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

PI计划研究与表示、线性和置换表示有关的有限群和代数群的一些基本问题，以及在算术代数几何问题中的上同调应用——特别是与多项式、有理函数、自同构和曲线覆盖有关的问题。利用Tiep和PI的工作，我们将研究简单群的小表示。这项工作已经应用于Katz， Kollar和Larsen在代数几何中的各种问题。最关键的开放情况是对模的外幂上不可约的所有闭子群进行分类。这也与有限简单群的极大子群的分类问题密切相关。后一个问题引出了许多基本问题。特别地，PI将研究关于各种简单群族的有界维的不可约表示的数量边界问题。PI和他的合作者最近表明，可能除了一个家族之外，每个有限单族都有最多50个关系的呈现。在许多情况下，可以做得更好（例如，无限数量的交替组可以用两个生成器和四个关系表示）。允许关系的数量增加一点，甚至可以生成简短的演示文稿（本质上是最好的）。最根本的问题是离散表示与无限表示和上同态之间的关系（考虑无限表示的一种方式是，如果一个是由生成器和关系给出的，并且已经知道这个群是有限的，那么就可以识别这个群）。这导致人们试图在一个简单模中给出有限和代数群的第一和第二上同群的大小的很好的界。一个目标是证明每一个有限单群都有一个有两个生成器和最多四个关系的无限表示（没有人能做得更好）。即使没有盈利条件，这也可能是正确的，但没有人知道如何解决这个问题。另一个主要问题是在有限域上完成异常多项式的分类。异常多项式就是假定场大小相对于度足够大的双射多项式，自20世纪90年代迪克森的论文以及舒尔、弗里德等人的论文以来，这种多项式已经得到了认真的研究。将深度群论与算术代数几何方法相结合，对次数不是特征次幂的不可分解异常多项式进行了分类。PI打算使用这些方法和新的结果来研究这些问题。这应该适用于密码学。

The PI plans to study some basic problems about finite and algebraic groups related to presentations, linear and permutation representations, and cohomology with applications to the problems in arithmetic algebraic geometry--particularly questions related to polynomials, rational functions, and automorphisms and coverings of curves. Using the work of Tiep and the PI, small representations of simple groups will be studied. This work has already had applications to various questions of Katz, Kollar, and Larsen in algebraic geometry. The most critical open case is to classify all closed subgroups which are irreducible on exterior powers of a module. This is also closely related to questions about the classification of maximal subgroups of the finite simple groups. This latter questions leads to many basic problems. In particular the PI will look at problems about bounding the number of irreducible representations of bounded dimension for various families of simple groups.The PI and his collaborators have shown recently that with the possible exception of one family, every finite simple group has a presentation with at most fifty relations. In many cases, one can do much better (for example, an infinite number of alternating groups can be presented with two generators and four relations). Allowing the number of relations to increase a bit, one can even produce short presentations (essentially best possible). The most fundamental problem is the relationship between discrete presentations and profinite presentations and cohomology (one way to think of a profinite presentation is that if one is given by generators and relations and one already knows the group is finite, then one can identify the group). This leads one to try to produce very good bounds on the size of the first and second cohomology groups of finite and algebraic groups with coefficients in a simple module. One goal is to prove that every finite simple group has a profinite presentation with two generators and at most four relations (one cannot do better). This may be even be true without the profiniteness condition but no one has any idea of how to approach this in general. Another major problem is to complete the classification of exceptional polynomials over finite fields. Exceptional polynomials are precisely the bijective polynomials assuming the field size is sufficiently large compared to the degree and have been studied seriously since the thesis of Dickson in the 1890's, as well as by Schur, Fried and others. The classification of indecomposable exceptional polynomials whose degree is not a power of the characteristic used a combination of deep group theory together with methods arithmetic algebraic geometry. The PI intends to use these methods and use new results to study these problems. This should have applications to cryptography.