Moduli spaces of meromorphic connections and the Fourier transform

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Stokes phenomena occur in the theory of complex differential equations with irregular singularities, and it is well-known that a differential system is determined by its Stokes data. The Fourier-Laplace transform is a prominent integral transform in all of mathematics and physics, and the behaviour of Stokes data under this transformation is an active field of research. In this project, we aim at understanding better the geometry of the Fourier-Laplace transformation by combining two questions that have been of great interest in the literature: On the one hand, the explicit computation of Fourier-Laplace transforms of Stokes data; on the other hand the construction of moduli spaces of meromorphic differential equations. Concretely, we first want to perform explicit computations of the Fourier-Laplace action on Stokes data, making use of recent developments which give us powerful tools for such considerations: the theory of enhanced ind-sheaves of D’Agnolo-Kashiwara. Secondly, we will use these results to describe the isomorphisms induced by Fourier-Laplace transform on moduli spaces of Stokes data and study their behaviour with respect to geometric structures on these spaces.