Cluster Algebras, Quiver Representations, and Rigid Curves

本计画旨在研究现代代数组合学中的一个核心对象：一类丛集代数。这些簇代数与代数几何、表示论和其他领域中出现的各种其他数学对象有关，但在粒子物理学中也很重要，它们与某些量子场论中的散射振幅有关。因此，更好地理解簇代数不仅本身很有趣，而且在数学的其他领域和更远的领域也有重要的潜在应用。该项目将通过参与研究为学生提供培训。 刚性模、不变曲线和纽结是许多数学分支中的基本对象。这个项目采取了集群代数的方法来研究这些对象。研究的目标，主要是出于同调镜像对称，是研究遗传代数上的不可分解模，Grassmannian和交替结的不变量。该项目旨在开发新的组合和几何对象，以揭示簇代数和其他数学领域之间更具体的关系。该项目的教育目标是培养本科生和研究生在代数，组合学，几何，拓扑学和表示论的交叉主题。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

This project aims to investigate a central object in modern algebraic combinatorics: a class of cluster algebras. These cluster algebras are related to a wide variety of other mathematical objects arising in algebraic geometry, representation theory, and other areas, but are also important in particle physics, where they are related to scattering amplitudes in certain quantum field theories. Hence a better understanding of cluster algebras is not only interesting in its own right, but also has significant potential application in other areas of mathematics and beyond. The project will provide training for students through involvement in the research. Rigid modules, invariant curves, and knots are fundamental objects in many branches of mathematics. This project takes a cluster-algebraic approach to studying these objects. The research goal, largely motivated by homological mirror symmetry, is to study invariants for indecomposable modules over hereditary algebras, for quiver Grassmannians, and for alternating knots. The project aims to develop new combinatorial and geometric objects to reveal more concrete relationships between cluster algebras and other areas of mathematics. The educational goal of this project is to train undergraduate and graduate students in topics at the intersection of algebra, combinatorics, geometry, topology, 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.