Determinantal representations, sums of squares and del Pezzo surfaces

A widespread principle in real algebraic geometry is to find and use algebraic certificates for geometric statements. This covers for example writing a globally nonnegative polynomial as a sum of squares or expressing a polynomial with only real zeros as the minimal polynomial of a symmetric matrix. In the first part of the talk I will survey some classical results in this direction. I will give a brief introduction to real del Pezzo surfaces and explain how they fit into the aforementioned context. In the second part, I will use the geometry of real del Pezzo surfaces to construct certain Ulrich bundles and explain how this can be used to obtain representations as sums of squares or characteristic polynomials.