Georgia Institute of Technology College of Sciences

A multivariate real polynomial p(x) is nonnegative if p(x) is at least 0 for all x in R^n. I will review the history and motivation behind the problem of representing nonnegative polynomials as sums of squares. Such representations are of interest for both theoretical and practical computational reasons, with many applications some of which I will present. I will explain how the problem of describing nonnegative polynomials fits into convex algebraic geometry: the study of convex sets with underlying algebraic structure, that brings together ideas of optimization, convex geometry and algebraic geometry. I will end by presenting current research problems in this area.