Riemannian Exponential Map on the Group of Volume-Preserving Diffeomorphisms

In 1966 V. Arnold showed how solutions of the Euler equations of hydrodynamics can be viewed as geodesics in the group of volume-preserving diffeomorphisms. This provided a motivation to study the geometry of this group equipped with the \(L^2\) metric. I will describe some recent work on the structure of singularities of the associated exponential map and related results