  We present a Hamiltonian framework for higher-dimensional vortex filaments
(or membranes) and vortex sheets as singular 2-forms with support of
codimensions 2 and 1, respectively, i.e. singular elements of the dual to the
Lie algebra of divergence-free vector fields. It turns out that the localized
induction approximation (LIA) of the hydrodynamical Euler equation describes
the skew-mean-curvature flow on vortex membranes of codimension 2 in any
dimension, which generalizes the classical binormal, or vortex filament,
equation in 3D. This framework also allows one to define the symplectic
structures on the spaces of vortex sheets, which interpolate between the
corresponding structures on vortex filaments and smooth vorticities.
