Dynamical systems with finite or infinite number of degrees of freedom, such as hydrodynamics, magnetohydrodynamics, and various interable systems, often have certain symmetry groups. Recent advances in the description of behaviour of those systems are related to the study of differential geometry of the corresponding groups. The geometric theory allows one to describe both quantitative and qualitative global properties of trajectories of such systems. For instance, the time evolution of a flow of an ideal incompressible fluid is governed by the geodesic equation on the group of volume-preserving diffeomorphisms.
Kinetic energy defines the corresponding right-invariant metric on that group.
Thus group theory and differential geometry provide a natural framework for the description of fluid motions. This approach, originally suggested by V.Arnold in 1966 for an ideal fluid, its later extensions to various media, magnetohydrodynamics, and integrable systems, as well as recent spectacular achievements in these areas, will be the subject of this workshop. This scope of mathematical ideas gives a profound insight on the nature of various physical systems, both finite and infinite-dimensional, and opens new areas of their studies. Moreover, numerous applications to incompressible and compressible, homogeneous and stratified fluids, MHD fluid, rigid bodies, and many other dynamical systems proved to be fruitful for mathematical theories involved.