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

基本信息

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

项目摘要

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.

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。小组是数学中的基本工具之一，在许多领域都出现，包括分析，几何学和数理论以及化学和物理学对称性的研究。有限简单组的分类于2006年完成，并导致了使用小组理论研究其他领域的革命。分类基本上说，有限的简单组是简单的谎言组的类似物，因此，要理解它们，必须研究简单的谎言和代数组。理解和使用群体理论的最佳方法是研究小组对不同对象的作用。该项目的一个方面是了解作用于Riemann表面（及其在有限领域的类似物）上的群体。这将导致对包括理性功能在内的基本对象的新基本理解，并应导致密码学和数字理论中的基本问题的进步。由于计算的进步，小组理论的实用性也大大扩展了。该项目的另一个方面是找到有限简单组的有用演示，这将提高计算效率。该项目中解决的第三个重要问题是极大地概括了所谓的山雀替代方案。这将导致结果显示扩展器图的存在（和构造）。这些图形相对于其中的边数高度连接。这在计算机科学中非常重要。研究生将通过研究培训。特别是，我们计划研究产生半密度代数群体的密度密度亚组的问题，并证明山雀替代方案的概括。这将为数字理论和扩展器图上的superstrong近似值提供一些新的结果。 Breuillard，Green和Tao的PI的早期结果将使用新的强方法进行概括。我们还想证明每个有限简单组都有两个发电机和最多四个关系的介绍。这应该导致计算数理论的进步。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。该项目的另一个目标是完全对低属的覆盖物的单型组进行分类，从而导致数量理论的基本突破，并从通用的Riemann表面进行分类（首次在Zariski的论文中研究）。最后，我们希望将简单代数组的通用稳定器分类为不可还原线性表示。这是以零特征进行的，但是在积极特征中需要新的想法。这将对基本维度产生后果，一些特殊情况将适合Bhargava计划，以解决代数家庭的有趣分类问题。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响审查标准来评估的支持。