# 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.