Fermat's Last Theorem : how does that proof really work anyway?

An integer relation of the form a^n + b^n = c^n generates a very special elliptic curve y^2 = x (x - a^n) (x + b^n) . This gadget is called a 'Frey curve'. Coming from an entirely different direction, the brilliant Japanese mathematician Yutaka Taniyama conjectured that all elliptic curves are 'modular' - which means they can be parametrized by a modular curve. That conjecture was essentially proved by Andrew Wiles. Frey curves are much too special to be modular. This contradiction proves Fermat's Last Theorem. Let's find out a few things about all this. It's certainly a magnificent part of mathematics, full of really beautiful and symmetric objects.

The meetup will consist of two parts: first the historical background: Hypatia and Diophantus Book 2, Question 8 where Fermat wrote his famous marginal note, Fermat's probable 'proof' from 1637, as well as the efforts by Lamé and Cauchy which failed, but spurred the development by Ernst Kummer of ideal theory for number fields around 1846.  We'll also mention the contributions of Sophie Germain to FLT.

The second part will focus on the structure of Wiles' proof of FLT.   It's a tour of the concepts involved:  elliptic curves, modular forms, and something about L functions and Galois representations.   It's a tough subject to encompass, and there is a tendency for the discussion to hit overflow errors, but we'll give it a try.   It's a great truly great subject, dense with ideas and results that are deeply surprising.

There's a nice popular exposition that I can recommend:  "Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem" by Simon Singh.

Discussion with Paul Pedersen of MongoDB

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  • archisman r.

    Mirroring Siddhartha, I would really like it if we can do follow ups on this.

    June 6

  • Siddhartha

    Is there is going to be a follow up to this event? I am interested in understanding the proof of Fermat's last theorem. I understand that this would take time. I have read a bit of number theory and the theory of elliptic curves. I would be happy to participate in a series of meetups / lectures during which we can work towards understanding the full proof with all the details.

    May 5

  • archisman r.

    Very nice selection of topics. The use of the group law to derive a descent was new (to me). Looking forward to the next ones in the series.

    March 20

  • Meghan

    just increased the rsvp limit, hope a few more can make it!

    March 20

  • Robert P.

    Cancelled then?

    February 13

    • Meghan

      It's rescheduled for March! Hope that you can make it.

      February 13

  • Robert P.

    For me it's just hopping in the subway. I probably can't go to the next one but you do need a minimum number of showups.

    February 13

  • paul p.

    I'm looking out the window and thinking "cancel". Let's move this meetup for some day when we're not buried in snow! I'll set up a new date.

    February 13

  • Matthew

    It's going to snow tomorrow like there is no tomorrow.
    I hope we get to meet-up in the future.

    February 12

    • paul p.

      Good point. Hopefully a few people will show up :) . We can certainly re-run in a month or so - for those who need to stay home tomorrow.

      February 12

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