Statistical and Number-Theoretical Aspects of Dynamical Systems

我的研究项目涉及动力学、概率论、遍历论、数论和数学物理的交叉领域。我的广泛目标是双重的：我计划了解大型动力系统的长期行为，以解决动力学和遍历理论中的开放性问题，同时我打算探索新的动力学工具的应用，以解决其他学科中的问题，从而开发一个动力学框架来研究动力学领域之外的几个数学对象。我的跨学科研究活动已经加强了动力学、数论、随机过程以及最近的数学物理领域之间的桥梁。(1)齐次空间上的单参数流，如双曲流形单位切线束中的测地流和环流，重点是环的有效均分。用拉特纳理论研究均匀流动，均匀动力学中的均匀分布结果在数论和数学物理中有重要的应用。例如，齐次空间上的流动可以用来描述量子系统中自相关函数的平均定量行为，甚至在超对称量子力学的设置中也是如此。***(2)在数论性质的确定性序列研究中出现的符号系统（如有限多个符号上的全移位的子移位）。Möbius随机启发式和Sarnak最近提出的相关猜想促进了对拓扑和测量理论动力系统等子转移的研究。这种对偶观点最近被证明是接近Chowla在数论中的经典猜想的关键。***(3)零熵的二维动力系统，如与Farey分数理论相关的Boca-Cobeli-Zaharescu图，或出现在三角台球研究中的Casati-Prosen图。****我的提案将概述几个项目，分为三类：(A)动力学的基础理论进展，(B)数论和量子力学的理论应用，(C)物理和化学的跨学科应用。****拟议的研究将以每年至少培训5名hqp为特色。参与的学生（包括研究生和本科生）和博士后将获得独特的技术技能，并将被训练成熟练的沟通者，从而获得许多潜在的数学科学职业。***我的研究议程，跨学科的应用，我的HQP培训，以及我为科学界服务的所有追求都将产生重大的国际影响，并将使动力系统和加拿大数学领域受益

My research program consists of several projects at the crossroad of dynamics, probability theory, ergodic theory, number theory, and mathematical physics. My broad aim is two-fold: I plan to understand the long-time behaviour of large classes of dynamical systems to address open questions in dynamics and ergodic theory, and at the same time I intend to explore the applications of novel dynamical tools to solve problems in other disciplines, thus developing a dynamical framework to study several mathematical objects outside the realm of dynamics.***My interdisciplinary research activity has already strengthened the bridges between the fields of dynamics, number theory, random processes, and recently also mathematical physics. In this proposal I intend to focus on the following classes of dynamical systems:****(1) One-parameter flows on homogeneous spaces, such as the geodesic and horocycle flow in the unit tangent bundle of a hyperbolic manifold, with emphasis on effective equidistribution of horocycles. Homogeneous flows are studied by means of Ratner's theory and equidistribution results in homogeneous dynamics yield important applications to number theory and mathematical physics. For example, flows on homogeneous spaces can be used to describe the average quantitative behavior of autocorrelation functions in quantum systems, even in the setting of supersymmetric quantum mechanics. ***(2) Symbolic systems (e.g. subshifts of the full shift on finitely many symbols) arising in the study of deterministic sequences of number-theoretical nature. The Möbius randomness heuristics and the related recent conjecture by Sarnak prompted the study of such subshifts as topological as well as measure-theoretical dynamical systems. This dual point of view has been recently shown to be crucial to approach a classical conjecture by Chowla in number theory.***(3) Two-dimensional dynamical systems with zero entropy, such as the Boca-Cobeli-Zaharescu map associated with the theory of Farey fractions, or the Casati-Prosen map appearing in the study of triangular billiards.****My proposal will outline several projects, divided into three categories: (A) fundamental theoretical progress in dynamics, (B) theoretical applications to number theory and quantum mechanics, (C) interdisciplinary applications to physics and chemistry.****The proposed research will feature the training of at least 5 HQPs per year. The students (both graduate and undergraduate) and postdoctoral fellows involved will will gain unique technical skills and will be trained to become proficient communicators, thus earning access to many potential careers in the mathematical sciences.***My research agenda, its interdisciplinary application, my HQP training, and all my pursuits in service of the scientific community will have significant international impact and will benefit the field of dynamical systems and Canadian mathematics at large.**