Room 208
Vivatsgasse 7
53111 Bonn
Vivatsgasse 7
53111 Bonn
Steven Sivek
Imperial College Department of MathematicsV5D3: 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, TidSymp(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]