Yilber Fabián Bautista, a physicist at the University of Edinburgh, presents this seminar at the Institute for Advanced Study as part of the program Black Holes from Theory to Observations, delivered in Wolfensohn Hall. He argues that the connection problem for the Heun equation, the mathematical object that governs wave perturbations around black holes, offers a single framework for computing scattering amplitudes that otherwise require separate techniques for different regimes. The talk walks through how solutions to the Heun equation connect across singular points, and how that connection data translates directly into amplitudes used to model gravitational wave signals from black hole mergers. Bautista contrasts this unified approach with the patchwork of post-Newtonian and numerical relativity methods currently used in the field, suggesting it can reproduce known results while extending to cases those methods struggle with. The audience is specialists in gravitational physics and amplitude methods, and the talk assumes familiarity with black hole perturbation theory.