CAREER: Algebraic Combinatorics and URE

PI打算研究一个集群代数的模型，在某些张量图称为网络。这个模型涉及到更高的Techmuller理论，特别是在三维向量空间的特殊线性群内的黎曼曲面的基本群的表示。这类簇代数允许人们对某些无限突变型的簇代数有一个确定的纯组合模型。 PI还打算研究一种称为Laurent现象代数的簇代数的推广。其中，交换多项式不一定必须是二项式。这种结构的例子出现在例如电网络的研究中，以及在不可定向曲面上的双曲度量的研究中。拟议的项目植根于两个数学领域：表示论和组合学。表示论是一个重要的领域，是量子力学和粒子物理研究的基础。簇代数为表示论和相关学科提供了一种新的方法，并在弦理论和量子引力中得到了应用。PI的研究重点是簇代数，它超出了我们目前的理解范围（所谓的无限突变类型）。另一部分研究集中在簇代数的推广上，称为Laurent现象代数。该提案的教育部分涉及为国际学生创建一个夏季本科研究计划（URE），该计划将与明尼苏达大学明尼阿波利斯分校现有的夏季REU组合学平行。

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.