Studies of the dynamics of symmetric systems which can both deform in shape and rotate in space have driven much of the development of theoretical mechanics since the time of Newton. Examples range from the N-body problems of celestial and atomic and molecular physics through the rotating fluid systems from astrophysics and geophysics. In the last two decades key concepts which emerged from the 18th and 19th century, such as the existence of conserved quantities and the technique of phase space reduction, have been reinterpreted and generalized in an increasingly sophisticated geometric and analytic framework which can be applied to a wide range of practical problems. These ideas have also led to equally significant improvements in numerical techniques for studying mechanical system, the emphasis being on the development of robust methods for long time simulations which make full use of the underlying geometric structures.
Very recent work has suggested a number of exciting new direction for future work. These include combining geometric and symmetry methods from classical mechanics with semi-classical techniques to obtain new explanations and predictions of distinctive qualitive feature of atomic and molecular spectra. They also include the application of geometric and analytical studies of diffeomorphism groups to rotating fluid systems, and applications of multi-sympletic' geometry to optical fibre design. The aim of the proposed Summer Schools is to introduce these new geometric, analytic and numerical techniques to young and prospective researchers, and to demonstrate a range of possible applications.