  This note describes some recent results about the homotopy properties of
Hamiltonian loops in various manifolds, including toric manifolds and one point
blow ups. We describe conditions under which a circle action does not contract
in the Hamiltonian group, and construct an example of a loop $\ga$ of
diffeomorphisms of a symplectic manifold M with the property that none of the
loops smoothly isotopic to $\ga$ preserve any symplectic form on M. We also
discuss some new conditions under which the Hamiltonian group has infinite
Hofer diameter. Some of the methods used are classical (Weinstein's action
homomorphism and volume calculations), while others use quantum methods (the
Seidel representation and spectral invariants).
