  Given a symplectic manifold, we ask in how many different ways can a torus
act on it. Classification theorems in equivariant symplectic geometry can
sometimes tell that two Hamiltonian torus actions are inequivalent, but often
they do not tell whether the underlying symplectic manifolds are
(non-equivariantly) symplectomorphic. For two dimensional torus actions on
closed symplectic four-manifolds, we reduce the counting question to
combinatorics, by expressing the manifold as a symplectic blowup in a way that
is compatible with all the torus actions simultaneously. For this we use the
theory of pseudoholomorphic curves.
