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: 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=678594
• 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).

该研究建议在于分析数理论与随机矩阵理论的相交。分析数理论是数字理论的一部分，它利用数学分析来研究诸如质数分布之类的主题。像这样的主题是纯数学的一部分，但例如密码学应用程序。随机矩阵理论是对已随机选择的条目的矩阵的研究。该地区的许多重要问题涉及随机矩阵特征值的分布。此类问题首先是由数学物理，统计和种群生物学的动机，而随机矩阵理论提供的答案在这些领域中产生了重要的见解。******分析数理论和随机矩阵理论是相当不同的领域，但存在着显着它们之间的联系。第一个这种联系在H. Montgomery在Riemann Zeta功能的零上的作品中产生。这些零很重要，因为它们表征了素数的分布。值得注意的是，至少从数字上讲，零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 - 但是没有人能证明这实际上是如此。其他复杂的系统似乎也显示出相同或相关的模式，以及为什么这种模式出现在这种不同的上下文中仍然是一个谜。 （另一个令人惊讶的例子是在墨西哥城市Cuernavaca到达的公共汽车到达之间的间距。）******该提案中概述的研究计划的一部分试图更好地理解为什么Zeta Zeros之间的间距类似于Eigenvalues之间的间距，而不是1 ）开发一个启发性的组合框架来理解这一事实，2）构建Riemann Zeta功能的随机模型，3）建立与随机点过程理论的联系。这三个方面的各个方面已被用来解决或阐明旧的未解决问题。该提案的第二部分涉及对伪随机矩阵的产品的研究 - 这在著名的Rudin-Shapiro多项式方面的分布中有应用，这对于他们自身对分析师和数字理论家的依据来说很有趣，但在分析师和数字理论家中也有应用。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旧开放问题。 ******将通过学习和发展概率的方面（包括随机矩阵理论和点过程理论），Combinatorics（包括组合代表理论）和数字理论，并最终通过学习概率的方面（包括随机矩阵理论和点过程理论），对高素质的人员进行培训。在学术界或行业中追求职业的目标（例如，数据科学，无线通信或数据安全）。