Symplectic and Poisson geometry (M-F512)
Symplectic and Poisson geometry (M-F512)
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 lectures typically take place weekly on Wednesday at 14:00-16:00 in room O7.214 on the 7th floor, with a few exceptions: see below. At the end of the course there will be an oral exam.
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, 14:00-17:00): Lie algebroids and Lie groupoids. Symplectic integrations of Poisson manifolds; Symplectic realizations of Poisson manifolds. Quantization.
Week 11 (December 11, 10:30-12:30): Singular foliations and almost-regular Poisson structures; Log- and elliptic Poisson manifolds. Tuesday
Lecture notes will be made available during the course and are available upon request. For further reading, we recommend:
Symplectic geometry:
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.
Poisson geometry:
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.