Daniel Plaumann

Hyperbolicity cones and spectrahedra: Introduction

Spectrahedra are the domains of semidefinite programs, contained in the a priori larger class of hyperbolicity regions and hyperbolic programs. In recent years, a lot of research in optimization and real algebraic geometry has been devoted to understanding the relation between these concepts. We will give an overview, focussing on algebraic techniques and on duality.

Simone Naldi

joint work with Mario Kummer, Daniel Plaumann

Spectrahedral representations of plane hyperbolic curves

We describe a new method for constructing a spectrahedral representation of the hyperbolicity region of a hyperbolic curve in the real projective plane. As a consequence, we show that if the curve is smooth and defined over the rational numbers, then there is a spectrahedral representation with rational matrices. This generalizes a classical construction for determinantal representations of plane curves due to Dixon and relies on the special properties of real hyperbolic curves that interlace the given curve.

Mario Kummer

joint work with Rainer Sinn

When is the conic hull of a curve a hyperbolicity cone?

Let C be a semi-algebraic curve in n-dimensional space. We will present obstructions and constructions for the conic hull of C to be a hyperbolicity cone. The tools we use are methods from complex and real algebraic geometry.