Applications of Random Matrix Theory to Analytic Number Theory

这个项目探索了随机矩阵理论和解析数理论相交的几个问题。从历史上看，这些领域中的第一个领域从数学物理中获得了很大的推动力，并广泛地涉及在随机选择条目的矩阵中发现的统计模式；来自这一领域的想法已在从流行病学到电信的大量学科中得到应用。第二个领域利用数学分析来研究整数的性质--例如，研究素数的分布或大整数倾向于因式的方式；这些是数学家长期以来一直感兴趣的自然问题，并可应用于数据安全。这两个领域最初是因为L函数的零点的局部分布和随机矩阵的特征值的分布之间令人惊讶的相似之处而联系在一起的，这些函数是解析数论中的中心兴趣。事实上，这种分布已经在各种情况下被发现，理解为什么这些分布如此普遍地出现是一个重要的问题。更详细地说，该项目由两个相关部分组成。一是利用算术函数的组合分解，研究了L函数的零点与随机矩阵的特征值之间的联系；类似的分解可以在函数域中找到，并揭示了函数域中的相关问题。第二部分是研究紧矩阵群上某些伪随机游动以与经典随机游动相同的方式均匀分布的程度；与第二个项目有关的结果最近被用来解决关于某些三角多项式的分布的公开问题。该项目的两个组成部分共同使用了组合表示理论，并研究了具有算术/组合考虑特征的弱相依随机变量序列。

This project explores several problems at the intersection of random matrix theory and analytic number theory. The first of these areas historically draws much of its impetus from mathematical physics and broadly concerns statistical patterns found in matrices where entries have been chosen at random; ideas from this area have found applications in a large number of subjects ranging from epidemiology to telecommunications. The second area makes use of mathematical analysis to study properties of the integers -- for instance to study the distribution of primes or the way large integers tend to factor; these are natural problems which mathematicians have been interested in for a long time and which have applications to data-security. The two areas were first linked by the surprising and still largely conjectural resemblance between the local distribution of zeros of L-functions -- these are certain functions which are of central interest in analytic number theory -- and the distribution of eigenvalues of a random matrix. In fact, distributions of this sort have been found in a diverse range of situations, and it is an important problem to understand why these distributions arise so universally. In more detail, the project consists of two related parts. The first is to study the link between zeros of L-functions and eigenvalues of random matrices by making use of combinatorial decompositions of arithmetic functions; analogous decompositions can be found in a function field setting and have shed light on related problems there. The second part is to investigate the extent to which certain pseudo-random walks on compact matrix groups equidistribute in the same fashion as classical random walks do; results pertaining to this second project have recently been used to resolve open questions about the distribution of certain trigonometric polynomials. The two components of the project share in common the use they make of combinatorial representation theory and also their study of sequences of random variables that are weakly dependent, with a dependence characterized by arithmetic/combinatorial considerations.