Applications of random matrix theory in analytic number theory

随机矩阵理论在解析数论中的应用

• 批准号: RGPIN-2019-04888
• 负责人: Rodgers, Bradley
• 金额: $ 1.6万
• 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=759111
• 关键词: Applications; random; matrix; theory; analytic

This research proposal lies at the intersection of analytic number theory and random matrix theory. Analytic number theory is the part of number theory that makes use of mathematical analysis to study topics like the distribution of prime numbers. Topics like these are a part of pure math but have applications to cryptography for instance. Random matrix theory is the study of matrices with entries that have been chosen randomly. Many important questions in the area concern the distribution of eigenvalues of random matrices. Such questions were first motivated by mathematical physics, statistics, and population biology and answers provided by random matrix theory yield important insights in these fields. Analytic number theory and random matrix theory are quite disparate fields, but there exist remarkable connections between them. The first such link arose in work of H. Montgomery on the zeros of the Riemann zeta-function. These zeros are important because they characterize the distribution of primes. Remarkably, at least numerically, the spacings between the zeros seem to resemble the spacings between eigenvalues of a wide variety of random matrices - but no one can prove that this is actually so. Other complex systems also seem to display the same or related patterns, and why this pattern appears in such disparate contexts remains a mystery. (Another surprising example is the spacing between bus arrival times in the Mexican city of Cuernavaca.) One part of the research program outlined in this proposal seeks to better understand why spacings between zeta zeros resemble spacings between eigenvalues by 1) developing an illuminating combinatorial framework for understanding this fact, 2) building random models of the Riemann zeta-function, and 3) developing links to the theory of stochastic point processes. Aspects of these three points have already been used to resolve or shed light on old unresolved problems. A second part of this proposal involves the study of products of pseudo-random matrices - this has applications to the distribution of the famous Rudin-Shapiro polynomials, which are interesting for their own sake to analysts and number theorists, but which also have applications in signal processing. Again, ideas related to this second part have also been used to resolve old open problems in mathematics. Highly qualified personnel will be trained throughout this proposal by learning and developing aspects of probability (including random matrix theory and the theory of point processes), combinatorics (including combinatorial representation theory), and number theory, with an eventual goal of pursuing careers in academia or industry (in for instance data science, wireless communications, or data security).

该研究提案位于解析数论和随机矩阵理论的交叉点。解析数论是数论的一部分，它利用数学分析来研究素数分布等主题。诸如此类的主题是纯数学的一部分，但也可应用于密码学等领域。随机矩阵理论是对具有随机选择的条目的矩阵的研究。该领域的许多重要问题都涉及随机矩阵特征值的分布。这些问题最初是由数学物理、统计学和群体生物学引发的，随机矩阵理论提供的答案在这些领域产生了重要的见解。解析数论和随机矩阵论是完全不同的领域，但它们之间存在着显着的联系。第一个这样的联系出现在 H. Montgomery 对黎曼 zeta 函数零点的研究中。这些零很重要，因为它们表征了素数的分布。值得注意的是，至少在数字上，零点之间的间距似乎类似于各种随机矩阵的特征值之间的间距 - 但没有人能证明事实确实如此。其他复杂的系统似乎也显示出相同或相关的模式，而为什么这种模式出现在如此不同的环境中仍然是个谜。 （另一个令人惊讶的例子是墨西哥库埃纳瓦卡市公交车到达时间之间的间隔。）本提案中概述的研究计划的一部分旨在更好地理解为什么 zeta 零点之间的间隔类似于特征值之间的间隔：1）开发一个具有启发性的组合框架为了理解这一事实，2) 构建黎曼 zeta 函数的随机模型，3) 开发与随机点过程理论的联系。这三点的各个方面已经被用来解决或阐明一些未解决的老问题。该提案的第二部分涉及伪随机矩阵乘积的研究 - 这适用于著名的 Rudin-Shapiro 多项式的分布，这些多项式本身对分析师和数论学家来说很有趣，但也适用于信号处理。同样，与第二部分相关的想法也被用来解决数学中的老开放问题。 在整个提案中，将通过学习和发展概率（包括随机矩阵理论和点过程理论）、组合学（包括组合表示理论）和数论方面来培训高素质人才，最终目标是在学术界谋求职业生涯或工业（例如数据科学、无线通信或数据安全）。