Semitoric systems, hypersemitoric systems, and the affine invariant of hypersemitoric systems

Many naturally occurring dynamical systems have symmetries or conserved quantities (just think of systems with preserved angles, invariance under rotation etc.). Roughly, integrable systems are Hamiltonian dynamical systems that admit a maximal number of independent symmetries/ conserved quantities.
In 1988, Delzant symplectically classified toric integrable systems by means of their momentum map image which is a very nice and special convex polytope, often referred to as ‘Delzant polytope’ of the toric system.
Semitoric systems are integrable systems of the form $F=(J,H): (M, \omega) \to \mathbb R$ where $(M, \omega)$ is a 4-dimensional connected symplectic manifold and $J$ is proper and induces an effective Hamiltonian torus action and $F$ admits only nondegenerate singularities and no hyperbolic components. Intuitively, semitoric systems generalize toric systems in dimension four by admitting in addition to elliptic-elliptic and elliptic-regular singularities also focus-focus singularities. In 2009-2011, Pelayo and Vu Ngoc symplecticaly classified semitoric integrable systems in terms of 5 invariants, among which a generalized semitoric polytope deduced from the momentum map image, i.e. generalizing the Delzant polytope.
When admitting also hyperbolic components for the nondegenerate singularities and possibly also mildly degenerate (so-called parabolic) singular points, then one generalizes semitoric systems to so-called hypersemitoric systems. The long term goal is to obtain a symplectic classification of hypersemitoric integrable systems on compact connected 4-dimensional symplectic manifolds.
This talk starts with an overview over the state of the art on research of semitoric and hypersemitoric systems and then focuses on one of the hypersemitoric invariants, namely the so-called ‘affine invariant’, which is the generalization of the semitoric polytope invariant.
This talk is based on ongoing work with N. Flamand (Antwerp) and a joint preprint (arXiv:2411.17509) with K. Efstathiou (Duke Kunshan University) and P. Santos (Antwerp).