Combinatorial Structures in Cluster Algebras

簇代数中的组合结构

基本信息

批准号：
2246570
负责人：
David Speyer
金额：
$ 30万
依托单位：
Regents of the University of Michigan - Ann Arbor
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2023
资助国家：
美国
起止时间：
2023-07-01 至 2026-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246570&HistoricalAwards=false
关键词：
Combinatorial Structures Cluster Algebras

项目摘要

Cluster varieties are certain highly symmetric and structured geometric spaces. Over the last twenty years, cluster varieties have turned up throughout geometry, representation theory and mathematical physics. Cluster algebras are algebraic structures which allow us to compute with cluster varieties and understand their structure. This research project pursues two lines of investigation in order to better understand cluster varieties. The project provides training opportunities for both undergraduate and graduate students.The first line of investigation is to construct cluster structures on braid varieties, Richardson varieties, and related spaces. These are geometric objects which originally arose in representation theory and have recently been discovered to play important roles in knot theory. Constructing cluster structures on these spaces will allow us to use all the techniques from cluster theory to explore these spaces. The second line of investigation is to develop relationships between cluster varieties and Coxeter groups, which describe the symmetries of collections of reflecting mirrors. This has been extensively done for cluster algebras "of finite type", which correspond to Coxeter groups with finitely many reflections, but an analogous story should exist for all cluster algebras, and this project proposes to develop it.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

集群品种是某些高度对称和结构化的几何空间。在过去的二十年中，群集品种在整个几何形状，表示理论和数学物理学中都出现了。群集代数是代数结构，使我们能够使用群集品种计算并了解其结构。该研究项目进行了两条调查，以便更好地了解集群品种。该项目为本科生和研究生提供了培训机会。第一线的调查是在编织品种，理查森品种和相关空间上构建集群结构。这些是最初在表示理论中出现的几何对象，最近被发现在结理论中起重要作用。在这些空间上构建群集结构将使我们能够使用聚类理论中的所有技术来探索这些空间。第二行调查是在集群品种和Coxeter组之间建立关系，这些关系描述了反射镜的集合的对称性。这是针对“有限类型”集群代数的广泛做到的，这与具有有限的反射的Coxeter组相对应，但是对于所有集群代数，应该存在一个类似的故事，并且该项目提议开发它。该奖项反映了NSF的法规使命，并认为通过基金会的知识优点和广泛的评论，它值得通过评估来进行评估。