Additive Combinatorics

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My primary field of research is additive combinatorics. This relatively new area of mathematics is concerned about the additive or multiplicative structure of subsets of groups with small sumset or product set. The sumset of a set is formed by the pairwise sums of elements of the set. (In a product set we take the pairwise products.) Additive combinatorics has strong connections to number theory, harmonic analysis, ergodic theory, and theoretical computer science. I am interested in the geometry behind some of the problems in additive combinatorics. It is often the case that a problem in additive combinatorics can be translated to a question about curve-point incidences in a plane over some field. A classical example is the so called "sum-product problem". An arithmetic progression has a small sumset and a geometric progression has small product set, but the arithmetic and geometric progressions have very different structure. Is it possible to find a set with small sumset and small product set at the same time? The answer very much depends on the underlying field or ring. A subfield of a finite field has very small sumset and product set but the same is not possible over fields with characteristic zero or when the set is "very far" from a subfield. The general problem is called the "sum-product phenomenon". Surprisingly the problem can be translated to bounding the number of incidences between points and lines in the plane defined over the field. Understanding such connections and giving efficient bounds on the number of point-curve incidences is a central part of this research proposal. Such results would have important applications not only to other fields of mathematics but to computer science as well.