CAREER: Long-time asymptotics of completely integrable systems with connections to random matrices and partial differential equations

该项目的研究方面侧重于从分析和数学物理角度理解各种模型的长期渐近性。特别是，首席研究员将集中在三个项目：（1）散焦非线性薛定谔方程的离散版本的继续研究，特别是，它与单位圆上的正交多项式，李泊松代数及其连续极限的联系;（2）性质的研究（例如，期望对角化时间）的某些随机矩阵系综时，被视为初始数据的可积流，如户田格和QR算法;和（3）研究，使用黎曼-希尔伯特方法，长时间渐近的可积系统，即户田格和Korteweg-deVries方程，并为某些不可积扰动这些系统。该项目的研究主题属于数学物理和偏微分方程的广泛领域，特别强调与完全可积系统，随机矩阵及其在某些数值算法中的应用等成熟数学领域相关的问题。偏微分方程出现在各种物理环境中，作为某些物理量随时间演化的数学模型。一类特殊的方程是所谓的完全可积系统，其特征在于它们满足足够大的（在某种意义上可以精确）数量的守恒律。完全可积系统的理论有着悠久而杰出的历史。特别是在过去的30年里，人们对这一领域的兴趣一方面是因为许多已知完全可积的方程都是作为流体动力学模型得到的，另一方面是因为人们发现它与许多纯数学领域（如李代数、辛几何和泊松几何）有密切的联系。当前项目的基本目标之一是扩展和加深对这些联系的理解，并扩大完全可积理论技术适用的模型类别。因此，我们的目标是尽可能充分地描述这些模型的解决方案，并希望借此获得对现实生活中现象的见解。该提案的一个重要组成部分是其教育部分，其中心是为高级研究生和最近的博士生组织年度暑期学校。的各种主题的当前研究兴趣的交界处的分析，偏微分方程和数值分析。暑期学校的所有讲座将由参与者根据预先指定的文章进行。除了其广泛的数学影响，该项目将允许首席研究员扩大其他教育活动以及。她将通过高级课程，研讨会和学生研究讲习班支持研究生的工作，她将与妇女在数学协会（AWM）在芝加哥地区的地方分会合作，以促进妇女在数学事业的进步。

The research side of this project focuses on understanding the long-time asymptotics for a diverse set of models from analysis and mathematical physics. In particular, the principal investigator will concentrate on three projects: (1) the continued study of a discrete version of the defocusing nonlinear Schroedinger equation, in particular, its connections with orthogonal polynomials on the unit circle, Lie-Poisson algebras, and its continuum limits; (2) the study of properties (e.g., expected diagonalization times) of certain random matrix ensembles when considered as initial data for integrable flows such as the Toda lattice and the QR-algorithm; and (3) the study, using Riemann-Hilbert methods, of long-time asymptotics for integrable systems, namely, the Toda lattice and the Korteweg-deVries equation, and for certain nonintegrable perturbations of these systems. The research topics of this project fall in the wide areas of mathematical physics and partial differential equations, with a particular emphasis on questions related to such well-established mathematical areas as completely integrable systems, random matrices, and their applications to certain numerical algorithms. Partial differential equations emerge in a variety of physical contexts as mathematical models for the time-evolution of certain physical quantities. A particular class of such equations are the so-called completely integrable systems, which are characterized by the fact they satisfy a sufficiently large (in a sense which can be made precise) number of conservation laws. The theory of completely integrable systems has a long and distinguished history. Over the past thirty years, in particular, interest in the field has been fueled on the one hand by the fact that many of the equations known to be completely integrable are obtained as models of fluid dynamics, and on the other hand by the close connections that have been discovered with many fields of pure mathematics, such as Lie algebras or symplectic and Poisson geometry. One of the fundamental goals of the current project is to expand and deepen the understanding of these connections and to widen the class of models to which the techniques of completely integrable theory apply. Hence one aims to describe as fully as possible the solutions to these models and hopefully to gain thereby insights into real-life phenomena. An essential part of the proposal is its educational component, which is centered around the organization of an annual summer school for advanced graduate students and recent Ph.D.'s on various topics of current research interest at the juncture of analysis, partial differential equations, and numerical analysis. All talks in the summer school will be delivered on preassigned articles by the participants. Beyond its broad mathematical impact, the project will allow the principal investigator to expand other educational activities as well. She will support the work of graduate students through advanced courses, seminars, and student research workshops, and she will work with local chapters of the Association for Women In Mathematics (AWM) in the Chicago area to promote the advancement of women in mathematical careers.