The session will be a rather informal overview about a problem that can roughly be phrased as: What are the feasible regions of semidefinite programming?
It is about the relation between hyperbolic polynomials and determinantal representations. Hyperbolic polynomials generalize the notion of polynomials with only real zeros to several variables. Originating in PDE they appear naturally in many areas (among them the Riemann-Hypothesis, as we will learn on Monday). They define a convex cone, which can be optimized over using so-called hyperbolic programming.
Since real symmetric matrices have only real eigenvalues it follows, that the characteristic polynomial of a symmetric matrix pencil is hyperbolic. In 1958, Lax conjectured the converse for ternary forms. It was proven by Helton and Vinnikov around 2005. For more than three variables several weaker versions have been formulated but later shown to be wrong. The current state of what has become known as the generalized Lax conjecture can best be phrased in geometric terms. It asserts that every hyperbolicity cone is spectrahadral. Or in other words, that every hyperbolic optimization problem is sectretly a semidefinite one.
I will present some of the history, and try to explain different points of view onto the problem, that have contributed to a couple of more recent results.
The talk as well as the nibbles that will be served are dedicated to Jurij Volčič as these are his last weeks before he is off to the desert. |