Universality and semi-classical behavior in 2+1 dimensional integrable systems and random matrices

对复杂现象的理解和最终控制是科学研究的首要目标。“普遍性”指的是某些现象的稳健性，以及相同现象在各种不同的物理情况和模型中违反直觉的普遍存在。例如，海洋中的波浪可以将自己组织成运输能量的“列车”，在光纤中传输的激光束中也观察到类似的列车；核实验中的统计波动导致了一种新型的普遍性，随后在各种随机模拟的情况下观察到了这种普遍性，就像对停放的汽车之间的间距的统计一样广泛！这项研究项目涉及对各种物理环境下的规范模型进行详细而严格的分析，这些模型的奇异行为是理解自然界中一些复杂现象的指南。该项目旨在开发了解、预测和控制系统行为的方法。这一研究计划的潜在长期影响源于普适性作为一种新的科学范式的出现：探索其适用范围在新兴领域和现有领域都是基本的。本研究项目涉及Riemann-Hilbert问题和d-bar问题的渐近分析的新方法的发展，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学在内的一系列领域的问题。在这些领域中的每一个领域，首要目标是提供系统的完整描述(无论是随机矩阵的特征值，还是非线性偏微分方程解)。研究中的两个例子是：(1)正态矩阵模型中的特征值统计。当特征值在二维区域累积时，寻找普遍的行为还处于初级阶段；这个项目的目标是在过去20年里发展起来的严格数学分析和物理直觉之间建立联系。(2)配分函数在两割区的渐近行为，其思想和猜想公式已经存在了一段时间。

The understanding and eventual control of complicated phenomena is a primary goal of scientific research. "Universality" refers to robustness of certain phenomena and to the counterintuitive prevalence of the same phenomena across a wide array of different physical situations and models. For example, waves in the ocean can organize themselves into "trains" transporting energy, and analogous trains are also observed in laser beams propagating in optical fibers; statistical fluctuations in nuclear experiments led to a new type of universality that subsequently has been observed in a wide variety of situations modeled with randomness, as far-flung as the statistics of spacings between parked cars! This research project involves the detailed rigorous analysis of canonical models for a wide variety of physical settings whose singular behavior is a guide for the understanding of some complicated phenomena in nature. The project aims to develop methods to understand, predict, and control system behavior. The potential long-term impacts of this research program stem from the emergence of universality as a new paradigm in science: probing its range of applicability is fundamental in emerging areas as well as established ones.This research project concerns the development of new methods for the asymptotic analysis of Riemann-Hilbert problems and d-bar problems and application of these techniques to problems in a range of fields including random matrix theory, nonlinear partial differential equations, orthogonal polynomials, and asymptotic combinatorics. In each of these areas, the overarching goal is to provide a complete description of the system (be it the eigenvalues of a random matrix, or the solution of a nonlinear partial differential equation). Two examples under study are: (1) Eigenvalue statistics in the normal matrix model. The quest for universal behavior when the eigenvalues are accumulating in a two-dimensional region is in its infancy; this project aims to create connections between rigorous mathematical analysis and physical intuition developed over the last 20 years. (2) The asymptotic behavior of the partition function in the two-cut regime, for which ideas and conjectural formulae have existed for some time.