The Hardy-Littlewood method is a well-known technique in analytic number
theory. Among its spectacular applications are Vinogradov's 1937 result that
every sufficiently large odd number is a sum of three primes, and a related
result of Chowla and Van der Corput giving an asymptotic for the number of
3-term progressions of primes, all less than N. This article surveys recent
developments of the author and T. Tao, in which the Hardy-Littlewood method has
been generalised to obtain, for example, an asymptotic for the number of 4-term
arithmetic progressions of primes less than N.Comment: 26 pages, submitted to Proceedings of ICM 200