Some prerequisites for this course are the notions taught in a first course onĀ  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.

Aim 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
  • symplectic, (co-)isotropic and Lagrangian submanifolds of a symplectic manifold
  • normal form theorem for a submanifold of a symplectic manifold
  • Darboux's theorem
  • Weinstein's neighbourhood theorem for a Lagrangian submanifold
  • Hamiltonian Lie group actions, momentum maps
  • symplectic reduction, Marsden-Weinstein quotient

We will also explain connections to classical mechanics, such as Noether's theorem and reduction of degrees of freedom. If time allows, we will also discuss one or two of the following additional topics:

  • coadjoint orbits, which are an important source of examples of symplectic manifolds with Hamiltonian Lie group actions;
  • the algebraic structure of the symplectomorphism group and the symplectic flux homomorphism;
  • contact geometry and its interactions with symplectic geometry.

The last lecture will be reserved for a panorama of recent results in the field of symplectic geometry, for instance the existence of symplectic capacities and the Arnol'd conjecture.

Fabian Ziltener (Universiteit Utrecht) & Federica Pasquotto (Universiteit Leiden)