Absolute Galois groups and Massey products

大约200年前，E.伽罗瓦发现了一个绝妙的想法，将对称作为一个对象来研究，并通过这种方式来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，在物理和化学的某些部分也可以找到。然而，伽罗瓦理论中的一些基本问题仍然没有解决。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，他们中的许多人聚集在一起，被称为绝对伽罗瓦群体，这给了我们找到一个模式的希望。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基本的方程求解问题，以及代数、几何、拓扑、物理、密码学等方面的进一步问题；以及大数据系统的问题。******然而，绝对伽罗瓦群是深刻的、基本的和神秘的物体，很难解决它们。过去一些最优秀的数学家；20世纪30年代的E. Artin和O. Schreier等数学家，以及最近40年来的J. Milnor、A. Merkurjev、M. Rost、A. Suslin、V. Voevodsky等数学家；发现了编码在上同调不变量中的绝对伽罗瓦群的显著而深刻的性质。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伽罗瓦群本身的结构性质的意义是一个巨大的挑战。******最近开辟了一条新的、新鲜的、创新的道路，与最初由拓扑学家引入的Massey产品有关的两个新猜想。事实证明，在拓扑学和物理学中使用的一些与结等图形形状有关的经典和新思想，在代数环境中表现得非常好，从而产生了非凡的新见解。*基于以前的工作，包括W. Dwyer， M. Hopkins和K. Wickelgren， I. Efrat和J. Minc的工作；与N. D. tn一起，我们提出了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. t<e:1>和其他合作者一起，我们现在有了一个令人兴奋的计划，第一个非常令人鼓舞的结果，可以推断出与解决这些猜想有关的绝对伽罗瓦群的基本性质，同时为可能的Bloch-Kato猜想的改进带来更多的光明。******加拿大数论和代数群的研究在国际上享有很高的声誉。这些研究的结果对当前数学的整个范围以及物理、化学和工业的重要部分都有影响。希望这个项目将有助于在加拿大维持这种高标准和传统。

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 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.