Cluster Algebras, Combinatorics, and Knot Theory

簇代数理论是数学中一个年轻的研究领域，由Fomin和Zelevinsky于2002年提出。他们最初的动机来自表示论，现代代数的一个分支，它考察代数结构的对称性，而不是直接考察结构。表示论在物理学、化学以及其他数学领域有着广泛的应用。簇代数为整个表示论中出现的基本模式提供了一个数学框架。令人惊讶的是，这些模式也在其他科学分支中观察到，先验地，与表征理论无关。这个项目将加强对簇代数的理解，并在一个已经非常活跃的研究领域提供新的研究方向。研究人员将建立和发展集群代数和其他数学领域之间的关系。这些新的连接将允许明确的计算结果以及结构的发展。该项目还将调查长期悬而未决的问题。本计画将邀请研究生参与研究，研究重点为丛集代数及其与组合学、纽结理论、数论及有限维代数表示理论的关系。调查员将追求几个目标。他将致力于建立明确的公式发电机的集群代数的任意类型。他还将开发一个基本的集群代数和结理论之间的联系，这将实现重要的结不变量作为专门的集群变量。此外，他将创建一个组合模型的范畴最大科恩-Macauley模一类有限维代数自然出现在添加剂的聚类代数的自同态代数集群。他还将使用一次穿孔环面的簇代数从一个新的角度研究马尔可夫数。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

The theory of cluster algebras is a young research area in mathematics that was set in motion by Fomin and Zelevinsky in 2002. Their original motivation came from representation theory, a branch of modern algebra, which examines the symmetries of an algebraic structure rather than examining the structure directly. Representation theory has numerous applications in physics and chemistry as well as other mathematical fields. The cluster algebras provide a mathematical framework for fundamental patterns that occur throughout representation theory. Surprisingly, these patterns are also observed in various other branches of science which, a priori, are not related to representation theory. This project will enhance the understanding of cluster algebras and provide new research directions in an already highly active research area. The investigator will establish and develop relations between cluster algebras and other areas of mathematics. These new connections will allow explicit computational results as well as structural development. The project will also investigate longstanding open questions. The project will involve graduate students in the proposed research.The research project focuses on cluster algebras and their relation to combinatorics, knot theory, number theory, and representation theory of finite-dimensional algebras. The investigator will pursue several objectives. He will work on establishing explicit formulas for the generators of a cluster algebra of arbitrary type. He will also develop a fundamental connection between cluster algebras and knot theory which will realize important knot invariants as specialized cluster variables. Furthermore, he will create a combinatorial model for the category of maximal Cohen-Macauley modules over a class of finite-dimensional algebras that arise naturally in additive categorifications of cluster algebras as the endomorphism algebras of clusters. He will also study Markov numbers from a novel point of view using the cluster algebra of the once-punctured torus.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.