Some Analytical Aspects of the Theory of Integrable Systems

术语“可积系统”通常指的是数学对象，最常见的是微分方程式，具有特殊的对称性，可以非常详细地研究它们，有时甚至可以用封闭的形式来求解它们。这类可积系统包括几个基本的自然方程，可积系统的数学基础可以追溯到刘维尔、高斯和庞加莱的经典著作。在我们的时代，可积系统理论已经成为一个不断扩展的领域，它作为现代数学和理论物理的许多分支的新的分析和代数思想的主要来源之一，发挥着越来越重要的作用。同时，它为研究现代非线性科学和技术中出现的一些基本数学模型提供了一种有效的分析工具。除了传统的微分方程域之外，可积技术在各种领域中变得普遍，如正交多项式、弦论、计数拓扑学、统计力学、随机过程、量子信息学和数论。这个项目中考虑的许多问题都与这些学科有直接的联系。该项目还包括几项教育活动，如在IUPUI培训博士生，以及共同组织关于随机矩阵理论普适性和可积性的国际研究/教育计划，该计划将于2021年秋季在伯克莱的MSRI举行。该项目延续了该研究所在可积系统理论方面的长期研究努力。这一研究计划的主要目标是解决可积系统理论的各种新的分析问题，这些问题是从随机矩阵理论和精确可解量子模型理论的最新发展中出现的。在这个项目中，PI将集中于三个方向的研究：(A)一般的Beta-系综和Calogero-Painleve系统；(B)研究等单形tau函数、它们的渐近性、Fredholm行列式表示及其与保形场论的关系；以及(C)从随机矩阵理论和统计力学中产生的Toeplitz+Hankel行列式的渐近分析。这些方向中的每一个都由一组具体的问题表示，它们将在相同的分析框架内进行研究，即Riemann-Hilbert方法。该项目(A)部分的成功将对发展随机矩阵理论和粒子相互作用理论以及整个现代非线性科学中的普遍性和可积性的一般概念产生重大影响。该项目的(B)部分将进一步提升Painleve Overcestions的“非线性特殊功能”地位，这些功能有时也被称为“21世纪的特殊功能”。(C)部分的成功将对Toeplitz和Hankel行列式的一般理论的发展做出重大贡献，这两个行列式几十年来一直在统计和量子力学的一些基本模型的研究中发挥核心作用。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The term "Integrable Systems" usually refers to mathematical objects, most often differential equations, with special symmetry properties which allow to study them in a very detailed way and sometimes even to solve them in a closed form. The class of integrable systems includes several fundamental equations of nature, and the mathematical foundations of integrable systems go back to classical works of Liouville, Gauss, and Poincare. In our days, the theory of integrable systems has become an expanding area which plays an increasingly important role as one of the principal sources of new analytical and algebraic ideas for many branches of modern mathematics and theoretical physics. Simultaneously, it provides an efficient analytical tool for the study of some of the fundamental mathematical models arising in modern nonlinear science and technology. In addition to the traditional domain of differential equations, integrable techniques are becoming common in such diverse fields as orthogonal polynomials, string theory, enumerative topology, statistical mechanics, random processes, quantum informatics, and number theory. Many of the problems considered in this project have direct connections with these disciplines. The project also includes several educational activities such as training of Ph.D. students at IUPUI and co-organizing of the international research/educational program on universality and integrability in random matrix theory which will be held in the Fall of 2021 at MSRI, Berkeley.This project continuous the PI’s long term research efforts in the theory of integrable systems. The principal goal of this research program is to address various new analytical questions of the theory of integrable systems which have emerged from recent developments in random matrix theory and in the theory of exactly solvable quantum models. In this project, the PI will concentrate on three directions of research: (a) The general beta-ensembles and the Calogero-Painleve system; (b) The study of the isomonodromic tau functions, their asymptotics, Fredholm determinant representations and their relations to conformal field theory; and (c) The asymptotic analysis of Toeplitz + Hankel determinants emerging from random matrix theory and statistical mechanics. Each of these directions is represented by a collection of concrete problems, and they will be investigated within the same analytical framework, viz., the Riemann-Hilbert method. Success in part (a) of the project would have a notable impact in the development of the general concept of universality and integrability in both the random matrix theory and in the theory of interacting particles as well as in the modern nonlinear science at large. The part (b) of the project will further enhance the "nonlinear special function" status of Painleve transcendences, which are also sometimes called "the Special Functions of 21st century". Success in part (c) would significantly contribute to the development of the general theory of Toeplitz and Hankel determinants which, for many decades, have been playing a central role in the study of some of the fundamental models of statistical and quantum mechanics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.