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

### This Week:

**Monday**. Schubert calculus.**Wednesday**. Schubert calculus.**Friday**. No class (thesis parade).

#### HW 12 due by noon Thursday, May 16

HW 12 (fancy solution template, plain solution template)## Course Outline

**Week**

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