Universality in random matrices and integrable systems: asymptotic analysis via Riemann-Hilbert and d-bar methods

McLaughlin的研究致力于开发Riemann-Hilbert问题和d-bar问题的渐近分析的新方法，并将这些技术应用于包括随机矩阵理论、非线性偏微分方程、正交多项式和渐近组合在内的一系列领域的前沿(以及经典)问题。驱动因素之一是探索这些领域的普遍行为。McLaughlin以前在非线性偏微分方程中的工作涉及到研究某种特殊的“可积”的非线性偏微分方程中的剧烈振荡。最近出现了一些猜想，类似于随机矩阵理论中著名的普适性猜想，它假定这些振荡的微观产生在整个非线性偏微分方程族中是普适的。在随机矩阵理论中，普适性是指当矩阵的大小增长到无穷大时，随机矩阵的特征值统计倾向于以通用的方式表现，而与用于生成随机矩阵的概率定律的确切形式无关。麦克劳克林建议在非线性偏微分方程式和随机矩阵理论中大幅扩展当前普适性的领域，并探索普适性的极限：V上的什么条件足以看到与典型普遍行为的偏离？各种研究项目也被提出，其目的是培养研究生。对复杂现象的理解和最终控制是科学研究的首要目标。通过这一根本目标，我们提高了技术进步的能力。复杂的非线性现象的物理模型通常归结为研究参数范围内的偏微分方程解表现出奇特的狂野行为。在其他情况下，发展具有大量随机性的统计理论来理解复杂的现象。“普遍性”指的是某些现象的稳健性，以及同一现象在各种不同的物理情况和模型中违反直觉的普遍现象。举两个例子：(1)海洋中的波可以将自己组织成传输能量的“列车”，在光纤中传输的激光也可以观察到类似的列车。(2)20世纪50年代测量的核共振水平的统计波动导致了一种新型的普遍性，在以随机性建模的各种情况下观察到了同样的统计行为，就像对停放的汽车之间的间距的统计一样遥远！科学家通过分析一般的非线性偏微分方程式或统计理论来预测戏剧性行为的能力是有限的。然而，有一类用于各种物理环境的规范模型。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，但这些模型的统一特点是，研究人员在分析方面取得了很大进展。麦克劳克林的研究涉及对这些模型的详细而严格的分析；他(与合作者)正在开发理解、预测和控制他们的行为的方法。这一研究计划的更广泛影响来自于普遍性作为非线性科学中一种新范式的出现：探索其适用范围在新兴领域和现有领域都是基本的。拟议研究的另一个非常重要的方面是在这些快速发展的领域继续培训研究生。

McLaughlin's research concerns the development of new methods for the asymptotic analysis of Riemann-Hilbert problems and d-bar problems, and to apply the full body of such techniques to cutting edge (as well as classical) problems in a range of fields including random matrix theory, nonlinear partial differential equations, orthogonal polynomials, and asymptotic combinatorics. One driving feature is the exploration of universal behavior in these areas. McLaughlin's previous work in nonlinear partial differential equations has concerned the study of violent oscillations in somewhat specialized ``integrable'' nonlinear partial differential equations. Very recently, conjectures have emerged, analogous to the famous universality conjectures in random matrix theory, which posit that the microscopic generation of these oscillations is universal over entire families of nonlinear partial differential equations. In random matrix theory, universality refers to the phenomenon that eigenvalue statistics of random matrices tend, as the size of the matrices grow to infinity, to behave in a universal way, independent of the exact form of the probabilistic laws used to generate the random matrices. McLaughlin proposes to extend the current realm of universality substantially, in nonlinear partial differential equations and random matrix theory, and to probe the limits of universality: what conditions on V are sufficient to see a deviation from the typical universal behavior? A variety of research projects are also proposed whose aim is the training of graduate students.The understanding and eventual control of complicated phenomena is a primary goal of scientific research. Through this fundamental goal we enhance our ability for technological advancement. Physical models for complex nonlinear phenomena often boil down to the study of partial differential equations in parameter regimes where their solutions exhibit singularly wild behavior. In other instances, statistical theories with great amount of randomness are developed to understand complex phenomena. ``Universality'' refers to robustness of certain phenomena, and to the counter-intuitive prevalence of the same phenomena across a wide array of different physical situations and models. Two examples: (1) Waves in the ocean can organize themselves into "trains" transporting energy, and analogous trains are also observed in laser beams propagating in optical fibers. (2) Statistical fluctuations of nuclear resonance levels measured in the 1950s led to a new type of universality, and the same statistical behavior has been observed in a wide variety of situations modeled with randomness, as far-flung as the statistics of spacings between parked cars! Scientists' ability to predict dramatic behavior through the analysis of such general nonlinear partial differential equations, or statistical theories, is limited. However, there is a class of canonical 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 models is that researchers are 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. The broader impacts of this research program stem from the emergence of universality as a new paradigm in nonlinear science: probing its range of applicability is fundamental in emerging areas as well as established ones. Another very important aspect of the proposed research is the continued training of graduate students in these rapidly developing areas.