Zoom-Meeting: https://wwu.zoom.us/j/97461739978 Abstract: We study the completion of the set of unit-volume Einstein 4-manifolds.Their degenerations were observed in the 70?s and understood in the 80?s in a Gromov-Hausdorff sense. We prove that any smooth Einstein 4-manifold close to a singular one in a mere Gromov-Hausdorff sense is the result of a gluing-perturbation procedure that we develop. This new description lets us extend Biquard?s obstruction to a general setting, allowing multiple singularities and trees of singularities and only assuming a Gromov-Hausdorff convergence. This also sheds light on the structure of the moduli space of Einstein 4-manifolds near its boundary.