TITLE: Motivation and unsolved problems of TQFT
QUESTION [13 upvotes]: I have been studying topological quantum field theory by mainly reading the Turaev's book.
I'd like to know if there are unsolved problems that motivate mathematicians to study TQFT, like Riemann's hypothesis for number theory.
I also would like to know if there is a paper or book that list big or small unsolved problems of TQFT. If not, could you suggest some problems here? I have been learning TQFT but I don't know what to do by myself as a graduate student.
Thank you.

REPLY [10 votes]: There are various open classification problems: classify modular tensor categories (the input for Reshetikhin-Turaev type theories), classify semisimple pivotal 2-categories (the input for Turaev-Viro type theories).  There is a vague conjecture, popular among physicists, that all examples of modular tensor categories are obtainable in some way from the standard $Rep(U_q(\mathfrak g))$ examples.