Small gaps between primes

We introduce a refinement of the GPY sieve method for studying prime k-tuples and small gaps between primes. This refinement avoids previous limitations of the method and allows us to show that for each k, the prime k-tuples conjecture holds for a positive proportion of admissible k-tuples. In particular, lim infn(pn+m -pn) < for every integer m. We also show that lim inf(pn+1 -pn) 600 and, if we assume the Elliott-Halberstam conjecture, that lim infn(pn+1 -pn) 12 and lim infn(pn+2 -pn) 600.