Cluster Algebras, Quantum Groups, and Decorated Character Varieties

这个研究项目处于代数和几何的交叉点。一个特别的焦点是关于聚类代数和模空间的理论。聚类代数是由Fomin和Zelevinsky在2001年发现的，是一类与整数偏对称矩阵相关的代数。自成立以来，聚类理论的快速发展与数学和物理的许多不同领域产生了巨大的相互作用，包括表示理论、结理论和高能物理。模空间是一种几何空间，用于对某些固定形状的物体或物理中特定系统的解进行分类。该项目将探索模空间和簇代数之间的联系，以进一步理解它们，并在不同领域之间建立相似之处。此外，PI将指导本科生研究项目，培训研究生，并支持研究研讨会。本课题将研究在簇代数框架下模空间的量子几何。它还探索了装饰角色品种、量子组和Legendrian结之间的丰富联系，在各个方向上都找到了新的结果。更详细地说，该项目将涉及四个主题。(1)将提供一个实现量子群的刚性簇模型，从簇代数的角度对量子群的许多性质进行新的解释。(2)研究量化纹饰品种的自然基，包括具体的网基图解构建。(3)探讨了Legendrian结的精确拉格朗日填充与其增广变种的簇种子之间的内在对应关系。作为一个应用，它将解决超越正编织的Legendrian结的无限填充问题。(4)利用Legendrian knot理论的工具引入广义Richardson变异上的簇结构，证实了Leclerc关于开放Richardson变异簇性质的一个猜想。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 research project lies at the crossroads of algebra and geometry. A particular focus is on the theory of cluster algebras and moduli spaces. Cluster algebras, discovered by Fomin and Zelevinsky in 2001, are a class of algebras associated with integer skew-symmetric matrices. Since its inception, the rapid developments of cluster theory have found tremendous interactions with many different areas of mathematics and physics, including representation theory, knot theory, and high energy physics. Moduli spaces are geometric spaces that classify objects of some fixed shapes or solutions to specific systems in physics. This project will explore the connections between moduli spaces and cluster algebras to further their understanding and build parallels between the different areas. In addition, the PI will mentor undergraduate research projects, train graduate students, and support a research seminar. This project will investigate the quantum geometry of moduli spaces under the framework of cluster algebras. It also explores the fruitful connections between decorated character varieties, quantum groups, and Legendrian knots, finding new results in all directions. In more detail, the project will touch on four topics. (1) It will provide a rigid cluster model realizing quantum groups, obtaining new interpretations of many properties of quantum groups from the perspective of cluster algebras. (2) It will study the natural bases of the quantized decorated character varieties, including a concrete diagrammatic construction of web bases. (3) It will explore an intrinsic correspondence between the exact Lagrangian fillings of Legendrian knots and the cluster seeds of their augmentation varieties. As an application, it will solve the infinite-filling problem for Legendrian knots beyond positive braids. (4) It will use tools from Legendrian knot theory to introduce a cluster structure on generalized Richardson varieties, confirming a conjecture of Leclerc on the cluster nature of open Richardson varieties.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.