A log/tropical take on Hurwitz numbers

I will present some joint work with Hannah Markwig and Dhruv Ranganathan, in which we interpret double Hurwitz numbers as intersection numbers of the double ramification cycle with a logarithmic boundary class on the moduli space of curves. This approach removes the "need" for a branch morphism and therefore allows the generalization to related enumerative problems on moduli spaces of pluricanonical divisors - which have a natural  combinatorial structure coming from their tropical interpretation. I will discuss some generalizations springing out from this approach that are currently being pursued in joint work with Hannah Markwig and Johannes Schmitt.