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

本项目探讨经典组合学、现代Teichmueller空间理论、实代数几何和全正性，以及快速发展的簇代数理论之间的联系。特别是，pi一方面利用修饰的Teichmueller空间与测地线代数之间的联系，另一方面利用聚类代数理论来研究聚类代数的对偶正则基的结构。此外，他们利用聚类代数的观点研究了曲面上的有向网络，描述了网络的相容泊松-李结构和相应逆问题的解，并研究了相关的可积层次。后者将应用于研究高属曲线覆盖球面的双Hurwitz数之间的新关系，另一方面，分析一种新的双矩阵和多矩阵模型及其相关的双正交多项式，并将其应用于双色嵌入图的枚举问题。在随机过程理论、数学物理（包括二维重力）、电势理论特别是电网络理论以及其他领域中，利用曲面上的网络进行空间离散是一个重要的研究方向。表面网络的组合特性捕捉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聚类代数结构的特征。在几年前由Fomin和Zelevinsky引入的聚类代数形式被证明广泛适用于研究具有对称性的代数和几何对象，这些对象通常与重要的物理系统有关。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.