Anatomy of a Plunge: Universality in the Last Moments of an EMRI

Accurate waveforms for a small black hole entering a much larger one remain difficult near merger. This talk presents a staged treatment of the first-order merger and ringdown that can isolate nonlinear contributions for later corrections. It also analyses the transition to plunge in rotating backgrounds, where catastrophe theory explains why equatorial and inclined trajectories share the same Painlevé equation. Together these approaches expose structure useful for extending self-force calculations.