Cluster Algebras, Quiver Representations, and Rigid Curves

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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.