The ACO Seminar (2024–2025)

Completely Log Concave, a.k.a., Lorentzian polynomials, were discovered a few years ago where they were used to relate seemingly distant areas of Math and CS such as geometry of polynomials, Hodge theory for combinatorial geometries, theory of high dimensional expanders and mixing time of Markov chains. Consequently, they lead to a resolution of several long-standing open problems on matroids such as the Mason's log-concavity conjecture and the Mihail-Vazirani conjecture on the expansion of the bases exchange graph.

Unfortunately, this family of polynomials are limited as their support corresponds to bases of a matroid or more generally vertices of a generalized permutahedra. I will explain a generalization of Lorentzian polynomial to convex cones in the positive orthant, called C-Lorentzian polynomials, and use them to study combinatorial objects such as distributive, modular, or geometric lattices and the corresponding sampling, and log-concavity problems. Enroute we will also discuss new local-to-global theorems for high-dimensional expanders called Trickledown theorems.

Based on a joint work with Jonathan Leake and Kasper Lindberg.