I don’t watch videotaped lectures — in general I’ve found the difficulty of seeing the board and hearing the lecturer makes it impossible for me to maintain enough focus to engage with the mathematics and take good notes. In fact, I think the only online video lecture I’ve ever viewed all the way through was one of my own, because I somehow lost the notes I’d used and needed to generate a new set so I could give the talk again.

But I was really sorry not to be able to make last week’s introductory workshop for the Newton Institute’s special semester on non-abelian fundamental groups in arithmetic geometry — so sorry that I decided to try watching the recorded lectures on my laptop. And they’re great! Crisp sound and visuals, appropriately timed close-ups on the board, and even a camera pointed at the audience so you can see the people asking questions. And you can download the talks to your iPod! Three cheers for the A/V team at the Newton Institute.

As of tonight just the Monday and Tuesday talks are up, which is already plenty to keep me busy. I just watched Deligne talk about counting l-adic local systems on curves over finite fields; highly recommended.

When I was first giving public lectures, someone gave me the hoary advice that I should quell nervousness by imagining the members of the audience in their underwear. Strange to think that, in this new broadband world, *most of them actually are.*

One-second precis of Deligne’s talk: starting with Drinfel’d in the early 80s, you can count the number of l-adic local systems on a curve over F_q by applying whatever version of the Langlands correspondence you have available and then using an appropriate trace formula to count automorphic forms. It turns out that the number of rank-d l-adic local systems “defined over F_{q^n}” seems to behave as if it were governed by a Lefschetz fixed point formula, i.e. as if it were the number of F_{q^n}-rational points on some variety. But what variety? Not the moduli space of rank-d vector bundles with connection on the curve; that has dimension twice as large as the dimension of the purported variety suggested by the result of the counting problem. But one still may hope — bolstered to some extent by recent work of Arinkin and Flicker — that the point count is reasonably legible and has *something* to do with the hyperkahler geometry of that moduli space. I don’t think that summary made tons of sense — so watch the video!

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