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Explore the mathematical foundations of homotopy quantum groups in this advanced seminar lecture from the BIMSA-Tsinghua Quantum Symmetry Seminar. Delve into the SymTFT paradigm in physics, which describes symmetries of (d+1)-dimensional quantum field theories through (D+1)-dimensional bulk topological quantum field theories with D-dimensional topological boundary theories called 'quiches'. Learn how quiches generalize higher categorical groups and topological spaces considered up to homotopy. Discover the process of assigning homotopy quantum groups (Hopf algebras in braided categories) to quiches, which extend the concept of homotopy groups from topological spaces. Examine the theoretical framework showing that for d ≥ 3 and sufficiently finite topological field theories valued in linear categories, these homotopy quantum groups are classical abelian groups. Understand computational methods for determining these groups in quiches arising from higher fusion categories. The presentation covers work in progress with Theo Johnson-Freyd and connects advanced topics in higher category theory, low-dimensional topology, homotopy theory, and quantum algebra.
