Matroids with coefficients and Lorentzian polynomials

In the first half of the talk, I will briefly survey the theory of matroids with coefficients, which was introduced by Andreas Dress and Walter Wenzel in the 1980s and refined by the speaker and Nathan Bowler in 2016. This theory provides a unification of vector subspaces, matroids, valuated matroids, and oriented matroids. Then, in the second half, I will outline an intriguing connection between Lorentzian polynomials, as defined by Petter Brändén and June Huh, and matroids with coefficients. The connection hinges on the relationship between valuated matroids and phylogenetic trees, which Dress made use of in both his earlier work on matroid theory and his later work on mathematical biology. This is joint work with June Huh, Mario Kummer, and Oliver Lorscheid.