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I posted on Mathematics Stack Exchange, but was encouraged to post on MathOverFlow instead.

It has been 20 years since Fermat's last theorem was proved by Andrew Wiles.
Has there been any simplification in proof in the last 20 years?
What I do know is only that different proofs of Faltings's theorem were given by Vojta and Bombieri
and Khare-Wintenberger's proof of Serre's Conjecture gives a different proof of FLT (the latter was told to me in a comment on the Mathematics Stack Exchange).

Thanks in advance.

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    $\begingroup$ Ken Ribet at the 2020 JMM in Denver, CO, "A 2020 View of Fermat's Last Theorem": youtube.com/watch?v=mq9BS6S2E2k $\endgroup$
    – Sam Hopkins
    Commented May 30, 2020 at 2:51
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    $\begingroup$ So maybe based on what you saw in the video, you might like to add an answer to your own question, for the benefit of those who have not watched it? $\endgroup$
    – auspicious99
    Commented May 30, 2020 at 7:51
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    $\begingroup$ The proof via Serre's conjecture allows one to avoid difficult results of Ribet and Langlands-Tunnel but requires other non-trivial input so it is not clear that it is really a simplification. But the proof is indeed conceptually simpler. $\endgroup$
    – naf
    Commented May 30, 2020 at 7:52
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    $\begingroup$ What I'd like to see is a serious technical discussion about how exactly newer approaches make things simpler. What is no longer needed if Serre's conjecture is used instead? Etc. Ribet's talk was for a general mathematical audience, not for people who want gory details. $\endgroup$
    – David Roberts ♦
    Commented May 30, 2020 at 10:39
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    $\begingroup$ @DavidRoberts: Serre's 1987 Duke paper contains a proof that Serre's conjecture implies Fermat's last theorem. So if you accept Serre's conjecture, then you don't need anything beyond Serre's 1987 Duke paper. Of course the proof of Serre's conjecture relies on the original ideas of Taylor and Wiles. $\endgroup$
    – GH from MO
    Commented May 31, 2020 at 15:15

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