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Explore advanced concepts in fusion categories through this third installment of a lecture series delivered at the 2025 SCGCS School on Topological Symmetries and Defects in Quantum Field Theory. Delve into cutting-edge research and recent breakthroughs in the mathematical framework of fusion categories, examining their structural properties, classification problems, and applications to quantum field theory. Investigate the intricate relationships between categorical symmetries and topological defects, building upon foundational knowledge to understand sophisticated theoretical developments in this rapidly evolving field. Gain insights into current research directions and open problems in fusion category theory as presented by leading experts in mathematical physics and category theory.
