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Monday, November 4, 2013

**Abstract:** The classification of completely integrable Hamiltonian systems on symplectic manifolds is a driving question in the study of Hamiltonian mechanics and symplectic geometry. From a symplectic perspective, such systems correspond to Hamiltonian R^n-actions which are locally toric. The class of integrable Hamiltonian systems on 4-dimensional symplectic manifolds corresponding to Hamiltonian S^1 x R actions (with some extra assumptions on the singularities) is known as semi-toric: it was introduced by Vu Ngoc, and Pelayo and Vu Ngoc obtained a classification for `generic' semi-toric systems. From such a system one obtains a 4-dimensional manifold with a Hamiltonian S^1-action by restricting the action: when the underlying symplectic manifold is closed, Karshon classified these spaces in terms of a labelled graph. This talk aims at explaining how, starting from a semi-toric system on a closed 4-dimensional symplectic manifold, Karshon's invariants of the underlying Hamiltonian S^1-space can be recovered using the notion of `polygons with monodromy' introduced by Vu Ngoc. This should be thought of as analogous to the procedure to obtain Karshon's invariants from Delzant polygons in the case of symplectic toric manifolds. This is joint work with Sonja Hohloch (EPFL) and Silvia Sabatini (IST Lisbon), and part of a longer term project to study Hamiltonian S^1 x R actions on closed 4-dimensional manifolds.