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Explore cutting-edge advances in fusion categories through this comprehensive lecture delivered by J. Plavnik from Indiana University as part of the 2025 SCGCS School on Topological Symmetries and Defects in Quantum Field Theory. Delve into the mathematical foundations and recent theoretical developments in fusion categories, which play a crucial role in understanding topological quantum field theories and their applications to quantum computing and condensed matter physics. Learn about the algebraic structures that govern the fusion of anyons and their representation theory, examining how these mathematical objects provide insights into topological phases of matter and quantum symmetries. Discover the latest research directions in this rapidly evolving field, including connections to defects in quantum field theory and their implications for understanding exotic quantum phenomena. This first part of the series establishes the fundamental concepts and prepares you for deeper exploration of advanced topics in subsequent lectures.
