Applications of random matrix theory in analytic number theory

本课题是解析数论与随机矩阵理论的交叉课题。解析数论是数论的一部分，它利用数学分析来研究素数的分布等问题。像这样的主题是纯数学的一部分，但在密码学中也有应用。随机矩阵理论研究的是随机选取条目的矩阵。该领域的许多重要问题都与随机矩阵的特征值分布有关。这些问题最初是由数学物理、统计学和种群生物学引起的，随机矩阵理论提供的答案在这些领域产生了重要的见解。解析数论与随机矩阵理论是两个完全不同的领域，但两者之间又存在着显著的联系。第一个这样的联系出现在H. Montgomery关于黎曼ζ函数零点的研究中。这些零很重要，因为它们表征了质数的分布。值得注意的是，至少在数值上，零之间的间隔似乎类似于各种随机矩阵的特征值之间的间隔——但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，为什么这种模式出现在如此不同的上下文中仍然是一个谜。（另一个令人惊讶的例子是墨西哥库埃纳瓦卡市公交车到站时间的间隔。）本提案中概述的研究计划的一部分旨在通过以下方式更好地理解为什么泽塔0之间的间隔类似于特征值之间的间隔：1)开发一个理解这一事实的启发性组合框架，2)建立黎曼泽塔函数的随机模型，以及3)开发与随机点过程理论的联系。这三点的各个方面已被用来解决或阐明过去未解决的问题。这个提议的第二部分涉及到伪随机矩阵的乘积的研究——这对著名的Rudin-Shapiro多项式的分布有应用，它本身对分析师和数论学家来说很有趣，但它也在信号处理中有应用。同样，与第二部分相关的思想也被用来解决数学中古老的开放性问题。通过学习和发展概率论（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，培养高素质的人才，最终目标是在学术界或工业界（例如数据科学、无线通信或数据安全）从事职业。

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).