Elena Giorgi, a mathematician at Columbia University, lays out the current state of the black hole stability problem in general relativity. Speaking at the kick-off conference for the new MPG-IAS-NTU Center, she surveys the mathematics behind proving that Kerr black holes remain stable under small perturbations of the Einstein equations, a question that sits at the center of modern mathematical relativity. She traces the line of results building toward the full non-linear stability theorem, touching on the techniques used to control gravitational radiation and curvature decay over long time scales, and situates her own work within this broader program. The talk is pitched as an overview rather than a technical deep dive, aimed at giving researchers from adjacent fields a map of where the problem stands, what has been proven, and what remains open. It runs just over forty minutes and assumes graduate-level familiarity with general relativity and partial differential equations.