Solitons and Geometry

This is an introduction to the geometry of Hamiltonian systems from the modern point of view where the basic structure is a Poisson bracket. Using this approach a mathematical analogue of the famous 'Dirac monopole' is obtained starting from the classical top in a gravity field. This approach is especially useful in physical applications in which a field theory appears; this is the subject of the second part of the lectures, which contains a theory of conservative hydrodynamic-type systems, based on Riemannian geometry, developed over the last decade. The theory has had success in solving problems in physics, such as ones associated with dispersive analogues of shock waves, and its development has led to the introduction of new notions in geometry. The book is based on lectures given by the author in Pisa and which were intended for a non-specialist audience. It provides an introduction from which to proceed to more advanced work in the area.