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Explore advanced concepts in topological quantum field theory through this 55-minute conference talk that examines the quantization of homotopy types. Delve into Kontsevich's topological quantization proposal from the 1990s for sigma-models into finite homotopy types in top dimensions (d, d+1) and its enhancement to a fully extended TQFT as described by Freed, Hopkins, Lurie and the speaker in the target category of iterated algebras. Examine the independently developed Chas-Sullivan construction of a partially defined 2-dimensional TQFT (d=1) with target compact oriented manifolds. Review the key features of finite homotopy theory and its boundary conditions, with special focus on Dirichlet conditions and their analogues in Chas-Sullivan theory from previous work by Blumberg, Cohen and the speaker. Discover a proposed generalization that combines these approaches into a higher-dimensional Chas-Sullivan theory, bridging concepts from algebraic topology, quantum field theory, and operator algebras.
