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Explore the mathematical foundations connecting fusion categories to anyon theories in this comprehensive lecture delivered at the International Centre for Theoretical Sciences. Delve into the sophisticated mathematical framework that bridges abstract categorical structures with physical theories of anyons, which are exotic quantum particles that exist in two-dimensional systems. Learn how fusion categories provide the mathematical language to describe the braiding and fusion properties of anyons, and understand the systematic approach to constructing anyon theories from categorical data. Examine the role of these mathematical structures in topological quantum computation and condensed matter physics, where anyons serve as building blocks for fault-tolerant quantum computing schemes. Discover how the categorical approach offers a unified perspective on various topological phases of matter and their associated quantum field theories. This lecture forms part of a broader program on generalized symmetries and anomalies in quantum phases of matter, providing essential mathematical tools for understanding exotic quantum states and their symmetry properties.
