Sylow-p subgroups of absolute galois groups, representation theory, and galois cohomology

大约200年前，E.伽罗瓦发现了一个绝妙的想法，把对称作为一个对象来研究，并通过这种方式来解决与给定数学结构相关的基本和看似棘手的问题。今天伽罗瓦理论是现代数学的核心部分。然而，伽罗瓦理论中的一些基本问题仍然没有解决。对称的组合如此之多，以至于人们渴望建造巨大的观测塔，从那里人们可以看到所有这些对称的组合。这些塔被称为绝对伽罗瓦群。如果一个人知道绝对伽罗瓦群的一些特殊性质，他就会得到一把打开现代数学许多密室的万能钥匙。在过去的30年中，A. Merkurjev， M. Rost， A. Suslin和V. Voevodsky在获得绝对伽罗瓦群的上同调不变量的非常具体的信息方面取得了显著的进展。因此，我们现在有了关于绝对伽罗瓦群的强有力的信息，但它是用上同调的语言编码的。这项研究的主要目的是将一些强大的信息解码成对绝对伽罗瓦群的实际描述。在最近与D. Benson， N. Lemire和J. Swallow的合作中，我们发现了绝对伽罗瓦群t群的某些商的完整描述。t群的分类提供了绝对伽罗瓦群的重要信息。t群可以被看作是我们塔楼的基础，现在是时候爬得更高，分类更大的伽罗瓦群商了。基于与M. Spira的共同工作，以及最近与A. Adem、D. Benson、N. Lemire、A. Schultz和J. Swallow在拓扑、模表示理论和伽罗瓦上同调的伽罗瓦模结构之间的相互作用方面的工作，我们计划提供关于绝对伽罗瓦群的进一步重要信息，并将其应用于纯数学基本问题的解决。加拿大数论及相关课题在国际上享有盛誉。希望这个项目将有助于在加拿大维持这种高标准和传统。

Almost 200 years ago E. Galois discovered a brilliant idea to study symmetries as one object, and in this way, to solve fundamental and seemingly intractible problems related to given mathematical structures. Today Galois theory is a central part of current mathematics. However some basic problems in Galois theory are still open. There are so many groups of symmetries that there became a desire to build enormous observing towers from which one could see all of these groups of symmetries. These towers are called absolute Galois groups. If one knew some special properties of absolute Galois groups one would obtain a master key to many secret rooms of current mathematics. Remarkable progress was completed during the last 30 years, by A. Merkurjev, M. Rost, A. Suslin, and V. Voevodsky, on obtaining very specific information in terms of cohomological invariants of absolute Galois groups. Thus we now have powerful information about absolute Galois groups, but it is encoded in the language of cohomology. The main objective of this proposed research is to decode some of the powerful information into down-to-earth descriptions of absolute Galois groups. In recent joint work with D. Benson, N. Lemire and J. Swallow we found complete descriptions of certain quotients of absolute Galois groups called T-groups. The classification of T-groups provides important information about absolute Galois groups. T-groups can be viewed as foundations of our towers and now it is time to climb higher and classify larger quotients of Galois groups. Based on joint work with M. Spira, and more recent work with A. Adem, D. Benson, N. Lemire, A. Schultz, and J. Swallow, on the interplay between topology, modular representation theory and the Galois module structure of Galois cohomology, we plan to provide further significant information about absolute Galois groups and to apply it to the solution of basic problems in pure mathematics. Number theory and related topics in Canada are internationally well regarded. It is hoped that this project will contribute to sustaining this high standard and tradition in Canada.