I thought that was a beautiful header. Really well done.
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.
I 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.
No, they mean elliptic curves defined over number fields that don't embed into the reals.
But 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.
You'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.
It'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.
I find it a little amusing that you describe yourself as one of Frank's classmates instead, e.g, as "I'm Bill Stein".
It's strange to see an article about the progress of modularity conjectures for elliptic curves that doesn't mention either Taniyama or Shimura, and instead mentions Langlands as the originator of the idea. This is bizarre historical revisionism.
Taniyama and Shimura only formulated a conjecture about elliptic curves over the rational numbers. They didn't come up with a statement about elliptic curves over number fields, and this article is about work to generalize modularity to number fields. The Langlands program, on the other hand does, help enormously with such generalizations. It's not trivial to formulate a correct conjectural generalization of modularity of elliptic curves over general number fields, and some naive analogues don't work at all...