Geometry and Cohomology of Arithmetic and Related Groups

算术及相关群的几何和上同调

• 批准号: 1509182
• 负责人: Kevin Wortman
• 金额: $ 18万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-15 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1509182&HistoricalAwards=false
• 关键词: Geometry; Cohomology; Arithmetic; Related; Groups

Matrices -- arrays of numbers -- are ubiquitous in science, engineering, and business. The study of the arithmetic governing the behavior of matrices has benefited all these application areas and has also led to advances in several branches of mathematics. This research project aims to advance work that is underway to bundle the equations that represent the algebra of matrices into a single geometric theory, thus allowing techniques from geometry to deepen our understanding of algebra. This research project aims to study the large-scale geometry, finiteness properties, and cohomological behavior of function-field-arithmetic groups such as the special linear groups SL(n,F[t]) where F is a finite field, and of related groups such as SL(n,Z[t]). Specifically, the principal investigator intends to investigate the cohomology with group ring coefficients of these families of groups. One would also like to understand when groups such as these special linear groups have projective resolutions or classifying complexes with finite skeleta through dimension m, and to determine coefficient modules M and natural numbers m for which the cohomology group of one of these special linear groups in degree m with coefficients in M is infinitely generated.

矩阵 - 数量数阵列 - 在科学，工程和商业中无处不在。 对矩阵行为的算术的研究使所有这些应用领域都受益，并导致了数学几个分支的进步。 该研究项目旨在推进正在进行的工作，以将代表矩阵代数的方程式捆绑到单个几何理论中，从而使技术从几何形状加深我们对代数的理解。该研究项目旨在研究功能场偏置基团的大规模几何形状，有限性能以及诸如特殊线性组SL（N，F [t]）之类的功能场偏置群的同谋行为，其中F是有限的领域，以及SL（N，Z [t]）等相关群体。具体而言，主要研究者打算使用这些组家族的组环系数研究同胞。人们还想了解诸如这些特殊线性群体诸如射影的分辨率或通过尺寸M进行有限骨架的复合物分类的群体何时确定系数模块M和自然数m，其中这些特殊线性组之一的同胞在M中与系数M中的一个特殊线性组中的一个组成。