# MA3402 Differential Forms on Manifolds - Fall 2019

Schedule | Room | ||
---|---|---|---|

Lectures: | Mondays 10:15–12:00 | Sentralbygg 2, 656 | |

Fridays 12:15–14:00 | Gamle fysikk: F4 | ||

Exam: | see here | ||

Lecturer | |||

Glen Wilson | |||

Office: | 1204 Sentralbygg 2 | ||

Email: | glen [dot] m [dot] wilson [at] ntnu [dot] no |

Rescheduled lecture: Wednesday November 20 at 14:15–16:00 in Realfagbygg R73.

Rescheduled lecture: Wednesday October 16 at 10:15–12:00 in Sentralbygg 2, S6.

Exercise session Tuesday October 15 at 14:15–15:30 in sentralbygg 1 265.

Exercise session Tuesday October 1 at 14:00–15:00 in sentralbygg 2 738.

Exercise session Tuesday September 3 at 14:15–15:15 in sentralbygg 2 656.

Please add your availability to the online poll here. A determination of the course time and meeting location will be made by Friday, 23 August.

## What this course is about

The main goal of this course is to develop the theory of differential forms, integration of forms, and de Rham cohomology on smooth manifolds. Building on the same foundational material from Guillemin and Pollack's book "Differential Topology", we will follow Guillemin and Haine's lecture notes (and now book) "Differential Forms".

An excellent companion to Guillemin and Haine's notes is the book by Loring W. Tu "An Introduction to Manifolds." This book covers the same material from a different perspective, using alternative (and equivalent) definitions and notation. Tu's book has an extensive collection of examples and counterexamples that are worth working through.

## Prerequisites

It is recommended that students have taken TMA 4190 "Introduction to Topology", however, this is not absolutely necessary. At the bare minimum, students should know the following concepts: open and closed subsets of n-dimensional Euclidean space, smooth maps between open subsets of Euclidean space, derivatives of smooth maps between Euclidean space, the inverse function theorem.

Students who have not taken TMA4190 will have time to learn about smooth manifolds before we begin to use them. Please study Guillemin and Pollack Chapter 1 §§1–5 and Chapter 2 §§1–2 on your own. Feel free to ask me questions about the material.

## Final Exam

The final exam is an oral exam, lasting around 30 minutes for each student. The final exams will take place week 50. Students will be able to select a time that suits them from a list of possible exam times.

You can sign up for an oral exam time by entering your name on this spreadsheet in the desired time.

The final exams will all take place in room 1246 (Gereon Quick's office).

## Lecture Plan

This is a rough (and optimistic) outline of the course structure that will be updated throughout the semester. All references are to the notes of Guillemin and Haine.

Week | Topic | Reference | Exercises | Notes |
---|---|---|---|---|

34.1 | Introduction | §1.1 | ||

34.2 | Multilinear forms | §1.2–1.3 | §1.2: i–iii, §1.3:vi–ix | Week 1 |

35.1 | Alternating tensors | §1.4 | §1.4:viii, §1.5:i–iv | Lecture 3 |

35.2 | Products | §1.5–1.6 | §1.6:i–iii | Lecture 4 |

36.1 | Interior product, pull backs | §1.7–1.8 | §1.7.iv, §1.8.vi | Lecture 5 |

36.2 | Tangent space, Vector fields | §1.9–2.1 | §1.9.iii, §2.1.vii,viii | Lecture 6 |

37.1 | Differential k-forms | §2.2–2.3 | §2.2.iv,§2.3.i–iv | Lecture 7 |

37.2 | Exterior differentiation, cohomology | §2.4 | §2.4i–iii | Lecture 8 |

38.1 | Canceled! | |||

38.2 | Interior Product, pull backs | §2.5–2.6 | Lecture 9 | |

39.1 | Poincare lemma | Bott&Tu Ch. 1, §4 | Lecture 10 | |

39.2 | Integration | Munkres Ch3 | Read proofs from Munkres | Lecture 11 (rough) |

40.1 | Integration, Change of variables | Munkres Ch3, 4 | Lecture 12 | |

40.2 | Manifolds, Tangent spaces | §4.1–4.2 | Lecture 13 | |

41.1 | Vector fields and differential forms | §4.3 | Lecture 14 | |

41.2 | Oriented manifolds, integration | Tu §21, 23 | Tu: §21: 1, 5, 10. | Lecture 15 |

42.1 | Stokes theorem | Tu §22, 23 | Tu §23: 1, 3, 5 | Lecture 16 |

42.2 | Oct 16 at 10:15, Sentralbygg 2, S6 | Tu §23 | Lecture 17 | |

43.1 | de Rham Cohomology | Tu §24 | §24.1 | |

43.2 | Mayer-Vietoris | Tu §25 | §25.3, §25.4 | Lecture 19 |

44.1 | Student presentation by Hannah | |||

44.2 | Cohomology calculations | Tu §26 | Lecture 20 | |

45.1 | More calculations | Tu §28 | Lecture 21 | |

45.2 | Cohomology properties | Bott&Tu §5 | Lecture 22 | |

46.1 | Poincare duality | Bott&Tu §5 | Lecture 23 | |

46.2 | Poincare duality | Bott&Tu §5 | Lecture 24 | |

47.1 | Summary and Exam prep | |||

47.2 | Lecture rescheduled |

## Reference Group

Tallak Manum

Ekaterina Poliakova

Johan Vik Mathisen

First meeting was September 27, 2019

## Course material

This course will follow:

- [GH] Guillemin and Haine,
*Differential Forms,*ISBN13: 9789813272774, World Scientific, 2019, Preprint here. - [T] L. W. Tu,
*An Introduction to Manifolds*, second edition, Springer Verlag, 2011.

Other interesting and related books (not required for the course):

- [GP] V. Guillemin and A. Pollack,
*Differential Topology*, Prentice Hall, 1974. - [M69] J. Milnor,
*Topology from the differentiable viewpoint*, The University Press of Virginia, 1969. - [M63] J. Milnor,
*Morse Theory*, Annals of Mathematic Studies No. 51, Princeton University Press, Princeton, N.J., 1963. - [D] B. Dundas, Differential Topology. Version January 2013
- [L] J.M. Lee,
*Introduction to Smooth manifolds*, Springer-Verlag. - [S] D. Spivak,
*Calculus on Manifolds*, Addison-Wesley, 1965. - [Mu91] J.R. Munkres,
*Analysis on Manifolds*, Addison-Wesley, 1991. - [Mu75] J.R. Munkres,
*Topology: a first course*, Prentice-Hall, 1975. - [RS] Robbin and Salamon,
*Introduction to Differential Topology*, 2018. Preprint Available Here