Collaborative Research: Generalized Cluster Structures of Geometric Type

合作研究：几何类型的广义簇结构

• 批准号: 1702115
• 负责人: Michael Shapiro
• 金额: $ 25万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-06-01 至 2022-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1702115&HistoricalAwards=false
• 关键词: Collaborative; Research; Generalized; Cluster; Structures

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.

该项目属于正在开发的代数领域，旨在数学和理论物理的应用。重点是 2001 年发明的一种称为“簇代数”的结构，该结构已被证明在广泛且仍在扩展的数学学科中非常有用，包括组合学、几何学和高能物理。该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开发。计划的活动将发展新的机构间和跨学科合作，包括会议、讲习班和研讨会系列。我们将特别关注从代表性不足的群体中招收具有不同教育背景的研究生，并通过社区外展吸引高中生参与数学研究。PI 将继续开展泊松几何在聚类理论中的应用工作代数。该项目的主要目标包括构建和研究（i）与 Belavin-Drinfeld 分类描述的 Poisson-Lie 括号兼容的简单李群上的奇异广义簇结构； (ii) 简单泊松-李群的德林菲尔德对偶和泊松-李对偶的广义簇结构； (iii) 作为更高属网的簇变换和初等变换序列而产生的离散可积系统； (iv) 有限突变型箭袋。

项目成果

Generalized cluster structures related to the Drinfeld double of ??_?

与 ??_? 的 Drinfeld 双相关的广义簇结构

• 发表时间: 2021
• 期刊: Journal of the London Mathematical Society
• 作者: Gekhtman, M.;Shapiro, M.;Vainshtein, A.
• 通讯作者: Vainshtein, A.

Secant Degeneracy Index of the Standard Strata in The Space of Binary Forms

二元形式空间中标准地层的正割简并指数

• DOI: 10.1007/s40598-017-0077-7
• 发表时间: 2017
• 期刊: Arnold Mathematical Journal
• 作者: Nenashev, Gleb;Shapiro, Boris;Shapiro, Michael
• 通讯作者: Shapiro, Michael

Log-canonical coordinates for symplectic groupoid and cluster algebras

辛群群和簇代数的对数正则坐标

• DOI: 10.1093/imrn/rnac101
• 发表时间: 2022
• 期刊: International mathematics research notices
• 影响因子: 1
• 作者: Chekhov, L.;Shapiro, M.
• 通讯作者: Shapiro, M.

Periodic Staircase Matrices and Generalized Cluster Structures

• DOI: 10.1093/imrn/rnaa148
• 发表时间: 2019-12
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: M. Gekhtman;M. Shapiro;A. Vainshtein

Upper cluster algebras and choice of ground ring

• DOI: 10.1007/s11425-018-9486-6
• 发表时间: 2018-02
• 期刊: Science China Mathematics
• 作者: Eric Bucher;John M. Machacek;M. Shapiro