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