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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=711252
• 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，将对称性研究为一个对象，并以这种方式解决了与给定的数学结构有关的基本和看似棘手的问题。如今，加洛伊斯理论已成为当前数学的核心部分，也可以在物理和化学的某些部分中找到。但是，加洛伊斯理论中的一些基本问题仍然开放。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，其中许多人都被称为绝对Galois群体，这使我们希望找到一种模式。如果我们能找到绝对Galois群体的结构和基本特性，我们可能会解决解决方程，代数和几何学，拓扑，物理学，密码学的进一步问题的许多基本问题；以及大型数据系统的问题。 但是，绝对的Galois团体是深层，基本和神秘的对象，很难解决它们。过去一些最好的数学家； 1930年代的E. Artin和O. Schreier等数学家，以及最近40年，J。Milnor，A。Merkurjev，M。Rost，A。Suslin，A。Suslin，V。Voevodsky等；发现了在共生不变式中编码的绝对Galois基团的显着深层特性。特别是他们解决了Bloch-Kato的猜想。 很好地了解绝对Galois群体本身的结构特性的含义是一个巨大的挑战。 最近，一条新的，新鲜的创新道路开放了两种与梅西产品有关的新猜想，这些猜想最初是由拓扑专家引入的。事实证明，拓扑和物理学中使用的一些古典和新想法与诸如结的形状有关，在代数环境中表现出色，从而带来了非凡的新见解。 根据先前的工作，包括W. Dwyer，M。Hopkins和K. Wickelgren，I。Efrat和J. Minc的工作；我们与N. D.Tân一起制定了N-Massey消失的概念和内核概念。这些概念已经导致了一系列活动，新的结果，新见解和新希望。 与N. D.Tân和其他各种合作者一起，我们现在制定了一个令人兴奋的计划，首次令人鼓舞的结果，以推论与解决这些猜想有关的绝对Galois团体的基本属性，同时为Bloch-Kato猜想提供了更多的光明。 对加拿大的数字理论和代数群体的研究在国际上是非常良好的。这些研究的结果在当前数学的整个范围内以及物理，化学和工业的重要部分都具有含义。希望该项目能够在加拿大维持这种高标准和传统。