##### November 26 - 28, 2004

 R. Wulkenhaar (Leipzig):   Renormalization of Noncommutative $\phi^4_4$-theory to All Orders   Abstract: The renormalisation of noncommutative quantum field theories was an open problem for a long time due to the mixing of ultraviolet and infrared divergences. Recently, I have proven with H. Grosse that the noncommutative $\phi^4$-model is renormalisable to all orders in perturbation theory, provided that the action is made covariant with respect to a duality between positions and momenta. I give the main ideas and techniques of the proof. This includes the reformulation of the model in the matrix base of the Moyal plane, the solution of the free theory by orthogonal Meixner polynomials as well as the integration of flow equations for ribbon graphs.