Statistical and Number-Theoretical Aspects of Dynamical Systems

动力系统的统计和数论方面

• 批准号: RGPIN-2016-03901
• 负责人: Cellarosi, Francesco
• 金额: $ 1.68万
• 依托单位: Queen's University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=679426
• 关键词: Statistical; Number; Theoretical; Aspects; Dynamical

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.**

我的研究计划包括在动态，概率理论，千古理论，数理论和数学物理学的十字路口的几个项目。我的广泛目的是两个折叠：我计划了解大量动态系统的长期行为，以解决动态和厄基理论中的开放问题，同时我打算探索新型动态工具在其他学科中解决问题的应用，从而开发一个动态框架，从而开发一个动态框架，研究了动态的数学对象。理论，随机过程以及最近的数学物理学。在此提案中，我打算专注于以下动态系统类别：****（1）在均质空间上流动的一参数流，例如单位过度歧管的单位切线束中的大地和肉圈流，重点是霍罗克氏菌的有效等分分布。通过Ratner的理论研究了均匀的流量，等均分配导致均质动力学产生重要的应用，从而为数字理论和数学物理学提供了重要的应用。例如，即使在超对称量子力学的情况下，均匀空间上的流量也可用于描述量子系统中自相关函数的平均定量行为。 ***（2）在数字理论性质的确定性序列的研究中产生的符号系统（例如，最终许多符号的全部转移的子缩短）。 Möbius随机性启发式方法和萨尔纳克（Sarnak）的最新签约促使研究了诸如拓扑和测量理论动态系统之类的子迁移。最近已经证明，这种双重观点对于乔拉（Chowla）的经典猜想至关重要。类别：（a）动力学的基本理论进步，（b）对数字理论和量子力学的理论应用，（c）对物理和化学的跨学科应用。学生（包括研究生和本科生）和参与的博士后研究员将获得独特的技术技能，并将接受培训以成为熟练的沟通者，从而在数学科学中获得许多潜在的职业。 大的。**