*Abstract:* Multiplication maps on linear series are among the most basic structures in algebraic geometry, encoding, for instance, the product structure on the homogeneous coordinate ring of a projective variety. I will discuss joint work with Dave Jensen, developing tropical and nonarchimedean analytic methods for studying multiplication maps of linear series on algebraic curves in terms of piecewise linear functions on graphs, with a view toward applications in classical complex algebraic geometry

This work is parallel in many ways to the limit linear series of Eisenbud and Harris. One key difference is that we focus on degenerations in which the special fiber is not of compact type. In this context, the tropical Riemann-Roch theory of Baker and Norine and Bakerâ€™s specialization lemma are starting points for sometimes intricate calculations in component groups of Neron models and on skeletons of Berkovich curves.