Fermat's Last Theorem - How does that proof really work, anyway?

This is a repeat of the last meetup. I thought the subject matter was exceptionally interesting, and while preparing and presenting I learned a vast amount of really cool math. I'm going to rerun the presentation with improvements. It's a brief historical summary, starting with Diophantus Book II, Question 8, and proceeding via Fermat, Euler, Lame and Cauchy, to the fundamental work of Kummer. Then switch gears and work forward from the key new idea introduced by Frey's elliptic curve, modular forms, the Modularity Theorem, Ribet's Theorem and finally the contradiction that FTL solutions create within the domain of elliptic curves.