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Explore advanced mathematical concepts in this colloquium lecture that challenges conventional group theory approaches to symmetry. Discover how anomalous, higher-group, and non-invertible symmetries—collectively termed algebraic higher symmetries—require a fundamentally different mathematical framework than traditional group theory. Learn how finite symmetries, when considered up to holo-equivalence, are precisely classified by topological orders in one higher dimension that can host gappable boundaries. Examine the paradigm shift from group-based descriptions to braided fusion higher categories as the natural mathematical language for describing symmetry. Gain insights into cutting-edge theoretical physics and mathematics that redefines our understanding of symmetry beyond classical frameworks.
