Enumerative and tropical aspects of Brill Noether loci

Brill--Noether loci concern the question: what are all the ways an algebraic curve can be mapped to projective space? An old theorem of Castelnuovo relates the enumerative geometry of Brill--Noether loci to the enumeration of standard Young tableaux. I will discuss a tropical analog of Castelnuovo's count, in which the role of Young tableaux can be made much more explicit, and ongoing work that refines this count to a computation of Ehrhart polynomials. I will speculate about what the Ehrhart theory on the tropical side may tell us about the algebraic side of the story.