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Geometry of Manifolds >> Content Detail



Syllabus



Syllabus

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Course Description


This is a second-semester graduate course on the geometry of manifolds. The main emphasis is on the geometry of symplectic manifolds, but the material also includes long digressions into complex geometry and the geometry of 4-manifolds, with special emphasis on topological considerations.



Main Topics


  • Review of differential forms and de Rham cohomology
  • Symplectic manifolds; symplectomorphisms; Lagrangian submanifolds
  • Darboux and Moser theorems, Lagrangian neighborhood theorem
  • Complex vector bundles
  • Almost-complex structures, compatibility, integrability
  • Kähler manifolds, Dolbeault cohomology, Hodge theory, projective embeddings
  • Smooth 4-manifolds, intersection pairing, topological invariants
  • Smooth 4-manifolds vs. symplectic 4-manifolds vs. complex surfaces

The symplectic geometry part of the course follows the book by Ana Cannas da Silva, Lectures on Symplectic Geometry (Lecture Notes in Mathematics 1764, Springer-Verlag); the discussion of Kähler geometry mostly follows the book by R. O. Wells, Differential Analysis on Complex Manifolds (Springer GTM 65).



Prerequisites


Geometry of Manifolds (18.965) or equivalent: manifolds, vector fields, differential forms, vector bundles, homology, cohomology.



Homework


Grading for this course is based on homework. Homework assignments are due every 3 weeks or so.



Texts


There is no required text. The following references are useful:

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.



Calendar



LEC #TOPICSKEY DATES
1Review of differential forms, Lie derivative, and de Rham cohomology.
2Cup-product and Poincaré duality in de Rham cohomology; symplectic vector spaces and linear algebra; symplectic manifolds, first examples; symplectomorphisms
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 mechanics
4Symplectic vector fields, flux; isotopy and deformation equivalence; Moser's theorem; Darboux's theorem
5Tubular neighborhoods; local version of Moser's theorem; Weinstein's neighborhood theorem
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 homology
7More Floer homology; almost-complex structures; compatibility with a symplectic structure; polar decomposition; compatible triplesHomework 1 due
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
9Horizontal distributions; metric connections; curvature of a connection: Intrinsic definition; expression in terms of connection 1-form
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 class
11Naturality properties of Chern classes and topological definition; equivalence between the two definitions; classification of complex line bundles
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 cohomologyHomework 2 due
13Nijenhuis tensor; integrability; square of the dbar operator; Newlander-Nirenberg theorem; Kähler manifolds; complex projective space
14Kähler forms; strictly plurisubharmonic functions; Kähler potentials; examples; Fubini-Study Kähler form; complex projective manifolds; Hodge decomposition theorem
15Hodge * operator on a Riemannian manifold; d* operator; Laplacian, harmonic forms; Hodge decomposition theorem; differential operators; symbol, ellipticity; existence of parametrix
16Elliptic regularity, Green's operator; Hodge * operator and complex Hodge theory on a Kähler manifold; relation between real and complex Laplacians
17Hodge diamond; hard Lefschetz theorem; holomorphic vector bundles; canonical connection and curvature
18Holomorphic sections and projective embeddings; ampleness; Donaldson's proof of the Kodaira embedding theorem: local model; concentrated approximately holomorphic sectionsHomework 3 due
19Donaldson's proof of the Kodaira embedding theorem: Estimates; concentrated sections; approximation lemma
20Proof of the approximation lemma; examples of compact 4-manifolds without almost-complex structures, without symplectic structures, without complex structures; Kodaira-Thurston manifold
21Symplectic fibrations; Thurston's construction of symplectic forms; symplectic Lefschetz fibrations, Gompf and Donaldson theorems
22Symplectic sum along codimension 2 symplectic submanifolds; Gompf's construction of symplectic 4-manifolds with arbitrary pi_1
23Symplectic branched covers of symplectic 4-manifolds.
24Homeomorphism classification of simply connected 4-manifolds; intersection pairings; spin^c structures; spin^c connections; Dirac operator
25Seiberg-Witten equations; gauge group; moduli space; linearized equations; compactness of moduli space
26Seiberg-Witten invariant; properties; vanishing for manifolds of positive scalar curvature; vanishing for connected sums; Taubes non-vanishing for symplectic manifolds; examples of non-symplectic 4-manifolds, of non-diffeomorphic homeomorphic manifolds

 








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