Cluster Algebras and Categorification

簇代数由Fomin和Zlevinsky在2001年提出，它具有一种复杂的结构，可以出现在许多其他看似无关的数学和理论物理领域。具体地说，它们提供了一个严格的框架，可以捕获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到不同的环境中，这允许我们使用新的工具和技术来研究它。利用这种方法，簇代数理论已经在数学上建立了许多有影响的结果。这个项目是一个非常活跃的研究领域，旨在回答关于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。这个项目将专注于获得明确的计算结果和开发一般性质。在整个奖项中，PI将通过与本科生合作、开发新课程、招收研究生以及促进女性之间的国际合作来促进数学界的教育和研究。该项目将探索簇代数与其各种分类之间的新联系，以加深他们对不同领域的理解和建立联系。它的目标是为某些重要的簇代数类建立新的组合模型，以便明确地描述它的生成元和递归建立的关系。解决这些问题的主要因素是来自结合代数表示理论的强大机制，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Grassmanian中的正核和Richardson变种的星团结构。此外，PI在某些Jacobian代数的模范类中提出了Cohen-Macauley子范畴的新的组合描述，该组合描述提供了对集群代数和表示理论中的各种主题的应用。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.