Geometric Theory of Hamiltonian Lie Systems. The theory of Lie systems of differential equations has been shown to be very efficient in dealing with many problems in physics and in mathematics. The usefulness of the existence of additional geometric structures in the manifold where the Lie system is defined, for instance Poisson structures, will be analysed and the theory will be illustrated with several examples as the Smorodinsky-Winternitz oscillator, the two photon system and the second order Riccati equation.

On the linearization of proper Lie groupoids. The linearization theorem for proper Lie groupoids is a generalization of the standard tube (slice) theorem from group actions. It is also closely related to Conn's linearization theorem in Poisson Geometry. It was conjectured by Weinstein, who proved it in the regular case and also reduced the statement to the case of fixed points. The fixed point case was proven by Zung. In this talk I will report on the recent joint work with Ivan Struchiner: in the fixed point case we give a shorter and more geometric proof than Zung's, based on a Moser deformation argument; then the passing to general orbits (Weinstein) is given a more conceptual interpretation: as a manifestation of Morita invariance. We also clarify the precise conditions needed for the theorem to hold (which often have been misstated in the literature).

On the nature of transverse Poisson structures to coadjoint orbits. Transverse Poisson structures to the symplectic leaves of a given Poisson structure were defined in 1983 by A. Weinstein, and are probably the simplest example of singular Poisson structures which are not linear (or Lie-Poisson) structures. Even when the original Poisson structure is linear, transverse Poisson structures (to its coadjoint orbits) can be rather complicated. We will focus on the nature (linear/polynomial) of transverse Poisson structures to coadjoint orbits, and try to see whether this notion is any different from the usual notion of linearizable/polynomializable Poisson structure.

Discrete dynamics in implicit form. We introduce the notion of implicit difference equation on a Lie groupoid and formulate an algorithm for extracting the integrable part (backward or/and forward). A motivation of this construction is that discrete Lagrangian dynamics on a Lie groupoid may be described in terms of Lagrangian implicit difference equations of the corresponding cotangent groupoid. Other situations include finite difference methods for time-dependent linear differential-algebraic equations and discrete nonholonomic Lagrangian systems, as particular examples. This is joint work with Juan Carlos Marrero, David Martín de Diego and Edith Padrón.

On Dirac structures and Dirac pairs. This is a story about pairs. In the first part we shall show how, on Courant algebroids that constitute a double, we can describe the characteristic pair which defines a Dirac structure in terms of Terashima's ``Poisson functions'' that generalize the ``Hamiltonian operators'' of Liu-Weinstein-Xu which themselves generalize the ``Poisson bivectors''. In the second part, we shall define Dirac pairs in terms of Nijenhuis relations, and we shall show that the notion of Dirac pairs unifies Hamiltonian pairs (bi-Hamiltonian structures), PΩ-structures, and a restricted class of ΩN-structures. We shall give explicit examples on 4-dimensional space.

Gerstenhaber structure on intersection cohomology. We explain why a structure of Gerstenhaber algebra that the intersection cohomology of two coisotropic submanifolds is endowed with (as shown by Behrend-Fantechi and Baranovsky-Ginsburg) is related to the L-infinity structure that the cotangent space of a coisotropic submanifold is endowed with. Also, the existence of quantizations of those is shown to play a role.

Global action-angle variables for non-commutative integrable systems. I will describe obstructions to the existence of global action-angle variables for non-commutative integrable systems in Poisson manifolds. This is ongoing joint work with Raquel Caseiro, Camille Laurent-Gengoux and Pol Vanhaecke.

Marsden-Weinstein reduction theory for the symplectic prolongation of a Lie algebroid. The natural example of a symplectic Lie algebroid is the prolongation T^A A* of a Lie algebroid over the vector bundle projection of the dual bundle A*. In this talk, I will present some results on the Marsden-Weinstein reduction of T^A A*. More precisely, I will discuss under what conditions the reduction of T^A A* is again the prolongation of a Lie algebroid over its dual vector bundle projection. It is the Lie algebroid counterpart of the cotangent bundle reduction theory.

