Cohomology, Representations, and Coverings of Curves

群的数学概念是一个集合，以及组合两个元素以产生另一个元素的运算，它在数学及其应用中发挥着核心作用，因为它以一种精确的方式对对称的直观概念进行了编码。对称为有限单群的有限群集合的分类是现代数学的一项不朽成就，它导致了使用群论来研究其他领域的一场革命。群论在计算方面的进步大大扩展了它的用途。该项目的一个目标是描述有限单群的有用表示，从而导致更高的计算效率。这将导致数论中基本问题的解决。这个项目的另一个目标是研究一个群论问题，这个问题将导致证明扩张图的存在和构造的结果，这在计算机科学中一直是非常重要的。更详细地说，这个项目的一个目标是证明了如果H是半单代数群的有限生成的Zariski稠密子群，那么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关于扩展图的一些结果。该项目的第二个目标是证明一个猜想，即每个有限单群都有一个有两代且至多有四个关系的表示。这将导致计算数论的进步。该项目的第三个目标是在不可约的线性表示中对简单代数群的通用稳定器进行分类。这将适用于Bhargava解决代数族分类问题的程序。该项目的第四个目标是对低亏格黎曼曲面覆盖的单色群进行分类。

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.