Cell decompositions of quiver Grassmannians

本课题将研究聚类代数，它是有理函数的一类代数。聚类代数是现代代数和几何组合学的中心对象，因为它们与代数几何、表示理论、拓扑学和其他领域中出现的各种其他数学对象有着深刻的联系。它们在粒子物理学中也很重要，在某些量子场论中它们与散射振幅有关。因此，更好地理解簇代数不仅本身就很有趣，而且在其他数学领域和其他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。更详细地说，当期望拓扑空间允许单元分解时，需要找到实际的分解。这个项目是寻找颤动格拉斯曼的细胞分解，可以用来给出簇代数正性的拓扑解释。一般格拉斯曼人的舒伯特细胞分解的基本概念将被推广。该项目将开发新的组合和几何对象，以揭示簇代数与其他数学领域之间更具体的关系。本项目由代数与数论项目、刺激竞争性研究的既定项目（EPSCoR）和组合学项目共同资助。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。

This project will investigate cluster algebras, which are certain algebras of rational functions. Cluster algebras are central objects in modern algebraic and geometric combinatorics because of their deep connections to a wide variety of other mathematical objects that arise in algebraic geometry, representation theory, topology, and other areas. They 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.In more detail, when a topological space is expected to admit a cell decomposition, it is desirable to find an actual decomposition. This project is to find cell decompositions of quiver Grassmannians that can be used to give a topological explanation for positivity in cluster algebras. The fundamental notion of Schubert cell decompositions for usual Grassmannians will be generalized. The project will develop new combinatorial and geometric objects to reveal more concrete relationships between cluster algebras and other areas of mathematics. This project is jointly funded by Algebra an Number Theory Program, the Established Program to Stimulate Competitive Research (EPSCoR), and the Combinatorics Program.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.