The harmonic maps parameterization identifies the Teichmüller space of a compact surface with the vector space of holomorphic quadratic differentials (on some base complex structure). There is an analogue of this picture for meromorphic quadratic differentials where the corresponding hyperbolic structures have additional moduli at each pole, which are called Stokes data.
We discuss computer experiments in which we compute Stokes data for polynomial quadratic differentials on the plane and compare the results to those predicted by a remarkable conjecture of Gaiotto-Moore-Neitzke. We also discuss analogous comparisons for cubic differentials. In all of these cases our results support the Gaiotto-Moore-Neitzke conjectures.