Some Analytical Aspects of the Theory of Integrable Systems

可积系统理论的一些分析方面

基本信息

  • 批准号:
    1955265
  • 负责人:
  • 金额:
    $ 29.93万
  • 依托单位:
  • 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

项目摘要

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“可积系统”通常指的是数学对象,最常见的是微分方程,具有特殊的对称性,允许以非常详细的方式研究它们,有时甚至以封闭的形式解决它们。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。在我们的日子里,理论的可积系统已成为一个不断扩大的领域,发挥着越来越重要的作用,作为一个主要来源的新的分析和代数思想的许多分支的现代数学和理论物理。同时,它为现代非线性科学技术中出现的一些基本数学模型的研究提供了有效的分析工具。除了传统的微分方程领域,可积技术在正交多项式、弦理论、枚举拓扑、统计力学、随机过程、量子信息学和数论等不同领域变得越来越普遍。这个项目中考虑的许多问题都与这些学科有直接联系。该项目还包括一些教育活动,如培养博士生。IUPUI的学生,并共同组织了关于随机矩阵理论的普适性和可积性的国际研究/教育计划,该计划将于2021年秋季在伯克利MSRI举行。该项目延续了PI在可积系统理论方面的长期研究成果。这个研究计划的主要目标是解决各种新的分析问题的理论已出现的随机矩阵理论和理论的最近发展的可积系统的精确可解量子模型。在这个项目中,PI将专注于三个研究方向:(a)一般β-系综和Calogero-Painleve系统;(B)研究等单道τ函数,它们的渐近性,Fredholm行列式表示及其与共形场论的关系;以及(c)随机矩阵理论和统计力学中出现的Toeplitz + Hankel行列式的渐近分析。这些方向中的每一个都由一系列具体问题代表,它们将在同一分析框架内进行研究,即,黎曼-希尔伯特方法该项目(a)部分的成功将对随机矩阵理论和相互作用粒子理论以及整个现代非线性科学中普遍性和可积性的一般概念的发展产生显著影响。项目的(B)部分将进一步提高Painleve超越的“非线性特殊函数”地位,有时也被称为“21世纪世纪的特殊函数”。部分(c)的成功将极大地促进Toeplitz和Hankel行列式的一般理论的发展,几十年来,这两个行列式在统计和量子力学的一些基本模型的研究中一直发挥着核心作用。该奖项反映了NSF的法定使命,并被认为值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估来支持。

项目成果

期刊论文数量(2)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
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:
  • 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:
  • 影响因子:
    1.2
  • 作者:
    Its, Alexander;Its, Elizabeth
  • 通讯作者:
    Its, Elizabeth
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Alexander Its其他文献

О методе задачи Римана для асимптотического анализа корреляционных функций квантового нелинейного уравнения Шредингера. Случай взаимодействующих фермионов@@@On the Riemann - Hilbert approach to asymptotic analysis of the correlation functions of the quantum nonlinear Schrödinger equation: Interacti
О 关于量子 ödinger 方程非线性相关函数渐近分析的黎曼 - 希尔伯特方法:Interacti
  • DOI:
    10.4213/tmf736
  • 发表时间:
    1999
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Александр Рудольфович Итс;Alexander Its;Никита Андреевич Славнов;N. A. Slavnov
  • 通讯作者:
    N. A. Slavnov

Alexander Its的其他文献

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{{ truncateString('Alexander Its', 18)}}的其他基金

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:
    1700261
  • 财政年份:
    2017
  • 资助金额:
    $ 29.93万
  • 项目类别:
    Continuing Grant
CRM 2015 Thematic Semester: AdS/CFT, Holography, Integrability
CRM 2015年主题学期:AdS/CFT、全息、可集成性
  • 批准号:
    1513526
  • 财政年份:
    2015
  • 资助金额:
    $ 29.93万
  • 项目类别:
    Standard Grant
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:
    1361856
  • 财政年份:
    2014
  • 资助金额:
    $ 29.93万
  • 项目类别:
    Continuing Grant
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:
    1001777
  • 财政年份:
    2010
  • 资助金额:
    $ 29.93万
  • 项目类别:
    Continuing Grant
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:
    0701768
  • 财政年份:
    2007
  • 资助金额:
    $ 29.93万
  • 项目类别:
    Continuing Grant
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:
    0401009
  • 财政年份:
    2004
  • 资助金额:
    $ 29.93万
  • 项目类别:
    Continuing Grant
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:
    0099812
  • 财政年份:
    2001
  • 资助金额:
    $ 29.93万
  • 项目类别:
    Continuing Grant
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:
    9801608
  • 财政年份:
    1998
  • 资助金额:
    $ 29.93万
  • 项目类别:
    Continuing Grant
Mathematical Sciences: Some Analytical Aspects of the Theory of the Integrable Systems
数学科学:可积系统理论的一些分析方面
  • 批准号:
    9501559
  • 财政年份:
    1995
  • 资助金额:
    $ 29.93万
  • 项目类别:
    Standard Grant
Mathematical Sciences: Some Analytical Aspects of the Theory of the Integrable Systems
数学科学:可积系统理论的一些分析方面
  • 批准号:
    9315964
  • 财政年份:
    1993
  • 资助金额:
    $ 29.93万
  • 项目类别:
    Standard Grant

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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:
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    2017
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    $ 29.93万
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可积系统理论的一些分析方面
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    1361856
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    $ 29.93万
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    Continuing Grant
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可积系统理论的一些分析方面
  • 批准号:
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可积系统理论的一些分析方面
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可积系统理论的一些分析方面
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