Asymptotic directions, Monge-Ampere equations and the geometry of   diffeomorphism groups
  In this note we obtain the characterization for asymptotic directions on various subgroups of the diffeomorphism group. We give a simple proof of non-existence of such directions for area-preserving diffeomorphisms of closed surfaces of non-zero curvature. Finally, we exhibit the common origin of the Monge-Ampere equations in 2D fluid dynamics and mass transport. 
Boris Khesin (U Toronto) and Gerard Misiolek (U Notre Dame)