"Sylow-p Subgroups of Absolute Galois Groups, their Natural Quotients, and Galois Cohomology"

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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 intractible problems related to given mathematical structures. Today Galois theory is a central part of current mathematics. However some basic problems in Galois theory are still open. There are so many groups of symmetries that there became a desire to build enormous observing towers from which one could see all of these groups of symmetries. These towers are called absolute Galois groups. If one knew some special properties of absolute Galois groups, one would obtain a master key to many secret rooms of current mathematics. Remarkable progress was completed during the last 30 years by A. Merkurjev, M. Rost, A. Suslin, and V. Voevodsky, on obtaining very specific information in terms of cohomological invariants of absolute Galois groups. Thus we now have powerful information about absolute Galois groups, but it is encoded in the language of cohomology. The main objective of this proposed research is to decode some of the powerful information into down-to-earth descriptions of absolute Galois groups. In recent joint work with D. Benson, S. Chebolu, I. Efrat, J. Labute, N. Lemire and J. Swallow; we have found certain and various small quotients of absolute Galois groups. These groups can be viewed as foundations of our towers and now it is time to climb higher and to classify larger quotients of Galois groups. Based on joint work with M. Spira, and more recent work with A. Adem, D. Benson, I. Efrat, N. Lemire, A. Schultz and J. Swallow, on the interplay between topology, modular representation theory and the Galois module structure of Galois cohomology, we plan to provide further significant information about absolute Galois groups, and to apply it to the solution of basic problems in pure mathematics. Number theory and related topics in Canada are internationally well regarded. It is hoped that this project will contribute to sustaining this high standard and tradition in Canada.