#### Learning Outcomes

Students should be able to demonstrate mastery of relevant vocabulary and use the vocabulary fluently in their work. They should know common examples and counterexamples, and be able prove that these examples and counterexamples have properties as claimed. Additionally, students should know the content (and limitations) of major theorems and the ideas of the proofs, and apply results of these theorems to solve suitable problems, or use techniques of the proofs to prove additional related results, or to make calculations and computations. For more detailed information visit the Math 655 Wiki page.

#### Overview

- Manifolds
- Topological and smooth manifolds
- Manifolds with boundary
- Tangent vectors
- Tangent bundles
- Vector bundles and bundle maps
- Cotangent bundles

- Submanifolds
- Submersions, immersions, embeddings
- Inverse and implicit function theorems
- Transversality
- Embedding and approximation theorems

- Differential forms and tensors
- Wedge product
- Exterior derivative
- Orientations
- Stoke's Theorem