Ambrus Pál and I organised a study group into the series of papers "Smooth Profinite Groups", by de Clercq and Florence, with the eventual goal being to understand their proof of the Norm Residue Isomorphism Theorem. A very neat summary of the idea behind the proof was written up by Wojtek Wawrów as entry 27 on the "Things I learnt this week" part of his website. A one sentence summary of the idea is that "We should be able to deduce the Norm-Residue Isomorphism Theorem using Galois Cohomology and Hilbert Theorem 90 type theorems, rather than having to build up the whole machinery of motivic homotopy".
The study group has now finished. A pdf overview of the original plan for the study group can be found here, with the only change being that the final two lectures were compressed into one. A huge thank you to all of our speakers.
Update January 2025: While doing this study group, it was known that the proof of "The Uplifting Theorem" contained an error, which we at the time believed was fixable. A recent preprint by Merkurjev-Scavia gives a counterexample to the statement of this theorem, needed to deduce the norm residue isomorphism theorem. The papers studied in this study group are still interesting and employ interesting and beautiful maths, but be aware that the final result can't hold.
(12/10) - Introduction and overview of the Norm-Residue isomorphism theorem- Ambrus Pál
(19/10) - G-categories, Witt vectors, and G-affine spaces: Sebastian Monnet : Notes
(26/10) - G-WtF modules and cyclotomic pairs: Dominik Bullach
(02/11) - Properties of (G, Wr(L)(1)) torsors: Matthew Honnor
(09/11) - ``Theorem A", lifting general torsors: Ashvni Narayanan
(16/11) - An introduction to flags: Alex Torzewski
(23/11) - Lifting and gluing bundles: Mads Christensen
(30/11) - No Talk (Strike action)
(07/12) - "The uplifting theorem" sketch and the first part of the Norm-Residue isomorphism theorem: Jesse Pajwani
(09/12)- The Norm Residue Isomorphism Theorem II: Wojtek Wawrów (Date changed, will be in Huxley 340)