On semi-toric integrable systems and their classification

Dr Christophe Wacheux

Frank Adams Room 1 ,

A classification of integrable Hamiltonian systems by objects that are both (relatively) easy to compute and meaningful from a physical point of view has been a long-standing goal of Hamiltonian geometry - a Holy Grail.

A first success in this direction in the 1980's was the classification of toric system (integrable systems whose flow are periodic of the same period, thus yielding a torus action). Atiyah, and Guillemin & Sternberg showed that the fibers of the momentum map were connected and its image was a convex polytope. Delzant showed that (under some natural extra hypotheses) this polytope describes entirely the integrable system.

A natural generalization of toric systems are the semi-toric systems. In these systems, occurrence of the so-called "focus-focus" singularities destroy one of the circle action, and the two theorems above do not apply anymore. In my talk I will discuss the problems that come along the focus-focus singularities, and how we can (hope to) recover a Delzant-like classification.


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