Applications of random matrix theory in analytic number theory

本研究方案介于解析数论和随机矩阵理论的交汇点上。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主题。这样的主题是纯数学的一部分，但也可以应用到密码学等领域。随机矩阵理论是对具有随机选择的条目的矩阵的研究。该领域中的许多重要问题都与随机矩阵的特征值的分布有关。这样的问题最初是由数学物理、统计学和种群生物学激发的，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理论是两个截然不同的领域，但它们之间存在着显著的联系。第一个这样的联系出现在H.Montgomery关于Riemann Zeta函数的零点的工作中。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--但没有人能证明这是真的。其他复杂的系统似乎也表现出相同或相关的模式，为什么这种模式出现在如此不同的背景下仍然是一个谜。(另一个令人惊讶的例子是墨西哥城市库埃纳瓦卡的公交车到达时间间隔。)这项建议中概述的研究计划的一部分试图通过以下方式更好地理解为什么Zeta零点之间的间距类似于本征值之间的间距：1)开发一个有启发性的组合框架来理解这一事实，2)建立Riemann Zeta函数的随机模型，以及3)发展到随机点过程理论的链接。这三点已经被用来解决或阐明老的悬而未决的问题。这项建议的第二部分涉及研究伪随机矩阵的乘积-这可以应用于著名的Rudin-Shapiro多项式的分布，这些多项式本身对分析师和数学家来说很有趣，但在信号处理中也有应用。同样，与第二部分相关的想法也被用来解决数学中的老问题。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，在整个提案中将培训高素质的人员，最终目标是在学术界或行业从事职业生涯(例如，数据科学、无线通信或数据安全)。

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).