Combinatorics in Cohomology and Computation

该研究计划分为三个项目，每个项目都以某种方式将组合学、上同调和计算结合在一起。第一个项目研究由离散凸多面体数据和连续参数组合而成的某些偏微分方程组。这些“超几何系统”为更一般的完整系统理论提供了丰富的例子来源，其目的是利用局部上同调的代数理论，阐明它们的解空间如何在连续族中变化。第二个项目将计算视角应用于内射分解的同调代数。它的目的是证明对多项式环上有限生成模的内射分解中的映射施加充分的组合控制可以使这些分解的有效计算和存储成为可能，即使内射模本身似乎是难处理的。期末课题给出了双射中某些泛上同调类的组合公式和代数群的轨道闭包的Grobner退化中的分量。这种退化技术应该为Buch-Fulton关于箭图系数的猜想提供几何上的正证明，该猜想推广了著名的Littlewood-Richardson系数。组合学，研究离散结构，作为一种组织原理出现在整个科学的广泛不同的背景下，包括数学，计算机科学，物理和生物学。组合学的应用不仅发生在原始问题本身是离散的时候，而且经常发生在原始问题处理连续现象时。例如，可以将一种单一类型的离散结构普遍地强加于各种连续系统。这种框架往往有助于深入了解这类系统的性质及其相互联系。组合框架还可以赋予某些特殊系统足够的阶数，从而使以前难以解决的问题在概念上或计算上都能被掌握。相反，在许多领域中，理解某些特定系统的参数是在组合上下文中定义的，可以得到普遍适用的方法。这里概述的项目将扩大对以这些方式产生的离散结构如何控制微分方程式、同调代数和代数几何领域中的现象的理解。

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.