This is the webpage for the course on symplectic and Poisson geometry during the fall of 2018 at the Université libre de Bruxelles. *Last update: November 21, 2018*.

The following is the tentative schedule of the course, subject to change.

**Week 1 (October 3):**Introduction. Symplectic linear algebra: symplectic vector spaces; (co)isotropic, Lagrangian and symplectic subspaces; the standard symplectic structure. Operations on symplectic vector spaces; linear coisotropic reduction.**Week 2 (October 10):**Symplectic manifolds: definition of symplectic structures; simple examples; cohomological obstructions. (Co)isotropic, Lagrangian and symplectic submanifolds. Hamiltonian and symplectic vector fields. The cotangent bundle and its Lagrangian submanifolds.**Week 3 (October 17):**Moser methods and Darboux's theorem: Moser isotopy theorem; symplectic vector bundles; Neighbourhood theorems around constant rank submanifolds: isotropic, coisotropic, Lagrangian, symplectic submanifolds. The origins and definition of the Poisson bracket for symplectic manifolds.**Week 4 (October 24):**Poisson manifolds: definition of almost-Poisson brackets; correspondence between k-derivations and k-multivector fields; the Schouten-Nijenhuis bracket. Integrability: almost-Poisson bivectors; Hamiltonian vector fields. Poisson maps and their reformulations. Simple examples: zero Poisson structure, symplectic manifolds.**Week 5 (October 31):**No lecture (semaine intermédiaire 1)**Week 6 (November 7, 14:00-17:00):**Further examples: constant, linear and quadratic Poisson structures; almost-symplectic and nondegenerate almost-Poisson structures. Geometry of Poisson manifolds: rank of the Poisson structure; regular points; regular Poisson manifolds. Poisson submanifolds; statement on the symplectic foliation.**Week 7 (November 14,****14:00-17:00):**Proof of existence of the symplectic foliation; the Weinstein splitting theorem. Further properties of the symplectic foliation: interaction with Poisson maps and Poisson submanifolds.**Week 8 (November 21):**Proof of the Weinstein splitting theorem. Poisson and coisotropic submanifolds; Poisson transversals and their properties.**Week 9 (November 27, 10:30-12:30):**More on Poisson transversals; Poisson cohomology. Linearizability; Poisson Moser trick; Gauge transformations. Poisson algebroid:**Tuesday****Week 10 (December 5):**(TBA): Lie algebroids and Lie groupoids. Symplectic integrations of Poisson manifolds; Symplectic realizations of Poisson manifolds. Quantization.**Week 11 (December 11, 10:30-12:30):**(TBA): Singular foliations and almost-regular Poisson structures; Log- and elliptic Poisson manifolds.**Tuesday**

Lecture notes will be made available during the course. For further reading, we recommend:

- A. Cannas da Silva -
**Lectures on symplectic geometry**, LNM 1764, Springer-Verlag, Berlin, 2001; - D. McDuff, D. Salamon -
**Introduction to symplectic topology**, Third Edition, Oxford University Press, 2017.

- J.-P. Dufour, N.T. Zung,
**Poisson structures and their normal forms**, Progress in Mathematics, Vol. 242, Birkhauser, Basel, 2005; - A. Cannas da Silva, A. Weinstein -
**Geometric models for noncommutative algebras**, Berkeley Mathematics Lecture Notes, 10. American Mathematical Society, Providence, RI, 1999; - I. Vaisman -
**Lectures on the geometry of Poisson manifolds**, Progress in Mathematics, Vol. 118, Birkhauser, Basel, 1994; - C. Laurent-Gengoux, A. Pichereau, P. Vanhaecke -
**Poisson structures**, GMW 347, Springer-Verlag, Berlin, 2013.