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.
Wednesday, April 03, 2013
Today the Alpha Magnetic Spectrometer (AMS) Collaboration announced the publication of its first physics result in Physical Review Letters. The AMS Experiment is the most powerful and sensitive particle physics spectrometer ever deployed in space. In the initial 18 month period of space operations, from May 19, 2011 to December 10, 2012, AMS analyzed 25 billion primary cosmic ray events. Of these, 6.8 million, were unambiguously identified as electrons and their antimatter counterpart, positrons. The positron to electron ratio shows no anisotropy indicating the energetic positrons are not coming from a preferred direction in space, but instead a common source, in support of new physics, e.g. dark matter.
AMS has measured the positron fraction (ratio of the positron flux to the combined flux of positrons and electrons) in the energy range 0.5 to 350 GeV At energies greater than 250 GeV, the spectrum appears to flatten but to study the behavior above 250 GeV requires more statistics – the data reported represents ~10% of the total expected.