Presentations, Cohomology, Representations of Finite Groups and Coverings of Curves

这个项目将涉及有限群和代数群的研究，特别是它们在线性空间和变异上的作用。该项目的一个重要方面是改进了低次上同群大小的界。这将导致具有两个生成器和非常少的关系的有限简单群的无限表示。也将有一个有限简单群的离散表示的研究建立在早期的工作提出者和其他人表明，除了一个族可能的例外，每个有限简单群有一个最多50个关系的表示。该项目还将考虑有限群的充分表示的概念。这是泰勒和怀尔斯用来证明某些表示是自同构的条件的弱化（并用于证明费马最后定理）。这个想法是为了表明许多表示确实满足这个条件（并构造其他不满足这个条件的表示）。这个概念已经被Clozel、Thorne、Gee、Dieulefait等人运用到很大的效果中。该项目的另一部分将是尝试推广Tits的替代方案。猜想是半简单代数群的任何有限生成的Zariski密子群都包含一个强密自由子群。这里强密集是指代数群中的每个非abel子群都是Zariski密集的。最后，我们将利用群理论的结果来研究光滑代数曲线的映射问题。这个项目的主要动机之一是计算群论的重大进展。这些进步不仅仅是基于计算能力的提高，而且是基于以关键方式使用该理论的非常创新的程序。上面提到的演示已经在MAGMA（一个非常强大的计算代数软件包）中实现，并用于识别组。关于充分表示的结果已经用于证明表示是自同构的一些重要进展，并且应该有助于基本的朗兰兹程序。关于强密子群的结果已经被用于证明关于展开图的新结果。这些图表相对稀疏，但也非常紧密相连。这些都在过去十年中引发了计算机科学的一场革命。新的结果将在这一领域有更多的应用。群论的深刻成果导致了有限域上双射多项式基本问题的重大进展（被视为光滑投影曲线上的映射），并已应用于密码学并解决了一个多世纪前的问题。

This project will involve the study of finite and algebraic groups and in particular their actions on linear spaces and varieties. One important aspect of the project is to improve bounds for the sizes of low degree cohomology groups. This should lead to profinite presentations of the finite simple groups with two generators and a very small number of relations. There will also be a study of discrete presentations of the finite simple groups building on earlier work of the proposer and others showing that with the possible exception of one family, every finite simple group has a presentation with at most 50 relations. The project will also consider the notion of adequate representations of finite groups. This is a weakening of a condition used by Taylor and Wiles to prove certain representations are automorphic (and used in the proof of Fermat's last theorem). The idea is to show that many representations do satisfy this condition (and also constructing others that do not). This notion has already been used to great effect by Clozel, Thorne, Gee, Dieulefait and others. Another part of the project will be to attempt to generalize the Tits alternative. The conjecture is that any finitely generated Zariski dense subgroup of a semisimple algebraic group contains a strongly dense free subgroup. Here strongly dense means every nonabelian subgroup is Zariski dense in the algebraic group. Finally, certain problems related to mappings of smooth algebraic curves will be studied using the group theory results.One of the major motivations for this project has been the major advances in computational group theory. These advances are not just based on improvements in computing power but very innovative programs which use the theory in a crucial way. The presentations mentioned above are already being implemented into MAGMA (a very powerful computational algebra software package) and is used to recognize groups. The results on adequate representations have already been used in some significant advances in showing representations are automorphic and should help in the fundamental Langlands program. The results on strongly dense subgroups have already been used in proving new results on expander graphs. These are graphs that are relatively sparse but are also very highly connected. These have sparked a revolution in computer science in the last decade. The new results will have even more applications to this field. Deep results in group theory have led to major advances in basic problems about bijective polynomials over finite fields (viewed as mappings on a smooth projective curve) and has had applications to cryptography and solved problems over a century old.