Combinatorics in Cohomology and Computation

上同调和计算中的组合学

• 批准号: 0304789
• 负责人: Ezra Miller
• 金额: $ 12.64万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0304789&HistoricalAwards=false
• 关键词: Combinatorics; Cohomology; Computation

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.

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与上同调和计算结合起来。第一个项目涉及由离散凸多面体数据和连续参数组合定义的某些偏微分方程系统。 这些“超几何系统”提供了一个肥沃的来源的例子，更一般的理论完整系统，其目标是阐明他们的解决方案空间如何在连续的家庭不同，使用代数理论的局部上同调。 第二个项目应用计算的角度来看，同调代数的内射决议。 它的目的是证明，施加足够的组合控制的映射在多项式环上的单射分解的生成模可以有效地计算和存储这些决议可能，即使内射模本身似乎是棘手的。 最后一个项目的地方summands在组合公式的某些普遍上同调类的双射与组件在Grobner退化的轨道闭包的代数群。 这种退化技术应该为Buch-Fulton猜想提供了一个几何上的正向证明，该猜想推广了著名的Littlewood-Richardson系数。组合学是研究离散结构的学科，作为一种组织原理出现在包括数学、计算机科学、物理学和生物学在内的各种科学背景中。 组合数学的应用不仅发生在原问题本身是离散的，而且经常也发生在原问题处理连续现象时。 例如，它可以发生，一个单一类型的离散结构可以普遍强加在各种连续系统。 这类推理往往有助于深入了解这些系统的本质及其相互联系。 组合框架还可以赋予某些特殊系统足够的秩序，使以前难以解决的问题在概念上或计算上都能得到解决。相反，在许多领域中，理解某些参数在组合环境中定义的特殊系统可以导致普遍适用的方法。 这里概述的项目将拓宽人们对以这些方式产生的离散结构如何控制微分方程、同调代数和代数几何领域的现象的理解。