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Counting toric actions on symplectic four-manifolds

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.