Riemann--Hilbert Problems in Random Matrix Theory, Approximation Theory, and Integrable Systems

主要研究者：Ken McLaughlin，University of北卡罗来纳州，Chapel HillDMS-0200749摘要：*McLaughlin的研究涉及黎曼-希尔伯特问题的应用以及为它们在（1）随机矩阵理论，（2）可积系统，（3）逼近理论和正交多项式中的经典问题的渐近分析而开发的新技术。在可积系统中，本文的研究将继续McLaughlin关于可积非线性偏微分方程奇异极限的工作。可积的非线性偏微分方程为各种各样的物理环境提供了规范模型。 对于一些可积模型，如聚焦非线性薛定谔方程的半经典极限、Korteweg-de弗里斯方程和连续极限中的户田晶格，McLaughlin（与合作者）正在开发理解、预测和控制其行为的方法。 在随机矩阵理论中，McLaughlin将继续他对随机Hermitian矩阵特征值的渐近行为的研究。他将研究随机矩阵理论的配分函数的渐近行为。这个基本量是一个配分函数在经典意义上的统计力学，为一个相互作用的对数气体。 McLaughlin建议研究的渐近性是随着粒子数量的增长。 这项研究将影响数学研究的几个领域，如70年前由Szego开创的Hankel行列式理论，二维量子引力理论和近似理论。在近似理论中，McLaughlin将研究Riemann-Hilbert问题的渐近分析与有理近似之间的新联系（最近由McLaughlin及其合作者发现）。 这包括经典的帕德近似问题。在相关的工作中，McLaughlin将研究离散正交多项式的渐近行为，这些多项式是关于Dirac质量之和的测度正交的多项式。科学研究的主要目标是理解和控制复杂的现象。 这种对物理过程的理解和控制增强了我们技术进步的能力。 复杂非线性现象的物理模型通常归结为研究参数区域中的偏微分方程，其中它们的解表现出奇异的野生行为。在其他情况下，具有大量随机性的统计理论被开发来理解复杂的现象。三个例子：光纤中的激光束可能爆炸，或者它们携带的信号可能由于噪音或剧烈振荡的发生而退化。海洋中的波浪可以组织成运输能量的“火车”。在20世纪50年代，核共振能级实验表明了一种新型的普遍性，其数学解释直到最近才得到解释。 科学家通过分析这些一般的非线性偏微分方程或统计理论来预测戏剧性行为的能力是有限的。然而，有一类典型的模型，标题下的“可积模型”，为各种各样的物理设置。它们的奇异行为是理解自然界中一些复杂现象的指南。其中一些是偏微分方程，另一些是统计模型，但这些可积模型的统一特征是研究人员在其分析方面取得了很大进展。McLaughlin的研究涉及对这些模型的详细严格分析;他（与合作者）正在开发理解，预测和控制他们行为的方法。

PI: Ken McLaughlin, University of North Carolina, Chapel HillDMS-0200749Abstract:*********************************************************McLaughlin's research concerns applications of Riemann-Hilbert problems and new techniques developed for their asymptotic analysis to classical problems in (1) random matrix theory, (2)integrable systems, and (3) approximation theory and orthogonal polynomials. In integrable systems, the proposed research will continue McLaughlin's work on singular limits of integrablenonlinear partial differential equations. Integrable nonlinear partial differential equations provide canonical models for a wide variety of physical settings. For some integrable models, such asthe semi-classical limit of the focusing nonlinear Schroedinger equation, the Korteweg-de Vries equation, and the Toda lattice in a continuum limit, McLaughlin (with collaborators) is developingmethods to understand, predict, and control their behavior. In random matrix theory, McLaughlin will continue his work on the asymptotic behavior of eigenvalues of random Hermitian matrices.He will study the asymptotic behavior of the partition function of random matrix theory. This basic quantity is a partition function in the classical sense of statistical mechanics, for aninteracting log-gas. The asymptotics McLaughlin proposes to study are as the number of particles grows. The research will impact upon several areas of mathematical research, such as the theory of Hankel determinants pioneered by Szego some 70 years ago, the theory of 2 dimensional quantum gravity, and approximation theory. In approximation theory, McLaughlin will investigate new connections (recently discovered by McLaughlin and collaborators) between the asymptotic analysis of Riemann-Hilbert problems and rational approximation. This includes the classical problems of Pade' approximation. In related work, McLaughlin will study the asymptotic behavior of discrete orthogonal polynomials, which are polynomials orthogonal with respect to a measure that is a sum of Dirac masses.A primary goal of scientific research is to understand and control complicated phenomena. Such understanding and control of a physical process enhances our ability for technologicaladvancement. Physical models for complex nonlinear phenomena often boil down to the study of partial differential equations in parameter regimes where their solutions exhibit singularly wildbehavior. In other instances, statistical theories with great amount of randomness are developed to understand complex phenomena. Three examples: laser beams in optical fibers canexplode, or signals they carry can degrade due to noise or the onset of violent oscillations. Waves in the ocean can organize themselves into "trains" transporting energy. In the 1950s, nuclear resonance level experiments indicated a new type of universality, whose mathematical explanation has only recently been explained. Scientists' ability to predict dramatic behaviorthrough the analysis of such general nonlinear partial differential equations, or statistical theories, is limited. However, there is a class of canonical models, under the heading "integrable models", for a wide variety of physical settings. Their singular behavior is a guide for the understanding of some complicated phenomena in nature. Some of these are partial differential equations, others are statistical models, but the unifying feature of these integrable models is that researchersare making great progress in their analysis. McLaughlin's research involves the detailed rigorous analysis of these models; he (with collaborators) is developing methods to understand, predict, and control their behavior.