Cluster Algebras, Combinatorics, and Knot Theory

聚类代数理论是一个年轻的数学研究领域，由佛明和泽列文斯基于2002年提出。他们最初的动机来自表征理论，这是现代代数的一个分支，它研究代数结构的对称性，而不是直接研究结构。表征理论在物理和化学以及其他数学领域有许多应用。聚类代数为整个表示理论中的基本模式提供了一个数学框架。令人惊讶的是，这些模式也在与表征理论无关的其他科学分支中被观察到。该项目将增强对簇代数的理解，并在一个已经非常活跃的研究领域提供新的研究方向。研究者将建立和发展簇代数与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发展。该项目还将调查长期悬而未决的问题。该项目将有研究生参与拟议的研究。该研究项目的重点是簇代数及其与组合学、结论、数论和有限维代数的表示理论的关系。调查员将追求几个目标。他将致力于为任意类型的聚类代数的生成器建立显式公式。他还将发展簇代数和结理论之间的基本联系，这将实现重要的结不变量作为专门的簇变量。此外，他将在一类有限维代数上创建极大Cohen-Macauley模范畴的组合模型，这些代数自然出现在簇代数的加性范畴中，作为簇的自同态代数。他还将从一个新颖的角度研究马尔可夫数，利用一次穿孔环面的聚类代数。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.