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
• 负责人: Alexander Its
• 金额: $ 18.08万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-08-01 至 2020-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700261&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 that allow them to be studied 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. Currently, 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 the project have direct connections with these disciplines. This research project continuous the principal investigator's long term research efforts in the theory of integrable systems. The principal goal of the project is to address the following two directions which have emerged from recent developments in random matrix theory and in the theory of exactly solvable quantum models: (a) The study of the isomonodromic tau functions, their asymptotics, Fredholm determinant representations and their relations to the conformal field theory; (b) The asymptotic analysis of Toeplitz, Hankel, and Fredholm determinants arising in the study of critical phenomena in random matrices 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. The main focus of the first direction is on the long standing question of evaluation the asymptotic connection formulae, including the constant pre-factors, for the generic families of Painleve tau functions. The principal concern of the second direction is various types of double scaling and transitional limits - topics which are among the principal questions in both the theory of random matrices and in statistical mechanics.

术语“集成系统”通常是指数学对象，通常是微分方程，具有特殊的对称属性，可以使它们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自然的几个基本方程，并且可集成系统的数学基础可以追溯到Liouville，Gauss和Poincare的古典作品。 当前，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，列举拓扑，统计力学，随机过程，量子信息和数理论等不同领域中，可集成的技术还变得普遍。 项目中考虑的许多问题与这些学科有直接的联系。该研究项目持续了主要研究者在可整合系统理论中的长期研究工作。该项目的主要目的是解决以下两个方向，这些方向是从随机矩阵理论的最新发展以及确切可解决的量子模型理论中出现的：（a）对等肌无质量tau函数的研究，它们的渐近学，弗雷德霍尔姆的决定性及其与正直现场理论的关系； （b）在随机矩阵和统计力学中对关键现象的研究中产生的Toeplitz，Hankel和Fredholm的渐近分析。 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。 第一个方向的主要重点是评估的长期存在问题，包括painleve tau功能的通用家族，包括恒定的prefters公式，包括恒定的前对应器。第二个方向的主要问题是各种各样的双重缩放和过渡限制 - 这些主题是随机矩阵和统计力学理论中的主要问题之一。

项目成果

Monodromy dependence and connection formulae for isomonodromic tau functions

• DOI: 10.1215/00127094-2017-0055
• 发表时间: 2016-04
• 期刊: Duke Mathematical Journal
• 影响因子: 2.5
• 作者: A. Its;O. Lisovyy;A. Prokhorov

A Riemann-Hilbert approach to asymptotic analysis of Toeplitz+Hankel determinants

Toeplitz Hankel 行列式渐近分析的黎曼-希尔伯特方法

• DOI: 10.3842/sigma.2020.100
• 发表时间: 2020
• 期刊: Symmetry integrability and geometry methods and applications
• 作者: Gharakhloo, R;Its, A
• 通讯作者: Its, A

Entanglement entropy of two disjoint intervals separated by one spin in a chain of free fermion*

自由费米子链中由一个自旋分隔的两个不相交区间的纠缠熵*

• DOI: 10.1088/1751-8121/ab9cf2
• 发表时间: 2020
• 期刊: Journal of Physics A: Mathematical and Theoretical
• 作者: Brightmore, L.;Gehér, G. P.;Its, A. R.;Korepin, V. E.;Mezzadri, F.;Mo, M. Y.;Virtanen, J. A.
• 通讯作者: Virtanen, J. A.