的尝试至关重要。

Proposal: DMS-9800783 Principal Investigator: Duong Phong Abstract: The influence of mathematical analysis has increased considerably over the last few decades. However, the field is at an important new juncture, where analytical methods have to be blended with techniques and insights from other fields, most particularly geometry, probability, and theoretical physics. This is a recurrent feature in the proposal. More specifically, the proposal addresses issues of bounds and stability for oscillatory integrals, regularity of Radon transforms and degenerate Fourier integral operators, asymptotic behavior of Green's functions, exact solutions of supersymmetric field and string theories, Hamiltonian theory of solitons and non-linear WKB methods. These are arguably among the most pressing problems in mathematics and theoretical physics today. Oscillatory integrals are, for example, a well-known feature of any physical scattering process, and progress along the lines suggested here can reduce greatly the extensive effort presently required for their numerical studies. On the other hand, the problems in the proposal concerning supersymmetric theories reflect very recent advances, indicating a deep and surprising correspondence between three seemingly different fields - namely, supersymmetric gauge theories, soliton equations, and Riemann surfaces. The origin of this correspondence is still mysterious, and it may well be essential to on-going attempts at understanding the laws of nature at their most fundamental level.