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

演示、上同调、有限群表示和曲线覆盖

• 批准号: 1302886
• 负责人: Robert Guralnick
• 金额: $ 23.1万
• 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: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1302886&HistoricalAwards=false
• 关键词: Presentations; Cohomology; Representations; Finite; Groups

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.

该项目将涉及对有限和代数群体的研究，尤其是它们对线性空间和品种的行为。 该项目的一个重要方面是改善低度同胞组的大小的界限。 这应该导致对有限的简单组的演示，并具有两个发电机和少量关系。 还将研究基于提议者早期工作的有限简单群体的离散演讲，而其他人则表明，除了一个家庭外，每个有限的简单小组都有最多有50个关系的介绍。 该项目还将考虑有限群体的适当表示的概念。这是对泰勒和威尔斯用来证明某些表示形式的条件的削弱（用于Fermat的最后一个定理证明）。 这个想法是表明许多表示确实满足了这种情况（并且还构建了其他情况）。 Clozel，Thorne，Gee，Dieulefait和其他人已经使用了这一概念。 该项目的另一部分是尝试概括山雀替代方案。 猜想是，任何有限生成的半神经代数群的Zariski密集的亚组都包含一个强大的自由亚组。 在这里，密集意味着每个非亚伯亚组在代数组中都是Zariski致密。 最后，将使用小组理论结果研究与平滑代数曲线映射有关的某些问题。该项目的主要动机之一是计算群体理论的主要进步。 这些进步不仅基于计算能力的改进，而且基于以至关重要的方式使用该理论的非常创新的程序。 上面提到的演示文稿已经实施到岩浆（一个非常强大的计算代数软件包）中，并用于识别组。 在表明表示形式的一些重大进展中，已经使用了足够代表的结果是自动形态的，应该有助于基本的Langlands计划。强烈密集的亚组的结果已经用于证明扩展器图的新结果。 这些图形相对稀疏，但也非常高度连接。 这些在过去十年中引发了计算机科学的革命。 新结果将在此领域中提供更多应用程序。 小组理论的深刻结果导致了关于有限领域的徒多项式基本问题的重大进展（视为平滑的投射曲线上的映射），并在一个世纪以上的密码学和解决问题上应用了问题。