Comment by zmgsabst
7 hours ago
I did an undergrad in math with a little research in number theory and recognized parts — eg, I myself worked through the proof for odd regular primes and that 37 is irregular, breaking the general case.
Wiles-Taylor-Wiles was the original proof by Andrew Wiles, and its corrections.
Galois representations is about vectors over Galois extensions, which are essentially adding roots to regular numbers (rationals, integers, etc). That ties into the Langlands program, which is a big area in number theory (that I don’t know much about).
Together with flat deformations and Frey curve, I think they’re talking about a topic in algebraic geometry as applied to number theory.
I also recognize the name Eisenstein from my time as an undergrad, though two decades out and not working in the field I’ve forgotten what his work on ideals implied here. Ideals are a well-known topic though, a sort of structure inside a ring (set with + and *) that is closed under operations — like evens in the integers are the 2Z ideal.
So I’d describe it as “sensible with an undergrad background”.
About the Langlands program, Nunberphile has an excellent episode with Edward Frenkel explaining what it's about: https://youtu.be/4dyytPboqvE.
Frenkel does a nice job explaining the Langlands program in general. But Buzzard's complaint about Langlands, I believe, refers specifically to the proof of a version of the Geometric Langlands Conjecture by Gaitsgory et al. The proo f is of order thousand pages of mathematical text and builds off of thousands of pages of higher-categorical algebraic geometry by Lurie & others. It's a ripe target for formalization because it's terrifically complicated, not well understood or thoroughly digested yet, and relatively important. A formal proof would be reassuring to mathematicians, whereas Fermat's Last Theorem is relatively unique in that so many mathematicians have examined the proof that it's not very likely to be wrong.