Jonny Evans : A First Course in Symplectic Topology
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Room 501, Department of Mathematics,
UCL, Gower Street, London, WC1E 6BT
j.d.evans •

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 3.0 Licence. That means you're free to use them as long as you give appropriate attribution and release derivatives under an isomorphic licence.

  1. Overview and motivation (summary, notes)
  2. Basics: dynamics; symplectic linear algebra (summary, notes)
  3. Neighbourhoods (summary, notes)
  4. Lagrangians I (summary, notes)
  5. Lagrangians II (summary, notes)
  6. Projective varieties I (summary, notes)
  7. Projective varieties II (summary, notes)
  8. Symplectic blow-up (summary, notes)
  9. Picard-Lefschetz I (summary, notes)
  10. Picard-Lefschetz II (summary, notes)
  11. The non-Kähler world (summary, notes)
  12. Hamiltonian group actions (summary, notes)
  13. Pseudoholomorphic curves I (summary, notes)
  14. Pseudoholomorphic curves II (summary, notes)

Lecture I: Overview and Motivation

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.

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Lecture II: Basics

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.

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Lecture III: Neighbourhoods

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).

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Lecture IV: Lagrangians I

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.

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Lecture V: Lagrangians II

Completion of Luttinger's proof of unknottedness; recap of Lagrangian Grassmannian, Maslov class; recap of Chern classes and adjunction.

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Lecture VI: Projective varieties I

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.

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Lecture VII: Projective varieties II

Quadrics, cubic surface; blow-ups, change in first Chern class, rationality of quadric and cubic surfaces, general position requirement for blow-up locus.

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Lecture VIII: Symplectic blow-up

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

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Lecture IX: Picard-Lefschetz I

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

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Lecture X: Picard-Lefschetz II

Parallel transport, vanishing cycles, Dehn twists, Picard-Lefschetz formula.

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Lecture XI: The non-Kähler world

Kodaira-Thurston manifold, McDuff's example. Symplectic fibre sum; Gompf's theorem on fundamental groups. **Comment on Kähler fundamental groups.**

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Lecture XII: Hamiltonian group actions

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.

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Lecture XIII: Pseudoholomorphic curves I

Definition. Area and energy. Outline of the analytical setting. Gromov compactness. Good properties in four dimensions. Example existence theorem.

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Lecture XIV: Pseudoholomorphic curves II

Symplectomorphism group of S^2 x S^2; McDuff's Hopf invariant example.

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