Schubert Structure Constants via Kohnert Combinatorics

Classical enumerative geometry asks questions such as how many points lie in the intersection of two lines in the plane, or how many lines in space intersect four given lines? These enumeration questions counting linear subspaces satisfying certain geometric conditions have been considered since the late 1800s and have sparked many advances in mathematics including the development of modern intersection theory. Solutions to these enumeration questions connect to areas outside of mathematics through Gromov-Witten invariants and quantum cohomology. The formalism for solving such problems, known as Schubert calculus, is now rigorous, though the original question of enumerating linear subspaces remains largely unsolved. This project will develop simple combinatorial rules for computing these numbers. The vertically integrated research program will incorporate training for undergraduates, masters and doctoral students, and postdoctoral scholars. The first milestone is to develop rules in the cohomology ring of the classical flag manifold that occur in the product of an arbitrary Schubert class by one pulled back from a Grassmannian projection. The second milestone will utilize combinatorial crystal graphs to extend this rule to products of Demazure characters. The ultimate goal of this ambitious project is to generalize these two cases to give rules for the product of any two Schubert classes, thereby resolving the fundamental Schubert problem for the complete flag manifold. 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.