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Geometry of Manifolds, Spring 2007, MIT
25 Papers
|
1 Member
| Created by
Bar tender
on
8/25/2008
| Category: Geometry
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.
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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
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
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Bar tender
August 25, 2008
Views: 474
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Geometry
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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
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
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Bar tender
August 25, 2008
Views: 278
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Geometry
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18- Holomorphic sections and projective embeddings; ampleness; Donaldson's proof of the Kodaira embedding theorem: local model; concentrated approximately holomorphic sections
18- Holomorphic sections and projective embeddings; ampleness; Donaldson's proof of the Kodaira embedding theorem: local model; concentrated approximately holomorphic sections
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August 25, 2008
Views: 245
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Geometry
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7- More Floer homology; almost-complex structures; compatibility with a symplectic structure; polar decomposition; compatible triples
7- More Floer homology; almost-complex structures; compatibility with a symplectic structure; polar decomposition; compatible triples
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Bar tender
August 25, 2008
Views: 285
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Geometry
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1 - Review of differential forms, Lie derivative, and de Rham cohomology
1 - Review of differential forms, Lie derivative, and de Rham cohomology
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By
Bar tender
August 25, 2008
Views: 1176
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Category:
Geometry
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Cohomology
de
Derivative
Differential
Forms
Lie
Rham
11- Naturality properties of Chern classes and topological definition; equivalence between the two definitions; classification of complex line bundles
11- Naturality properties of Chern classes and topological definition; equivalence between the two definitions; classification of complex line bundles
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Bar tender
August 25, 2008
Views: 418
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Category:
Geometry
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Bundles
Chern
classification
complexl
ine
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
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
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August 25, 2008
Views: 298
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Geometry
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bundle
Chern
Classes
cohomological
Complex
criterion
Tangent
13 - Nijenhuis tensor; integrability; square of the dbar operator; Newlander-Nirenberg theorem; Kähler manifolds; complex projective space
13 - Nijenhuis tensor; integrability; square of the dbar operator; Newlander-Nirenberg theorem; Kähler manifolds; complex projective space
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Bar tender
August 25, 2008
Views: 1362
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Category:
Geometry
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dbar
Integrability
Kähler
Newlander-Nirenberg
Nijenhuis
tensor
14 - Kähler forms; strictly plurisubharmonic functions; Kähler potentials; examples; Fubini-Study Kähler form; complex projective manifolds; Hodge decomposition theorem
14 - Kähler forms; strictly plurisubharmonic functions; Kähler potentials; examples; Fubini-Study Kähler form; complex projective manifolds; Hodge decomposition theorem
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Bar tender
August 25, 2008
Views: 768
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Category:
Geometry
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Decomposition
Hodge
Kähler
plurisubharmonic
15 - Hodge * operator on a Riemannian manifold; d* operator; Laplacian, harmonic forms; Hodge decomposition theorem; differential operators; symbol, ellipticity; existence of parametrix
15 - Hodge * operator on a Riemannian manifold; d* operator; Laplacian, harmonic forms; Hodge decomposition theorem; differential operators; symbol, ellipticity; existence of parametrix
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Bar tender
August 25, 2008
Views: 268
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Category:
Geometry
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Bar tender
New York USA
Massachusetts Institute of Technology, Massachusetts
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