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

PI计划研究有关有限和代数群的一些基本问题介绍，线性和置换表示，并与算术代数几何中的问题应用上同调-特别是有关多项式，有理函数，自同构和覆盖曲线的问题。利用Tiep和PI的工作，将研究简单群的小表示。 这项工作已经应用到各种问题的卡茨，Kollar和拉森在代数几何。 最关键的开情形是对模的所有外幂上不可约的闭子群进行分类。 这也与有限单群的极大子群的分类问题密切相关。后一个问题导致了许多基本问题。特别是PI将着眼于问题的界限的数量不可约表示有界的维度为各种家庭的简单groups.The PI和他的合作者最近表明，可能的例外之一家庭，每一个有限的简单组有一个介绍，最多50个关系。在许多情况下，我们可以做得更好（例如，可以用两个生成元和四个关系来表示无穷多个交错群）。 允许关系的数量增加一点，甚至可以制作简短的演示文稿（基本上是最好的）。最基本的问题是离散表示与profinite表示和上同调之间的关系（一种思考profinite表示的方法是，如果一个表示由生成元和关系给出，并且已经知道群是有限的，那么可以识别群）。这导致人们试图产生非常好的界限的大小的第一和第二上同调群的有限和代数群的系数在一个简单的模块。 一个目标是证明每个有限单群都有一个profinite表示，其中有两个生成元和至多四个关系（不能做得更好）。在没有无穷性条件的情况下，这甚至可能是真的，但是没有人知道如何一般地处理这个问题。 另一个主要问题是完成有限域上例外多项式的分类。例外多项式正是双射多项式假设领域的大小是足够大的程度相比，并已认真研究以来的论文迪克森在19世纪90年代，以及舒尔，弗里德和其他人。利用深群论和算术代数几何相结合的方法，对次数不是特征幂的不可分解例外多项式进行了分类。PI打算使用这些方法并使用新的结果来研究这些问题。 这应该可以应用到密码学中。

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.