Sonja Hohloch, From Hamiltonian S^1-actions to integrable systems with mild degeneracies
Channel: Scrambled Maths
Date: 2021-09-08

Sonja Hohloch, From Hamiltonian $$\mathbb S^1$$-actions to integrable systems with mild degeneracies

Abstract:
Let $$(M, \omega)$$ be a 4-dimensional compact symplectic manifold and assume that the Hamiltonian $$L: M \to \mathbb R$$ induces an effective Hamiltonian $$\mathbb S^1$$-action on $$(M, \omega)$$. Such a triple $$(M, \omega, L)$$ was called a {\em Hamiltonian $$\mathbb S^1$$-space} by Karshon who classified these spaces by means of labeled directed graphs. Karshon also showed that some $$\mathbb S^1$$-actions can be extended to a Hamiltonian $$\mathbb S^1 \times \mathbb S^1$$-action on $$(M, \omega)$$ induced by a smooth integrable `toric' system $$(L, T): (M, \omega) \to \mathbb R^2$$ and she gave the precise conditions under which this is possible. 
Thus, if one wants to extend a larger class of Hamiltonian $$\mathbb S^1$$-spaces to an integrable system one needs to admit a larger class than toric systems as target.
Hohloch $$\&$$ Sabatini $$\&$$ Sepe $$\&$$ Symington's work in progress removes part of Karshon's conditions and then they can extend to so-called {\em semitoric integrable systems} $$(L, S): (M, \omega) \to \mathbb R^2$$ which are, roughly, integrable systems that only allow elliptic-regular, elliptic-elliptic, and/or focus-focus points as singularities. These systems induce in fact an $$\mathbb S^1 \times \mathbb R$$-action on $$M$$, thus the name.
The aim of this talk is the class of so-called {\em hypersemitoric integrable systems} (see the joint work Hohloch $$\&$$ Palmer available at {\em arXiv:2105.00523}), which is the `easiest, smallest, and most reasonable' set of smooth integrable systems $$(L, H): (M, \omega) \to \mathbb R^2$$ to which {\em all} Hamiltonian $$\mathbb S^1$$-spaces $$(M, \omega, L)$$ can be extended. These systems induce an $$\mathbb S^1 \times \mathbb R$$-action on $$M$$ and the {\em nondegenerate} singularities of $$(L,H)$$ may have components of regular, elliptic, focus-focus, and/or hyperbolic type, thus the name. The occurrence of (finitely many) degenerate singular points cannot always be avoided, but there is a certain choice of what kind of degeneracies one wants to admit in the extension. Since parabolic (`cuspidal') points are a very natural class of degenerate points we opted to admit this type of degeneracies.
Eventually, we will outline the construction of extending a Hamiltonian $$\mathbb S^1$$-space to a hypersemitoric system by means of an example.

Recording of the workshop "Learning from Insulators: New Trends in the Study of Conductivity of Metals". For additional material and information, please consult the workshop page https://www.lorentzcenter.nl/learning-from-insulators-new-trends-in-the-study-of-conductivity-of-metals.html