Georgia Institute of Technology College of Sciences

In this talk I will discuss a new technique discovered by myself and Olof Sisask which produces many new insights in additive combinatorics, not to mention new proofs of classical theorems previously proved only using harmonic analysis. Among these new proofs is one for Roth's theorem on three-term arithmetic progressions, which gives the best bounds so far achieved by any combinatorial method. And another is a new proof that positive density subsets of the integers mod p contain very long arithmetic progressions, first proved by Bourgain, and improved upon by Ben Green and Tom Sanders. If time permits, I will discuss how the method can be applied to the 2D corners problem.