Abstract: RiemannRoch theory for graphs, as developed in the work of BakerNorine, points to the existence of deep combinatorial
structures related to the classical algebraic geometry of linear series on algebraic curves. In this talk I will survey joint work with
CoolsDraismaRobeva, and with Jensen, pursuing this line of reasoning to give tropical proofs of the BrillNoether and GiesekerPetri
theorems, which describe the dimension and local structure of the moduli spaces parametrizing linear series of given degree and rank on a
general curve, as well as new progress toward the Maximal Rank Conjecture.
