By the Riemann-Hilbert correspondence, complex linear ODEs with polynomial coefficients, or more geometrically connections with regular singularities on curves, are characterized by their monodromy data. This  admits an generalization to the case of irregular singularities, involving generalized monodromy data known as Stokes data.

On the other hand, there is a notion of Fourier transform for irregular connections on the Riemann sphere: it acts in a nontrivial way, typically changing the rank, number of singularities and pole orders of the connections.

In this talk, I will present a topological way to compute the Stokes data of the Fourier transform of a connection in terms of its Stokes data in a new class of cases, relying on work of T. Mochizuki. In particular, this gives explicit isomorphisms between the corresponding wild character varieties. This is joint work with Andreas Hohl.