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.
|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.|