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This is an amazingly well written article, given how difficult the topic is to explain. It's about massive generalizations over the last few decades of the bas
by williamstein 7y ago
This is an amazingly well written article, given how difficult the topic is to explain. It's about massive generalizations over the last few decades of the basic idea that Wiles proof of FLT depended on, and does a great job explaining the key role of collaboration in mathematical research. This is everything Mochizuki's "proof" of the ABC conjecture is not.
- QuesnayJr 7y agoI think it's basically impossible to write a popular article on the Langlands program, but I really struggled with what they were trying to say. For example, when they were talking about "complex numbers", did they literally mean non-real fields, or did they mean elliptic curves with complex multiplication? I think the author meant the latter, but I'm not really sure.
- leanuser57 7y agoNo, they mean elliptic curves defined over number fields that don't embed into the reals.
- QuesnayJr 7y agoBut they don't mean the general case, do they? If I look at the abstract for the 10-author paper, they talk about CM fields.
- williamstein 7y agoYou're right -- they do not and are instead restricting to certain specific classes of number fields. Proving some notion of modularity of all elliptic curves over all number fields is surely completely out of reach at present.
- williamstein 7y agoIt's an extremely deep topic. I have a Ph.D. in number theory (Frank Calegari was one of my classmates), so maybe that's why the article was so readable to me. Also, fortunately, it's been known for a long long time that elliptic curves with complex multiplication arise from automorphic forms, since their Galois representations are relatively easy to understand compared to general elliptic curves.
- auntienomen 7y agoI find it a little amusing that you describe yourself as one of Frank's classmates instead, e.g, as "I'm Bill Stein".