CAREER: Algebraic Combinatorics and URE

PI的目的是研究一种基于特定张量图的簇代数模型，称为网络。这个模型涉及到高等Techmuller理论，特别是关于三维向量空间的特殊线性群内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类型的簇代数有一个猜想的纯组合模型。PI还打算研究一个称为Laurent现象代数的簇代数的推广。其中，交换多项式不一定是二项式的。例如，这种结构的例子出现在电网络的研究中，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。表象理论是量子力学和粒子物理研究的重要领域。簇代数为表示论及相关学科提供了一种新的途径，并在弦论和量子引力中得到了应用。PI的研究重点是位于我们目前理解的前沿之外的簇代数(所谓的无限突变类型)。研究的另一部分重点是簇代数的推广，称为Laurent现象代数。该提案的教育部分涉及为国际学生创建暑期本科生研究项目(URE)，该项目将与明尼苏达大学明尼阿波利斯分校现有的组合数学暑期本科研究项目并行。

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.