Cohomology and Representations of Finite and Algebraic Groups with Applications

有限代数群的上同调和表示及其应用

• 批准号: 1901595
• 负责人: Robert Guralnick
• 金额: $ 31.5万
• 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: 2019
• 资助国家: 美国
• 起止时间: 2019-08-01 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1901595&HistoricalAwards=false
• 关键词: Cohomology; Representations; Finite; Algebraic; Groups

This project will involve the study of finite and algebraic groups and in particular their actions on linear spaces and varieties. Groups are one of the fundamental tools in mathematics and arise in many areas including analysis, geometry and number theory as well as in the study of symmetries in chemistry and physics. The classification of finite simple groups was completed in 2006 and has led to a revolution in using group theory to study other fields. The classification basically says that the finite simple groups are analogs of the simple Lie groups and so to understand them, one must study simple Lie and algebraic groups. The best way to understand and use group theory is to study the action of groups on different objects. One aspect of this project is to understand groups acting on Riemann surfaces (and their analog over finite fields). This will lead to a new fundamental understanding of basic objects including rational functions and should lead to advances in cryptography and fundamental problems in number theory. The utility of group theory has also been greatly expanded due to advances in computation. Another aspect of this project is to find useful presentations of the finite simple groups which will lead to more computational efficiency. A third important problem addressed in this project is to greatly generalize what is called the Tits alternative. This will lead to results showing the existence (and construction) of expander graphs. These are graphs that are highly connected relative to the number of edges in them. This has been of great importance in computer science. Graduate students will be trained through research. In particular, we plan to study the problem of producing strongly dense subgroups of semisimple algebraic groups and proving a generalization of the Tits alternative. This will give some new results about superstrong approximation in number theory and results on expander graphs. Earlier results of the PI, with Breuillard, Green, and Tao, will be generalized using new stronger methods. We also want to prove the conjecture that every finite simple group has a presentation with two generators and at most four relations. This should lead to advances in computational number theory. 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. Another goal of the project is to completely classify monodromy groups of coverings of low genus Riemann surfaces leading to fundamental breakthroughs in number theory and also to classify monodromy groups of mappings from generic Riemann surfaces (first studied in Zariski's thesis). Finally, we want to classify generic stabilizers for simple algebraic groups in irreducible linear representations. This has been done in characteristic zero but new ideas are required in positive characteristic. This will have consequences for essential dimension and some special cases will fit into the program of Bhargava to solve interesting classification problems of algebraic families.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

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。群是数学的基本工具之一，出现在许多领域，包括分析、几何和数论，以及化学和物理中的对称性研究。有限单群的分类于2006年完成，并引发了利用群论研究其他领域的一场革命。分类基本上是说，有限单群是单李群的类似物，因此要理解它们，必须研究单李群和代数群。理解和运用群论的最好方法是研究群在不同物体上的作用。这个项目的一个方面是了解作用在黎曼曲面上的群（以及它们在有限域上的模拟）。这将导致对包括有理函数在内的基本对象的新的基本理解，并将导致密码学和数论基本问题的进步。由于计算技术的进步，群论的应用也得到了极大的扩展。这个项目的另一个方面是找到有限简单群的有用表示，这将导致更高的计算效率。在这个项目中要解决的第三个重要问题是大大推广所谓的Tits替代方案。这将导致结果显示扩展图的存在性（和构造）。这些图是高度连接的相对于边的数量。这在计算机科学中是非常重要的。研究生将通过研究进行训练。特别地，我们计划研究半简单代数群的强密子群的产生问题，并证明Tits替代的推广。这将给出数论中关于超强逼近的一些新结果和关于展开图的一些新结果。布鲁拉德、格林和陶的早期PI结果将使用新的更强的方法进行推广。我们还想证明一个猜想，即每一个有限单群都有两个生成子和最多四个关系的表示。这将导致计算数论的进步。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。该项目的另一个目标是对低格黎曼曲面覆盖的单群进行完全分类，从而在数论方面取得根本性突破，并对一般黎曼曲面的映射的单群进行分类（在Zariski的论文中首次研究）。最后，我们想要对不可约线性表示中的简单代数群的一般稳定器进行分类。这已经在特征零中完成了，但在正特征中需要新的想法。这将对基本维数产生影响，一些特殊情况将适用于巴尔加瓦的程序，以解决代数族的有趣分类问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Generic Stabilizers for Simple Algebraic Groups

• DOI: 10.1307/mmj/20217216
• 发表时间: 2021-05
• 期刊: Michigan Mathematical Journal
• 影响因子: 0.9
• 作者: S. Garibaldi;R. Guralnick

GENERICALLY FREE REPRESENTATIONS III: EXTREMELY BAD CHARACTERISTIC

一般免费的表现 III：极其糟糕的特征

• DOI: 10.1007/s00031-020-09590-4
• 发表时间: 2020
• 期刊: Transformation groups
• 影响因子: 0.7
• 作者: Garibaldi, S.;Guralnick, R.
• 通讯作者: Guralnick, R.

The spread of a finite group

• DOI: 10.4007/annals.2021.193.2.5
• 发表时间: 2020-06
• 期刊: arXiv: Group Theory
• 作者: Timothy C. Burness;R. Guralnick;Scott Harper

Topological generation of exceptional algebraic groups

特殊代数群的拓扑生成

• DOI: 10.1016/j.aim.2020.107177
• 发表时间: 2020
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Burness, Timothy C.;Gerhardt, Spencer;Guralnick, Robert M.
• 通讯作者: Guralnick, Robert M.

GENERICALLY FREE REPRESENTATIONS II: IRREDUCIBLE REPRESENTATIONS

一般自由表示 II：不可约表示

• DOI: 10.1007/s00031-020-09591-3
• 发表时间: 2020
• 期刊: Transformation groups
• 影响因子: 0.7
• 作者: Garibaldi, S.;Guralnick, R.
• 通讯作者: Guralnick, R.