Conférences
# Integrable Systems and Geometry of Complex Foliations

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Europe/Paris

Salle 318 (Institut de Mathématiques de Bourgogne)
### Salle 318

#### Institut de Mathématiques de Bourgogne

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Le programme de la conférence est le suivant :

**9h-10h** - San Vu Ngoc

Title: On semi-global invariants for focus-focus singularities**Abstract**: In this talk, I will describe a classification, up to symplectic equivalence, of singular Lagrangian foliations given by a completely integrable system of a 4-dimensional symplectic manifold, in a full neighborhood of a singular leaf of focus-focus type.

**10h15-11h15** - Daniel Panazzolo

**Title:** Classification of Dulac series and invariants of homoclinic loops.**Abstract**: The Dulac series is a class of generalized power series appearing,

for instance, as asymptotic expansions of the first return map in the vicinity of homoclinic loops.

We will discuss the problem of analytic classification of germs of ramified analytic functions which expand in Dulac series and show a quite intriguing rigidity phenomenon, where the formal classification implies the analytic classification.

Joint work with M. Resman and L. Teyssier.

**Déjeuner 12h-13h45**

**14h-15h** - Olga Lukina

**Title**: Dynamical and ergodic properties of flows on an infinite translation surface**Abstract**: A translation surface is a two-dimensional (often non-compact) topological manifold that admits an atlas where all transition maps are translations.

The simplest way to construct such a surface is by gluing the sides of a polygon (minus at most a countable collection of points) along parallel sides using translations.

Our interest is in the case when the resulting surface is an infinite translation surface, in particular, a surface of infinite genus with a wild singularity in its metric completion.

In the talk, we will study the dynamical and ergodic properties of parallel flows on one realization of such a surface.

We will model the first return maps of such flows by interval exchange transformations with an infinite number of intervals and by symbolic dynamical systems. Using these tools, we uncover some properties of such flows.

Joint work with Henk Bruin.

**15h30-16h30** - Konstantinos Efstathiou

**Title**: Maslov $S^1$ bundles**Abstract**: We introduce the Maslov $S^1$ bundles over symplectic manifolds $(M, \omega)$, that is, the determinant bundle $\Gamma$ of the unitary frame bundle over $M$, and the bundle $\Gamma^2 = \Gamma / \{ \pm 1 \}$.

The usual Maslov index is defined when the bundles are trivial. We discuss the properties of the Maslov bundles focusing on the interplay between their geometry and the dynamics of symplectic group actions on $M$.

Symplectic group actions can be lifted to group actions on the Maslov bundles. When $M$ is a homogeneous $G$-space, then so are $\Gamma$ and $\Gamma^2$.

Moreover, we provide an alternative proof of the fact that when $M$ is a monotone symplectic manifold then the symplectic action is Hamiltonian.

Finally, in the particular case of symplectic circle actions, we define the notion of Maslov data which generalizes the notion of Maslov index to the case where the Maslov bundle is not trivial.

Joint work with Bohuan Lin and Holger Waalkens.