Quantum field theories (QFT) in four dimensions tend to be trivial or difficult, often both. QFT on noncommutative geometries provide new examples to try. They are not admissible examples in the strict sense of axioms, but they share very similar challenges such as renormalisation and construction of products of distributions. Since there is a finite-dimensional approximation in terms of matrices, QFTs on noncommutative geometries come with a topological grading by the Euler characteristic of a Riemann surface. We give examples where the formal expansion in the Euler characteristic, together with topological recursion, permit to solve non-commutative QFT-models exactly. We also discuss how this topological expansion relates to questions in free probability.

Bio: Raimar Wulkenhaar is a German mathematical physicist. He earned his doctorate form the University of Leipzig in 1997 under supervision of Gerd Rudolph. After postdoc positions at the Centre de Physique Théorique in Marseille, the University of Vienna and the Max Planck Institute for Mathematics in the Sciences in Leipzig, Wulkenhaar obtained a professorship in Mathematics at the University of Münster in 2005. His research is concerned with quantum fields on noncommutative geometries and more recently with topological recursion.