Abstract: Introduced by Dervan-McCarthy-Sektnan, the Z-critical equations are conjectural complex differential-geometric analogues of Bridgeland stability conditions, and seek to strengthen the classical Hitchin-Kobayashi correspondence: a holomorphic vector bundle over a smooth projective variety is slope polystable if and only if it admits a Hermite-Einstein metric. In this talk, I will briefly discuss the motivation and setting of the Z-critical equations, before specialising to the case of line bundles on projective surfaces. In this special case, I will attempt to explain how certain facts from the classical geometry of surfaces give a PDE analogue of a wall-and-chamber decomposition. This is joint work with Sjöström-Dyrefelt.