
‘Amazing’ Math Bridge Extended Beyond Fermat’s Last Theorem - pseudolus
https://www.quantamagazine.org/amazing-math-bridge-extended-beyond-fermats-last-theorem-20200406/
======
floatingatoll
The art at the top of the article was commissioned from this artist:
[https://www.dcmaia.art/portfolio/a-bridge-beyond-fermats-
las...](https://www.dcmaia.art/portfolio/a-bridge-beyond-fermats-last-theorem)

~~~
bt1a
I thought that was a beautiful header. Really well done.

------
williamstein
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
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.

~~~
williamstein
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.

~~~
auntienomen
I find it a little amusing that you describe yourself as one of Frank's
classmates instead, e.g, as "I'm Bill Stein".

------
cwzwarich
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.

~~~
williamstein
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...

