Determinantal representations via sums of squares of the Hermite matrix

Spectrahedra are the feasible sets of semidefinite programming. An important step in classifying spectrahedra consists of writing polynomials as determinants of symmetric linear matrix polynomials. It turns out that there is an interesting relationship between such determinantal representations, and sums of squares decompositions of the Hermite matrix of the polynomial. I will explain this fact, which is recent work with Daniel Plaumann and Andreas Thom.