The study of hyperbolic polynomials began with the
works of Grading, who studied them in
connection with the theory of hyperbolic PDEs
(hence the name). Grading has revealed that
hyperbolic polynomials possess remarkable
convexity properties and thus opened the way to
applications of hyperbolic polynomials to other
fields in mathematics. In particular, hyperbolic
polynomials are used in optimization and were
recently applied to solve the celebrated
Kadison-Singer problem. An example of a hyperbolic
polynomial is the determinant of a matrix of
self-adjoint matrices. In this talk, I will describe
real fibered morphisms between real
projective varieties. This notion abstracts the notion
of hyperbolicity for polynomials. I will
discuss the properties of real fibered morphisms,
the constraints on topology of the real
points of a smooth variety, induced by such a
morphism to a projective space and connect it to
Ulrich sheaves. I will show in what precise way
do Ulrich sheaves arise as determinant
representations. I will show that if the
Ulrich sheaf possesses a positive bilinear form, then
the variety is hyperbolic. Time permitting,
I will discuss deformations of hyperbolic varieties
and some open problems.
This talk is based on joint work with M. Kummer.
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