Collaborative Research: Cluster Algebras, Canonical Bases and Nets on Surfaces of Higher Genus

合作研究：簇代数、规范基和更高属面上的网络

• 批准号: 0801204
• 负责人: Michael Gekhtman
• 金额: $ 12万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-15 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0801204&HistoricalAwards=false
• 关键词: Collaborative; Research; Cluster; Algebras; Canonical

This project explores links between classical combinatorics, modern theory of Teichmueller spaces, real algebraic geometry and total positivity, and the rapidly developing theory of cluster algebras. In particlular, PIs utilize the link between decorated Teichmueller spaces and the algebra of geodesics on the one hand and the theory of cluster algebras on the other hand to investigate the structure of the dual canonical basis of a cluster algebra. Furthermore, they use cluster algebra point of view to study directed nets on surfaces and describe compatible Poisson-Lie structures for nets and solutions of corresponding inverse problems and to study associated integrable hierarchies. The latter will be applied to investigate new relations between double Hurwitz numbers of coverings of the sphere by higher genus curves and, on the other hand, to analyze a new two- and multi-matrix models and associated biorthogonal polynomials and apply them to problems of enumeration of bicolored embedded graphs.Space discretization using networks on surfaces is an important in the theory of random processes, in mathematical physics, including 2D gravity, the theory of electric potential and especially theory of electrical networks and in other fields.Combinatorial properties of surface networks capture crucial features of complex mathematical and physical structures.Recently it was observed that surface networks also exhibit many features that are typical for cluster algebra structures.Introduced only a few years ago by Fomin and Zelevinsky, the cluster algebra formalism is proved to be widely applicable in investigation of algebraic and geometric objects with symmetries often associated with important physical systems. Interplay between the two concepts will be instrumental in the study of combinatorial quantities and geometric phenomena of physical relevance and classical and quantum exactly solvable models.

该项目探讨了古典组合学，Teichmueller空间的现代理论，实际代数几何学和总阳性以及群集代数的快速发展理论之间的联系。在特定的情况下，PIS一方面利用了装饰的Teichmueller空间与大地测量代数之间的联系，另一方面，群集代数理论来研究集群代数的双重规范基础的结构。此外，他们使用群集代数的观点来研究表面上的定向网，并描述相应的逆问题的网和解决方案的兼容泊松结构，并研究相关的可集成层次结构。后者将应用于通过较高的属曲线来调查球体覆盖率的新关系，另一方面，分析了新的两种和多矩阵模型以及相关的生物双歧式多项式，并将其应用于双色嵌入式图的枚举。在表面上使用网络的空间离散化在随机过程的理论中，在数学物理学的理论中很重要，包括2D重力，电势理论，电位理论，尤其是电网和其他领域的理论。网络捕获了复杂数学和物理结构的关键特征。很明显，表面网络还表现出许多对于群集代数结构而言是典型的特征。几年前，Fomin和Zelevinsky仅引入了群集的形式，被证明是广泛的。适用于研究具有与重要物理系统相关的对称性的代数和几何对象。这两个概念之间的相互作用将在组合数量和物理相关性以及经典和量子恰好可解决模型的几何现象的研究中发挥作用。