The proposer plans to work on a variety of problems in additive combinatorics, specifically: Continuing his work on the Polymath4 project to produce a deterministic algorithm to find large prime numbers quickly; exploring the ramifications of a ground-breaking probabilistic approach that he has developed to attack certain additive combinatorial problems (such as the 2D corners problem); and continuing the development of some fresh ideas on sum-product inequalities. A good example here is the proposer's work in the Polymath4 project: To date he, and other participants, have produced an algorithm to compute the parity of the prime counting function faster than any previous method; and furthermore, the proposer has produced a ``polynomial ring analogue'' of this algorithm which shows great promise towards completing the project (when generalized further, perhaps).
Understanding the structure of prime numbers (2,3,5,7,11, and so) is one of the enduring legacies of the ancient Greeks, and in recent years enormous progress has been made. Polymath4, an online collaborative research program initiated by Timothy Gowers, Gil Kalai, and Terrence Tao, addresses a related, fundamental question: How quickly can one even produce large prime numbers? The proposer has been a major participant in this project, and has thus far made significant strides in addressing this problem. He plans to continue working on it with other Polymath4 researchers and with some of his students. In addition, he plans to continue his work on developing some ground-breaking methods in a relatively new field called ``additive combinatorics'', which has become a hot area of late due to works of Gowers, Green, Tao, and others.
The proposer accomplished some of the goals outlined in the proposal, one of the most fruitful of which led to the publication of the paper, "A Probabilistic Technique for Finding Almost-Periods of Convolutions," which is joint with Olof Sisask. This paper has led to many powerful results in Additive Combinatorics, and is an ingredient in at least two major breakthroughs of Tom Sanders, that resulted in the solution of some long-standing open problems. To give a flavor of the kind of problem addressed in these papers, consider the following: suppose you take the numbers from 1 to a million, and are allowed to remove 90% of them. Can you choose that remaining 10% of the numbers in such a way as to avoid a triple of numbers of the form x, x+d, x+2d (where d > 0)? -- in other words, can you ensure that your numbers can't be in arithmetic progression? Other papers of the proposer have, since the time of the award, appeared in such journals as the Annals of Mathematics and SIAM Journal of Discrete Mathematics; and the most recent publication of the proposer is to appear in the Proceedings of the American Mathematical Society. The proposer has also worked with students on a number of occasions, particularly REU students (every summer except last summer); and he helped organized a mini-coference on Additive Combinatorics where a number of younger students were encouraged to present their work. The proposer has worked with his graduate students, and co-supervised a third student, on a number of projects. One of these has led to the resolution of a difficult conjecture of Jozsef Solymosi on "Rich Lines in Grids". The paper has not yet been submitted; but it seems likely it will be published in a top-rated journal.
