Abstract: Let M be a closed symplectic manifold and denote by Ham its group of Hamiltonian diffeomorphisms. When equipped with the standard smooth topology, this is an infinite dimensional Fréchet Lie group. It is generally believed that Ham is "tamer" than the diffeomorphism group Diff(M) and constitutes an intermediate object between compact Lie groups and more general diffeomorphism groups. To develop a better understanding of this principle, one may look at maximal Hamiltonian actions by tori or, in other words, to classify symplectic conjugacy classes of maximal compact tori in Ham. In this talk, we will show that for 4-dimensional symplectic manifolds, there are at most finitely many of those conjugacy classes. As a by-product, we will also prove that the rational cohomology algebra of the symplectomorphism group of a generic blow-up is not finitely generated.