BPS invariants of certain physical theories correspond to Donaldson-Thomas (DT) invariants of an associated Calabi-Yau geometry. BPS structures refer to the data of the DT invariants together with their wall-crossing structure. On the same Calabi-Yau geometry another set of invariants are the Gromov-Witten (GW) invariants. These are organized in the GW potential, which is an asymptotic series in a formal parameter and can be obtained from topological string theory. Bridgeland showed that the GW potential can be extracted from a Tau-function which solves a Riemann-Hilbert problem associated to BPS structures. I will show that there is also a path going in the other direction. Studying the resurgence of the GW potential leads to DT invariants. DT wall-crossing phenomena correspond to Stokes jumps of the Borel resummation of the GW potential. In an explicit example, I will furthermore discuss a corresponding hyperkähler geometry and integrable hierarchy. This is based on joint work with Arpan Saha, Iván Tulli and Jörg Teschner.

The complexity of the singular fibers of the Hitchin system stems from the diversity of singularities of spectral curves. Already for G=GL(2,C) all singularities of type A appear. In light of the Deligne-Mumford compactification of the moduli space of smooth projective curves, it is a natural idea to compactify the family of smooth spectral curves over the regular locus of the Hitchin base to a family of nodal curves over a modified Hitchin base. In the talk, I will compare several approaches to achieve this goal in the GL(2,C)-case. These motivate the consideration of the moduli space of quadratic multi-scale differentials with simple zeroes as modified Hitchin base. I will conclude by formulating a spectral correspondence in this context. This is joint work with Martin Möller.

In 2009 Gaiotto, Moore and Neitzke introduced a blueprint for the twistorial construction of Hyperkähler metrics for a wide range of integrable systems. For the space of rank 2 parabolic Higgs bundles with diagonalizable residues at fixed punctures the construction becomes very concrete: One can build the metric out of sections of the Higgs bundles which are solutions of a certain flatness equation build out of the corresponding Higgs pair. The success of the procedure relies heavily on an asymptotic property of these sections, which Gaiotto, Moore and Neitzke originally conjectured based on the WKB method. I will outline the construction of the Hyperkähler metric and, using recent results of Fredrickson, Mazzeo, Swoboda and Weiß, show how one can circumvent the use of the WKB method to obtain a rigorous derivation of the necessary asymptotic properties. Finally I will explain how these properties yield the conjectured optimal decay rate of the asymptotic approximation of two different Hyperkähler metrics, by solving an associated Riemann-Hilbert problem.