Collaborative Research: Generalized Cluster Structures of Geometric Type

合作研究：几何类型的广义簇结构

基本信息

批准号：
1702054
负责人：
Michael Gekhtman
金额：
$ 30万
依托单位：
University of Notre Dame
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=1702054&HistoricalAwards=false
关键词：
Collaborative Research Generalized Cluster Structures

项目摘要

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.

该项目位于一个代数的区域，该区域正在开发，以数学和理论物理学的应用。重点放在一种称为“集群代数”的结构上，该结构已被证明在广泛而仍在扩展的数学学科范围内很有用，包括组合剂，几何学和高能量物理学。该项目自然会导致有关本科和研究生级别的课程和研究项目的发展。计划活动将开发新的机构间和跨学科合作，包括会议，研讨会和研讨会系列。将特别关注来自代表性不足的群体，具有多种教育背景的研究生，并通过社区宣传，吸引高中生从事数学研究。该项目的主要目标包括对（i）与Belavin-Drinfeld分类所描述的Poisson-lie括号兼容的简单谎言组上的（i）构造和研究；（ii）Drinfeld Double上的广义集群结构和一个简单的泊松型组的泊松式二重奏；（iii）作为集群变换的序列和较高属网的基本变换而产生的离散集成系统；（iv）有限突变类型的颤动。