Orthogonal Polynomials and Random Matrices

正交多项式和随机矩阵

• 批准号: 1362208
• 负责人: Doron Lubinsky
• 金额: $ 24万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-15 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1362208&HistoricalAwards=false
• 关键词: Orthogonal; Polynomials; Random; Matrices

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.

它是物理学家尤金·威格（Eugene Wigner），他在1950年的第一次使用随机矩阵的特征值来对中子的相互作用对重核进行建模。此后，随机矩阵已成为与数学物理学，概率理论，数量理论，数值分析和正交多项式的联系的主要研究领域。的确，在1970年代初期的物理学家弗里曼·戴森（Freeman Dyson）和普林斯顿大学的数字理论家休·蒙哥马利（Hugh Montgomery）之间的互动中，有一个众所周知的轶事，他们的讨论导致人们意识到随机矩阵与riemann zeta zeta Zeta Zeta函数的数字理论之间存在联系。 PI的重点是这些随机矩阵的“通用”行为：某些特征似乎与几乎所有基本假设无关，因此非常普遍地保持。这已经闻名了很长时间，并且已经被数学物理学家和纯数学家探索。研究这种“普遍性”的技术在许多其他数学领域都有用。该建议将从正交多项式和经典分析中开发出适当的工具，并使用这些工具来建立尽可能伟大的“通用”特征。该项目还将有一个教育组成部分，涉及与其他研究人员的合作，组织，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最近确定的各种财产的后果和概括。希望这将使人们能够在最小的条件下对紧凑的支持以及各种措施的措施建立普遍的限制。衡量标准的单调性，正交多项式的技术是这种方法的关键工具。更雄心勃勃的是将各种原理扩展到β浓度的目标。这将包括涉及多个变量中的多项式的广义基督教函数的渐近学。也将研究离散类似物。另一个主要目标是开发迪里奇（Dirichlet）正交多项式的理论，并研究了其在林德（Riemann）Zeta函数中的应用中的应用。其他目标包括研究生物三相的多项式和相关的数值二次；平面曲线上的正交多项式，并逐渐近似。