I will discuss some recent progress on the Donaldson Segal programme, and in particular how calibrated cycles (coassociative submanifolds, special Lagrangians) arise from the large mass limit of $G_2$ and Calabi Yau monopoles.

S-duality is an intriguing symmetry of (twisted) N=4 supersymmetric Yang-Mills theory on a four-manifold. When the four-manifold underlies a complex projective surface, it leads to the Vafa-Witten invariants defined by Tanaka-Thomas in 2017. I will discuss some developments related to Azumaya algebras, universality, Seiberg-Witten invariants, wall-crossing for Nakajima quiver varieties, the structure of S-duality, and modular curves (including relations to the Rogers-Ramanujan continued fraction and Klein quartic).

We apply techniques of exponential asymptotics to the KdV equation to derive the small-time behaviour for dispersive waves that propagate in one direction.  The results demonstrate how the amplitude, wavelength and speed of these waves depend on the strength and location of complex-plane singularities of the initial condition.  Using matched asymptotic expansions, we show how the small-time dynamics of complex singularities of the time-dependent solution are dictated by a Painlevé II problem with decreasing tritronquée solutions.  We relate these dynamics to the solution on the real line.