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Last updated:

November 7, 2022


Unlimited Duration


This course includes:

Unlimited Duration

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Unlimited Duration


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.

Course Curriculum

  • 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

About the instructor


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Massachusetts Institute of Technology
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