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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

基本信息

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

项目摘要

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.

“可积系统”这个术语通常指的是数学对象，最常见的是微分方程，它们具有特殊的对称性，可以用非常详细的方式研究它们，有时甚至可以用封闭形式求解它们。可积系统包括自然界的几个基本方程，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。目前，可积系统理论已成为一个不断扩大的领域，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领域外，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也变得越来越普遍。项目中考虑的许多问题与这些学科有直接的联系。本研究项目延续了首席研究员在可积系统理论方面的长期研究努力。该项目的主要目标是研究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以下两个方向：(a)研究等同构tau函数及其渐近性、Fredholm行列式表示及其与共形场论的关系；(b)随机矩阵和统计力学中临界现象研究中出现的Toeplitz、Hankel和Fredholm行列式的渐近分析。这些方向中的每一个都由一组具体问题来表示，它们将在相同的分析框架内进行研究，即黎曼-希尔伯特方法。第一个方向的主要焦点是对Painleve函数一般族的渐近连接公式（包括常数前因子）的评估这一长期存在的问题。第二个方向的主要关注点是各种类型的双标度和过渡极限-这些主题是随机矩阵理论和统计力学中的主要问题之一。

项目成果

期刊论文数量（4）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

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
通讯作者：
A. Its;O. Lisovyy;A. Prokhorov

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
影响因子：
0
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.

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