Combinatorics of Cluster Varieties

簇簇组合学

• 批准号: 1600223
• 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: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600223&HistoricalAwards=false
• 关键词: Combinatorics; Cluster; Varieties

Cluster algebras are a fundamental mathematical structure employed to describe highly symmetric mathematical objects. In the last decade, cluster structures have been constructed on spaces throughout representation theory and mathematical physics, most recently in the theory of scattering amplitudes in high-energy particle physics. This project comprises fundamental mathematical research into the combinatorial structure of several crucial cluster algebras, and the geometric spaces related to them. This research project addresses several questions concerning the combinatorial structure of various cluster algebras. Most of these cluster algebras come from natural spaces in algebraic geometry, such as open positroid varieties, which occur in stratifying the Grassmannian and generalized Teichmuller spaces. The first part of the project investigates the coefficients of the Laurent polynomials that occur in changing from one cluster to another in the Grassmannian (and other related spaces). The second part of the project aims to determine the relation between the parametrizations of positroid cells by dimer configurations on tori and the values of Plucker coordinates as charts on those cells. The third part of the project investigates the cohomology of cluster varieties, trying to find presentations for the cohomology ring that are as explicit and combinatorial as the presentations already known for the coordinate rings.

簇代数是用于描述高度对称数学对象的基本数学结构。在过去的十年中，簇结构已经在整个表示理论和数学物理的空间上构建，最近是在高能粒子物理的散射振幅理论中。该项目包括对几个关键簇代数的组合结构以及与之相关的几何空间的基础数学研究。该研究项目解决了有关各种簇代数的组合结构的几个问题。这些簇代数大多数来自代数几何中的自然空间，例如开正类簇，它出现在格拉斯曼空间和广义泰希米勒空间的分层中。该项目的第一部分研究了格拉斯曼空间（和其他相关空间）中从一个簇到另一个簇变化时发生的洛朗多项式的系数。该项目的第二部分旨在确定圆环上二聚体配置的正样细胞参数化与这些细胞上图表的 Plucker 坐标值之间的关系。该项目的第三部分研究簇簇的上同调，试图找到与已知的坐标环表示一样明确和组合的上同调环表示。