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

大约200年前，E.Galois发现了一个绝妙的想法，将对称性作为一个对象来研究，并以这种方式解决与给定的数学结构相关的基本和看似复杂的问题。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悬而未决。有如此多的对称组，以至于人们渴望建造巨大的观察塔，从那里人们可以看到所有这些对称组。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，他就会获得一把通向当前数学的许多密室的万能钥匙。在过去的30年里，A.Merkurjev、M.Rost、A.Suslin和V.Voevodsky在获得绝对Galois群的上同调不变量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大信息，但它是用上同调语言编码的。这项拟议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切合实际的描述。在最近与D.Benson，N.Lemire和J.Swallow的联合工作中，我们发现了称为T-群的绝对Galois群的某些商的完整刻画。T-群的分类提供了关于绝对Galois群的重要信息。T-群可以被视为我们的塔的基础，现在是时候爬得更高，并对更大的伽罗瓦群的商进行分类。基于与M.Spira的合作，以及最近与A.Adem、D.Benson、N.Lemire、A.Schultz和J.Swallow关于拓扑、模表示理论和Galois上同调的Galois模结构之间的相互作用的工作，我们计划提供关于绝对Galois群的更多有意义的信息，并将其应用于解决纯数学中的基本问题。加拿大的数论及相关话题在国际上备受推崇。希望该项目将有助于在加拿大保持这一高标准和传统。

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.