Giulia Isabella, a physicist at UCLA, presents this research talk at the Institute for Advanced Study as part of the Black Holes from Theory to Observations seminar series. She works through the mathematics of gravitational tidal matching, the technique used to calculate how a black hole or neutron star's internal structure deforms under the tidal pull of a companion in a binary system, a quantity that leaves an imprint on gravitational wave signals. The talk centers on an unexpected connection to elliptic curves, the same algebraic objects that appear in number theory, and shows how this structure simplifies calculations relevant to interpreting gravitational wave observations. Delivered in Wolfensohn Hall to a specialist audience, the talk assumes familiarity with general relativity and black hole perturbation theory, walking through derivations on the board rather than offering a general-audience overview.