
Twenty Years of Number Theory (1998) [video] - espeed
https://www.youtube.com/watch?v=EDMJElLpm1U
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dsacco
Fun fact: Wiles’ proof of Fermat’s Last Theorem took him six years to complete
in isolation, during which time he essentially told no one in the mathematical
community that he was working on it. To prove it, he had to utilize results
from number theory, algebraic geometry and category theory. He largely proved
it in 1993, save for one error that he did not correct until 1995. One of my
favorite videos on YouTube is a ~30 second clip of Wiles describing the moment
he discovered the solution to the last remaining error in his proof:
[https://youtube.com/watch?v=SccDUpIPXM0](https://youtube.com/watch?v=SccDUpIPXM0).
Wiles was completely enamored with the problem since his childhood, and
pursued a proof in earnest from his early 30s.

His proof was exceptionally understated when it was presented. In 1993, he
gave lectures on his proof of the Taniyama-Shimura-Weil conjecture (the
modularity theorem), which was the bulk of the machinery of the proof (the
modularity theorem for semistable elliptic curves implies Fermat’s Last
Theorem). He simply stated at the end of the lecture that his proof also
happened to imply Fermat’s Last Theorem.

Up until Wiles did it, much of the mathematical community at the time didn’t
think Fermat’s Last Theorem was “accessible” to prove with contemporary
techniques. Incidentally, Fermat himself wrote that he had a “truly marvelous
proof” of the theorem in the margin next to it in one of his books, but which
he couldn’t write down becuase the margin was apparently too small to contain
it. There’s pretty broad skepticism Fermat actually had a proof however,
considering that most of Wiles’ (very long, and very complex) proof borrowed
results from large swaths of mathematics not developed until the 20th century.

If anyone is interested in trying to learn how the proof works at the formal
level, the book _Modular Forms and Fermat’s Last Theorem_ by Cornell,
Silverman and Stevens provides a rigorous presentation that builds up the
necessary mathematics to understand the entire thing. It’s easily a graduate-
level textbook, but it’s probably accessible (with effort) to anyone who has
worked through undergraduate abstract algebra (up to finite fields and Galois
theory) and number theory (up to elliptic curves).

~~~
williamstein
Good summary, except that you are understating the fatal and very significant
mistake in Wiles' original argument (which involved Euler systems), and the
difficulty in fixing it. Also, Richard Taylor helped enormously in coming up
with a new argument (using deformation theory) that actually worked...

~~~
dsacco
Thanks for that insight, now I’ve got some more reading material to work
through!

