The readings in this section are not required.
The recommended texts are:
Cannas da Silva, A. Lectures on Symplectic Geometry (Lecture Notes in Mathematics). New York City, NY: Springer, 2001. ISBN: 9783540421955.
Wells, R. O. Differential Analysis on Complex Manifolds. New York City, NY: Springer, 1980. ISBN: 9780387904191.
McDuff, D., and D. Salamon. Introduction to Symplectic Topology. New York City, NY: Oxford University Press, 1999. ISBN: 9780198504511.
Morgan, J. W. The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds (Mathematical Notes 44). Princeton, NJ: Princeton University Press, 1995, ISBN: 9780691025971.
| LEC # | TOPICS | READINGS |
|---|---|---|
| 2 | Cup-product and Poincaré duality in de Rham cohomology; symplectic vector spaces and linear algebra; symplectic manifolds, first examples; symplectomorphisms | Cannas. pp. 3-7. |
| 3 | Symplectic form on the cotangent bundle; symplectic and Lagrangian submanifolds; conormal bundles; graphs of symplectomorphisms as Lagrangian submanifolds in products; isotopies and vector fields; Hamiltonian vector fields; classical mechanics | Cannas. pp. 9-19, 35-37, and 105-107. |
| 4 | Symplectic vector fields, flux; isotopy and deformation equivalence; Moser’s theorem; Darboux’s theorem | Cannas. pp. 106 and 42-46. |
| 5 | Tubular neighborhoods; local version of Moser’s theorem; Weinstein’s neighborhood theorem | Cannas. pp. 37-40 and 45-52. |
| 6 | Tangent space to the group of symplectomorphisms; fixed points of symplectomorphisms; Arnold’s conjecture; Morse theory: Gradient trajectories, Morse complex, homology; action functional on the loop space, and the basic idea of Floer homology | Cannas. pp. 53-56. |
| 7 | More Floer homology; almost-complex structures; compatibility with a symplectic structure; polar decomposition; compatible triples | Cannas. pp. 67-70. |
| 8 | Almost-complex structures: Existence and contractibility; almost-complex submanifolds vs. symplectic submanifolds; Sp(2n), O(2n), GL(n,C), and U(n); connections: definition, connection 1-form |
Cannas. pp. 71-76 Wells. pp. 65-70. |
| 9 | Horizontal distributions; metric connections; curvature of a connection: Intrinsic definition; expression in terms of connection 1-form | Wells. pp. 70-74. |
| 10 | Twisted de Rham operator; Levi-Civita connection on (TM,g); Chern classes of complex vector bundles (via curvature and Chern-Weil); Euler class and top Chern class | Wells. pp. 73-77 and 84-91. |
| 11 | Naturality properties of Chern classes and topological definition; equivalence between the two definitions; classification of complex line bundles | Wells. pp. 91-96. |
| 12 | Chern classes of the tangent bundle; cohomological criterion for existence of almost-complex structures on a 4-manifold, examples; splitting of tangent and cotangent bundles of (M,J), types; complex manifolds, Dolbeault cohomology | Cannas. pp. 78-81 and 83-87. |
| 13 | Nijenhuis tensor; integrability; square of the dbar operator; Newlander-Nirenberg theorem; Kähler manifolds; complex projective space | Cannas. pp. 82 and 88-89. |
| 14 | Kähler forms; strictly plurisubharmonic functions; Kähler potentials; examples; Fubini-Study Kähler form; complex projective manifolds; Hodge decomposition theorem | Cannas. pp. 90-97. |
| 15 | Hodge * operator on a Riemannian manifold; d* operator; Laplacian, harmonic forms; Hodge decomposition theorem; differential operators; symbol, ellipticity; existence of parametrix |
Cannas. pp. 98-99. Wells. pp. 114-116 and 136. |
| 16 | Elliptic regularity, Green’s operator; Hodge * operator and complex Hodge theory on a Kähler manifold; relation between real and complex Laplacians |
Cannas. pp. 99-100 Wells. pp. 136-141, 154-163, and 191-199. |
| 17 | Hodge diamond; hard Lefschetz theorem; holomorphic vector bundles; canonical connection and curvature | Wells. pp. 77-80. |
| 20 | Proof of the approximation lemma; examples of compact 4-manifolds without almost-complex structures, without symplectic structures, without complex structures; Kodaira-Thurston manifold | Cannas. pp. 101-103. |
| 21 | Symplectic fibrations; Thurston’s construction of symplectic forms; symplectic Lefschetz fibrations, Gompf and Donaldson theorems | McDuff-Salamon. pp. 197-203. |
| 22 | Symplectic sum along codimension 2 symplectic submanifolds; Gompf’s construction of symplectic 4-manifolds with arbitrary pi_1 | McDuff-Salamon. pp. 253-256. |
| 24 | Homeomorphism classification of simply connected 4-manifolds; intersection pairings; spin^c structures; spin^c connections; Dirac operator | Morgan. |
