Stability data, irregular connections and tropical curves

I will outline the construction of isomonodromic families of irregular meromorphic connections
on $\mathbb{P}^1$ with values in the derivations of a class of infinite-dimensional Poisson
algebras, and describe two of their scaling limits. In the ``conformal limit'' we recover a version
of the connections introduced by Bridgeland and Toledano-Laredo, while in the ''large complex
structure limit" the connections relate to tropical curves in the plane and, through work of
Gross, Pandharipande and Siebert, to tropical/GW invariants. This is joint work with
M. Garcia-Fernandez and J. Stoppa.