Positive Gorenstein ideals

A multivariate real polynomial $p$ is nonnegative if $p(x)\geq0$ for all $x\in\mathbb R^n$. 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. I will explain that the difference between nonnegative polynomials and sums of squares comes from Gorenstein ideals with special structure, which I will call positive Gorenstein ideals. Although the problem of nonnegative polynomials and sums of squares retains real flavor, this makes it amenable to the more familiar tools from complex algebraic geometry and commutative algebra. I will present some applications of these ideas.