Graham Denham (University of London Ontario) - Distances in trees and inequalities for matroids

The distance matrix of a tree appears in a range of contexts, from phylogenetics to physical chemistry. I will describe a new result about the spectrum of this matrix, one that gives affirmative answer to two questions about matroid positivity properties. These are both strengthenings of Mason's conjectures about the log-concavity of sequence of numbers of independent sets of a matroid, proposed by Igor Pak and by Giansiracusa, Rincón, Schleis, Ulirsch, respectively.

At the heart is a characterization of the gross substitutes property from economics in terms of the Lorentzian property of a generating function.

This is joint work with Federico Ardila-Mantilla, Sergio Cristancho, Chris Eur, June Huh, and Botong Wang.