Abstract: The cobordism hypothesis of Baez--Dolan, whose proof
was sketched by Lurie, provides a beautiful classification of
topological field theories: for every fully dualizable object in a
symmetric monoidal (∞,d)-category, there is a unique (up to a
contractible choice) topological field theory whose value at the point
coincides with this object. As beautiful as this classification is, it
fails to include non-topological field theories. Such theories are
important not just in physics, but also in pure mathematics (for
example, Yang-Mills). In this talk, I will survey recent work with
Dmitri Pavlov, which proves a geometric enhancement of the cobordism
hypothesis. In the special case of topological structures, our theorem
reduces to the first complete proof of the topological cobordism
hypothesis, after the 2009 sketch of Lurie.