I will present a theory of linear series on tropical curves that balances two notions of rank: the Baker-Norine rank from the tropical Riemann-Roch theorem and another idea of rank based on tropical independence. The resulting theory is surprisingly closely related to matroids. Every tropical linear series contains an open dense subset of nondegenerate divisors in a neighborhood of which the tropical linear series of dimension r is locally isomorphic to the Bergman fan of a matroid of rank r + 1. Moreover, every matroid appears as such a local matroid of a tropical linear series at a nondegenerate divisor. For instance, cographic matroids appear as local matroids of canonical linear series at the canonical divisor.

Based on joint work with Chih-Wei Chang, Matthew Dupraz, Hernán Iriarte, David Jensen, Dagan Karp, and Jayden Wang. https://arxiv.org/abs/2508.20062