Structural stability and general relativity

We review recent developments in structural stability as applied to key topics in general relativity. For a nonlinear dynamical system arising from the Einstein equations by a symmetry reduction, bifurcation theory fully characterizes the set of all stable perturbations of the system, known as the `versal unfolding'. This construction yields a comprehensive classification of qualitatively distinct solutions and their metamorphoses into new topological forms, parametrized by the codimension of the bifurcation in each case. We illustrate these ideas through bifurcations in the simplest Friedmann models, the Oppenheimer-Snyder black hole, the evolution of causal geodesic congruences in cosmology and black-hole spacetimes, crease flow on event horizons, and the Friedmann-Lemaître equations. Finally, we list open problems and briefly discuss emerging aspects such as partial differential equation stability of versal families, the general relativity landscape, and potential connections between gravitational versal unfoldings and those of the Maxwell, Dirac, and Schrödinger equations.