Abstract: I will review the notion of a topological quantum field theory for manifolds with corners (staying somewhat informal for higher dimensions, where higher categorical structures appear), as well as Lurie's classification of these. Next, we will focus on invertible field theories (a.k.a anomaly theories). They seem able to reconcile an older classification result of Reshetikhin-Turaev in 3 dimensions, in terms of modular tensor categories, with Lurie's general classification. Reconciliation proceeds via a (new) observation that invertibility of a theory can be detected on spheres of half the dimension, as well as an (older) theorem of M. Mueger's characterizing modular tensor categories. This is joint work with Dan Freed, much of it still in progress, and is inspired by ideas of K. Walker on Chern-Simons theory.
