### Symplectic Geometry - M2 - 8EC

1. Prerequisites:

Some prerequisites for this course are the notions taught in a first course on differential geometry, such as: manifold, smooth map, immersion, submersion, tangent vector, Lie derivative along a vector field, the flow of a vector field, tangent bundle, differential form, de Rham cohomology.

Basic understanding of Lie groups and Lie algebras will also be useful, but not strictly necessary. The mastermath course Lie groups'' covers considerably more material from Lie theory than what we will use in this course.

In addition, we will use the notion of a smooth vector bundle over a manifold and some basic operations involving vector bundles, such as dualization and direct sum. The mastermath course Differential Geometry'' covers considerably more material from differential geometry than what we will use in this course.

Some knowledge of classical mechanics can be useful in understanding the context and some examples.

2. Aim and content of the course:

This course is aimed at master students with some background in differential geometry. This course will focus on the foundations of symplectic geometry:

• linear symplectic geometry
• symplectic manifolds
• canonical symplectic form on a cotangent bundle
• symplectomorphisms, Hamiltonian diffeomorphisms
• Poisson bracket
• Moser's isotopy method
• Darboux' theorem
• symplectic, (co-)isotropic and Lagrangian submanifolds of a symplectic manifold
• neighbourhood theorems for Lagrangian and symplectic submanifolds
• construction of symplectic manifolds (blow-up, symplectic sum)
• Hamiltonian Lie group actions, momentum maps
• symplectic reduction, Marsden-Weinstein quotient
• toric symplectic manifolds: convexity and Delzant's theorem

At the very beginning of the course, we will also explain some connections to classical mechanics, in order to provide the students with some context and motivation. If time allows, at the end of the course we might also discuss one of the following, more advanced topics: coadjoint orbits (as an important source of examples of symplectic manifolds with Hamiltonian Lie group actions), or the algebraic structure of the symplectomorphism group and the symplectic flux homomorphism, or existence of symplectic capacities and the Arnol'd conjecture.

There will be biweekly assignments, a group project whose outcome will be presented to the class and at the end of the course there will be an exam. There will be a retake exam, which every student may take. If the grade for the exam is at least 5 then the final grade is computed as