Cluster Algebras and Categorification

簇代数和分类

• 批准号: 2054255
• 负责人: Khrystyna Serhiyenko
• 金额: $ 19.32万
• 依托单位: University of Kentucky Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-08-15 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054255&HistoricalAwards=false
• 关键词: Cluster; Algebras; Categorification

Cluster algebras, introduced by Fomin and Zelevinsky in 2001, have an intricate structure that can appear in many other, seemingly unrelated, areas of mathematics and theoretical physics. In particular, they provide a rigid framework that captures various patterns formed by a given system of elements. Using cluster algebras, this system can then be translated into a different setting, which allows us to use new tools and techniques for its study. Using this approach, the theory of cluster algebras has led to the establishment of many influential results in mathematics. This project is in a highly active research area aimed at answering fundamental questions about the structure of cluster algebras, as well as investigating new connections to other fields of mathematics. This project will focus both on obtaining explicit computational results and developing general properties. Throughout this award, the PI will contribute to the advancement of education and research in the mathematical community by working with undergraduates, developing new courses, recruiting graduate students, and fostering international collaborations among women.The project will explore new connections between cluster algebras and their various categorifications to further their understanding and build parallels between the different areas. Its goal is to develop novel combinatorial models for certain important classes of cluster algebras in order to get an explicit description of its generators and relations that are built recursively. The main ingredient behind solving these questions is the powerful machinery coming from the representation theory of associative algebras, which encode the structure of cluster algebras and constitute an invaluable tool in their study. In particular, the PI will study the recently-found cluster structures coming from positroid and Richardson varieties in the Grassmannian. Moreover, the PI proposes a new combinatorial description of Cohen-Macauley subcategories in the module category of certain Jacobian algebras which offers applications to various topics in cluster algebras and representation theory.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.

由Fomin和Zelevinsky在2001年引入的簇代数具有复杂的结构，可以出现在许多其他看似无关的数学和理论物理领域。特别是，它们提供了一个严格的框架，捕捉由给定的元素系统形成的各种模式。 使用簇代数，这个系统可以被转换成不同的设置，这使我们能够使用新的工具和技术进行研究。 使用这种方法，簇代数理论导致了数学中许多有影响力的结果的建立。该项目是一个高度活跃的研究领域，旨在回答有关簇代数结构的基本问题，以及调查与其他数学领域的新联系。 这个项目将集中在获得明确的计算结果和开发一般属性。在这个奖项中，PI将通过与本科生合作，开发新课程，招收研究生，促进女性之间的国际合作，为数学界的教育和研究进步做出贡献。该项目将探索簇代数及其各种代数之间的新联系，以加深对它们的理解，并在不同领域之间建立相似之处。 它的目标是为某些重要的簇代数类开发新的组合模型，以明确描述其生成元和递归建立的关系。 解决这些问题背后的主要成分是来自结合代数表示论的强大机制，它编码了簇代数的结构，并构成了研究它们的宝贵工具。 特别是，PI将研究最近发现的来自格拉斯曼正素和理查森品种的集群结构。 此外，PI提出了一种新的组合描述的科恩-Macauley子类别的模块类别的某些雅可比代数，它提供了应用到各种主题的集群代数和表示理论。这个奖项反映了NSF的法定使命，并已被认为是值得支持的评估使用基金会的智力价值和更广泛的影响审查标准。