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——数字数组——在科学、工程和商业中无处不在。 对控制矩阵行为的算术的研究使所有这些应用领域受益，并且也导致了数学的几个分支的进步。 该研究项目旨在推进正在进行的工作，将表示矩阵代数的方程捆绑到一个单一的几何理论中，从而使几何技术能够加深我们对代数的理解。本研究项目旨在研究函数域算术群的大规模几何、有限性和上同调行为，例如特殊线性群 SL(n,F[t])，其中 F 是有限域，以及相关的组如 SL(n,Z[t])。具体来说，主要研究者打算研究这些群族的群环系数的上同调性。人们还想了解诸如这些特殊线性群之类的群何时具有射影分辨率或通过维度 m 对具有有限骨架的复合体进行分类，并确定系数模 M 和自然数 m，其中这些特殊线性群之一的上同调群在系数为 M 的度 m 是无限生成的。