Cell decompositions of quiver Grassmannians

这个项目将研究簇代数，它是有理函数的某些代数。簇代数是现代代数和几何组合学中的中心对象，因为它们与代数几何、表示论、拓扑学和其他领域中出现的各种其他数学对象有着密切的联系。它们在粒子物理学中也很重要，在粒子物理学中，它们与某些量子场理论中的散射幅度有关。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来为学生提供培训。更详细地说，当一个拓扑空间被期望允许单元分解时，希望找到一个实际的分解。本课题的目的是寻找箭图Grassmannians的胞元分解，它可以用来对簇代数的正性给出一个拓扑解释。对于通常的Grassmannians来说，Schubert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，以揭示簇代数和其他数学领域之间更具体的关系。该项目由代数数论计划、既定的激励竞争研究计划(EPSCoR)和组合数学计划联合资助。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。

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.