Math 653 — Differentiable Manifolds II:

A Course in Symplectic Geometry

Spring 2003

Instructor: Patrick Iglesias

Time: MWF 11:15-12:05

Room: MT 230

This course assumes a minimal knowledge of basic differential geometry: manifolds, differential calculus, differential forms, action of groups, Lie derivative...

1) Symplectic algebra:

- definition of a symplectic linear form
- theorem of the symplectic base, canonical expression of the symplectic form, space of symplectic structures
- isotropic and lagrangian sub-spaces, orthogonality and dimension, linear symplectic reduction
- the lagrangian grassmannian, homotopy, covering
- Examples in finite and infinite dimension

2) Symplectic geometry

- Symplectic manifolds, definition
- Moser and Darboux theorems, Darboux Charts, Darboux Atlases
- Classical examples
- Lagrangian submanifolds
- Symplectomorphisms, symplectic action of Lie groups, moment map, symplectic reduction, examples
- Canonical symplectic structure on coadjoint orbits
- Some results in classification of G-symplectic manifolds
- Some results on the group of conformal symplectic transformations

3) Some application to mechanics

- Variational problems and symplectic geometry
- Space of geodesics
- the Buffon theorem for Hadamard manifolds
- Geometrical optics as a field of application of symplectic geometry
- Caustics, Chekanov theorem and all that
- Principle of geometric quantization, prequantization, polarizations,
- The quantization of the harmonic oscillator and the metaplectic representation
- The regularisation of the two-bodies problem

4) An introduction to symplectic topology

- what is symplectic topology, intersection of lagrangian submanifolds, some results
- Various invariants, Gromov capacity and theorem of symplectic embeding
- Existence and type of lagrangian submanifolds in projective spaces, recent results
- Pseudo holomorphic curves, some aspects