Explore the concepts of chaos and ergodicity through the lens of adiabatic transformations in this comprehensive lecture by Anatoli Polkovnikov. Delve into the stability of quantum matter in and out of equilibrium at various scales, examining coherent time evolution in many-body quantum systems. Investigate mechanisms constraining many-body dynamics in Hilbert space, including Many-body localization (MBL), Floquet MBL, Dynamical Freezing, and Hilbert Space Fragmentation. Gain insights into novel phases of quantum matter and their potential applications in quantum technologies. Drawing from diverse disciplines such as Condensed Matter Physics, Statistical Mechanics, and Quantum Field Theory, this lecture offers a deep dive into the frontiers of non-equilibrium physics and the many-body stability problem.