Statistical and Number-Theoretical Aspects of Dynamical Systems

我的研究计划包括几个项目，涉及动力学、概率论、遍历理论、数论和数学物理的交叉学科。我的主要目标有两个：我计划了解大类动力系统的长期行为，以解决动力学和遍历理论中的开放问题，同时我打算探索新的动力学工具在其他学科中解决问题的应用，从而开发一个动力学框架来研究动力学领域以外的几个数学对象。 我的跨学科研究活动已经加强了动力学、数论、随机过程以及最近数学物理领域之间的桥梁。在这项提议中，我打算集中讨论以下几类动力系统： (1)齐次空间上的单参数流，如双曲流形的单位切丛中的测地线流和角线流，强调角线的有效均匀分布。用Ratner理论研究均质流动，均质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，均匀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量化行为，即使在超对称量子力学的背景下也是如此。 (2)在研究具有数论性质的确定性序列时产生的符号系统(例如有限多个符号上的全移位的子移位)。Möbius随机性启发式和Sarnak最近的相关猜想促进了对拓扑动力系统和测度论动力系统等子转移的研究。这种双重观点最近被证明是接近乔拉在数论中的一个经典猜想的关键。 (3)具有零熵的二维动力系统，如与Farey分式理论有关的Boca-Cobeli-Zaharesu映射，或三角台球研究中出现的Casati-Prosen映射。 我的提案将概述几个项目，分为三类：(A)动力学的基本理论进展，(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名总部基地医生为特色。参与的学生(包括研究生和本科生)和博士后研究员将获得独特的技术技能，并将接受培训，成为熟练的沟通者，从而获得在数学科学领域的许多潜在职业。 我的研究议程，它的跨学科应用，我的HQP培训，以及我在科学界服务的所有追求将产生重大的国际影响，并将造福于动力系统领域和整个加拿大数学。

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.