Some Analytical Aspects of the Theory of Integrable Systems

可积系统理论的一些分析方面

• 批准号: 1955265
• 负责人: Alexander Its
• 金额: $ 29.93万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1955265&HistoricalAwards=false
• 关键词: Some; Analytical; Aspects; Theory; Integrable

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.

术语“可集成系统”通常是指具有特殊对称属性的数学对象，最常见的是微分方程，允许以非常详细的方式进行研究，有时甚至可以以封闭的形式解决它们。一类可集成的系统包括几种基本股票的自然股票，而整合系统的数学基础可以追溯到liouville，高斯和庞加雷的古典作品。在我们时代，综合系统的理论已成为一个不断扩展的领域，它是现代数学和理论物理学许多分支的新分析和代数思想的主要来源之一。同时，它为研究现代非线性科学和技术产生的一些基本数学模型提供了有效的分析工具。除了传统的微分方程领域外，在正交多项式，弦理论，枚举拓扑，统计力学，随机过程，量子信息和数字理论等潜水领域中，可集成的技术也变得普遍。该项目中考虑的许多问题与这些学科有直接的联系。该项目还包括几项教育活动，例如培训博士学位。 IUPUI的学生以及关于随机矩阵理论的国际研究/融合的国际研究/教育计划，该计划将于2021年秋季在伯克利市的MSRI举行。该项目继续PI在可集成系统理论中的长期研究工作。该研究计划的主要目标是解决可集成系统理论的各种新分析问题，这些问题来自随机矩阵理论的最新发展以及确切可解决的量子模型的理论。在该项目中，PI将集中在研究的三个方向上：（a）一般β-元素和Calogero-Painleve系统； （b）对异构果etau函数的研究，它们的渐近学，弗雷德尔姆确定的表示及其与保形场理论的关系； （c）Toeplitz + Hankel的渐近分析决定了从随机基质理论和统计力学中出现的。这些方向中的每一个都由一系列具体问题表示，并且将在同一分析框架中进行研究，即Riemann-Hilbert方法。该项目（a）部分的成功将对随机矩阵理论以及相互作用粒子以及整个现代非线性科学的相互作用理论的普遍性和整合的一般概念的发展产生显着影响。该项目的（b）部分将进一步增强潘莱夫超越的“非线性特殊功能”状态，有时也称为“ 21世纪的特殊功能”。 （c）部分中的成功将对Toeplitz和Hankel Checenters的一般理论的发展做出重大贡献，这些理论在数十年中一直在研究一些统计和量子力学的一些基本模型中发挥了核心作用。该奖项反映了NSF的法定任务，反映了通过使用基金会的智力效果和广阔的范围进行评估，并通过评估了支持。

项目成果

Asymptotics of Bordered Toeplitz Determinants and Next-to-Diagonal Ising Correlations

有界Toeplitz行列式的渐近性和邻对角Ising相关性

• DOI: 10.1007/s10955-022-02894-7
• 发表时间: 2022
• 期刊: Journal of Statistical Physics
• 影响因子: 1.6
• 作者: Basor, Estelle;Ehrhardt, Torsten;Gharakhloo, Roozbeh;Its, Alexander;Li, Yuqi
• 通讯作者: Li, Yuqi

Riemann–Hilbert approach to the elastodynamic equation: half plane

弹动力学方程的黎曼-希尔伯特方法：半平面

• DOI: 10.1007/s11005-021-01390-5
• 发表时间: 2021
• 期刊: Letters in Mathematical Physics
• 影响因子: 1.2
• 作者: Its, Alexander;Its, Elizabeth
• 通讯作者: Its, Elizabeth