Abstract: We present two algebraic methods of constructing invariants for smooth closed 4-manifolds. The first one is defined as a state-sum model on triangulations of 4-manifolds and the input data are certain tensor categories and more generally certain higher categories. The invariant can be extended to a topological quantum field theory (TQFT) and extends some of the previously known ones such as Dijkgraaf-Witten and Crane-Yetter/Walker-Wang TQFTs. The second is defined by contracting a tensor diagram assigned to trisections of 4-manifolds and the input data are three Hopf algebras satisfying some consistency conditions. It is not known if this invariant has a TQFT extension, though it includes the Crane-Yetter as special cases. Time permitting, we also mention some possible generalizations of these constructions to obtain potentially more powerful invariants.
