Orthogonal Polynomials and Random Matrices

物理学家尤金·维格纳在20世纪50年代首先使用随机矩阵的本征值来模拟重核中子的相互作用。随机矩阵已经成为数学物理、概率论、数论、数值分析和正交多项式的一个主要研究领域。事实上，有一个关于20世纪70年代初物理学家弗里曼·戴森（Freeman Dyson）和数论家休·蒙哥马利（Hugh Montgomery）在普林斯顿的一次互动的著名轶事，他们的讨论导致了随机矩阵和数论的黎曼泽塔函数之间存在联系的认识。PI的重点是这些随机矩阵的“普遍”行为：某些特征似乎与几乎任何潜在的假设无关，因此非常普遍。这已经知道很长一段时间了，并且已经被数学物理学家和纯数学家所探索。研究这种“普遍性”的技术在数学的许多其他领域都很有用。该提案将从正交多项式和经典分析中开发适当的工具，并使用这些工具尽可能普遍地建立“通用”特征。该项目还将包括教育部分，涉及与其他研究人员的合作，组织会议，编辑职责以及对本科生和/或研究生的监督。该项目的具体目标包括调查PI最近建立的变分性质的分支和概括。人们希望这将使人们能够建立普遍性的限制厄米合奏措施的最低条件下紧支持，也为不同的措施。测度的单调性和正交多项式技术是这种方法的关键工具。更雄心勃勃的目标是将变分原理扩展到β系综。这将包括渐近的广义Christoffel函数涉及交替多项式在几个变量。离散的类似物也将进行研究。另一个主要目标是发展狄利克雷正交多项式的理论，并研究它们在黎曼Zeta函数的林德洛夫假设中的应用。其他目标包括调查双正交多项式，和相关的数值求积;正交多项式的曲线在平面上，和帕德近似。

It was the physicist Eugene Wigner who in the 1950's first used eigenvalues of random matrices to model the interactions of neutrons for heavy nuclei. Random matrices have since become a major research area with connections to mathematical physics, probability theory, number theory, numerical analysis, and orthogonal polynomials. Indeed, there is a well known anecdote about an interaction in the early 1970's between the physicist Freeman Dyson, and number theorist Hugh Montgomery, at Princeton, where their discussions led to the realization that there is a link between random matrices and the Riemann Zeta function of number theory. The PI's focus is on "universal" behavior of these random matrices: certain features seem to be independent of almost any underlying assumption, and consequently hold very generally. This has been known for a long time, and has been explored by both mathematical physicists and pure mathematicians. The techniques that have been developed to study this "universality" have been useful in many other areas of mathematics. This proposal will develop appropriate tools from orthogonal polynomials and classical analysis, and use these to establish "universal" features in as great a generality as possible. There will be also be an educational component to the project, involving collaboration with other researchers, organization of conferences, editorial duties, and supervision of undergraduate and/or graduate students.The specific goals of the project include investigating the ramifications and generalizations of a variational property recently established by the PI. It is hoped that this will enable one to establish universality limits for Hermitian ensembles under minimal conditions on measures with compact support, and also for varying measures. Monotonicity in the measure, and techniques of orthogonal polynomials are key tools in this approach. Somewhat more ambitious is the goal of extending the variational principle to beta-ensembles. This will include asymptotics of generalized Christoffel functions involving alternating polynomials in several variables. Discrete analogues will also be studied. Another major goal is developing the theory of Dirichlet orthogonal polynomials, and investigating their application to the Lindelof hypothesis for the Riemann Zeta function. Additional goals include investigating biorthogonal polynomials, and related numerical quadratures; orthogonal polynomials on curves in the plane, and Pade approximations.