We discuss the geometry underlying the difference between
nonnegative polynomials and sums of squares. The hypersurfaces that
discriminate these two cones for ternary sextics and quaternary quartics
are NoetherLefschetz loci of K3 surfaces. We compute their degrees using
numerical algebraic geometry, thereby verifying results due to Maulik and
Pandharipande. Other algebraic geometry classics to appear naturally are
Severi varieties, low rank Hankel matrices, and quartic symmetroids. This
lecture is based on work with Blekherman, Hauenstein, Ottem and Ranestad.
