Geometry of hyperbolic polynomials

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At the heart of this project are hyperbolic polynomials and their hyperbolicity cones. These are real polynomials in several variables bounding a convex cone, with a particular reality condition on the roots. They originate in complex analysis and the theory of partial differential equations, but have more recently arisen in optimization, combinatorics, and probability theory. Hyperbolic polynomials can be seen as generalizations of characteristic polynomials of real symmetric matrix pencils. On the one hand, one can try to express them as such (determinantal representations) and compare hyperbolicity cones to the resulting cones of matrices. It is an open question whether this is always possible in a certain sense. On the other hand, one can also imitate matrix theory within the framework of hyperbolicity. This project will focus on the latter and study hyperbolicity cones from the point of view of convex algebraic geometry, with a particular emphasis on the interplay of duality theories, namely (1) duality of algebraic varieties in projective space, (2) duality of convex cones, and (3) convex programming duality. The last is well understood and heavily exploited in semidefinite programming, but remains largely incomplete for hyperbolic programming. Along with these fundamental questions, we will also apply such techniques to concrete problems supplied by the above connections to other fields.