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Math 411: Topics in Advanced Analysis

Course description, text, HW policy, etc. (click)

This Week:

HW 12 due by noon Thursday, May 16

HW 12 (fancy solution template, plain solution template)

Course Outline

  1. Examples of manifolds, especially projective space; definition of the category of manifolds.
  2. Tangent space: three equivalent characterizations.
  3. Multilinear algebra.
  4. Multilinear algebra on manifolds. Orientations.
  5. Integration; manifolds with boundary.
  6. Tangent space for manifolds with boundary; the Cartan derivative; Stokes’ theorem.
  7. De Rham cohomology; homotopy invariance.
  8. Mayer-Vietoris, vector fields on spheres, scalar products.
  9. Semi-Riemannian manifolds; the Hodge \(*\)-operator. Poincaré duality.
  10. Toric varieties: basic constructions, cohomology, homogeneous coordinates.
  11. Embeddings of toric varieties. Grassmannians.
  12. Plücker embedding. Schubert varieties.
  13. The Schubert calculus.