Microlocal Analysis and Hyperbolic Dynamics

这个项目研究了微局部分析和双曲动力学交叉点的广泛主题。微局域分析（英语：Microlocal analysis）是一种强大的数学理论，它将经典哈密顿动力学与波和量子态的奇点联系起来。双曲动力学是强混沌系统的数学理论，其中初始数据的小扰动导致长时间后的指数发散轨迹。该项目利用这两个领域之间的相互作用，研究波和量子态在基本动力学强烈混沌的情况下的行为，并探索微局部方法在纯动力学问题中的应用。该项目为研究生提供研究培训机会。该项目的一个方向是在量子混沌的高度活跃的领域，研究量子系统的光谱特性，其中底层经典系统具有混沌行为。主要研究者（PI）介绍了来自调和分析，分形几何，加法组合学和拉特纳理论，结合在一起的分形不确定性原理的概念在该领域的新方法。该项目的具体目标包括：（1）理解封闭混沌系统的高能量本征函数的宏观集中，例如负弯曲的黎曼流形和量子猫映射;（2）证明具有分形双曲陷集的开放系统的基本谱隙（特别是指波的指数局部能量衰减）。第二个研究方向是分层流体中的强迫波研究（旋转流体也出现类似的问题），其动机是实验观察到的水族馆内波和海洋学应用。第三个方向是应用最初为双曲型偏微分方程理论开发的微局部方法来研究经典对象，如动态zeta函数，这是量子/经典对应反转的罕见例子。特别地，PI和他的合作者研究了（1）负弯曲流形的动力学zeta函数的特殊值如何与流形的拓扑关系;以及（2）该奖项反映了美国国家科学基金会的法定使命，并已被认为是值得支持的，通过评估使用基金会的知识价值和更广泛的影响审查标准。

This project investigates a broad range of topics at the intersection of microlocal analysis and hyperbolic dynamics. Microlocal analysis, with its roots in physical phenomena such as geometric optics and quantum/classical correspondence, is a powerful mathematical theory relating classical Hamiltonian dynamics to singularities of waves and quantum states. Hyperbolic dynamics is the mathematical theory of strongly chaotic systems, where a small perturbation of the initial data leads to exponentially divergent trajectories after a long time. The project takes advantage of the interplay between these two fields, studying the behavior of waves and quantum states in situations where the underlying dynamics is strongly chaotic, and also exploring the applications of microlocal methods to purely dynamical questions. The project provides research training opportunities for graduate students.One direction of this project is in the highly active field of quantum chaos, the study of spectral properties of quantum systems where the underlying classical system has chaotic behavior. The Principal Investigator (PI) has introduced new methods in the field coming from harmonic analysis, fractal geometry, additive combinatorics, and Ratner theory, combined together in the concept of fractal uncertainty principle. The specific goals of the project include: (1) understanding the macroscopic concentration of high energy eigenfunctions of closed chaotic systems, such as negatively curved Riemannian manifolds and quantum cat maps; and (2) proving essential spectral gaps (implying in particular exponential local energy decay of waves) for open systems with fractal hyperbolic trapped sets. A second research direction is the study of forced waves in stratified fluids (with similar problems appearing also for rotating fluids), motivated by experimentally observed internal waves in aquaria and by applications to oceanography. A third direction is to apply microlocal methods originally developed for the theory of hyperbolic partial differential equations to study classical objects such as dynamical zeta functions, which is a rare example of the reversal of quantum/classical correspondence. In particular, the PI and his collaborators study (1) how the special values of the dynamical zeta function for a negatively curved manifold relate to the topology of the manifold; and (2) whether dynamical zeta functions can be meromorphically continued for systems with singularities such as dispersive billiards.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.