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Monday, January 26, 2015

**Abstract:** The classification of completely integrable Hamiltonian systems is a driving question in Hamiltonian mechanics and symplectic geometry as these can be thought of as (examples of) complexity zero Hamiltonian R^n-actions on symplectic 2n-dimensional manifolds. Away from the locus of singularities (which, mechanically, correspond to equilibria of the system), Duistermaat's approach has revealed the importance of integral affine structures in the study of such systems. A natural question is whether the relation between integral affine geometry and completely integrable Hamiltonian systems can be extended to singularities, which contain important geometric and dynamical information. The aim of this talk is to present a differential-geometric notion of singular integral affine structures by illustrating its role in the classification of neighbourhoods of focus-focus singular fibres, which are analogous to nodal fibres in Lefschetz fibrations. This is ongoing joint work with Rui Loja Fernandes.