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.

群集代数是一种基本的数学结构，用于描述高度对称的数学对象。在过去的十年中，群集结构在整个代表理论和数学物理学的空间上建立，最近一次是在高能粒子物理学中散射幅度的理论中。该项目包括对几个关键集群代数的组合结构以及与之相关的几何空间的基本数学研究。该研究项目解决了有关各个集群代数的组合结构的几个问题。这些群集代数中的大多数来自代数几何形状中的自然空间，例如开放式阳性品种，它们发生在分层和广义的teichmuller空间中。该项目的第一部分研究了从一个从一个群集变为格拉斯曼尼亚（及其他相关空间）的laurent多项式的系数。该项目的第二部分旨在通过Tori上的二聚体构型和Plucker坐标的值作为这些细胞的图表来确定阳性细胞的参数化之间的关系。该项目的第三部分调查了集群品种的共同体，试图找到与坐标环已经知道的表现一样明确和组合的同种学环的演示文稿。