Grants and Contributions:

Grant or Award spanning more than one fiscal year. (2017-2018 to 2022-2023)

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.