Cohomology, Representations, and Coverings of Curves

曲线的上同调、表示和覆盖

• 批准号: 1600056
• 负责人: Robert Guralnick
• 金额: $ 19.5万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-15 至 2019-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600056&HistoricalAwards=false
• 关键词: Cohomology; Representations; Coverings; Curves

The mathematical concept of a group, a set together with an operation for combining two elements to produce another element, plays a central role in mathematics and its applications because it encodes intuitive notions of symmetry in a precise manner. The classification of the collection of finite groups known as finite simple groups is a monumental achievement of modern mathematics that has led to a revolution in using group theory to study other fields. The utility of group theory has been greatly expanded by advances in computation. One goal of the project is to describe useful presentations of finite simple groups that lead to more computational efficiency. This will lead to solutions of fundamental problems in number theory. Another goal of the project is to study a group-theoretical problem that will lead will lead to results showing the existence and construction of expander graphs, which have been of great importance in computer science.In more detail, one goal of the project is to prove a generalization of the fact that if H is a finitely generated Zariski dense subgroup of a semisimple algebraic group, then H contains a strongly dense subgroup. This will give some new results about superstrong approximation in number theory and results on expander graphs. A second goal of the project is to prove the conjecture that every finite simple group has a presentation with two generations and at most four relations. This will lead to advances in computational number theory. A third goal of the project is to classify generic stabilizers for simple algebraic groups in irreducible linear representations. This will fit into the program of Bhargava to solve classification problems of algebraic families. A fourth goal of the project is to classify monodromy groups of coverings of low genus Riemann surfaces.

一个组的数学概念，即与将两个元素组合在一起产生另一个元素的操作的组合，在数学及其应用中起着核心作用，因为它以精确的方式编码了对称性的直觉概念。被称为有限简单群体的有限群体集合的分类是现代数学的巨大成就，这导致了使用群体理论研究其他领域的革命。群体理论的实用性已随着计算的进步而大大扩展。该项目的一个目标是描述有限简单组的有用演示，从而提高计算效率。这将导致数量理论中基本问题的解决方案。 该项目的另一个目标是研究一个将导致的群体理论问题，将导致结果表明扩展器图的存在和构建在计算机科学中至关重要。更详细的是，该项目的一个目标是证明h是有限生成的Zariski密集产生的Zariski密集的AlgeBraic Groups，然后cy comptect的Zariski密集型组，然后comply是一个强大的小组。这将为数字理论和扩展器图上的superstrong近似值提供一些新的结果。该项目的第二个目标是证明每个有限的简单组都有两代和最多四个关系的介绍。 这将导致计算数理论的进步。该项目的第三个目标是将简单代数组中的通用稳定器分类为不可还原线性表示。 这将适合Bhargava计划，以解决代数家庭的分类问题。该项目的第四个目标是对低属riemann表面的覆盖物进行分类。