It is a work in collaboration with E. Padrón and M. Rodríguez-Olmos.

Hamiltonian systems on Lie algebroids: Variational description and momentum conservation.

From Statics to Dynamics: equations which govern equilibria and motions of mechanical systems. More than two hundred years before J.C., Archimedes understood the basic principles of Statics. The mathematical formulation of the laws of Dynamics was developed much later, during the XVI, XVII and XVIII centuries, and reached a state of maturity at the end of the XIX century. New views about Space and Time appeared at the beginning of the XX century, with the Special and General Relativity theories. Their integration in the mathematical description of the motion of mechanical systems was surprisingly easy. In this lecture, after a short historical introduction, I will present the main ideas which allowed the transition from Statics to Dynamics and the development of a usable mathematical formulation of the motion of mechanical systems. The principle of inertia, virtual displacements and virtual work, the Lagrange differential, Lagrangian and Hamiltonian formulations of dynamics and the equivalence principle will be discussed.

Poisson structures with prescribed Casimirs. We consider the problem of constructing Poisson brackets on smooth manifolds M with prescribed Casimir functions. If M is of even dimension, we achieve our construction by considering a suitable almost symplectic structure on M, while, in the case where M is of odd dimension, our objective is achieved by using a convenient almost cosymplectic structure. Several examples and applications are presented.

Circle valued momentum maps. The talk will present a proof of two 'folklore' results: every symplectic circle action on a compact manifold admits a circle valued momentum map which is Morse-Bott-Novikov and each connected component of the fixed point set has even index. If time permits, the following problem will be also discussed: when Hodge theory survives on non-compact symplectic manifolds, a compact symplectic Lie group action having fixed points is necessarily Hamiltonian, provided the associated almost complex structure preserves the space of harmonic one-forms.

Quadratic non-commutative Poisson structures. We study some general non-abelian quadratic Poisson brackets. Their study was motivated by integrable systems on associative algebras. We give full classification results in the case of the free associative algebra with 2 generators. Relations with double Poisson structure of M. Van den Bergh and structures of W. Crawley-Boevey are discussed.

Lie theory for representations up to homotopy. I give an overview of recent developments regarding the Lie theory for representations up to homotopy and applications thereof. The talk is partly based on joint work with Camilo Arias Abad.

Unifying hypercomplex and holomorphic symplectic structures. We will describe a new type of structure on Courant algebroids and explain how it can be seen as a simultaneous/dual generalization of two seemingly unrelated classical geometric structures on manifolds: hypercomplex triples and holomorphic symplectic 2-forms. Two equivalent characterizations of these structures will be presented: one in terms of Nijenhuis concomitants and the other in terms of (almost) torsion free connections. Several classes of examples will be discussed and classical results recovered.

Simultaneous deformations of Maurer-Cartan elements. Given an $L_{\infty}$-algebra, we prove a theorem about the simultaneous deformation of the $L_{\infty}$-algebra structure and of its Maurer-Cartan elements. The main tool is Ted Voronov's derived bracket construction. The above theorem allows to make statements about the deformation theory of a wide class of algebraic and geometric objects. We will discuss, among others, the example of deformations of twisted Poisson structures. This is joint work with Yael Fregier (Univ. Lens).

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Paulo Antunes, Camille Laurent-Gengoux and Joana Nunes da Costa.
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Compatibility on Courant algebroids.

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Mònica Aymerich.
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On the local structure of coisotropic submanifolds of linear Poisson manifolds.

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Ángel Ballesteros.
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q-Poisson algebras as Poisson-Lie groups.

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Alfonso Blasco.
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Integrable deformations of Lotka-Volterra systems from
Poisson-Lie structures.

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Leonardo Colombo.
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On the geometry of variational problems on Lie groups.

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Joseph Dongho.
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On logarithmic Poisson cohomology and applications.

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Chiara Esposito.
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Poisson reduction.

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Ignazio Lacirasella.
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Symplectic AV-bundle reduction.

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Miguel Vaquero.
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On poly-Dirac manifolds.