Collaborative Research: Generalized Cluster Structures of Geometric Type

这个项目是一个正在开发的代数领域，以期在数学和理论物理中应用。重点是2001年发明的一种被称为“簇代数”的结构，自那以来，它在包括组合数学、几何和高能物理在内的广泛且仍在不断扩大的数学学科中被证明是有用的。这个项目自然会导致本科生和研究生水平的课程和研究项目的发展。计划开展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。将特别注意从代表不足的群体和具有不同教育背景的学生中招收研究生，并通过社区推广吸引高中生从事数学研究。私人投资机构将继续他们在泊松几何应用于簇代数理论方面的工作。该项目的主要目标包括：(I)构造和研究(I)与Belavin-Drinfeld分类所描述的Poisson-Lie括号相容的单李群上的奇异广义簇结构；(Ii)单Poisson-Lie群的Drinfeld对偶和Poisson-Lie对偶上的广义簇结构；(Iii)作为簇变换序列和更高亏格网的初等变换序列出现的离散可积系统；以及(Iv)有限突变型箭图。

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.