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

这项研究建立在PI先前在泊松几何应用于簇代数方面的合作基础上。在本项目中，我们将系统地研究在代数几何、表示论和数学物理中具有重要意义的若干种坐标环上的多重团簇结构，并研究相应的簇代数之间的相互作用。重要的例子包括单李群，齐次空间，点的配置空间，并与离散和连续可积系统有关。要考虑的问题包括单李群上的团结构，与Belavin-Drinfeld分类有关的Poisson-Lie结构，高亏格曲面上有向网的反问题，有向网络的连续极限及其在平坦连通的模空间中的应用，以及簇代数的增长率分类。近年来簇代数理论的快速发展揭示了簇代数与许多领域之间的关系，其中包括交换和非交换代数几何，箭图表示和Teichmuller理论。拟议的研究与本科生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合作，吸引来自代表性不足群体和具有不同教育背景的研究生，并通过社区外联，使高中生在早期接触数学研究。

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.