This award will provide support for a conference in Contemporary Combinatorics, in Memphis. In the past decade or two, combinatorics has gone from strength to strength. It has always been rich in problems and conjectures but, until a few years ago, there had been rather few tools to attack many important conjectures. That this situation has been improving rapidly is due, to a large extent, to the influx of methods from probability theory, ergodic theory, Fourier analysis, and algebra; in particular, the interaction between combinatorics and probability theory has greatly benefited both communities.
The major mathematical event of 2004 was the deep result proved by Ben Green and Terry Tao about arithmetic progressions in primes. Their ingenious proof makes use of pure combinatorics, ergodic theory, number theory, and probabilistic methods. A surprising aspect of their proof is that it does not need Szemeredi's theorem with a much smaller density, as had been expected: it suffices to transfer the original theorem to a new setting. It is likely that the methods of this great achievement will have implications for quite a while to come. The main speaker at the conference is Terry Tao. There will be several other exciting speakers. There are not many high-level mathematics conferences in the Mid-South. The Contemporary Combinatorics 2006 meeting in Memphis will have considerable impact. Most of the NSF grant would be used to help graduate students and senior mathematicians attend the meeting and so benefit from the personal contact with the speakers.
