Members' Colloquium

Lorentzian polynomials serve as a bridge between continuous and discrete convex analysis, with tropical geometry providing the critical link. The tropical connection is used to produce Lorentzian polynomials from discrete convex functions, leading for example to a short proof of Mason's conjecture on the number of independent sets of a matroid. In this talk, we will delve into the intricate relationships between Grassmannians over hyperfields, dequantization processes, and the theory of Lorentzian polynomials. In ongoing collaborative work with Matt Baker, Mario Kummer, and Oliver Lorscheid, we explore how Lorentzian polynomials relate to matroids over triangular hyperfields, a concept introduced by Viro. This connection enhances our understanding of the space of Lorentzian polynomials, uncovering a rich interplay between analysis, combinatorics, and geometry. The talk is designed to be accessible to a wide audience.