Special Year Seminar II

The theory of stable polynomials features a key notion called proper position, which generalizes interlacing of real roots to higher dimensions. I will show how a Lorentzian analog of proper position connects the structure of spaces of Lorentzian polynomials and the subtle incidence geometry of tropical linear spaces. The central object is the moduli space of codimension-1 tropical linear subspaces of a given tropical linear space. In one direction, the geometry of this moduli space can be detected using Lorentzian polynomials; conversely, by answering certain incidence problems, one discovers non-obvious convex cones of Lorentzian polynomials.