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Explore advanced representation theory of extended operators through the lens of Tannaka duality and Koszul duality in quantum field theory in this third part of a specialized lecture series. Delve into the mathematical frameworks that connect categorical structures with physical theories, examining how these dualities provide powerful tools for understanding topological symmetries and defects. Learn how Tannaka duality relates representation categories to their underlying algebraic structures, while investigating the role of Koszul duality in organizing the homological properties of quantum field theories. Analyze the interplay between these mathematical concepts and their applications to modern theoretical physics, particularly in the context of topological quantum field theories and their defect structures. Gain insights into how these sophisticated mathematical tools illuminate the deep connections between algebra, topology, and quantum physics through concrete examples and theoretical developments presented by a leading expert in the field.
