Renormalisation of \phi^4-theory on noncommutative R^4 to all orders
  We present the main ideas and techniques of the proof that the duality-covariant four-dimensional noncommutative \phi^4-model is renormalisable to all orders. This includes the reformulation as a dynamical matrix model, the solution of the free theory by orthogonal polynomials as well as the renormalisation by flow equations involving power-counting theorems for ribbon graphs drawn on Riemann surfaces. 
Harald Grosse (Vienna), Raimar Wulkenhaar (Leipzig)