Seminar in Real and Complex Geometry

Thursday, December 13, 2018, 16:00-18:00, Schreiber building, room 209

Eli Shamovich (University of Waterloo)

Hyperbolic varieties, real fibered morphisms, and Ulrich sheaves


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.