Statistical and Number-Theoretical Aspects of Dynamical Systems

我的研究计划包括几个项目的交叉动力学，概率论，遍历理论，数论和数学物理。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未决问题，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，最近也数学物理领域之间的桥梁。在这个建议中，我打算集中在以下几类动力系统：*（1）齐性空间上的单参数流，例如双曲流形的单位切丛中的测地线和horocycle流，重点是horocycle的有效等分布。均匀流的研究通过拉特纳的理论和均匀动力学的均匀分布的结果产生重要的应用数论和数学物理。例如，齐次空间上的流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背景下。 * （2）符号系统（例如，全移位的子移位在200个符号上），产生于数论性质的确定性序列的研究。莫比乌斯随机性理论和最近由萨纳克提出的相关猜想促进了对拓扑和测度论动力系统等子移位的研究。这种对偶的观点最近被证明是至关重要的接近一个经典猜想Chowla在数论中。(3)熵为零的二维动力系统，例如与Farey分数理论相关的Boca-Cobeli-Zaharescu映射，或在三角台球研究中出现的Casati-Prosen映射。我的建议将概述几个项目，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。拟议的研究将包括每年至少培训5名HQP。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，从而获得许多潜在的数学科学职业。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，并将有利于动力系统和加拿大数学领域。

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.**