Absolute Galois groups and Massey products

绝对伽罗瓦群和梅西积

• 批准号: RGPIN-2017-05344
• 负责人: Minac, Jan
• 金额: $ 2.19万
• 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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=659843
• 关键词: Absolute; Galois; groups; Massey; products

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 intractable problems related to given mathematical structures. Today Galois theory is a central part of current mathematics, and is also found in some parts of physics and chemistry. However some basic problems in Galois theory are still open. There are many different symmetries of polynomial equations. It is a daunting task to study them. Yet there are assemblies of many of them which are known as absolute Galois groups, which gives us hope to find a pattern. If we could find the structures and basic properties of absolute Galois groups, we could possibly solve a number of fundamental problems of solving equations, further problems in algebra and geometry, topology, physics, cryptography; and problems with large data systems.******However, absolute Galois groups are deep, fundamental and mysterious objects, and it is hard to tackle them. Some of the best mathematicians in the past; mathematicians such as E. Artin and O. Schreier in the 1930s, and more recently in the last 40 years, J. Milnor, A. Merkurjev, M. Rost, A. Suslin, V. Voevodsky, and others; found remarkable, deep properties of absolute Galois groups encoded in cohomological invariants. In particular they solved the Bloch-Kato conjecture. ******It is a great challenge to well understand the meaning of this progress for the structural properties of absolute Galois groups themselves. ******Very recently a new, fresh, innovative road was opened up with two new conjectures related to Massey products, which were originally introduced by topologists. It has turned out that some classical and new ideas used in topology and physics, related to the shape of figures like knots, work extraordinarily well in an algebraic setting leading to remarkable new insights.*Based on previous work, including the work of W. Dwyer, M. Hopkins and K. Wickelgren, I. Efrat and J. Minác; together with N. D. Tân we formulated the n-Massey vanishing conjecture and the kernel conjecture. These conjectures have already led to a flurry of activity, new results, new insights, and new hopes. ******Thus together with N. D. Tân and various other collaborators, we now have an exciting program with the first very encouraging results for deducing the fundamental properties of the absolute Galois groups related to solving these conjectures, and at the same time bringing more light to a possible refinement of the Bloch-Kato conjecture. ******Studies of number theory and algebraic groups in Canada are very well-regarded internationally. The results of these studies have implications throughout the whole spectrum of current mathematics and significant parts of physics, chemistry and industry. It is hoped that this project will contribute to sustaining this high standard and tradition in Canada.

大约200年前，E。伽罗瓦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，也在物理学和化学的某些部分中找到。然而，伽罗瓦理论中的一些基本问题仍然是开放的。多项式方程有许多不同的对称性。研究它们是一项艰巨的任务。然而，有许多集合体被称为绝对伽罗瓦群，这给了我们找到模式的希望。如果我们能找到绝对伽罗瓦群的结构和基本性质，我们就有可能解决一些解方程的基本问题，代数和几何、拓扑、物理、密码学中的进一步问题;以及大数据系统的问题。然而，绝对伽罗瓦群是深刻的，基本的和神秘的对象，很难处理它们。过去一些最好的数学家，如E。Artin和O. Schreier在20世纪30年代，以及最近的40年里，J. Milnor，A. Merkurjev，M. Rost，A. Suslin，V. Voevodsky等人;发现了编码在上同调不变量中的绝对伽罗瓦群的显著的、深层的性质。特别是他们解决了布洛赫-加藤猜想。** 很好地理解这一进展对于绝对伽罗瓦群本身的结构性质的意义是一个巨大的挑战。** 最近，一个新的，新鲜的，创新的道路开辟了两个新的Massey产品，这是最初介绍的拓扑学家。事实证明，拓扑学和物理学中使用的一些经典和新的思想，与像结这样的图形的形状有关，在代数环境中工作得非常好，导致了引人注目的新见解。在前人工作的基础上，包括W. Dwyer，M.霍普金斯和K.威克尔格伦岛Efrat and J. Minác; together with N. D.我们用公式表示了n-梅西消失猜想和核猜想。这些成果已经带来了一系列的活动、新的成果、新的见解和新的希望。****** D. Tân和其他各种合作者，我们现在有一个令人兴奋的计划，第一个非常令人鼓舞的结果，用于推导与解决这些问题有关的绝对伽罗瓦群的基本性质，同时为布洛赫-加藤猜想的可能改进带来更多的光。** 加拿大的数论和代数群研究在国际上非常受欢迎。这些研究的结果对整个当前数学领域以及物理、化学和工业的重要部分都有影响。希望这个项目将有助于保持加拿大的高标准和传统。