Cohomology and Representations of Finite and Algebraic Groups with Applications

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簇上的作用。群是数学中的基本工具之一，它产生于许多领域，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，并在使用群论研究其他领域方面引发了一场革命。这种分类基本上是说，有限单群是单李群的类比，因此要理解它们，就必须研究单李群和代数群。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在黎曼曲面上的群(以及它们在有限域上的类似物)。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的基本问题的进步。由于计算的进步，群论的用途也得到了极大的扩展。这个项目的另一个方面是找到有限单群的有用表示，这将导致更高的计算效率。这个项目解决的第三个重要问题是极大地推广所谓的Tits替代方案。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数是高度相连的。这在计算机科学中一直是非常重要的。研究生将通过研究来培养。特别地，我们计划研究半单代数群的强稠密子群的生成问题，并证明了Tits择优的推广。这将给出数论中关于超强逼近的一些新结果和关于扩展图的一些结果。早期的PI与Breuillard、Green和Tao的结果将使用新的更强大的方法进行推广。我们还想证明这样一个猜想：每个有限单群都有一个表示，有两个生成元，至多有四个关系。这应该会导致计算数论的进步。群论的深刻结果导致了有限域上双射多项式(被视为光滑射影曲线上的映射)基本问题的重大进展，并在密码学中得到了应用，解决了一个多世纪以来的问题。该项目的另一个目标是完全分类低亏格Riemann曲面的覆盖的单元群，从而导致数论的根本突破，并从一般Riemann曲面(最先在Zariski的论文中研究)分类映射的单元群。最后，我们想要在不可约线性表示中对单代数群的一般稳定子进行分类。这已经在特征零中完成了，但在正特征中需要新的想法。这将对基本维度产生影响，一些特殊情况将适合Bhargava解决代数家族的有趣分类问题的计划。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.