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Explore the mathematical foundations of black hole stability in this advanced lecture examining the nonlinear stability of Kerr and Kerr–Newman black holes as solutions to Einstein's field equations. Delve into the complex mathematical framework surrounding asymptotically flat stationary solutions that exhibit horizons, trapping phenomena, and dispersive dynamics. Learn about groundbreaking recent progress in proving the full nonlinear stability of slowly rotating Kerr black holes within vacuum Einstein equations, achieved through collaborative research by Klainerman, Szeftel, Shen, and the presenter. Discover how the introduction of electromagnetic fields and charge fundamentally alters the dynamical behavior in Kerr–Newman spacetimes. Gain insight into how these results build upon Sergiu Klainerman's pioneering contributions to nonlinear wave equation analysis and the global dynamics of Einstein's equations. This presentation provides a comprehensive survey of cutting-edge research in mathematical relativity, offering deep insights into the stability properties of some of the most important solutions in general relativity.
