John Carlos Baez on Nostr: My friend James Dolan wants to understand the Langlands program and - as usual for ...
My friend James Dolan wants to understand the Langlands program and - as usual for him - refashion it in his own language. It's going slowly, because first he had to redo all the preliminary concepts of algebraic geometry, like schemes and algebraic stacks and elliptic curves and modular curves and Artin reciprocity. But I really enjoy this process of "learning by redoing": it's a lot more fun than just reading books and absorbing the usual viewpoint on everything. So I've been talking to him about this stuff every week on Zoom, and putting videos of the conversations on YouTube. Alas, I'm way behind on uploading them - about a year behind - because I want to write a summary of each conversation.
I'm more willing than Jim to just absorb the usual viewpoint by reading stuff, but it's fairly hard to get a simple statement of some chunk of what people want to prove in the Langlands program. One reason is that it's a hydra-headed program: while popularizations calling it a "grand unified theory of mathematics" are hopelessly exaggerated, it does cover a lot of ground. Another is that math always seems confusing before it's actually done; later people come and clean it up.
One piece of the Langlands program says something like "every motivic L-function is also an automorphic L-function, and vice versa".
Huh? 🤔
An L-function is a function like
L(s) = 1 -2⁻ˢ - 3⁻ˢ + 6⁻ˢ + 8⁻ˢ - 13⁻ˢ - 16⁻ˢ + 23⁻ˢ - 24⁻ˢ + ....
where the list of numbers here is not random: it encodes interesting information about a mathematical object. For a fun and elementrary introduction to some examples, aimed at people who were good at math in high school, watch the video here!
(1/n)
https://www.youtube.com/watch?v=4bzSFNCiKrk&list=PL300nJOfVNKZH3H-2J_g1bZRYOx-dQUp3&index=2
I'm more willing than Jim to just absorb the usual viewpoint by reading stuff, but it's fairly hard to get a simple statement of some chunk of what people want to prove in the Langlands program. One reason is that it's a hydra-headed program: while popularizations calling it a "grand unified theory of mathematics" are hopelessly exaggerated, it does cover a lot of ground. Another is that math always seems confusing before it's actually done; later people come and clean it up.
One piece of the Langlands program says something like "every motivic L-function is also an automorphic L-function, and vice versa".
Huh? 🤔
An L-function is a function like
L(s) = 1 -2⁻ˢ - 3⁻ˢ + 6⁻ˢ + 8⁻ˢ - 13⁻ˢ - 16⁻ˢ + 23⁻ˢ - 24⁻ˢ + ....
where the list of numbers here is not random: it encodes interesting information about a mathematical object. For a fun and elementrary introduction to some examples, aimed at people who were good at math in high school, watch the video here!
(1/n)
https://www.youtube.com/watch?v=4bzSFNCiKrk&list=PL300nJOfVNKZH3H-2J_g1bZRYOx-dQUp3&index=2