Henri Darmon: Andrew Wiles' marvelous proof

Abstract:
Pierre de Fermat famously claimed to have discovered “a truly marvelous proof” of his last theorem, which the margin in his copy of Diophantus' Arithmetica was too narrow to contain. Fermat's proof (if it ever existed!) is probably lost to posterity forever, while Andrew Wiles' proof has been part of the mathematical landscape for over two decades. This lecture will describe a few of the new ideas in this marvelous proof, and the remarkable impact they have had on number theory.

Professor Henri Darmon is a French Canadian mathematician specializing in number theory. He works on Hilbert's 12th problem and its relation with the Birch-Swinnerton-Dyer conjecture. He is currently a James McGill Professor of Mathematics at McGill University.

This lecture was held at The University of Oslo, May 25, 2016 and was part of the Abel Prize Lectures in connection with the Abel Prize Week celebrations.

Program for the Abel Lecture 2016
1. "Fermat's Last Theorem: abelian and non-abelian approaches" by Abel Laureate Sir Andrew Wiles, University of Oxford
2. "Andrew Wiles' marvelous proof" by professor Henri Darmon, McGill University
3. "What is the Birch--Swinnerton-Dyer Conjecture, and what is known about it?" by professor Manjul Bhargava, Princeton University
4. "From Fermat's Last Theorem to Homer's Last Theorem" - a popular lecture by Simon Singh, author of Fermat's Last Theorem among other achievements. This lecture will never be published because the presentation contained material protected by intellectual property.

[parrotraiser]

As an interested non-mathematician, I haven't learnt any new skills from this talk, but the message seems clear. Wiles' proof is not just the answer to a particular question, but a technical advance that will enable more useful work by others. It also introduced me to Abel and his work, which was accomplished in a tragically short life, under difficult circumstance.

[robert-skibelo]

The camera work is very unhelpful. Too often, when the lecturer is talking about a slide, we are shown the lecturer standing in the dark (very uninteresting) rather than the slide (vital). We are apparently supposed to have committed the complicated slide to memory in the few seconds before the camera shifted away from it!

[NothingMaster]

One should keep in mind that Wiles's proof of the Fermat's Last Theorem is in reality a proof of a special case of the modularity theorem for elliptic curves. It is only in conjunction with Ribet's theorem, that it provides a proof for the Fermat's Last Theorem. In other words, Wile’s proof is not a wholly self-contained proof of the Fermat’s Last Theorem and could not stand alone, or be decoupled, from Ribet’s theorem. This in turn leads us to the key question: Is there an alternative path to a proof of the Fermat’s Last Theorem that is completely self-contained and independent of Ribet’s theorem?

[algebra]

Wow these Zeta functions are very interesting!

[richardslater]

I’m very interested in Fermat but at this level, I understood 2% of your very well, presented lecture. Could you help, please? What level does one have to be to reasonably understand your talk. Not to prove it, as Wiles did but to understand what is going on. I did A Level maths and this is way way beyond me.

[sixpooI]

i wanted to watch this, but unfortunately the audio is unbareable for me

[christiansmakingmusic]

Sit down with 3blue1brown or aleph0 and do an in depth series of videos about each step in the proof. This is the problem with higher maths, no one is able to bridge the communication gap with “laymen”. There should be a “Great Communicator” which recognizes the advanced scientist who took the time to really help people understand what they accomplished.

[joetursi]

Ribet should be included as solution contributor.

[unbeatableox]

There a lot of screwed-up people, in the comment section. Where the planet are you'll from? I don't get their logic. Wft?!

[Monster-nn.]

Zeta function it's really interesting

[paillote]

I don't understand the follow: If "x^2-x-1" has no integer solutions, and if N_p is the amount of solutions (mod p), wouldn't that be 0 or 'undefined'.

I'm sure I'm wrong, but I understood that N_p is the amount of solutions (mod P) of some equation.

[willk]

I almost came up with that same proof ... but I forgot to carry the 1.

[georgeice-um]

i have two elementary proofs of wiles theorem sorry he is going to see how much he is a genius and how much is a nerd.iam sort for the mathematicians which were unable to find a proof that mens that the system of teaching mathematics got to get a revolution.

[poetlaureate]

Smoking my pipe and saying to myself 'indeed'....

[corkkyle]

Id assume that Andrew has been working on the Riemann hypothesis for quite a few years now. Im guessing that is the great beast he is currently trying to slay.

[ionmurgu]

BIGGEST EVER AND FOREVER #HSCUT #COMPONENTS .

[gilberttheisen]

Il est très intéressant de voir des mathématiciens de renom exposer cette théorie fort compliquée alors que la démonstration de la conjecture est toute simple et se démontre en moins de 8 lignes. Les calculs avancés étaient inconnus du temps de FERMAT. Pour ne pas me répéter, allez sur d'autres sites concernant ce sujet où je traite en détail cette conjecture tout simplement et avec des calculs plus que basiques. Cette conjecture se révèle, à l'analyse, être le plus grand CANULAR mathématiques de tous les temps. La plupart des mathématiciens, pour sauver les apparences, n'accepteront JAMAIS de s'être fait piéger par FERMAT, méprisé par ses confrères, et par entêtement, persisteront à affirmer que la démonstration de WILES est la SEULE VALABLE et à douter même de la démonstration prétendue réelle de FERMAT !.Mais, laissons le temps faire son oeuvre et les mathématiques n'en sortiront que grandies d'une pareille histoire.

[utuonebrown]

Mission impossible for you I'm afraid. Too much time spent lecturing within the corridors of academia.

[odinedin]

Fermat's Great Theorem NEVER! and nobody! NOT! HAS BEEN PROVEN !!! / - except me!!!

[odinedin]

NOT! " -wileS" = he IS the LiEZ !!!