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主要目标。 “普遍性”是指某些现象的稳健性，以及同一现象在各种不同的物理情况和模型中的违反直觉的普遍性。 例如，海洋中的波浪可以组织成传输能量的“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各样的随机建模的情况下被观察到，就像停放的汽车之间的间距统计一样广泛！ 该研究项目涉及对各种物理环境的规范模型进行详细严格的分析，这些物理环境的奇异行为是理解自然界中一些复杂现象的指南。 该项目旨在开发理解，预测和控制系统行为的方法。 这项研究计划的潜在长期影响源于普遍性作为科学新范式的出现：探索其适用范围是基本的新兴领域，以及既定的。本研究项目涉及的发展的新方法的渐近分析黎曼希尔伯特问题和d-杆问题和应用这些技术的问题，在一系列领域，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的解）。 研究的两个例子是：（1）正规矩阵模型中的特征值统计。当特征值在二维区域中积累时，对普遍行为的探索尚处于起步阶段;该项目旨在建立严格的数学分析与过去20年来发展的物理直觉之间的联系。(2)配分函数在两次切割区域的渐近行为，关于这一点的思想和几何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.