From compact semi-toric systems to Hamiltonian $S^1$-actions and back

Vendredi 20 novembre 2015 14:00-15:00 - Sonja Hohloch - Anvers

Résumé : Roughly, a semi-toric integrable Hamiltonian system (briefly, a semi-toric system) on a compact 4-dimensional manifold consists of two commuting Hamiltonian flows one of which is periodic. Thus the flow parameters induce an $S^1 $x R-action on the manifold. Under certain assumptions on the singularities, semi-toric systems have been classified by Pelayo & Vu Ngoc by means of 5 invariants.
Every semi-toric system induces a Hamiltonian $S^1$-action on the manifold by `forgetting’ the R-valued flow parameter. Effective Hamiltonian $S^1$-actions on compact 4-manifolds have been classified by Karshon by means of so-called `labeled directed graphs’.
In a joint work with S. Sabatini and D. Sepe, we linked Pelayo & Vu Ngoc’s classification of semi-toric systems to Karshon’s classification of Hamiltonian $S^1$-actions. More precisely, we show that only 2 of the 5 invariants are necessary to deduce the Karshon graph of the underlying $S^1$-action.
In an ongoing work with S. Sabatini, D. Sepe and M. Symington, we study how to `lift’ an effective Hamiltonian S^1-action on a compact 4-manifold to a semi-toric system.
In this talk, we give an introduction to semi-toric systems and Hamiltonian $S^1$-actions and sketch parts of our constructions.

Lieu : Bât 440, salle 228

From compact semi-toric systems to Hamiltonian $S^1$-actions and back  Version PDF