Spectral Theory and Microlocal Analysis

谱理论和微局域分析

基本信息

批准号：
1952939
负责人：
Maciej Zworski
金额：
$ 32.99万
依托单位：
University of California-Berkeley
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1952939&HistoricalAwards=false
关键词：
Spectral Theory Microlocal Analysis

项目摘要

The PI investigates manifestations of the classical/quantum (particle/wave) correspondence in mathematics. The quantum states or waves are described as solutions of partial differential equations and their properties are often determined by the properties of underlying classical (particle) systems. The subject has its origins in geometric optics (going back to the 17th century) and quantum mechanics (going back to the first half of the 20th century) but the numerical, experimental and mathematical advances provide a new range of challenges and research opportunities. For instance, quantum resonances, which in chemistry can describe transitional states in chemical reactions, are now more accessible experimentally, and mathematically, compared to the time when they were introduced. Quasinormal modes, which are an analogue of these resonances in general relativity now have a chance of being observed for the first time, thanks to the LIGO experiments. At the same time, the methods originally developed to study differential equations using insights from classical dynamics, are now successfully used to answer questions about chaotic systems or geometry of geodesics. The project provides research training opportunities for graduate students. Among the specific problems studied by the PI are: (1) distribution of scattering resonances for classically chaotic systems; (2) understand dynamical zeta function (generating function for periods of closed orbits in much the same way as the Riemann zeta function is a generating function of prime numbers); and (3) spectral problems arising in fluid mechanics, specifically in the formation of internal waves. The concrete problem about chaotic scattering concerns the existence of a spectral gap for any (hyperbolic) configuration of convex obstacles in the plane. Since the late 80s it was proposed in the mathematics and physics literature that the gap is determined by the "topological pressure" of the trapped reflected rays. Recent advances on the fractal uncertainty principle should imply that there always is a spectral gap. For dynamical zeta functions, one of the goals is to understand the Fried conjecture which proposes a relation between dynamical (value of the zeta function at 0), spectral and topological quantities (corresponding torsions) for general manifolds with chaotic flows. The microlocal tools developed, among others by the PI, are particularly promising here. Internal waves in fluids, theoretically described by spectral methods, have only been observed, in a controlled experiment, 25 years ago. The importance of viscosity and nonlinear effects (on both classical and wave level) is still to be fully understood. The PI and his collaborators made some advances here but many questions, such as the analysis of the physically relevant boundary value problems, remain.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

PI研究了数学中经典/量子（粒子/波）对应的表现。量子状态或波浪被描述为部分微分方程的解，其性能通常取决于基础经典（粒子）系统的特性。该主题起源于几何光学（可以追溯到17世纪）和量子力学（可以追溯到20世纪上半叶），但数值，实验性和数学进步提供了新的挑战和研究机会。例如，与引入的时间相比，在化学中可以描述化学反应中的过渡状态的量子共振现在更容易获得。由于Ligo实验，现在是总体相对论中这些共振的类似物的类似模式。同时，最初开发的用于使用经典动力学的见解来研究微分方程的方法现在已成功地回答有关混乱系统或地球学几何形状的问题。该项目为研究生提供了研究培训机会。 PI研究的特定问题包括：（1）经典混乱系统的散射共振分布；（2）了解动态ZETA函数（以与Riemann Zeta函数相同的方式生成封闭轨道周期的生成函数是素数的生成函数）；（3）流体力学中引起的光谱问题，特别是在内部波的形成中。关于混乱散射的具体问题涉及存在光谱差距，用于平面中凸障碍的任何（双曲线）构型。自80年代后期以来，在数学和物理学文献中提出了差距是由被困的反射射线的“拓扑压力”决定的。分形不确定性原理的最新进展应暗示始终存在光谱差距。对于动力学Zeta函数，目标之一是了解炸毁的猜想，该猜想提出了动态（Zeta函数在0处的值），光谱和拓扑数量（相应的扭转），用于混乱的一般流向。 PI开发的微局部工具在这里特别有希望。在25年前的一项受控实验中，仅观察到流体中的内部波，理论上用光谱方法描述了。粘度和非线性效应的重要性（在经典和波动水平上）仍待完全了解。 PI和他的合作者在这里取得了一些进步，但仍然存在许多问题，例如对物理相关的边界价值问题的分析。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的评估来支持的。审查标准。