Spectral Theory and Microlocal Analysis

PI研究数学中经典/量子（粒子/波）对应的表现。量子态或波被描述为偏微分方程的解，它们的性质通常由底层经典（粒子）系统的性质决定。该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，量子共振在化学中可以描述化学反应中的过渡状态，与它们刚被引入的时候相比，现在在实验和数学上都更容易获得。准正态模式是广义相对论中这些共振的类似物，多亏了LIGO实验，现在有机会第一次被观察到。与此同时，最初开发的方法是利用经典动力学的见解来研究微分方程，现在成功地用于回答有关混沌系统或测地线几何的问题。本项目为研究生提供研究训练机会。PI研究的具体问题有：(1)经典混沌系统的散射共振分布；(2)理解动态zeta函数（闭合轨道周期的生成函数与黎曼zeta函数是素数生成函数的方式大致相同）；(3)流体力学中出现的谱问题，特别是内波的形成。关于混沌散射的具体问题涉及平面上任何（双曲）凸障碍物构型的谱隙的存在。自80年代末以来，数学和物理文献中提出，间隙是由被捕获反射光线的“拓扑压力”决定的。分形测不准原理的最新进展表明，谱间隙总是存在的。对于动态zeta函数，目标之一是理解Fried猜想，该猜想提出了具有混沌流的一般流形的动态量（zeta函数在0处的值）、谱量和拓扑量（相应的扭转量）之间的关系。由PI开发的微本地工具在这里尤其有前景。理论上用光谱方法描述的流体内波，直到25年前才在一个对照实验中被观察到。粘度和非线性效应（在经典和波浪水平上）的重要性仍有待充分认识。PI和他的合作者在这里取得了一些进展，但许多问题，如物理相关的边值问题的分析，仍然存在。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.