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.

这个项目调查了微观局部分析和双曲动力学交叉的广泛的主题。微域分析起源于几何光学和量子/经典对应等物理现象，是一种将经典哈密顿动力学与波和量子态的奇异性联系起来的强大的数学理论。双曲动力学是强混沌系统的数学理论，初始数据的微小扰动会导致在很长一段时间后轨迹呈指数发散。该项目利用这两个场之间的相互作用，研究了在潜在动力学强烈混沌的情况下波和量子态的行为，并探索了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沌领域，研究量子系统的光谱性质，其中潜在的经典系统具有混沌行为。首席调查员(PI)引入了调和分析、分形几何、加性组合学和Ratner理论的新方法，并结合了分形测不准原理的概念。该项目的具体目标包括：(1)了解闭合混沌系统高能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阱集的开放系统的基本谱间隙(特别是波的指数局部能量衰减)。第二个研究方向是研究分层流体中的强迫波(旋转流体也出现了类似的问题)，其动机是在水族箱中通过实验观察到的内波以及在海洋学中的应用。第三个方向是应用最初为双曲型偏微分方程组理论发展起来的微局域方法来研究经典对象，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和他的合作者研究了(1)负曲线流形的动态Zeta函数的特殊值如何与流形的拓扑相关；以及(2)对于具有奇点的系统(如色散台球)，动态Zeta函数是否可以亚纯连续。这一奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。