Department of Mathematics Syllabus
This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.
MAT 239: Differential Topology
Approved: 2015-08-21,
Suggested Textbook: (actual textbook varies by instructor; check your
instructor)
Prerequisites:
MAT 201A; or Consent of Instructor.
Suggested Schedule:
| Lectures | Sections (from Tu's book) |
| Week 1 |
§1 Smooth Functions on a Euclidean Space §2 Tangent Vectors in R^n as Derivations §3 The Exterior Algebra of Multicovectors |
| Week 2 |
§4 Differential Forms on R^n §5 Manifolds |
| Week 3 | §6 Smooth Maps on a Manifold §8 The Tangent Space |
| Week 4 | §9 Submanifolds §12 The Tangent Bundle |
| Week 5 | §13 Bump Functions and Partitions of Unity §14 Vector Fields |
| Week 6 | §17 Differential 1-Forms §18 Differential k-Forms §19 The Exterior Derivative |
| Week 7 | §21 Orientations §22 Manifolds with Boundary |
| Week 8 | §23 Integration on Manifolds |
For the remaining time, instructors are welcome to cover some of the following topics. Also, some instructors might go faster, and leave time for the following time-permitting topics.
- §15 Lie Groups
- §16 Lie Algebras
- §24 De Rham Cohomology
- Sard's theorem
- Morse theory
Additional Notes:
Other references:
- Introduction to Smooth Manifolds, by Jack Lee.
- Notes on Manifolds, by Dmitry Fuchs.