Title:  Non-negative polynomials and Gorenstein ideals.

Abstract:  A homogenous polynomial of degree d in n variables is called non-negative if it is at least zero when evaluated at any point with real coordinates. The cone of such non-negative polynomials contains the cone of the homogeneous polynomials that are sums of squares. Hilbert characterized the pairs (n,d) such that the two cones are the same.
Recently, Blekherman strengthened Hilbert's results by describing the extremal rays of the cone that is dual to the cone of non-negative polynomials. These rays correspond to certain extremal Gorenstein ideals.
We will discuss these results.