Studying Randomness in Number Theory and Quantum Mechanics via Dynamics

通过动力学研究数论和量子力学中的随机性

• 批准号: RGPIN-2022-04330
• 负责人: Cellarosi, Francesco
• 金额: $ 1.31万
• 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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758926
• 关键词: Studying; Randomness; Number; Theory; Quantum

My research explores the intersection of dynamics, probability theory, ergodic theory, number theory, and mathematical physics. My long-term goal is to study the extent to which classical objects from number theory and mathematical physics can be regarded as random; while the results are often of probabilistic nature, the tools I use are dynamical. Number Theory supplies us with a multitude of deterministic sequences that depend on a small number of parameters. It is of great interest to understand the degree to which these sequences exhibit random features. In other words, I aim to study whether familiar results from probability theory (or variations thereof) hold for sequences of number-theoretical origin, and explore the range of applicability of the new results. The same holds for functions arising in Quantum Mechanics, such as correlation functions. My broad aim is therefore 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. The proposal includes the following projects: A) Equidistribution results in homogeneous dynamics. B) The distribution of exponential sums and processes of number-theoretical origin.  C) Bounds for theta sums. D) Autocorrelation functions in quantum mechanics. E) Berry-Tabor conjecture for nilmanifolds. F) Advances toward Chowla's and Sarnak's conjectures. G) Limit theorems for ergodic translations on compact abelian groups.

我的研究探索了动力学、概率论、遍历理论、数论和数学物理学的交叉点。我的长期目标是研究数论和数学物理学中的经典对象在多大程度上可以被视为随机的。通常具有概率性质，我使用的工具是动态的，数论为我们提供了大量依赖于少量参数的确定性序列，了解这些序列表现出随机特征的程度非常有趣。换句话说，我的目的是研究是否概率论（或其变体）的熟悉结果适用于数论起源的序列，并探索新结果的适用范围，这同样适用于量子力学中出现的函数，例如相关函数。有两个方面：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，同时我打算探索新颖的动力工具的应用来解决其他学科的问题，从而开发出一个该提案包括以下项目：A) 均匀动力学的结果 B) 指数和的分布以及数论起源的过程。 C) θ 和的界限。 D) 量子力学中的自相关函数 E) 尼尔流形的 Berry-Tabor 猜想 F) Chowla 和 Sarnak 猜想的进展 G) 极限。紧阿贝尔群的遍历翻译定理。