Cartan-Lie groupoids and geometric structures

Cartan-Lie groupoids and geometric structures The goal is to demonstrate how Cartan-Lie groupoids naturally appear in geometry and mathematical physics with the main focus on bi-invariant Riemannian metrics.
Hide abstract

{σ, τ}-Rota-Baxter operators, infinitesimal Hom-bialgebras
and the associative Hom-Yang-Baxter equation

{σ, τ}-Rota-Baxter operators, infinitesimal Hom-bialgebras
and the associative Hom-Yang-Baxter equation We present the concept of {sigma, tau}-Rota-Baxter operator, a twisted version of a Rota-Baxter operator of weight zero. We show how to obtain a certain {sigma, tau}-Rota-Baxter operator from a solution of the associative Hom-Yang-Baxter equation, and, in a compatible way, a Hom-pre-Lie algebra from an infinitesimal Hom-bialgebra.
The talk is based on recent joint work with L. Liu, A. Makhlouf and C. Menini.
Hide abstract

Integrable Hamiltonian systems with degenerate fibers

Integrable Hamiltonian systems with degenerate fibers After reviewing the classical definitions and properties of Hamiltonian systems and their momentum maps, I will explain some recent results on degenerate singularities. These results concern three specific families of Hamiltonian systems coming up with the surprising property that their fibers, despite being degenerate, are isotropic manifolds with nice geometric descriptions.
Hide abstract

Supersymmetric Poisson sigma model and graded geometry around it

Supersymmetric Poisson sigma model and graded geometry around it In this talk I will recall various instances of graded geometry, appearing naturally in some fields of classical differential geometry (Poisson, equivariant theory...). I will also explain that this is a useful language for theoretical physics (sigma models, gauging, symmetries of functionals...). The main goal would be to discuss possible extensions of the formalism to the supersymmetric setting. I will explain what I mean by "supersymmetric", how the above mentioned contructions can be useful, and what is complicated in the problem. The Poisson sigma model wil be the key example for all the constructions.
This talk is mostly based on the preprint: arXiv:1608.07457
Hide abstract

Pre-Lie algebras : algebraic and combinatorial approach

Pre-Lie algebras : algebraic and combinatorial approach Pre-Lie algebras were independently discovered by M. Gerstenhaber and E. B. Vinberg in 1963. They were already in germ more than one century ago in the works of A. Cayley on vector fields and rooted trees: this fact became clear in 2001 when F. Chapoton and M. Livernet gave an explicit description of the pre-Lie operad.
After recalling this historical context, I will present M. Al Kaabi's work on monomial bases, and a non-freeness result on the pre-Lie algebra associated with the free operad generated by one binary operation (joint work with E. Burgunder et B. Delcroix-Oger)
Hide abstract

Modular class of a singular foliation

Modular class of a singular foliation La classe modulaire est une généralisation de la divergence d'un champ de vecteur à d'autres structures géométriques comme la géométrie de Poisson ou les algébroides de Lie. La définition d'une classe modulaire pour un feuilletage régulier implique une forme volume du fibré normal ainsi que la connexion de Bott, et elle est horizontalement fermée (le long des feuilles). Dans le cas singulier, la définition classique ne marche plus puisque le fibré normal n'est plus nécessairement un fibré vectoriel. Nous montrerons qu'une manière naturelle de définir la classe modulaire d'un feuilletage singulier est d'utiliser un algébroide de Lie universel qui lui est associé.
Hide abstract

Smooth linearization of Poisson structures

Smooth linearization of Poisson structures In this talk, I will review some classical results of smooth linearization of Poisson structures. In particular, when the Lie algebra corresponding to the linear part is semisimple and compact, then the linearization is possible (J. Conn). If this Lie algebra has a real rank larger than 2, then one can find nonlinearizable cases (A. Weinstein). The real-rank 1 is still open ; we will give ideas to try giving some answers.
Hide abstract