Sam Mansfield

Sums, Products, Convexity, Squeezing, Expansion, Generalizations.

Convex functions are of particular interest in additive combinatorics due to their proclivity for destroying additive structure. Results about the growth of sets defined using convex functions have found applications in, for example, a recent lower-bound for the number of dot products determined by a set of points R^2—due to Hanson, Roche-Newton and Senger—and to many different sum-product type inequalities. In this talk, I will survey some of the results and techniques in this area and discuss generalizations to higher-dimensional sets.