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Explore the mathematical framework and theoretical foundations underlying integrable quantum field theories in this 57-minute conference talk delivered at ICBS2025. Delve into the geometric and algebraic structures that characterize the space of integrable quantum field theories, examining how these special classes of field theories maintain exact solvability through their underlying symmetries and conservation laws. Learn about the classification schemes, moduli spaces, and parameter families that organize different integrable models, while discovering the connections between integrability conditions and the geometric properties of theory space. Investigate the role of Bethe ansatz techniques, Yang-Baxter equations, and other mathematical tools in understanding how integrable theories are embedded within the broader landscape of quantum field theories. Gain insights into current research directions in mathematical physics that seek to map and understand the complete structure of integrable quantum field theory space, including discussions of deformation theory, scaling limits, and the relationship between classical and quantum integrability.
