This talk starts with motivating the definition of hypersemitoric systems which are two degree of freedom integrable Hamiltonian systems on 4-dimensional symplectic manifolds with possibly mild degeneracies where one of the integrals gives rise to an effective Hamiltonian S^1-action. Then we give a brief overview of their main features and some examples before we sketch a topological classification of their fibers using 'labeled graphs'. This talk starts with motivating the definition of hypersemitoric systems which are two degree of freedom integrable Hamiltonian systems on 4-dimensional symplectic manifolds with possibly mild degeneracies where one of the integrals gives rise to an effective Hamiltonian S^1-action. Then we give a brief overview of their main features and some examples before we sketch a topological classification of their fibers using 'labeled graphs'.
