Statistical and Number-Theoretical Aspects of Dynamical Systems

点击翻译按钮获取中文摘要

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.