Representations and Presentations of Finite Groups and Coverings of Curves

有限群和曲线覆盖的表示和展示

• 批准号: 0653873
• 负责人: Robert Guralnick
• 金额: $ 17.64万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0653873&HistoricalAwards=false
• 关键词: Representations; Presentations; Finite; Groups; Coverings

The proposer will 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 coverings of curves. In some recent work with Kantor, Kassabov and Lubotzky, it was shown that every finite simple group, with the possible of one family, have short and/or bounded presentations. This was a very surprising result and much better than conjectured. It does lead to some important and basic problems including removing the exception, understanding the differences (if any) between discrete presentations and profinite presentations and getting a good bound on cohomology. More generally, the proposer will consider linear and permutation representations of finite and algebraic groups and use these results to study problems in number theory and algebraic geometry. One wants to obtain fairly precise information about finite groups acting on curves and higher dimensional varieties. In the case of curves, one wants to know about possibilities for ramification groups. In higher dimensions, the non-irreducibility of certain modules leads to some interesting properties of varieties of families of curves as shown by earlier work of the proposer with Tiep answering questions of Katz and Kollar.Since the classification of finite simple groups was completed, many problems about finite groups and related questions in other areas of mathematics have been solved that would have been unimaginable without this classification. The classification manifests itself in two ways. First of all, it is a list and so to prove properties about all finite groups, one can often reduce it to questions about simple groups and then attack the families. For example, one can show that the group of all permutations of a set of a finite set has a presentation with 4 generators and 10 relations. It seems likely that 2 generations and 3 relations suffice (that would be the best possible answer). The standard presentations, known for a century, have a linear number of generators and quadratic number of relations (in the size of the set). Secondly, the classification also gives information about the subgroup structure of the simple groups. This is closely related to understanding how simple groups can act on sets as permutations and on vectors by linear transformations. Rather surprisingly, such information can lead to breakthroughs in seemingly unrelated topics. For example, one aspect of the proposal is to classify exceptional polynomials over a finite field. These are polynomials which are bijective (with degree large enough compared to the size of the field). Using knowledge of simple groups and their subgroups has lead to a complete classification aside from the case where the degree is a power of the characteristic of the field. The proposer expects to complete this classification. These polynomials have been studied since the late 1800's but it has been only in the last 15 years that there has been major progress. The proposer will study relatedproblems in number theory and geometry and translate them to questions in group theory and then apply this powerful theory.

该提案者将研究有关有限和代数群体与演示，线性和置换表示以及共同体相关的一些基本问题，并在算术代数几何学中应用了问题，尤其是与多项式，理性功能和曲线覆盖有关的问题。 在与Kantor，Kassabov和Lubotzky最近的一些工作中，表明每个有限的简单群体，有一个家庭的可能性，都有简短和/或有限的演讲。 这是一个非常令人惊讶的结果，比猜想要好得多。它确实导致了一些重要和基本的问题，包括删除例外，了解离散演示和涂鸦演示之间的差异（如果有的话），并构成了共同体的良好束缚。 更一般而言，提议者将考虑有限和代数群体的线性和置换表示，并使用这些结果来研究数字理论和代数几何形状中的问题。 一个人希望获得有关作用于曲线和更高维度品种的有限群体的相当精确的信息。 在曲线的情况下，人们想了解后果小组的可能性。 在较高的维度中，某些模块的不可回答性导致了一些有趣的曲线家族的有趣特性，如提议者的早期工作所示，TIEP回答了Katz和Kollar的问题。由于有限简单组的分类已经完成，因此，在数学的其他领域中，许多有限的问题和相关问题都无法解决这种分类，这是可以解决的。 分类以两种方式表现出来。 首先，这是一个列表，因此要证明所有有限群体的属性，人们通常可以将其减少到有关简单组的问题，然后攻击家庭。 例如，可以证明一组有限集的所有排列的组都有一个带有4个发电机和10个关系的演示文稿。似乎有2代和3个关系就足够了（这是最好的答案）。 一个世纪已知的标准演示文稿具有线性数量的发电机和二次关系数（以集合的大小）。 其次，分类还提供了有关简单组的亚组结构的信息。 这与理解简单组如何通过线性变换对置换和向量作用密切相关。令人惊讶的是，这些信息可能会导致看似无关的话题的突破。例如，该提案的一个方面是在有限字段上对特殊的多项式进行分类。 这些是两种族的多项式（与田间的大小相比，程度足够大）。 使用简单组及其子组的知识，除了该学位是该领域特征的力量外，还导致了一个完整的分类。 提议者希望完成此分类。自1800年代后期以来，这些多项式已经进行了研究，但直到过去15年才取得了重大进展。 提议者将研究数字理论和几何形状的相关问题，并将其转化为群体理论中的问题，然后应用这一强大的理论。