Microlocal Analysis and Hyperbolic Dynamics

微局域分析和双曲动力学

• 批准号: 2400090
• 负责人: Semyon Dyatlov
• 金额: $ 42.25万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2400090&HistoricalAwards=false
• 关键词: Microlocal; Analysis; Hyperbolic; Dynamics

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.

该项目在微局部分析和双曲动力学的交集中研究了广泛的主题。微局部分析及其根源在物理现象（例如几何光学和量子/经典对应关系）中，是一种强大的数学理论，将经典的哈密顿动力学与波浪和量子状态的奇异性有关。双曲动力学是强烈混乱的系统的数学理论，其中很长一段时间后，初始数据的少量扰动会导致指数差异。该项目利用了这两个字段之间的相互作用，研究了基础动力学强烈混乱的情况，研究波和量子状态的行为，还探索了微局部方法在纯动力学问题上的应用。该项目为研究生提供了研究培训机会。该项目的一个方向是在高度活跃的量子混乱领域，这是对基础古典系统具有混乱行为的量子系统光谱特性的研究。主要研究者（PI）在谐波分析，分形几何形状，添加剂组合学和Ratner理论的领域中引入了新方法，将分形不确定性原理的概念结合在一起。该项目的具体目标包括：（1）了解封闭的混沌系统高能量本征函数的宏观浓度，例如负弯曲的Riemannian歧管和量子猫图； （2）证明了具有分形双曲线捕获套件的开放系统的必需光谱差距（特别意味着波浪的指数局部能量衰减）。第二个研究方向是研究分层流体中强制波的研究（旋转流体也出现了类似的问题），这是由在水族箱中实验观察到的内部波和海洋学应用的。第三个方向是应用最初针对双曲偏微分方程理论开发的微局部方法来研究经典对象，例如动态Zeta函数，这是量子/经典对应关系逆转的罕见例子。特别是，PI及其合作者研究（1）动态Zeta功能的特殊值如何与歧管的拓扑层学相关； （2）对于具有分散台球等奇异性的系统，是否可以继续进行动力学Zeta功能。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智力优点和更广泛影响的评估评估标准的评估值得支持的。