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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研究经典/量子(粒子/波)对应在数学中的表现。量子态或波被描述为偏微分方程解，它们的性质通常由潜在的经典(粒子)系统的性质决定。这门学科起源于几何光学(可追溯到17世纪)和量子力学(可追溯到20世纪上半叶)，但数值、实验和数学方面的进步提供了一系列新的挑战和研究机会。例如，量子共振在化学上可以描述化学反应中的过渡态，与它们被引入时相比，现在从实验和数学上都更容易获得。准正常模是广义相对论中这些共振的类似物，由于LIGO实验，现在有机会第一次被观测到。与此同时，最初利用经典动力学的见解研究微分方程的方法，现在成功地用于回答关于混沌系统或测地线几何的问题。该项目为研究生提供了研究培训机会。PI研究的具体问题包括：(1)经典混沌系统的散射共振分布；(2)了解动态Zeta函数(闭合轨道周期的生成函数，与Riemann Zeta函数是素数的生成函数的方式大致相同)；以及(3)流体力学中出现的频谱问题，特别是在内波的形成中。关于混沌散射的具体问题涉及到平面上任何(双曲)凸形障碍物的谱间隙的存在。自80年代末以来，数学和物理文献中就提出了这种带隙是由被捕获的反射光线的“拓扑力”决定的。最近关于分维测不准原理的进展应该意味着光谱总是有间隙的。对于动态Zeta函数，目标之一是理解Fry猜想，该猜想提出了具有混沌流的一般流形的动力学(Zeta函数在0处的值)、谱和拓扑量(对应的挠率)之间的关系。除其他工具外，PI开发的微本地工具在这里特别有前途。理论上用频谱方法描述的流体中的内波，只在25年前的一次对照实验中被观测到。粘性和非线性效应(在经典水平和波动水平上)的重要性仍有待充分理解。PI和他的合作者在这方面取得了一些进展，但仍然存在许多问题，如物理相关边值问题的分析。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。