Mathematical Sciences: Cohomology and Actions of Finite Groups

这个项目将结合和应用的想法和技术 从表示论和群作用到 组主要目标如下： 1)扩展正在进行的项目，以计算和解释mod-2 有限群的上同调 希望能找到一个 一组有用的例子，这将导致进一步的理论 这一领域的发展，就像过去发生的那样。 2)分析某些离散群的上同调和K-理论 例如算术组和映射类组，使用 模块化表示理论的方法以及最近的 K理论中的分裂结果。 3)将等变K-理论的方法应用于某些G-复形 与有限群G相关联。 这是由一个猜想激发的 在模表示论中，涉及到 复杂的。 群是对称性的抽象数学体现。 它们在数学中无处不在， 应用于其他科学，几乎是不可或缺的 在量子力学中。 因此，人们不能低估 努力更好地了解其属性的重要性 比我们现在做的-没有告诉在哪里这样的理解将 但数学物理是一个合理的假设。 这 项目在某种意义上是一个引导项目，因为上同调是 已经有了一个基于群的理论，各种上同调群。 调查人员在应用 上同调群在研究其他 群体，特别是某些所谓的有限零星的 他们击败了其他人的努力。 他的项目将 通过进行困难的计算和不断发展 理论，使这些计算的结果。

This project will incorporate and apply ideas and techniques from representation theory and group actions to the cohomology of groups. The main objectives are as follows: 1) Extend an ongoing project to calculate and interpret the mod-2 cohomology of certain key finite groups. The hope is to find a set of useful examples which will lead to further theoretical developments in the field, as has happened in the past. 2) Analyze the cohomology and K-theory of certain discrete groups such as arithmetic groups and mapping class groups, using methods from modular representation theory as well as a recent splitting result in K-theory. 3) Apply methods from equivariant K-theory to certain G-complexes associated to a finite group G. This is motivated by a conjecture in modular representation theory involving the isotopy groups of the complex. Groups are the abstract mathematical embodiment of symmetry. They are ubiquitous in mathematics and in many mathematical applications to the other sciences, being well nigh indispensable in quantum mechanics. As such, one cannot underestimate the importance of efforts to understand their properties even better than we do now - there is no telling where such understanding will pay off, but mathematical physics is a plausible assumption. This project is in a sense a bootstrap project, for cohomology is already a theory based on groups, the various cohomology groups. The investigator has been very successful in applying the cohomology groups in studying the structure and properties of other groups, particularly certain of the so-called finite sporadic groups which had defeated the efforts of others. His project will advance both by making difficult calculations and by evolving theory to bring order to the results of these calculations.