Georgia Institute of Technology College of Sciences

In recent years, the theories of Lorentzian polynomials and combinatorial Hodge theory have been developed and utilized to resolve long-standing conjectures in matroid theory, related to log-concavity inequalities and sampling algorithms. The overarching idea in these theories is to extract the conjectured results from basic eigenvalue bounds on certain natural matrices associated to matroids. Since then, Lorentzian polynomials have been generalized beyond matroids to simplicial complexes of various types, implying old and new results on various combinatorial structures such as linear extensions of posets. That said, many questions remain open. In this talk, we will describe this generalized theory and discuss how it can be used to prove various combinatorial results. No knowledge of matroid theory will be assumed. Joint work with Kasper Lindberg and Shayan Oveis Gharan, and also with Petter Brändén.