Description
Quasi-normal modes (QNMs) govern the ringdown of perturbed black holes and encode key information about the remnant's mass, spin, and strong-field geometry. However, their non-orthogonality makes unambiguous mode decomposition a persistent challenge. In the first part of this talk, I present two agnostic, data-driven methods — the Matrix Pencil Method (MPM) and AAA rational approximation — that extract QNM frequencies and amplitudes directly from time-domain waveforms without prior assumptions about which modes are present. Visualized in the complex frequency plane, these tools give an immediate overview of the mode content, achieve high resolution capable of resolving closely-spaced overtones, and are sensitive enough to detect Price-law tails. Crucially, the extracted amplitudes remain stable under variation of the analysis time window, making them reliable for precision spectroscopy. In the second part, I discuss nonlinear QNM resonances in Kerr black holes. While linear perturbation theory already reveals a near-resonance between distinct overtone branches, the nonlinear sector is richer: at third perturbative order, quasi-normal modes incident on the black hole source new modes through Absorption-Induced Mode Excitation (AIME). I show that the first overtone of the dominant AIME mode satisfies a near-resonance condition with the quadratic source frequency, leading to a significant amplification of its amplitude. Although the resonant contributions partially cancel when summed, the residual imprint on the waveform morphology is observable — with potential implications for ringdown modeling and gravitational-wave data analysis.