"Sylow-p Subgroups of Absolute Galois Groups, their Natural Quotients, and Galois Cohomology"

“绝对伽罗瓦群的 Sylow-p 子群、它们的自然商和伽罗瓦上同调”

• 批准号: 41981-2012
• 负责人: Minac, Jan
• 金额: $ 2.55万
• 依托单位: University of Western Ontario
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2014
• 资助国家: 加拿大
• 起止时间: 2014-01-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=544905
• 关键词: Sylow; Subgroups; Absolute; Galois; Groups

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, S. Chebolu, I. Efrat, J. Labute, N. Lemire and J. Swallow; we have found certain and various small quotients of absolute Galois groups. These groups can be viewed as foundations of our towers and now it is time to climb higher and to classify larger quotients of Galois groups. Based on joint work with M. Spira, and more recent work with A. Adem, D. Benson, I. Efrat, 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.

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进行研究，并通过这种方式解决与给定数学结构相关的基本且看似棘手的问题。今天，伽罗瓦理论是当前数学的核心部分。然而，伽罗瓦理论中的一些基本问题仍然悬而未决。对称群的数量如此之多，以至于人们渴望建造巨大的观测塔，从中可以看到所有这些对称群。这些塔称为绝对伽罗瓦群。如果一个人知道绝对伽罗瓦群的一些特殊性质，就可以获得一把打开当前数学许多秘密房间的万能钥匙。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方面的非常具体的信息方面取得了显着的进展。因此，我们现在拥有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。这项研究的主要目标是将一些强大的信息解码为绝对伽罗瓦群的实际描述。最近与 D. Benson、S. Chebolu、I. Efrat、J. Labute、N. Lemire 和 J. Swallow 合作；我们已经找到了绝对伽罗瓦群的某些和各种小商。 这些群可以被视为我们塔的基础，现在是时候爬得更高并对伽罗瓦群的更大商进行分类了。基于与 M. Spira 的合作以及最近与 A. Adem、D. Benson、I. Efrat、N. Lemire、A. Schultz 和 J. Swallow 的合作，关于拓扑、模表示理论和伽罗瓦之间的相互作用伽罗瓦上同调的模结构，我们计划提供有关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。加拿大的数论及相关主题在国际上享有盛誉。希望该项目将有助于维持加拿大的这一高标准和传统。