Course Description
This is a course on manifolds including Stokes' theorem, de Rham cohomology, and Poincaré duality.

Text
Vector Analysis, by Klaus Jänich.

Course Summary

• Week 1: Examples of manifolds, especially projective space; definition of the category of manifolds.
• Week 2: Tangent space: three equivalent characterizations.
• Week 3: Multilinear algebra.
• Week 4: Multilinear algebra on manifolds. Orientations.
• Week 5: Integration.
• Week 6: Integration; quiz; manifolds with boundary.
• Week 7: Tangent space for manifolds with boundary; the Cartan derivative; Stokes' theorem.
• Week 8: De Rham cohomology.
• Week 9: Homotopy invariance theorem; combing the hair on a tennis ball.
• Week 10: semi-Riemannian manifolds; the star operator.
• Week 11: Poincaré duality.
• Week 12: Toric varieties: basic constructions.
• Week 13: Toric varieties: cohomology, homogeneous coordinates, embeddings.