Graham Denham (University Of Western Ontario)
Title:Â Distances in trees and inequalities for matroids
Abstract: 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 answers to some questions about matroid positivity properties. Those are both strengthenings of Mason’s conjectures about the log-concavity of sequence of numbers of independent sets of a matroid, proposed by Dowling in 1980 and much more recently by Giansiracusa, Rincón, Schleis, Ulirsch. At the heart is a characterization of the gross substitutes property from economics in terms of the Lorentzian property of a certain generating function.
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