Multiplication maps on linear series are among the most basic structures in algebraic geometry, encoding, for instance, the product structure on the graded pieces of the homogeneous coordinate ring of a projective variety. In this talk, 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.