Geometry of the generalized SQG equations

The generalized SQG family of equations interpolate between the Euler equations of ideal hydrodynamics and the inviscid surface quasi-geostrophic equation; a well-known model for the three-dimensional Euler equations. These equations can be realised as geodesic equations on groups of diffeomorphisms. I will present some results pertaining to Fredholmness of the corresponding exponential maps, as well as the distribution of conjugate points. This is joint work with Martin Bauer, Gerard Misiołek and Stephen Preston.