Boundary conditions for extended TQFTs and anomalies

In this talk I will present recent work together with Domenico Fiorenza on topological quantum field theories(TQFTs) on manifolds with boundaries. Using infinity-category theory, we develop a formalism to relate extended TQFTs with boundary conditions to lower dimensional theories with anomalies, which in turn give rise to projective representations of the mapping class group of closed manifolds. The case of fully extended TQFTs will be discussed as well.