Problems at the Interface of Analysis with Geometry and Physics

摘要：在过去的几十年里，数学分析的影响显著增加。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、概率论和理论物理学。这是提案中反复出现的特点。更具体地说，该提案解决了振荡积分的边界和稳定性，Radon变换的正则性和退化傅立叶积分算子，格林函数的渐近行为，超对称场和弦理论的精确解，孤子的哈密顿理论和非线性WKB方法等问题。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，沿着这里所建议的路线的进展可以大大减少目前对它们的数值研究所需要的大量努力。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，表明了三个看似不同的领域之间深刻而令人惊讶的对应关系-即超对称规范理论，孤子方程和黎曼曲面。这种通信的起源仍然是神秘的，它很可能是在最基本的层面上理解自然规律的持续尝试所必需的。

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.