Moduli spaces in algebraic geometry parametrise stable objects (bundles, varieties,…), and hence depend on a choice of stability condition. As one varies the stability condition, the moduli spaces vary in a well-behaved manner, through what is known as wall-crossing. As a general principle, moduli spaces admit natural Weil-Petersson metrics; I will state conjectures around the metric behaviour of moduli spaces as one varies the stability condition.

I will then discuss analogues of these results in the model setting of symplectic quotients of complex manifolds, or equivalently geometric invariant theory. As one varies the input that determines a quotient, I will state results which explain the metric geometry of the resulting quotients (more precisely: Gromov-Hausdorff convergence towards walls, and metric flips across walls). As a byproduct of the approach, I will extend variation of geometric invariant theory to the setting of non-projective complex manifolds. This is all work in progress.