## Final Activity Report Summary - GEASSIS (Geometry and stability of singularities in integrable systems)

We have studied the singularities of integrable systems in symplectic and Poisson manifolds, including their structural stability aspects, normal forms, entropy, geometric quantisation and equivariant aspects.

As a previous aproach to the problem of integrable systems on Poisson manifolds, one has to consider the problem of normal forms for the Poisson structure itself. The first local normal form results for Poisson structures was given by Weinstein in 83. Weinstein proved that locally a Poisson manifold is a product of a symplectic manifold which has as dimension the rank of the Poisson manifold together with a transversal Poisson manifold where the Poisson structure vanishes. We (the grantholder together with the Scientific supervisor) have studied the equivariant versions of this result in the paper entitled "A note on equivariant normal forms of Poisson structures"(Mathematics Research Letters, 2006 , vol 13-6, pages 1001-1012.)

Namely, we prove that given a Poisson structure which is tame at a point and a compact group action that fixes the point and preserves the Poisson structure there exists coordinates in which the action is linear and the Poisson structure is split as in Weinstein's theorem. As a consequence of this result the authors obtain a linearization result for the Poisson structure and the group action itself when the transversal part is of compact semisimple type. In order to obtain this result, the authors combine the linearization result of Conn which previous linearization theorems of Viktor Ginzburg. The result obtained by Eva Miranda together with Nguyen Tien Zung generalises other results of linearization as Bochner to the Poisson context. Equivariant versions are related to integrable systems in the sense that many examples of integrable systems are given by group actions.

These problems of linearization lead naturally to the study of rigidity problems for group actions in the Poisson context. Eva Miranda has used Moser's path method in the paper (Some rigidity results for symplectic and Poisson group actions, to appear in Proceedings of the XV International Conference on Geometry and Physics) to prove that symplectic compact group actions are rigid on compact symplectic manifolds. Eva Miranda has also established similar linearization results for semisimple Lie algebra actions in the analytic setting thus extending results of Guillemin and Sternberg.

In the Poisson case, Eva Miranda and Nguyen Tien Zung together with Philippe Monnier have studied the problem of global and local rigidity in the Poisson case and have proved a rigidity result for the case of Hamiltonian group actions (these results are contained in a preprint).

Finally, as far as the problem of singularities and normal forms for integrable systems is concerned. Eva Miranda in a recent work with Camille Laurent-Gengoux and Pol Vanhaecke have studied the case of normal forms for integrable systems in a Poisson manifold in the local and semilocal (neighbourhood of an orbit) cases. In the case the Poisson manifold is regular they prove the existence of action-angle coordinates for Poisson manifolds. In the general case, they prove a partial splitting theorem for a subset of first integrals and prove that those integrable systems admit no splitting theorem in general.

We have also started new directions related to the subject and its applications to perturbation theory in Dynamical systems. Namely, the grantholder has been working on entropy of integrable Hamiltonian systems, proving that the kind of singularities play a key role in the computation of entropy. Thus, if the system is analytical we have proved that the entropy is zero. If the system is smooth the existence of flat functions can make entropy be non-zero. This is related to chaotic behaviour.

Last but not least, Eva Miranda has also studied together with Victor Guillemin and Mark Hamilton the geometric quantisation of integrable systems with non-degenerate singularities. In particular, we obtain a result where it is seen that quantization depends strongly on the polarization and the kind of singularities.

As a previous aproach to the problem of integrable systems on Poisson manifolds, one has to consider the problem of normal forms for the Poisson structure itself. The first local normal form results for Poisson structures was given by Weinstein in 83. Weinstein proved that locally a Poisson manifold is a product of a symplectic manifold which has as dimension the rank of the Poisson manifold together with a transversal Poisson manifold where the Poisson structure vanishes. We (the grantholder together with the Scientific supervisor) have studied the equivariant versions of this result in the paper entitled "A note on equivariant normal forms of Poisson structures"(Mathematics Research Letters, 2006 , vol 13-6, pages 1001-1012.)

Namely, we prove that given a Poisson structure which is tame at a point and a compact group action that fixes the point and preserves the Poisson structure there exists coordinates in which the action is linear and the Poisson structure is split as in Weinstein's theorem. As a consequence of this result the authors obtain a linearization result for the Poisson structure and the group action itself when the transversal part is of compact semisimple type. In order to obtain this result, the authors combine the linearization result of Conn which previous linearization theorems of Viktor Ginzburg. The result obtained by Eva Miranda together with Nguyen Tien Zung generalises other results of linearization as Bochner to the Poisson context. Equivariant versions are related to integrable systems in the sense that many examples of integrable systems are given by group actions.

These problems of linearization lead naturally to the study of rigidity problems for group actions in the Poisson context. Eva Miranda has used Moser's path method in the paper (Some rigidity results for symplectic and Poisson group actions, to appear in Proceedings of the XV International Conference on Geometry and Physics) to prove that symplectic compact group actions are rigid on compact symplectic manifolds. Eva Miranda has also established similar linearization results for semisimple Lie algebra actions in the analytic setting thus extending results of Guillemin and Sternberg.

In the Poisson case, Eva Miranda and Nguyen Tien Zung together with Philippe Monnier have studied the problem of global and local rigidity in the Poisson case and have proved a rigidity result for the case of Hamiltonian group actions (these results are contained in a preprint).

Finally, as far as the problem of singularities and normal forms for integrable systems is concerned. Eva Miranda in a recent work with Camille Laurent-Gengoux and Pol Vanhaecke have studied the case of normal forms for integrable systems in a Poisson manifold in the local and semilocal (neighbourhood of an orbit) cases. In the case the Poisson manifold is regular they prove the existence of action-angle coordinates for Poisson manifolds. In the general case, they prove a partial splitting theorem for a subset of first integrals and prove that those integrable systems admit no splitting theorem in general.

We have also started new directions related to the subject and its applications to perturbation theory in Dynamical systems. Namely, the grantholder has been working on entropy of integrable Hamiltonian systems, proving that the kind of singularities play a key role in the computation of entropy. Thus, if the system is analytical we have proved that the entropy is zero. If the system is smooth the existence of flat functions can make entropy be non-zero. This is related to chaotic behaviour.

Last but not least, Eva Miranda has also studied together with Victor Guillemin and Mark Hamilton the geometric quantisation of integrable systems with non-degenerate singularities. In particular, we obtain a result where it is seen that quantization depends strongly on the polarization and the kind of singularities.