Ehrhart Rings, Invariants and Koszulity

This project uses algebra to enhance famous counting formulas by (insightfully) throwing in an extra variable "q". These q-counts have connections to quantum theory and to the theory of error-correcting codes, which help reduce errors in noisy transmissions from objects in space. A beautiful feature of these "q-counts" is that they respect the symmetries present in the objects being counted. The project will involve undergraduate and graduate students in the research. The PI proposes a suite of problems, conjectures and questions concerning the "point orbit method" for producing q-analogues in various counting problems. These include q-analogues of Ehrhart's theory of lattice point counting in dilated polytopes, and the Crapo-Rota "finite field" method for analyzing hyperplane arrangements. The point orbit method originated in the invariant theory of polynomial rings, and a second thread of the project concerns cases where deformations of these invariant rings turn out to be Koszul algebras. 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.