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

理解并最终控制复杂现象是科学研究的首要目标。“普适性”指的是某些现象的稳健性，以及同一现象在一系列不同的物理情况和模型中反直觉的普遍性。例如，海洋中的波浪可以组织成传递能量的“火车”，在光纤中传播的激光束中也可以观察到类似的火车；核实验中的统计波动导致了一种新型的普遍性，这种普遍性随后在各种随机建模的情况下被观察到，就像停车间隔的统计一样广泛！本研究项目包括对各种物理环境的规范模型进行详细严格的分析，这些模型的单一行为是理解自然界中一些复杂现象的指南。该项目旨在开发理解、预测和控制系统行为的方法。这个研究项目的潜在长期影响源于普适性作为一种科学新范式的出现：探索其适用范围在新兴领域和已建立的领域都是至关重要的。本研究项目涉及发展Riemann-Hilbert问题和d-bar问题的渐近分析新方法，并将这些技术应用于随机矩阵理论、非线性偏微分方程、正交多项式和渐近组合等一系列领域的问题。在这些领域中，首要目标是提供系统的完整描述（可能是随机矩阵的特征值，也可能是非线性偏微分方程的解）。研究的两个例子是：(1)正态矩阵模型中的特征值统计。当特征值在二维区域中累积时，对普遍行为的追求尚处于起步阶段；该项目旨在建立严谨的数学分析和过去20年来发展起来的物理直觉之间的联系。(2)双切区配分函数的渐近性态，其思想和推测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.