On a symplectic manifold M, the group of Hamiltonian diffeomorphisms Ham(M) is a natural object of study. In classical mechanics, when M models the phase space of a system, Ham(M) arises as the group describing all admissible motions. In 1990, Hofer introduced a bi-invariant metric on this group. Intuitively, the Hofer distance between a Hamiltonian diffeomorphism and the identity can be thought of as the minimal energy necessary to generate the diffeomorphism. One can also extend this notion to a metric between Lagrangian submanifolds in the same Hamiltonian isotopy class. The Hofer distance between them should be thought of as the minimal amount of energy that has to spent to deform one into the other. The metric properties of these spaces remain largely elusive. However, they are very sensitive to the symplectic topology of the ambient space and the Lagrangian. This can often be seen through the large-scale behaviour of the Hofer metric. Here one usually looks at the space of all Lagrangians in a specific Hamiltonian isotopy class. In some situations, the diameter of this space endowed with the Hofer metric is finite. In other situations, it admits quasi-isometric embeddings of infinite dimensional normed vector spaces. We will outline several examples of these large-scale phenomena. If time permits, we will discuss the construction of such quasi-isometric embeddings for a specific example.
