Quaternionic Monge-Ampere equations and HKT-geometry

A notion of quaternionic Monge-Ampere equation will be introduced. These are non-linear second order elliptic equations which make sense on so called hypercomplex manifolds, in particular on the flat quaternionic space. They admit an interpretation in the framework of Hyper Kahler with Torsion (HKT) geometry (to be explained in the talk). We formulate a quaternionic version of the Calabi conjecture, and state a number of partial results towards its proof. Part of the results are joint with M. Verbitsky.