Isomonodromy Transformations of Difference Equations

差分方程的等单变换

• 批准号: 0402047
• 负责人: Alexei Borodin
• 金额: $ 15万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2007-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0402047&HistoricalAwards=false
• 关键词: Isomonodromy; Transformations; Difference; Equations

ABSTRACTThe project has two major parts. First, we take a rathernon-obvious definition of "monodromy" for difference equations dueto Birkhoff and aim to develop a general theory of transformationsof difference equations which preserve the monodromy. Theresulting structure bears some similarity to the classical theoryof isomonodromy deformations of differential equations anddegenerates to it in a certain limit. Second, we apply theisomonodromy transformations of difference equations to evaluationof the so-called gap probabilities in discrete probabilisticmodels of random-matrix type which arise in many different domainsincluding combinatorics, representation theory, percolationtheory, tiling models, etc.The theory of isomonodromy deformations of ordinary differentialequations with rational coefficients is a classical subjectdeveloped in the end of the XIXth -- beginning of the XXth centuryby Riemann, Schlesinger, Fuchs, and Garnier. Since then theisomonodromy deformations have found numerous applications in verydifferent domains of mathematics and mathematical physics, fromalgebraic and differential geometry to random matrices andrepresentation theory. On the other hand, in recent years therehas been considerable interest in analyzing a certain class ofdiscrete probabilistic models which in appropriate limits convergeto well-known models of random matrix theory. The sources of thesemodels are quite diverse, they include combinatorics,representation theory, percolation theory, random growthprocesses, tiling models and others. The goal of the researchpresented in this proposal is to develop a general theory of"isomonodromy transformations" for linear systems of DIFFERENCEequations with rational coefficients. In these probabilisticmodels mentioned above, we often see how two problems which seemto be unrelated to each other, and which even come from parts ofmathematics that do not seem to have any overlap, lead to the samefinal result. We believe that the project will provide a visiblecommon ground for many of such coincidences and thus will promoteand accelerate the exchange of methods and ideas accumulated bydifferent groups of researchers.

本项目分为两个主要部分。首先，我们采用Birkhoff对差分方程的“monodromy”的一个非显而易见的定义，旨在发展一个保持monodromy的差分方程变换的一般理论。这种结构与经典的微分方程的等单点形变理论有相似之处，并在一定的极限下退化为等单点形变理论。第二，我们应用差分方程的isomonodromy变换来评估随机矩阵类型的离散概率模型中所谓的间隙概率，这些模型出现在许多不同的领域，包括组合学，表示论，简化理论，平铺模型，有理系数常微分方程的等距变形理论是十九世纪末发展起来的一个经典课题，二十世纪初由黎曼，施莱辛格，富克斯和卡尼尔。从那以后，等单值形变在数学和数学物理的许多不同领域都得到了广泛的应用，从几何学和微分几何到随机矩阵和表示论。另一方面，近年来，人们对分析一类离散概率模型产生了相当大的兴趣，这些模型在适当的限制下收敛于随机矩阵理论的著名模型。这些模型的来源是相当多样化的，它们包括组合数学，表示理论，渗流理论，随机增长过程，平铺模型和其他。本文提出的研究目标是发展有理系数差分方程线性系统的“isomonodromy变换”的一般理论。在上面提到的这些概率模型中，我们经常看到两个看似不相关的问题，甚至来自似乎没有任何重叠的数学部分，如何导致相同的最终结果。我们相信，该项目将为许多这样的巧合提供一个共同的基础，从而将促进和加速不同研究小组积累的方法和思想的交流。