Continuing on the theme of the last post (Buzzard related viral videos), you can now view Buzzard’s MSRI course (in progress at the time of this post) online here. Having previously excoriated MSRI for restricting how many people can attend such workshops, I must now congratulate them on doing an excellent job in the audio-visual department and making the lectures available to everyone. Students at many levels could learn a lot by watching these and making an honest effort to think about the (implicit) exercises. Even if you know the material, it is still fun to watch; a little like your cool uncle telling you a familiar bedside story but with his own subversive twist. For various psychological reasons, I suspect that those watching the videos now as they come out will have a lower dropout rate than those who watch them later. So go watch them now! (Unless you are a student at Brown, of course.)
Kevin is always refreshingly honest about things he was confused by as a student (or is still confused by now), although sometimes it is reminiscent of Volodya “reminding” almost every speaker at the start of his seminar that his is a beginner and so the speaker will have to go very slowly. Along those lines, here are some (very) tangential remarks on the lectures so far.
When I was a student, I always got very confused when someone talked about the “closure” of the commutator subgroup [G,G]. The basic problem was that I couldn’t conceive of taking the quotient of by 1 and getting anything other than the trivial group. Of course that is what you should get unless you are doing it wrong, because anyone who thinks about profinite groups as abstract groups are probably crazy.
That said, here’s an idle question: is the commutator subgroup [W,W] of the the Weil group of a local field K actually already closed? I believe that the corresponding result for the (local) Galois group G itself is positive (essentially as a consequence of the fact that G is a finitely generated pro-finite group), but W has a distinctly non-compact quotient so I’m not sure. Maybe this is an easy question, I don’t know.
Another random fact: I was a graduate student at Berkeley in 2000 when Richard gave a colloquium on the local Langlands conjectures for GL(n). One aspect of the talk I remember was Richard defining the p-adic numbers, to which Mariusz Wodzicki cried out: “excuse me, this is Berkeley, do you really think you need to define the p-adic numbers?.” At this point, someone else cried out “Yes!” and the talk continued as planned. But the part of this story that is relevant here is that I somehow internalized (either at this talk or before) the fact that, long before Harris-Taylor, the local Langlands conjectures had been proved for GL(n) when p > n (which mirrors the story for n = 2). But I was surpised to find out recently (i.e. this week) that this result was not something from the distant past, but rather was a theorem of 1998 from (friend of the blog) Michael Harris in Inventiones.