This is a summary of the content of a course given at ETH Zürich in the Herbstsemester 2010. It was aimed at giving Masters and PhD students a broad overview of this subject with much emphasis on examples and computations and less on general theory. It owes a lot to this course taught by Ivan Smith at DPMMS in 2006 from which I learned most of the material.

All teaching materials available from this site are released under a CC-BY-SA 4.0 Licence. That means you're free to use them as long as you give appropriate attribution and release derivatives under an isomorphic licence.

- Overview and motivation (notes)
- Complex and symplectic manifolds, integrability conditions and atlases. Examples from algebraic geometry, dynamics, gauge theory. Symplectomorphism group, rigidity, non-squeezing. Pseudoholomorphic curves. Taubes's theorem and applications to low-dimensional topology.

- Basics (notes)
- Hamiltonian dynamics: Review the Hamiltonian formulation of classical dynamics in Euclidean space; understand this formulation from the point of view of symplectic geometry. Generalise this to cotangent bundles to illustrate the passing from linear to nonlinear symplectic manifolds; geodesic flow as an example.
- Linear algebra: Alternating forms; compatible complex structures; the linear symplectic group; the unitary subgroup as a retract; homogeneous spaces and their topology: compatible complex structures, the Lagrangian Grassmannian and the Maslov class; symplectic manifolds and compatible almost complex structures; contractibility of the space of almost complex structures. First Chern class.

- Neighbourhoods (notes)
- Moser's argument, Darboux's theorem, symplectic submanifolds: their normal bundles, symplectic neighbourhood theorem; Banyaga's symplectic isotopy extension theorem (and Auroux's version for symplectic submanifolds).

- Lagrangians I (notes)
- Lagrangian submanifolds: zero-sections, graphs of closed forms,
Weinstein's neighbourhood theorem (some of its corollaries,
e.g. orientable embedded Lagrangians in
**C**^{2}are tori); Luttinger surgery, unknottedness of Lagrangian tori in**C**^{2}.

- Lagrangian submanifolds: zero-sections, graphs of closed forms,
Weinstein's neighbourhood theorem (some of its corollaries,
e.g. orientable embedded Lagrangians in
- Lagrangians II (notes)
- Completion of Luttinger's proof of unknottedness; recap of Lagrangian Grassmannian, Maslov class; recap of Chern classes and adjunction.

- Projective varieties I (notes)
- The Fubini-Study form on
**CP**^{n}, complex projective varieties as symplectic manifolds, adjunction and Chern classes for projective hypersurfaces; topology of surfaces of low degree in**CP**^{3}.

- The Fubini-Study form on
- Projective varieties II (notes)
- Quadrics, cubic surface; blow-ups, change in first Chern class, rationality of quadric and cubic surfaces, general position requirement for blow-up locus.

- Symplectic blow-up (notes)
- Symplectic blow-up of a point, formula for change in cohomology class of the symplectic form. Compatibility. Sketch of Gromov's nonsqueezing theorem.

- Picard-Lefschetz I (notes)
- Lefschetz hyperplane theorem, sketch via plurisubharmonic Morse theory, holomorphic curves and the maximum principle, Lefschetz pencils (examples).

- Picard-Lefschetz II (notes)
- Parallel transport, vanishing cycles, Dehn twists, Picard-Lefschetz formula.

- The non-Kähler world (notes)
- Kodaira-Thurston manifold, McDuff's example. Symplectic fibre sum; Gompf's theorem on fundamental groups. **Comment on Kähler fundamental groups.**

- Hamiltonian group actions (notes)
- Symplectic cut along a Hamiltonian circle action, blow-up as an
example (connection with fibre sum). Torus actions and the moment
polytope. Examples:
**CP**^{2}, blow-up. Reading off geometry from the moment polytope. Convexity. Delzant's theorem.

- Symplectic cut along a Hamiltonian circle action, blow-up as an
example (connection with fibre sum). Torus actions and the moment
polytope. Examples:
- Pseudoholomorphic curves I (notes)
- Definition. Area and energy. Outline of the analytical setting. Gromov compactness. Good properties in four dimensions. Example existence theorem.

- Pseudoholomorphic curves II (notes)
- Symplectomorphism group of S
^{2}x S^{2}; McDuff's Hopf invariant example.

- Symplectomorphism group of S