I'm giving a talk at a postgrad seminar on the topic of topological quantum field theories (TQFTs) with a mixed audience of pure and applied mathematicians. As such, I'd like to be able to offer some applications of TQFTs within both disciplines beyond the obvious foundations in theoretical physics.
I'm aware of some uses in knot theory in terms of knot/link invariants in the form of Khovanov homology and also the induced map of the knot compliment under the TQFT.
I'm also aware that the functor category of $2$d-TQFTs is equivalent to the category of commutative Frobenius algebras.
Beyond these examples my reading has not gone far and I would appreciate any references and suggestions for further reading, or just links between TQFTs and other areas of mathematics that I can mention.