Many-body quantum systems with a coherent time evolution are now experimentally realizable at various scales of size and durations. The behavior of such quantum matter holds many deep questions. One of these regards the stability of such systems, especially under when subjected to external drives. A branch of non-equilibrium physics, this is a vast and deep subject, harboring much unchartered territory.Mechanisms of constraining the many-body dynamics in the Hilbert space range from highly debated and richly explored conundra like Many-body localization (MBL), to more recent ones like Floquet MBL, Dynamical Freezing, and Hilbert Space Fragmentation. These mechanisms can potentially underpin novel phases of quantum matter in and out of equilibrium.Investigating these mechanisms is likely to draw heavily on the moving fronts of diverse disciplines like Condensed Matter Physics, Statistical Mechanics, Quantum Field Theory, and even frontline Mathematics. The aim of this program is to take...
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