Recent results on black hole interiors suggest a failure of strong cosmic censorship for charged black holes in the presence of a positive cosmological constant. In this talk we show that, in the context of the Einstein-Maxwell-real scalar field system, such violations are non-generic in a larger moduli space of non-smooth (spherically symmetric) initial data.

I will begin with a concise review of the fundamental ideas underlying optimal transport theory, before illustrating how these concepts naturally emerge in cosmological contexts and provide a powerful framework for addressing the problem of cosmological reconstruction. I will then highlight how optimal transport extends beyond its analytical utility in handling big data, and demonstrate intriguing connections with certain theories of modified gravity.

The Vaidya spacetime is a spherically symmetric solution of the Einstein equations with a null dust source. This can be used to model the gravitational collapse of a thick shell of radiation: a flat interior region is matched at an inner boundary to the null dust filled region, which is then matched at an outer boundary to Schwarzschild spacetime. A central singularity inevitably forms, and depending on the profile of the energy density of the null dust, this singularity can be globally naked. Motivated by the cosmic censorship hypothesis, we consider perturbations of this configuration. We review previous work, and describe recent work where the perturbation of the inner boundary — the past null cone of the central singularity — is analysed using a framework for studying perturbations of general hypersurfaces. This sets boundary conditions for perturbations at the past null cone, and we then consider the 3+1 evolutionary problem, focussing on the question of the stability of the Cauchy horizon of the naked singularity.

A rigorous understanding of the dynamical nature of spacelike singularities remains an open problem in mathematical cosmology. Since the heuristic work of Belinski–Khalatnikov–Lifshitz and Misner's Mixmaster construction, vacuum spatially homogeneous cosmological models are expected to play a key role for generic singularities. We therefore focus on this class of models. The most general cases are the Bianchi type VIII, type IX, and type VI$_{-1/9}$, each with a four-dimensional Hubble-normalized state space.

On one hand, we embed the types VIII and IX models into modified gravity theories and show that general relativity (GR) arises as a bifurcation point where chaotic dynamics become generic, suggesting a new approximation scheme for GR. On the other hand, we analyze the type VI$_{-1/9}$ oscillatory regime and show that only a subset of its structure is dynamically relevant.

I will discuss recent work on estimates for spinor fields obeying first-order equations, formulated within the space–spinor framework. The approach is motivated by the positive commutator method presented in the work of P. Hintz and A. Vasy, originally designed for tensor fields satisfying second-order equations. In this talk, I will outline how these ideas can be adapted to the spinorial setting and how the resulting estimates fit naturally into the first-order structure of the equations.