12/12 - Wednesday, 12/14, will be the last office hour for the semester.

12/9 - The exam has been corrected as stated in the email.

12/7 - There was a slight gap in the argument before. This has been corrected and an updated version is now posted.

12/7 - A small typo has been fixed in the notes posted earlier today. ("p to q in M" in the footnote should have been "p to q in C".)

12/7 - Here is a proof of de Rham's Theorem for those of you with exposure to singular cohomology.

12/2 - The final exam is now up. I decided to add an easy problem, so there are now 7 problems.

11/30 - The final exam will be posted Friday, December 2 by 6pm. It will be due Thursday, December 15 at 5pm. Collaboration and use of outside resources are prohibited, except for the textbook and your personal course notes.

11/28 - Notes on The Mayer-Vietoris sequence in de Rham cohomology for those interested.

11/16 - Assignment #6
Complete the following exercises in Spivak:
8-3, 8-4, 8-7, 8-8, 8-10, 8-14, 8-23, 8-31.
Due December 2 at 5pm.

11/9 - The deadline for Assignment #5 is extended to Wednesday, 11/16 at 5pm due to the lost office hour for Veteran's Day.

11/1 - Tomorrow's office hour (Wednesday, 11/2) is postponed to Monday, 11/7 at 3-4pm.

10/31 - Assignment #5
Complete the following exercises in Spivak:
6-6, 6-8, 7-2, 7-4, 7-5, 7-6, 7-16, 7-27.
Due November 14 at 5pm.

10/17 - Assignment #4
Complete the following exercises in Spivak:
5-4, 5-9, 5-10, 5-12, 5-15, 5-19, 6-1, 6-2.
Due October 31 at 5pm. The assignment with the best costume wins first grading rights.

10/7 - The N in 4-2 for Assignment #3 should be replaced by R.

10/7 - I will also mention this briefly on Monday, but here is a patch for today's notes:
In the proof of Claim 2, we assumed WLOG that g had non-vanishing derivative on all of U. This is justified by the same sort of covering argument used multiple times previously in the proof. g therefore has no critical points, and every value is regular. In particular, V=g^{-1}(0) is the level set of a regular value and thus a smooth manifold of smaller dimension (n-1).

9/30 - Assignment #3 is due October 17 at 5pm.

9/28 - The easiest way to finish the argument from the end of today's lecture is a two step process:
1) Every proper map is a closed map.
2) Every bijective closed map is a homeomorphism.

9/21 - The Tuesday, September 27th office hour will be postponed to Thursday, September 29th at 11am.

9/18 - Assignment #2 is due October 3 at 5pm.

9/9 - Bates's C^{k-1,1} version of Sard's Theorem is available here from the AMS. Whitney's classic counterexample is here, but it must be accessed from campus.

8/29 - The polls have closed and office hours will be Tu 11-12, W 3-4, and F 12-1. I was able to reduce to only one hour with a known conflict, so hopefully these times will work well for everyone.

8/29 - Assignment #1 - Due September 16 at 5pm. You may turn it in during class or slide it under my door (Evans 805).

8/26 - Regarding the shortage of desks: 9/12 is the earliest a room change can be requested. If there are still too many bodies for Evans 5 at that point, we should be able to obtain a new room in about a week. For now, unfortunately, the seats will go to the early arrivals unless some of you can bring portable furniture.

8/26 - S^n should be replaced by S^{n-1} in the obvious places in your notes from today.

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