update

Stream: FLT

[Kevin Buzzard]

For the first year of the project, we focussed a lot on a bottom-up theorem, that certain spaces of automorphic forms are finite-dimensional, and we are largely done with the proof. I have been quite bad at reviewing PRs recently (I'll get to why later on) but I had a go over the weekend and I've realised that we have done almost all of the adeles part of the argument and almost all of the finiteness part of the argument; what's left is some technical results about haar characters.

More recently I have turned to thinking about a top-down argument, which involves a lot of technical Galois representation stuff and modularity lifting theorems, and I've mostly been keeping myself to myself. I'm going to give 8 2-hour lectures at Imperial next term (Jan-March 2026) where I'm really going to push forwards with this stuff and actually write a blueprint; it has somehow taken me a very long time to flesh out the proof into something coherent which I understood all the details of, but I think I am there now modulo a modularity lifting theorem which is well-documented in the literature.

The thing that has been a big distraction for me recently is that I want to prove the main theorems of local class field theory and I've been working on this in private with a group of people who attended the Oxford Clay workshop on the topic. We have nearly finished the proof and this has become my main priority. I really hope to get back to bottom-up FLT soon and get the finiteness result sorry-free. I am naively hoping that I can get bottom-up FLT back on track and class field theory over the line by Christmas. Then in the new year I'm going to focus on top-down FLT.

There might be some money for jobs at Imperial, perhaps for people who are experts in the number theory related to top-down FLT and/or number theorists interested in AI. If there's anyone out there who would be able to move to Imperial quickly (e.g. by Jan) then could they let me know by DM? It's a pretty niche job description so I am not optimistic I'll find anyone appropriate but I thought I'd ask.

[Mario Carneiro]

Kevin Buzzard said:

I'm going to give 8 2-hour lectures at Imperial next term (Jan-March 2026) where I'm really going to push forwards with this stuff and actually write a blueprint

Will these lectures be recorded?

[Kevin Buzzard]

Only if I secretly record them, which I could do...