CAREER: Algebraic Combinatorics and URE

职业：代数组合和 URE

• 批准号: 1351590
• 负责人: Pavlo Pylyavskyy
• 金额: $ 40万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-04-01 至 2019-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1351590&HistoricalAwards=false
• 关键词: CAREER; Algebraic; Combinatorics; URE

The PI intends to study a model for cluster algebras in terms of certain tensor diagrams called webs. This model relates to higher Techmuller theory, specifically to representations of fundamental groups of Riemann surfaces inside special linear group of a three dimensional vector space. This class of cluster algebras allows one to have a conjectured purely combinatorial model for some cluster algebras of infinite mutation type. The PI also intends to study a generalization of cluster algebras called Laurent phenomenon algebras. In it, the exchange polynomials do not necessarily have to be binomial. The examples of such structures appear for example in the study of electrical networks, and in the study of hyperbolic metrics on non-orientable surfaces.The proposed project is rooted in two areas of mathematics: representation theory and combinatorics. Representation theory is an important area that is fundamental for the study of quantum mechanics and particle physics. Cluster algebras provide a new approach to representation theory and related disciplines, and have found their application in string theory and quantum gravity. The PI's research focuses on cluster algebras that lie just beyond the current frontier of our understanding (the so called infinite mutation type). Another part of the proposed research focuses on a generalization of cluster algebras, called Laurent phenomenon algebras. The educational part of the proposal pertains to creating a summer undergraduate research program (URE) for international students which would run parallel to the already existing summer REU in combinatorics at University of Minnesota, Minneapolis.

PI打算根据称为网的某些张量图研究集群代数的模型。该模型涉及更高的TechMuller理论，特别是在三维矢量空间的特殊线性组内的基本riemann表面基本组的表示。这类群集代数允许人们具有一个纯粹的组合模型，用于某些无限突变类型的群集代数。 PI还打算研究称为月桂现象代数的群集代数的概括。在其中，交换多项式不一定必须是二项式。这些结构的示例例如在电网络的研究中以及在不可取向表面上双曲线指标的研究中出现。拟议的项目植根于数学的两个领域：表示理论和组合学。代表理论是一个重要的领域，是研究量子力学和粒子物理学的基础。集群代数为表示理论和相关学科提供了一种新的方法，并在弦理论和量子重力中找到了它们的应用。 PI的研究集中于群集代数，该群集位于我们当前理解的前沿（所谓的无限突变类型）之外。拟议的研究的另一部分侧重于群集代数的概括，称为月桂树现象代数。该提案的教育部分涉及为国际学生创建夏季本科研究计划（URE），这将与明尼阿波利斯明尼苏达大学的联合会现有的夏季REU平行。