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Geometry of Manifolds >> Content Detail



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The readings in this section are not required.

The recommended texts are:

Amazon logo Cannas da Silva, A. Lectures on Symplectic Geometry (Lecture Notes in Mathematics). New York City, NY: Springer, 2001. ISBN: 9783540421955.

Amazon logo Wells, R. O. Differential Analysis on Complex Manifolds. New York City, NY: Springer, 1980. ISBN: 9780387904191.

Amazon logo McDuff, D., and D. Salamon. Introduction to Symplectic Topology. New York City, NY: Oxford University Press, 1999. ISBN: 9780198504511.

Amazon logo 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 #TOPICSREADINGS
2Cup-product and Poincaré duality in de Rham cohomology; symplectic vector spaces and linear algebra; symplectic manifolds, first examples; symplectomorphismsCannas. pp. 3-7.
3Symplectic 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 mechanicsCannas. pp. 9-19, 35-37, and 105-107.
4Symplectic vector fields, flux; isotopy and deformation equivalence; Moser's theorem; Darboux's theoremCannas. pp. 106 and 42-46.
5Tubular neighborhoods; local version of Moser's theorem; Weinstein's neighborhood theoremCannas. pp. 37-40 and 45-52.
6Tangent 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 homologyCannas. pp. 53-56.
7More Floer homology; almost-complex structures; compatibility with a symplectic structure; polar decomposition; compatible triplesCannas. pp. 67-70.
8Almost-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.

9Horizontal distributions; metric connections; curvature of a connection: Intrinsic definition; expression in terms of connection 1-formWells. pp. 70-74.
10Twisted 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 classWells. pp. 73-77 and 84-91.
11Naturality properties of Chern classes and topological definition; equivalence between the two definitions; classification of complex line bundlesWells. pp. 91-96.
12Chern 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 cohomologyCannas. pp. 78-81 and 83-87.
13Nijenhuis tensor; integrability; square of the dbar operator; Newlander-Nirenberg theorem; Kähler manifolds; complex projective spaceCannas. pp. 82 and 88-89.
14Kähler forms; strictly plurisubharmonic functions; Kähler potentials; examples; Fubini-Study Kähler form; complex projective manifolds; Hodge decomposition theoremCannas. pp. 90-97.
15Hodge * 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.

16Elliptic 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.

17Hodge diamond; hard Lefschetz theorem; holomorphic vector bundles; canonical connection and curvatureWells. pp. 77-80.
20Proof of the approximation lemma; examples of compact 4-manifolds without almost-complex structures, without symplectic structures, without complex structures; Kodaira-Thurston manifoldCannas. pp. 101-103.
21Symplectic fibrations; Thurston's construction of symplectic forms; symplectic Lefschetz fibrations, Gompf and Donaldson theoremsMcDuff-Salamon. pp. 197-203.
22Symplectic sum along codimension 2 symplectic submanifolds; Gompf's construction of symplectic 4-manifolds with arbitrary pi_1McDuff-Salamon. pp. 253-256.
24Homeomorphism classification of simply connected 4-manifolds; intersection pairings; spin^c structures; spin^c connections; Dirac operatorMorgan.

 








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