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Department Colloquium

Speaker: Graham Denham (University of Western Ontario)

Title: 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. In the context of telephone switching networks, Graham and Pollak (1971) computed the signature of the distance matrix in order to bound length of addresses in Pierce's loop switching model. I will describe a recent refinement of Graham and Pollak's result, together with its use in establishing new inequalities for matroids, including one that Dowling conjectured in 1980. I will highlight the role of Lorentzian polynomials, which were introduced by Brändén and Huh in 2020 and are part of an ongoing close relationship between discrete convexity and polynomial positivity properties. This is joint work with Federico Ardila, Sergio Cristancho, Chris Eur, June Huh, and Botong Wang.