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.

簇簇是某些高度对称且结构化的几何空间。在过去的二十年里，簇的变种出现在几何学、表示论和数学物理中。簇代数是代数结构，它使我们能够计算簇簇并理解它们的结构。该研究项目进行两条线的调查，以便更好地了解集群品种。该项目为本科生和研究生提供培训机会。研究的第一线是构建辫子品种、理查森品种和相关空间的簇结构。这些几何对象最初出现在表示论中，最近被发现在纽结理论中发挥着重要作用。在这些空间上构建簇结构将使我们能够使用簇理论中的所有技术来探索这些空间。研究的第二条线是发展簇簇和考克塞特群之间的关系，它描述了反射镜集合的对称性。这已经在“有限类型”簇代数中得到了广泛的应用，它对应于具有有限多次反射的 Coxeter 群，但是所有簇代数都应该存在类似的故事，并且该项目建议开发它。该奖项反映了 NSF 的法定使命和通过使用基金会的智力优点和更广泛的影响审查标准进行评估，该项目被认为值得支持。