Course Identification

Familiarity with the multidimensional calculus and firm knowledge of the linear algebra.

• Notion of a manifold: topological manifolds and smooth manifolds.
  One-dimensional and two-dimensional manifolds: classification.
  Other examples: spheres, real projective spaces, Grassmanian varieties,
  Lie groups.
• Notion of a submanifold. Examples of submanifolds in R^n. Implicit function theorem. Real plane curves. Real surfaces in R^3.
• Example: smooth complex analytic (and/or algebraic) varieties in a
  complex projective space CP^n as smooth manifolds.
• Homology groups: intuitive definition. Examples: homology of 2-dimensional oriented manifolds. Intersection in homology. Pairing in middle homology. Possible examples: vanishing cycles; curves on complex surfaces.
• Infinitesimal theory: Tangent space to a manifold. Vector bundles on manifolds. Tangent and cotangent bundle. Their sections: vector fields and 1-forms. k-forms on a manifold.
  Lie derivative on functions, vector fields and 1-dorms.
• Integration: change of variable formula suggests that the natural object to integrate are n-forms. Stokes theorem.
• de Rham cohomology: Poincare lemma and de Rham complex. Global sections of the de Rham complex and definition of the de Rham cohomology group.
  Computation of de Rham cohomology of a circle and of a torus.

Upon successful completion of this course students will be able to:

1. Demonstrate knowledge of the invariant language of analysis in the coordinate-free form that is used in modern mathematics from differential geometry to mathematical physics.