Representation theory, topological field theory, and the Andrews-Curtis   conjecture
  We pose a representation-theoretic question motivated by an attempt to resolve the Andrews-Curtis conjecture. Roughly, is there a triangular Hopf algebra with a collection of self-dual irreducible representations $V_i$ so that the product of any two decomposes as a sum of copies of the $V_i$, and $\sum (\rank V_i)^2=0$? This data can be used to construct a `topological quantum field theory' on 2-complexes which stands a good chance of detecting counterexamples to the conjecture. 
Frank Quinn