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Explore the fundamental role of symmetries in quantum many-body systems through this comprehensive lecture delivered at the International Centre for Theoretical Sciences. Delve into the profound transformation occurring in the study of quantum symmetries, moving beyond traditional notions to encompass higher-form, non-invertible, and subsystem symmetries that are revolutionizing our understanding of quantum phases of matter. Examine how these generalized symmetries provide powerful new frameworks for classifying quantum phases, including those that fall outside conventional paradigms. Investigate the concept of anomalies and their transition from high-energy physics to condensed matter settings, where they now play crucial roles in understanding microscopic symmetry realizations and protecting exotic quantum phases such as symmetry-protected topological states. Discover how these new insights are reshaping our comprehension of quantum entanglement, conservation laws, hydrodynamics in interacting systems, and their emerging influence on mixed states and open quantum systems. Learn about the cutting-edge developments in fractons, categories in condensed matter physics, defects and symmetries, emergent symmetries, Hilbert space fragmentation, and numerical methods for topological phases. Gain conceptual clarity and technical grounding in this rapidly evolving field that continues to reveal unexpected ways symmetry shapes the quantum world.
