V5D3: Advanced topics in geometry

• Wintersemester 2016/17, Di 8-10 in 0.006, Do 8-10 in 1.007
• Office hours: by appointment, 1.029
• Course information: PDF
• Suggested exercises: PDF (updated 31.01)

Suggested references:

• Michèle Audin and Mihai Damian, Morse theory and Floer homology
• Denis Auroux, 18.966 Geometry of Manifolds, MIT OCW lecture notes
• Ana Cannas da Silva, Lectures on symplectic geometry (available online here)
• Robert E. Gompf and András I. Stipsicz, 4-manifolds and Kirby calculus
• Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology
• Dusa McDuff and Dietmar Salamon, J-holomorphic curves and symplectic topology
• Chris Wendl, From ruled surfaces to planar open books: holomorphic curves in symplectic and contact manifolds of low dimension
• Chris Wendl, Lectures on holomorphic curves in symplectic and contact geometry

Material covered in class:

• 18.10: Symplectic linear algebra, symplectic forms [CdS §1]
• 20.10: Hamiltonian vector fields, mechanics, cotangent bundles [CdS §2, 18.1–18.2]
• 25.10: Lagrangians, graphs of symplectomorphisms, Moser's theorem [CdS §3, 7.1–7.2]
• 27.10: Relative Moser, Darboux, and Lagrangian neighborhood theorems [CdS §6.2, 7.3–8]
• 3.11: Weinstein tubular neighborhood thm, T_idSymp(M,ω), almost complex structures [CdS §9, 12]
• 8.11: Almost complex structures and compatibility, almost complex 4-manifolds [CdS §13, Aur §12]
• 10.11: Forms of type (l,m), Nijenhuis tensor, integrability, complex manifolds, ℂℙⁿ [CdS §14–15]
• 15.11: Dolbeault cohomology, Kähler forms, Kähler potentials [CdS §16]
• 17.11: Fubini-Study form on ℂℙⁿ, topology of projective hypersurfaces [MS1 §4.3]
• 22.11: Hodge theory, Kodaira-Thurston manifold [CdS §17]
• 24.11: Symplectic fiber bundles, ruled surfaces [MS1 §6.1–6.2]
• 29.11: Blowing up and down, Lefschetz pencils, ℂℙ²#ℂℙ² [MS1 §7.1]
• 1.12: Lefschetz fibrations, fiber sums, symplectic 4-manifolds with arbitrary π₁ [MS1 §7.2]
• 6.12: Topology of Lefschetz fibrations, monodromy, mapping class groups [GS §8.2]
• 8.12: Hurwitz moves, rational blow-down (see [EG]); Morse theory [AD §1.2–1.3]
• 12.12: Overview of Morse theory and Morse homology [AD §2.1, 3.1–3.2]
• 15.12: The h-principle and symplectic forms on open manifolds [MS1 §7.3]
• 20.12: The Arnold conjecture, the action functional [AD §6.1–6.3]
• 22.12: The Floer equation and its finite-energy solutions [AD §6.4–6.5]
• 10.1: Maslov indices, transversality [AD §7–8]
• 12.1: Hamiltonian Floer homology, invariance, and comparison to Morse homology [AD §10–11]
• 17.1: J-holomorphic curves, outline of Gromov's non-squeezing theorem [W2 §5.1]
• 19.1: Gromov non-squeezing part 2: monotonicity [W2 §5.2]
• 24.1: Gromov non-squeezing part 3: compactness [W2 §5.3]
• 31.1: Intersection positivity [MS2 §E.1–E.2]
• 2.2: Rational and ruled 4-manifolds part 1: outline, automatic transversality [W1 §2,4,5]
• 7.2: Rational and ruled 4-manifolds part 2: constructing Lefschetz pencils [W1 §6]
• 9.2: The topology of some symplectomorphism groups [MS2 §9.5]