Today Andrew Wiles turns 60 (Born: April 11th 1953) and Lubos Motl at TRF has posted a very nice BBC documentary on the solution of Fermat's last theorem.

The video features the chain of conjectures, that when proven, leads to the proof Fermat's last theorem. The key component of this reasoning involved the proof of the Taniyama-Shimura conjecture, which states that every elliptic curve is really a modular form in disguise. The problem is doing the proper counting on each side to solidify the correspondence. The key insight is to count Galois representations, which can be associated to the elliptic curves. Wiles accomplished this after seven years of solitary work and a last repair of his initial proof.

The Taniyama-Shimura conjecture is, in a sense, a two-dimensional special case of the more general Langlands correspondence. In the string theory context, Taniyama-Shimura is to the worldsheet, as the more general Langlands is to worldvolumes. Further work on the Langlands should shed some light on M-theory, and vice versa, as string theory/M-theory so far has shown quantum gravity prefers to work in certain, special dimensions. To fully understand this, one must go on a journey from number theory to geometry, and back to number theory. So far, Witten and Kapustin have suggested that Langlands duality and S-duality are related, so it is tempting to conjecture further about the role of U-duality in the general Langlands correspondence.

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