Symplectic Geometry - M2 - 8EC

  1. Prerequisites:
 
Notions taught in a bachelor 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, exterior derivative, de Rham cohomology, integration and Stokes theorem.

Notions taught at the Mastermath course on differential geometry such as smooth vector bundles over manifolds, transition functions for vector bundles with trivializations, operations involving vector bundles, such as dualization and direct sum, connections, curvature, simple computations of the first Chern class. We expect students to have successfully completed the Mastermath course on differential geometry.


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.

Some knowledge of classical mechanics can be useful in understanding the context and some examples.
  
  1. 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
  •         complex structures and Kähler manifolds
  •         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
  •         Duistermaat-Heckman theorem

  1. Rules about homework/exam:
 
There will be biweekly homework assignments 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
If (grade of the exam) < 5, then
final grade = grade for the exam, 
If (grade of the exam) >= 5, then
final grade = 0.25*(grade for the assignments) + 0.75*(grade for exam).
In case of a retake, the same formula applies to the retake exam:
final grade after retake  = 0.25*(grade for the assignments) + 0.75*(grade for retake exam).
    4.  Literature:
      
        Course lecture notes
 
        Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology.
 
        Ana Cannas da Silva, Lectures on symplectic geometry.
 
       
         5.   Lecturers:
 
Gil Cavalcanti (Universiteit Utrecht) & Federica Pasquotto (Universiteit Leiden)