Collaborative Research: Cluster Algebras Approach to Poisson-Lie Groups and Higher Genus Directed Networks

提议的研究建立在pi先前在泊松几何应用于聚类代数方面的合作基础上。在当前的项目中，我们将系统地研究在代数几何、表示理论和数学物理中具有多种重要意义的坐标环中的多个簇结构，并研究相应簇代数之间的相互作用。重要的例子包括单李群，齐次空间，点的构形空间，以及与离散和连续可积系统相关的问题。考虑的问题包括与泊松-李结构相容的简单李群上的簇结构与Belavin-Drinfeld分类、高属曲面上有向网的逆问题、平面连接模空间上有向网络的连续极限以及簇代数的生长速率分类。近年来聚类代数理论的迅速发展揭示了聚类代数与许多领域的关系，其中包括交换代数几何和非交换代数几何、颤振表示和Teichmuller理论。拟议的研究与本科和研究生课程和研究项目的发展有关。计划开展协同活动，目的是促进机构间和部门间的合作，吸引来自代表性不足和教育背景不同的群体的研究生，并通过社区外展，使高中学生尽早接触数学研究。

The proposed research builds upon PIs' previous collaboration on applications of Poisson geometry to cluster algebras. In the current project we will undertake a systematic study of multiple cluster structures in coordinate rings of a number of varieties of importance in algebraic geometry, representation theory and mathematical physics and study an interaction between corresponding cluster algebras. Important examples include simple Lie groups, homogeneous spaces, configuration spaces of points, and are related to discrete and continuous integrable systems. The problems to be considered include cluster structures on simple Lie groups compatible with Poisson-Lie structures associated with the Belavin-Drinfeld classification, inverse problems for directed nets on surfaces of higher genus, continuous limits for directed networks with applications to moduli spaces of flat connections and growth rate classification of cluster algebras.The rapid development of the cluster algebra theory in recent years revealed relations between cluster algebras and a variety of areas including, among others, commutative and non-commutative algebraic geometry, quiver representations and Teichmuller theory. The proposed research is linked to the development of undergraduate and graduate courses and research projects. Synergistic activities are planned with the goal to promote inter-institutional and inter-departmental cooperation, to attract graduate students from underrepresented groups and with diverse educational backgrounds, and, through a community outreach, at the early exposure of high school students to mathematical research.