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

这个项目将涉及有限和代数群的研究，特别是他们对线性空间和品种的行动。 该项目的一个重要方面是提高低程度上同调群的大小的界限。 这将导致profinite介绍了有限简单的群体与两个发电机和一个非常小的数量的关系。 也将有一个研究的离散介绍有限的简单群体的建设早期工作的提议者和其他人表明，与可能的例外，一个家庭，每一个有限的简单群体有一个介绍，最多50个关系。 该项目还将考虑有限群的适当表示的概念。这是泰勒和怀尔斯用来证明某些表示是自守的条件的弱化（并用于证明费马最后定理）。 这个想法是为了证明许多表示确实满足这个条件（也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和其他人使用了很大的效果。 该项目的另一部分将是试图推广山雀的替代品。 猜想是半单代数群的任意一个生成的Zebraki稠密子群都含有一个强稠密自由子群。 这里的强稠密是指代数群中的每个非交换子群都是Zebriki稠密的。 最后，将利用群论的结果研究与光滑代数曲线的映射有关的某些问题。这个项目的主要动机之一是计算群论的重大进展。 这些进步不仅是基于计算能力的提高，而且是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，并用于识别群。 充分表示的结果已经被用于一些显着的进展，显示表示是自守的，应该有助于基本的朗兰兹计划。强稠密子群的结果已经被用来证明扩张图的新结果。 这些图是相对稀疏的，但也是高度连通的。 在过去的十年里，这些引发了计算机科学的革命。 新的结果将在这个领域有更多的应用。 群论中的深入结果导致了有限域上的双射多项式（被视为光滑射影曲线上的映射）基本问题的重大进展，并已应用于密码学并解决了世纪的问题。

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.