Combinatorics in Cohomology and Computation

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This research plan is divided into three projects, each of whichcombines 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 effectivecomputation and storage of these resolutions possible, even thoughinjective modules are themselves seemingly intractable. The finalproject places summands in combinatorial formulae for certainuniversal cohomology classes in bijection with components inGrobner degenerations of orbit closures for algebraic groups. Thisdegeneration technique should provide a geometrically positiveproof of the Buch-Fulton conjecture for quiver coefficients, whichgeneralize the famous Littlewood-Richardson coefficients.Combinatorics, the study of discrete structures, arises as anorganizing principle in widely varying contexts throughout thesciences, including mathematics, computer science, physics, andbiology. Applications of combinatorics occur not only when theoriginal problem is itself discrete, but frequently also when theoriginal problem deals with continuous phenomena. For instance, itcan happen that a single type of discrete structure can be imposeduniversally upon a variety of continuous systems. This kind offramework often lends deep insight into the nature of such systemsand their interconnections. Combinatorial frameworks can alsoendow certain special systems with enough order to bring previouslyintractable problems within grasp, conceptually or computationally.Conversely, within many fields, understanding certain specialsystems whose parameters are defined in a combinatorial context canlead to methods applicable in general. The projects outlined herewill broaden the understanding of how discrete structures arisingin these ways can control phenomena in the areas of differentialequations, homological algebra, and algebraic geometry.