Cell decompositions of quiver Grassmannians

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.