First course in symplectic topologyThis 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.
Lecture I: Overview and MotivationComplex 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.Lecture II: BasicsHamiltonian 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. Lecture III: NeighbourhoodsMoser'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). Lecture IV: Lagrangians ILagrangian 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. Lecture V: Lagrangians IICompletion of Luttinger's proof of unknottedness; recap of Lagrangian Grassmannian, Maslov class; recap of Chern classes and adjunction. Lecture VI: Projective varieties IThe 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. Lecture VII: Projective varieties IIQuadrics, cubic surface; blow-ups, change in first Chern class, rationality of quadric and cubic surfaces, general position requirement for blow-up locus. Lecture VIII: Symplectic blow-upSymplectic blow-up of a point, formula for change in cohomology class of the symplectic form. Compatibility. Sketch of Gromov's nonsqueezing theorem.Lecture IX: Picard-Lefschetz ILefschetz hyperplane theorem, sketch via plurisubharmonic Morse theory, holomorphic curves and the maximum principle, Lefschetz pencils (examples). Lecture X: Picard-Lefschetz IIParallel transport, vanishing cycles, Dehn twists, Picard-Lefschetz formula. Lecture XI: The non-Kähler worldKodaira-Thurston manifold, McDuff's example. Symplectic fibre sum; Gompf's theorem on fundamental groups. **Comment on Kähler fundamental groups.** Lecture XII: Hamiltonian group actionsSymplectic 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. Lecture XIII: Pseudoholomorphic curves IDefinition. Area and energy. Outline of the analytical setting. Gromov compactness. Good properties in four dimensions. Example existence theorem. Lecture XIV: Pseudoholomorphic curves IISymplectomorphism group of S^2 x S^2; McDuff's Hopf invariant example. |
