# Seminar on Symplectic and Poisson Geometry

Takes place on Thursdays following the following schedule

Place: Big room of CRM
Morning session 1130-1230
Afternoon session 1500-1600

(from 11 on, there will be cookies and coffee in front of the conference room)

Exceptions for the second week of April when we also have sessions on Monday and Tuesday (see the schedule below for details)

#### April, 2008   May 2008  June 2008

April 3, Thursday

11:30 am: Philippe Monnier  (Université de Toulouse III) "Rigidity of Poisson group action"

We study the rigidity of Hamiltonian actions on a Poisson manifold when the Lie algebra is semisimple of compact type. We first give a result in the local case using an iterative process. Then we explain how to use a Nash-Moser theorem to try to get a similar result in the global case, when the manifold is compact.

15:00 pm: Anne Pichereau (CRM) "Formal deformations of Poisson structures in low dimension"

We consider formal deformations of Poisson structures in low dimensions. In fact, to each (weight) homogeneous polynomial $\varphi\in\C[x,y,z]$, one can associate a Poisson structure $\{.,.\}_\varphi$ on $\C^3$ and a singular Poisson surface in $\C^3$. Using some results about the Poisson cohomology of these Poisson affine varieties, we obtain a formula for all formal deformations of the Poisson structures considered, up to equivalence. In the generic case, we give a system of representatives for all formal deformations of $\{.,.\}_\varphi$, modulo equivalence.

April 7, Monday

12:00: Alberto Cattaneo (Universität Zürich) "Reduction via graded geometry"

After Roytenberg, Courant algebroids may be regarded as graded symplectic manifolds endowed with a Poisson self-commuting function. A generalized complex manifold may be regarded as such an object endowed with a second function satisfying one condition  also expressible in terms of Poisson brackets.As a consequence, reduction of Courant algebroids and generalized complex structures may be regarded as the usual symplectic reduction (now in the graded sense). In this talk (based on work in progress with Bursztyn, Mehta and Zambon) I will explain how to reformulate this in terms of classical (i.e., nongraded)  differential geometry, recovering and extending the existing reduction procedures.

April 8, Tuesday

12:00: David E. Blair (Michigan State University), "A complex geodesic flow", abstract.

April  10, Thursday

11:30 am: Marco Zambon (CRM)"A revisitation of the Marsden-Ratiu reduction theorem"

We will discuss the Marsden-Ratiu reduction theorem in Poisson geometry, which can be regarded as a version without group actions of the symplectic reduction by moment maps. We argue that the assumptions of the theorem are too restrictive and give a more general reduction theorem. The proof is of algebraic nature; we will discuss briefly other approaches.

April 17 Thursday

11:30 am Ignasi Mundet i Riera (Universitat de Barcelona) "The splitting formula for Hamiltonian Gromov-Witten invariants"

I will first briefly recall the definition of Hamiltonian Gromov-Witten invariants (which are invariants of compact symplectic manifolds endowed with a Hamiltonian action of the circle) and I will then explain that these invariants satisfy a splitting formula similar to the one in Gromov-Witten  theory, with the difference that the role of the class of the diagonal is played by the so called biinvariant diagonal class, a class in equivariant cohomology which seems to be interesting by itself.

15:00 pm: Raquel Caseiro (Universidade da Coimbra) "Modular classes of Poisson maps"

If $M$ and $N$ are Poisson manifolds and $\phi: M\rightarrow N$ is a Poisson map, what can one say about the relationship between the modular classes of $M$ and $N$? In order to answer this question, we will introduce the notion of the modular class of a Poisson map. Several examples will be discussed.

April 24 Thursday

11:30 am Jean Pierre Marco (Université de Paris 6), "Dynamical complexity and symplectic integrability''

IMPORTANT: This talk has a second part entitled "Freedom indices for strongly integrable hamiltonian systems" which will take place on MONDAY 28 at the seminar of the group of Dynamical Systems of UAB. The talk will take place at: CRM AT 1500h (click here for abstracts of both talks)

Completely integrable systems should be dynamically simple. Nevertheless there exist $C^\infty$--integrable geodesic flows with positive topological entropy. On the other hand integrable systems in action-angle form (that is systems on the annulus $T^*(T^l)$ defined by Hamiltonian functions which depend only on the action variable) obviously have zero topological entropy. By the $\sigma$--union property, one easily deduces that the topological
entropy of a completely integrable system is localized on the singular set of its first integral map. We will first give a new definition of integrability (which we call strong integrability), which gives enough control on the singular to guarantee that the topological entropy of a strongly integrable system vanishes.

In the class on strongly integrable systems, it is then possible to define a slower entropy (the freedom index), which amounts to considering the polynomial growth rate of the characteristic covering numbers instead of the exponential ones. The freedom index is infinite when the topological entropy is positive. We will give some natural properties of that index and investigate its behaviour on some very simple examples (action-angle systems and Hamiltonian systems on the two dimensional annulus).

15:00 pm:  Juan Pablo Ortega (CNRS, Université de Franche-Comté), "Stochastic Hamiltonian systems. Symmetries and skew-products"

In this talk we will start by presenting a global generalization of the stochastic Hamiltonian systems introduced by Bismut and we will see how they can be characterized via a variational principle. We will also see that that in the particular case in which a symmetry is present, the standard reduction and reconstruction techniques for the invariant dynamics yield skew-products in the sense of the theory developed for stochatic processes. This is a joint work with Joan-Andreu Lázaro Camí.