Cohomology and Representations of Finite and Algebraic Groups with Applications

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。群是数学中的基本工具之一，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，并导致了使用群论研究其他领域的革命。分类基本上说，有限单群是简单李群的类似物，因此要理解它们，必须学习简单李群和代数群。理解和使用群论的最好方法是研究群对不同对象的作用。这个项目的一个方面是理解作用于黎曼曲面的群（以及它们在有限域上的类似物）。这将导致对基本对象（包括有理函数）的新的基本理解，并应导致密码学和数论基本问题的进步。群论的效用也由于计算的进步而大大扩展。这个项目的另一个方面是找到有用的有限单群，这将导致更多的计算效率的介绍。第三个重要的问题，在这个项目中解决的是大大推广什么是所谓的山雀替代。这将导致显示扩展图的存在（和构造）的结果。这些图相对于其中的边的数量是高度连通的。这在计算机科学中非常重要。研究生将通过研究进行培训。特别是，我们计划研究的问题，生产强稠密的子群的半单代数群，并证明了推广的山雀替代。这将给出数论中关于超强逼近的一些新结果和扩张图上的结果。PI的早期结果，与Breuillard，绿色，陶，将推广使用新的更强的方法。我们还想证明一个猜想，即每个有限单群都有一个表示有两个生成元和至多四个关系。这将导致计算数论的进步。群论的深入成果导致了有限域上双射多项式（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一个多世纪的老问题。该项目的另一个目标是完全分类单值群覆盖低属黎曼曲面导致根本突破数论，并分类单值群映射从一般黎曼曲面（首先研究Zenkiki的论文）。最后，我们想对不可约线性表示中的简单代数群的一般稳定子进行分类。这是在零特性中完成的，但在积极特性中需要新的想法。这将对基本维度产生影响，一些特殊情况将适合Bhargava计划，以解决代数族的有趣分类问题。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.