The sequence Math 521-522-621 has the objective of conveying an understanding of the structure of analysis in itself as well as its role as a tool for other disciplines. This sequence is highly recommended for any math major and it is essential for students preparing for graduate studies in mathematics; also it should be taken by students in physics and engineering who intend to do graduate work in their areas.

Graduate courses in analysis, geometry and topology

- Review of differential calculus in normed spaces.
- Integration in Euclidean spaces.
- Basic definitions
- Measure and content zero, and characterization of Riemann integrability (optional).
- Fubini's theorem.
- Partition of unity.
- Changes of variables.

- Some multilinear algebra.
- Review of determinants.
- Multilinear maps, tensors, alternating tensors, wedge product.

- Manifolds
- Basic concept of a manifold, definitions of tangent space.
- Fields and forms on manifolds, orientation.

- Integration
- Stokes' theorem on manifolds.
- Euclidean measure for submanifolds.
- Grad, curl, and div, the Laplacian.
- The classical theorems in vector analysis by Green, Gauss and Stokes.
- Cauchy's integral theorem and formula.

- Other optional topcs.