Realizability of tropical abelian differentials

The realizability problem for tropical abelian differentials can be stated as follows: Given a pair $(\mathrm{\Gamma },D)$ consisting of a stable tropical curve $\mathrm{\Gamma }$ and a divisor $D$ in the canonical linear system on $\mathrm{\Gamma }$, we give a purely combinatorial condition to decide whether there is a smooth curve realizing $\mathrm{\Gamma }$ together with a canonical divisor that specializes to $D$. In this talk I am going to introduce the basic notions needed to understand this problem and outline a comprehensive solution based on recent work of Bainbridge-Chen-Gendron-Grushevsky-Möller on compactifcations of strata of abelian differentials. Along the way, I will also develop a moduli-theoretic framework to understand the specialization of divisors to tropical curves as a natural tropicalization map in the sense of Abramovich-Caporaso-Payne.

This talk is based on joint work with Bo Lin, as well as on an ongoing project with Martin Möller and Annette Werner.