Tropical Independence I: Shapes of Divisors and a Proof of the   Gieseker-Petri Theorem
  We develop a framework to apply tropical and nonarchimedean analytic techniques to multiplication maps on linear series and study degenerations of these multiplications maps when the special fiber is not of compact type. As an application, we give a tropical criterion for a curve over a valued field to be Gieseker-Petri general. 
David Jensen and Sam Payne
math.AG