Mathematical Sciences: Cohomology and Actions of Finite Groups

这个项目将结合和应用从表象理论和群行动到群的上同调的思想和技术。主要目的如下：1)扩展一个正在进行的项目以计算和解释某些关键有限群的mod-2上同调。我们希望找到一组有用的例子，这些例子将导致该领域的进一步理论发展，就像过去发生的那样。2)利用模表示理论中的方法以及最近K-理论中的一个分裂结果，分析了算术群和映射类群等离散群的上同调和K-理论。3)将等变K-理论的方法应用于与有限群G相关的某些G-复形。这是由模表示理论中的一个猜想引起的，该猜想涉及复形的同调群。群是对称性的抽象数学体现。它们在数学中无处不在，在其他科学的许多数学应用中，在量子力学中几乎是不可或缺的。因此，人们不能低估比我们现在更好地理解它们的性质的努力的重要性--目前还不清楚这种理解将在哪里得到回报，但数学物理是一个看似合理的假设。这个项目在某种意义上是一个自举项目，因为上同调已经是一种基于群的理论，各种上同调群。这位研究者非常成功地将上同调群应用于其他群的结构和性质的研究，特别是某些所谓的有限零星群，这些群已经击败了其他人的努力。他的项目将通过进行困难的计算和进化理论来推动这些计算结果的有序。

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.