Computing Semitoric Invariants of Focus-Focus Singularities




2018.11.09 15:00-16:00


Middle Lecture Room, Math Building


A semitoric integrable Hamiltonian system, briefly a semitoric system, is given by two autonomous Hamiltonian systems on a 4-dimensional manifold whose flows Poissoncommute and induce an (S 1×R)-action that has only nondegenerate, nonhyperbolic singularities. Semitoric systems have been symplectically classified a few years ago by Pelayo & Vu Ngoc by means of five invariants. Three of these five invariants are the so-called Taylor series invariant, the height invariant, and the twisting index. Roughly, the first one describes the behaviour of the system near the focus-focus singular fibre, the second one the position of the focus-focus value, and the third one compares the ‘rotation’ within regular fibres close to one focus-focus singularity to that close to another focus-focus singularity. Recently there has been made considerable progress in understanding and computing these invariants and, in this talk, we present the (results of the) finished and ongoing projects with J. Alonso (Antwerp), H. Dullin (Sydney), and J. Palmer (Rutgers):

• Taylor series and twisting index for coupled spin oscillators. • Taylor series, height invariant, and twisting index for coupled angular momenta.

• Putting the Taylor series and twisting index in relation with wellknown notions from classical dynamical systems like rotation number and rotation vector etc.

• Change of the Taylor series and twisting index when varying the parameters of these systems.


• Symplectic classification of coupled angular momenta (with J. Alonso and H. Dullin), 59p., arXiv:1808.05849.

• Taylor series and twisting-index invariants of coupled spin-oscillators (with J. Alonso and H. Dullin), to appear in Journal of Geometry and Physics.

• A family of compact semitoric systems with two focus-focus singularities (with J. Palmer), Journal of Geometric Mechanics 2018, 10(3): 3