Collaborative Research: Generalized Cluster Structures of Geometric Type

这个项目位于代数的一个领域，它正在发展，以期应用于数学和理论物理。重点是2001年发明的一种称为“簇代数”的结构类型，这种结构已被证明在包括组合学、几何和高能物理在内的广泛且仍在扩展的数学学科中很有用。这个项目自然会带动本科和研究生阶段的课程和研究项目的发展。计划的活动将发展新的机构间和跨学科合作，包括会议、讲习班和系列研讨会。将特别注意从代表性不足的群体和具有不同教育背景的群体中招收研究生，并通过社区外展吸引高中学生从事数学研究。他们将继续研究泊松几何在聚类代数理论中的应用。项目的主要目标包括构建和研究(i)与Belavin-Drinfeld分类描述的泊松-李括号相容的简单李群上的奇异广义簇结构；（ii）单泊松-李群的Drinfeld对偶和泊松-李对偶上的广义团簇结构；(3)由高属网的聚类变换和初等变换序列产生的离散可积系统；有限突变型颤振。

This project lies in an area of algebra that is being developed with a view to applications in mathematical and theoretical physics. The emphasis is on a type of structure called "cluster algebras" invented in 2001 that have since proven to be useful in a wide and still expanding range of mathematical subjects including combinatorics, geometry, and high energy physics. This project will naturally lead to the development of courses and research projects on both undergraduate and graduate levels. Planned activities will develop new inter-institutional and cross-disciplinary collaborations include conferences, workshops and seminar series. Particular attention will be paid to recruitment of graduate students from underrepresented groups and with diverse educational backgrounds, and, through community outreach, to attracting high school students to mathematical research.The PIs will continue their work on applications of Poisson Geometry to the theory of cluster algebras. The main goals of the project include construction and study of (i) exotic generalized cluster structures on simple Lie groups compatible with Poisson-Lie brackets described by the Belavin-Drinfeld classification; (ii) generalized cluster structures on the Drinfeld double and the Poisson-Lie dual of a simple Poisson-Lie group; (iii) discrete integrable systems arising as sequences of cluster transformations and elementary transformations of higher genus nets; and (iv) quivers of finite mutation type.