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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=683335
• 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. Minc; 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. 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。然而，伽罗瓦理论中的一些基本问题仍然存在，研究它们中的许多不同的对称性是一项艰巨的任务。它们被称为绝对伽罗瓦群，这给我们找到一种模式带来了希望。如果我们能够找到绝对伽罗瓦群的结构和基本性质，我们就有可能解决许多求解方程的基本问题，以及代数和几何中的进一步问题。 、拓扑、物理、密码学；以及大数据系统的问题。******然而，绝对伽罗瓦群是深奥的、基本的、神秘的对象，过去一些最好的数学家很难解决它们；数学家，例如 E. Artin 和 O. Schreier 在 20 世纪 30 年代，以及最近 40 年间，J. Milnor、A. Merkurjev、M. Rost、A. Suslin、V. Voevodsky 等人发现了绝对伽罗瓦群的显着而深刻的性质；特别是他们解决了布洛赫-加藤猜想，要很好地理解这一进展对于绝对结构性质的意义是一个巨大的挑战。伽罗瓦群本身。 ****** 最近，与梅西积有关的两个新猜想开辟了一条新的、新鲜的、创新的道路，这两个猜想最初是由拓扑学家提出的。拓扑学和物理学，与结等图形的形状相关，在代数环境中工作得非常好，从而带来非凡的新见解。*基于以前的工作，包括 W. Dwyer、M. Hopkins 和 K. Wickelgren 的工作， I. Efrat 和 J. Minc；与 N. D. Tân 一起提出了 n-Massey 消失猜想和核猜想，这些猜想已经引发了一系列的活动、新的结果、新的见解和新的希望。 **因此，与 N. D. Tân 和其他各种合作者一起，我们现在有了一个令人兴奋的计划，并取得了第一个非常令人鼓舞的结果，用于推导与解决这些问题相关的绝对伽罗瓦群的基本属性。猜想，同时为布洛赫-加藤猜想的​​可能改进提供更多线索 ******加拿大对数论和代数群的研究在国际上非常受重视，这些研究的结果具有重要意义。涵盖当前数学的整个领域以及物理、化学和工业的重要部分，希望该项目将有助于维持加拿大的这一高标准和传统。