How a closed quantum system reaches thermal equilibrium is a fundamental question in statistical physics. Recent work
has uncovered surprising richness in this process, leading to a new classification of quantum many-body systems into
distinct .dynamical phases.. On one extreme are many-body localized states, which fail to thermalize and can retain
accessible quantum correlations indefinitely. At the other end are maximally chaotic systems, which scramble quantum
information rapidly. I will review progress made in understanding many-body localized phases, both theoretically and
experimentally. I will then discuss a novel approach for computing the time evolution of quantum many-body systems,
which captures the emergence of chaos and hydrodynamic behavior.
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