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A central ingredient in quantum field theory that is usually expected, or demanded, is that it is local (meaning there is no ``spooky action at a distance" and all large-scale phenomena are determined by their behavior at small scales). It has long been expected that the formalism of fully extended field theories naturally leads to this type of locality, however no proof of this fact has emerged in the literature. In this talk, I will discuss joint work with Dmitri Pavlov in which we formulate the notion of locality for extended field theories and prove that all extended field theories are local in this sense. Since we do not restrict to the case of topological field theories, this requires defining a smooth variant of the fully extended bordism category, which is amenable to geometric structures. I will also discuss applications to the Stolz--Teichner program, including a classifying space construction for field theories and the construction of power operations from field theories (following the recent work of Barthel, Berwick-Evans, and Stapleton).

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