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.

群的数学概念，即一个集合与一个将两个元素组合成另一个元素的运算，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观概念。有限单群的分类是现代数学的一个里程碑式的成就，它导致了一场使用群论研究其他领域的革命。群论的效用因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群的有用表示，从而提高计算效率。这将导致解决数论中的基本问题。 该项目的另一个目标是研究一个群论问题，该问题将导致将导致结果显示扩展图的存在性和构造，这在计算机科学中一直非常重要。更详细地说，该项目的一个目标是证明一个推广的事实，即如果H是一个半单代数群的一个生成Zebraki稠密子群，则H包含一个强稠密子群。这将给出数论中关于超强逼近的一些新结果和扩张图上的结果。该项目的第二个目标是证明猜想，即每个有限单群都有一个两代和最多四个关系的表示。 这将导致计算数论的进步。该项目的第三个目标是对不可约线性表示中简单代数群的通用稳定器进行分类。 这将适合Bhargava的程序来解决代数族的分类问题。该项目的第四个目标是对低亏格黎曼曲面的单值覆盖群进行分类。