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

这个项目探讨了经典组合学，现代Teichmueller空间理论，真实的代数几何和全正性，以及快速发展的簇代数理论之间的联系。特别地，PI一方面利用修饰Teichmueller空间与测地线代数之间的联系，另一方面利用簇代数理论来研究簇代数的对偶标准基的结构.此外，他们使用簇代数的观点来研究曲面上的有向网，描述网的相容Poisson-Lie结构和相应的反问题的解，并研究相关的可积族。后者将被应用于研究球面上高亏格曲线覆盖的双Hurwitz数之间的新关系，另一方面，将被应用于分析一种新的二矩阵和多矩阵模型以及相关的双正交多项式，并将其应用于双色嵌入图的计数问题。包括二维引力，电势理论，特别是电网络理论和其他领域。表面网络的组合性质捕获了复杂数学和物理结构的关键特征。最近观察到表面网络也表现出许多典型的簇代数结构的特征。仅在几年前由Fomin和Zelevinsky介绍，簇代数形式主义被证明是广泛适用于研究代数和几何对象的对称性往往与重要的物理系统。这两个概念之间的相互作用将有助于研究组合量和物理相关的几何现象以及经典和量子精确可解模型。

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.