Studying Randomness in Number Theory and Quantum Mechanics via Dynamics

我的研究探索的交叉动力学，概率论，遍历理论，数论和数学物理。我的长期目标是研究数论和数学物理中的经典对象在多大程度上可以被视为随机的;虽然结果通常是概率性的，但我使用的工具是动态的。数论为我们提供了依赖于少量参数的大量确定性序列。了解这些序列表现出随机特征的程度是非常有趣的。换句话说，我的目标是研究概率论（或其变体）的熟悉结果是否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。因此，我的广泛目标是双重的：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。 该提案包括以下项目：A）均匀分布导致均匀动力学。 B）指数和的分布和数论起源的过程。 C）theta和的界。量子力学中的自相关函数。E）诣零流形的Berry-Tabor猜想。F）朝Chowla和Sarnak的方向前进。G）紧交换群上遍历平移的极限定理。

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.