Combinatorics in Cohomology and Computation

University Of Minnesota-Twin Cities, Minneapolis MN

Investigators

Abstract

This research plan is divided into three projects, each of which combines combinatorics with cohomology and computation in some way. The first project deals with certain systems of partial differential equations defined by a combination of discrete convex polyhedral data and continuous parameters. These `hypergeometric systems' provide a fertile source of examples for the more general theory of holonomic systems, and the goal is to shed light on how their solution spaces vary in continuous families, using the algebraic theory of local cohomology. The second project applies a computational perspective to the homological algebra of injective resolutions. It aims to demonstrate that exerting sufficient combinatorial control over the maps in injective resolutions of finitely generated modules over polynomial rings can make effective computation and storage of these resolutions possible, even though injective modules are themselves seemingly intractable. The final project places summands in combinatorial formulae for certain universal cohomology classes in bijection with components in Grobner degenerations of orbit closures for algebraic groups. This degeneration technique should provide a geometrically positive proof of the Buch-Fulton conjecture for quiver coefficients, which generalize the famous Littlewood-Richardson coefficients. Combinatorics, the study of discrete structures, arises as an organizing principle in widely varying contexts throughout the sciences, including mathematics, computer science, physics, and biology. Applications of combinatorics occur not only when the original problem is itself discrete, but frequently also when the original problem deals with continuous phenomena. For instance, it can happen that a single type of discrete structure can be imposed universally upon a variety of continuous systems. This kind of framework often lends deep insight into the nature of such systems and their interconnections. Combinatorial frameworks can also endow certain special systems with enough order to bring previously intractable problems within grasp, conceptually or computationally. Conversely, within many fields, understanding certain special systems whose parameters are defined in a combinatorial context can lead to methods applicable in general. The projects outlined here will broaden the understanding of how discrete structures arising in these ways can control phenomena in the areas of differential equations, homological algebra, and algebraic geometry.