Cluster Algebras and Categorification

由Fomin和Zelevinsky于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，可以捕获由给定元素系统形成的各种模式。使用聚类代数，这个系统可以转换成不同的设置，这使我们能够使用新的工具和技术来研究它。利用这种方法，聚类代数理论在数学中建立了许多有影响的结果。这个项目是一个高度活跃的研究领域，旨在回答关于簇代数结构的基本问题，以及研究与其他数学领域的新联系。该项目将侧重于获得明确的计算结果和发展一般性质。通过这项奖励，PI将通过与本科生合作、开发新课程、招收研究生以及促进妇女之间的国际合作，为数学界的教育和研究的进步做出贡献。该项目将探索簇代数及其各种分类之间的新联系，以进一步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，以便对其生成器和递归构建的关系进行显式描述。解决这些问题背后的主要因素是来自联想代数表示理论的强大机制，它编码了簇代数的结构，并构成了研究簇代数的宝贵工具。特别是，PI将研究最近发现的来自格拉斯曼系正电子和理查德森品种的簇状结构。此外，本文还提出了一类Jacobian代数的模范畴中Cohen-Macauley子范畴的一种新的组合描述，该描述在聚类代数和表示理论中具有广泛的应用。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。

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.