I will explain my recent work with Daniel Grady on the locality of functorial field theories (arXiv:2011.01208) and the geometric cobordism hypothesis (arXiv:2111.01095). The latter generalizes the Baez–Dolan cobordism hypothesis to nontopological field theories, in which bordisms can be equipped with geometric structure, such as smooth maps to a fixed target manifold, Riemannian metrics, conformal structures, principal bundles with connection, or geometric string structures.

Applications include

• a generalization of the Galatius–Madsen–Tillmann–Weiss theorem;
• a solution to a conjecture of Stolz and Teichner on representability of concordance classes of functorial field theories;
• a construction of power operations on the level of field theories (extending the recent work of Barthel–Berwick-Evans–Stapleton);
• and a recent solution by Grady of a conjecture by Freed and Hopkins on deformation classes of reflection positive invertible field theories.

If time permits, I will talk about planned work on the nonperturbative quantization of functorial field theories and generalized Atiyah–Singer-style index theorems.