A postview of Bellairs/Barbados

I am just recovering from my trip to Barbados for the McGill sponsored conference at the Bellairs institute (which I previously discussed here). I thought it was a wonderfully enjoyable conference, for many reasons. The first is that I got to give 14 hours or so of talks, and I like the sound of my own voice. What was unique, however, was the really high level of the audience, not just in terms of technical strength, but in terms of their knowledge of the particular topics which were being discussed. Usually when you have a chance to talk to a specialized audience, you only have 50 minutes to speak, and for at least for the first 20 minutes or so you should not assume that your audience is au fait with all the latest technical developments in the subject. On the other hand, the contexts in which one has multiple hours to give details (such as a mini-course or graduate class) it’s often the case that the target audience is graduate students first encountering the material. At this conference, practically half the audience had written papers proving modularity lifting theorems! I surveyed some participants beforehand on how long I should spend reviewing the basic theory of Galois deformations, and the answers typically ranged from 1 to 5 minutes. In reality, I gave a 150 minute “background” talk on the first morning, although by background here I really mean Wiles’ proof of minimal modularity lifting for irreducible modular Galois representations of \(G_{\mathbf{Q}}\).

I broke the mold of previous Bellairs conferences by scheduling an additional talk in the afternoon, so typically we had some 6-7 hours of lectures per day. This sounds a lot, but when it is divided up into only three speakers and spread out from early morning to late evening, it didn’t seem so much at all. (We still had plenty of time every day to snorkel at the reef, and even one free afternoon to go on a boat tour and swim with the turtles. Even Sug Woo’s 200+ minute talk just flew by, although it was accompanied by rum drinks.) In addition to the background talks I mentioned previously, there were also research talks by Peter Scholze, Jack Thorne, George Boxer, Ila Varma, and David Geraghty (I may blog about some of these talks later). I think this was the first conference in which I learned something from every single talk. Of course, I did get to suggest many of the participants, so in a way this conference was designed for me.

Speaking of great theses by Richard Taylor students (George and Ila), it’s kind of amazing what is required/expected of a graduate student in number theory nowadays. It certainly makes me feel positive towards the future of our subject. Speaking of Richard, I heard (although have no confirmation) that he thought the conference sounded interesting, and so it is somewhat embarrassing that I didn’t suggest his name as someone to be invited. On the other hand, it would have been even more embarrassing for him to have actually come and then had to share a room with someone (the accommodations were fairly spartan) while I was in a room by myself. Along those same lines, I’m 100% certain that Mark (“I don’t get out of bed for less than $10,000 a day”) Kisin would not have come. (Full disclosure, Mark claims that Toby and I exaggerate spread false rumours concerning his demands for luxury accommodations at conferences.)

One outcome of the conference is that I feel confident that we will have unconditional modularity lifting theorems for \(\mathrm{GL}(n)/\mathbf{Q}\) in the next five years. Of course, it’s always dangerous to make predictions.

Finally, apropos of nothing, I hope to have more posts in the future whose keywords include both “Richard Taylor” and “Turtles.”