The ACO Seminar (2026–2027)

There is a remarkable connection between spectral properties of tree metrics and log-concavity in matroid theory, bridged by the theory of Lorentzian polynomials of Brändén and Huh. Classical results on distance matrices of metric trees both by Graham and Pollack, and Schoenberg serve as a fundamental ingredient for this link. In this talk I want to share joint work with Ardila, Denham, Eur, Huh and Wang, where we obtain novel log-concativity inequalities for matroids which answer two questions: one posed by Giansiracusa, Rincón, Schleis, and Ulirsch for valuated matroids, and another by Dowling and Zhao for ordinary matroids. We achieve this using new results about tree metrics that allow us to characterize M♮-concave functions on sets in terms of Lorentzian polynomials.