Non-negative polynomials versus sums of squares

We discuss the geometry underlying the difference between non-negative
polynomials and sums of squares. The hypersurfaces that discriminate these
two cones for ternary sextics and quaternary quartics are shown to be
Noether-Lefschetz loci of K3 surfaces. The projective duals of these
hypersurfaces are defined by rank constraints on Hankel matrices. We
compute their degrees using numerical algebraic geometry, thereby verifying
results due to Maulik and Pandharipande. The non-SOS extreme rays of the two
cones of non-negative forms are parametrized respectively by the Severi
variety of plane rational sextics and by the variety of quartic symmetroids.
This lecture is based on work of Greg Blekherman, and on our joint paper
with Jonathan Hauenstein, John Christian Ottem and Kristian Ranestad.