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This is a second-semester graduate course on the geometry of manifolds.
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English
English [CC]
FREE
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Description
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.
Course content
- Review of differential forms Unlimited
- Cup-product and Poincaré duality in de Rham cohomology Unlimited
- Symplectic form on the cotangent bundle Unlimited
- Symplectic vector fields, flux Unlimited
- Tubular neighborhoods Unlimited
- Tangent space to the group of symplectomorphisms Unlimited
- More Floer homology Unlimited
- Almost-complex structures Unlimited
- Horizontal distributions Unlimited
- Twisted de Rham operator; Levi-Civita connection on (TM,g) Unlimited
- Naturality properties of Chern classes and topological definition Unlimited
- Chern classes of the tangent bundle Unlimited
- Nijenhuis tensor; integrability Unlimited
- Kähler forms; strictly plurisubharmonic functions Unlimited
- Hodge * operator on a Riemannian manifold Unlimited
- Elliptic regularity, Green’s operator Unlimited
- Hodge diamond; hard Lefschetz theorem Unlimited
- Holomorphic sections and projective embeddings Unlimited
- Donaldson’s proof of the Kodaira embedding theorem Unlimited
- Proof of the approximation lemma Unlimited
- Symplectic fibrations Unlimited
- Symplectic sum along codimension 2 symplectic submanifolds Unlimited
- Symplectic branched covers of symplectic 4-manifolds Unlimited
- Homeomorphism classification of simply connected 4-manifolds Unlimited
- Seiberg-Witten equations Unlimited
Instructor
Massachusetts Institute of Technology
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