Math 141 (Differential Topology) with Dmitry Vaintrob

Welcome to Math 141!

Syllabus. Highlights: •class meets at Etcheverry 3109, TTh 3:30-5pm. •Homeworks due Tuesdays. •OH Thursday 1:30-2:30 and Monday, 2-4. •Midterms in class on 10/10 and 11/21. •Final on 12/20, 7-10 (☹) PM.

Piazza (question-and-answer site for students). Please email me directly (mylastname@math.berkeley) with important/personal questions.

Announcements/resources

Final exam with solutions from a previous course and Midterm with solutions from the same course.

Final prep problems: see midterm prep problems together with Guillemin-Pollack: 2.2.5, 2.2.7, 2.3.5, 2.3.7, 2.4.4, 2.4.7, 2.4.12, 2.6.1.

Solutions. Also, Solutions to homeworks 10 and 11 (sorry for the late update on these!). See also: Chris Eur's solutions to the starred problems for solutions to starred problems in previous chapters.

Midterm 2 problems and solutions.

Midterm 2 practice problems and Solutions.

Midterm 1 and solutions

Homework set # 11, and LaTeX template. Due Tuesday 12/03.

Homework set # 10, and LaTeX template. EXTENDED: due Tuesday 12/03.

Homework solutions. (Thank you to everyone who submitted solutions!) 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11

See Chapter 1 of these notes for a nice resource on Lie groups.

Book: Guillemin and Pollack, "Differential Topology" (there is only one edition, with two different covers).

Resources for point set topology:

"What is a Manifold?" -- a fun and extremely informal sequence of youtube videos that covers the basics in the first five 40-minute lectures. Recommended resource for beginners. Warning: lots of minor mistakes, good for getting intuition for the subject but double-check one of the three sources below for precise statements.
A well-motivated lecture from a physics conference which gives an elementary, self-contained and rigorous introduction to topology (no physics required).
Hatcher's notes (first 2-3 chapters out of 4 total) -- complete, concise and readable, a short introduction to all we will need from the subject and more.
May's "An outline summary of basic point set topology" -- a super-dense bullet point list of relevant results. Contains way more than what we need: most of what we will use is on the first 3 pages. Useful mainly as a reference.

Supplementary Resources. In addition to the textbook, you can read:

Chapter I from A mathematical gift, III: first few pages (visible in linked preview) provide a fantastic historical introduction to manifolds. Free but hard-to-read copy available here.
L. Christine Kinsey, Topology of Surfaces
Good, high-level youtube lectures on differential geometry from a physicist's perspective: # 1 # 2, # 3, # 4 and # 5 (no physics required).

Readings

For best results, please do the readings before the class they are assigned for. Keep in mind: one page of math takes me as long to read as 5-10 pages of ordinary text.

Date	Readings	Supplementary Resources
9/24	Chapter 3
9/26	Chapter 4, p. 20-middle 23	See These notes for a nice and concrete list of examples and properties of immersions and submersions between Euclidean spaces. Notice the alternative definition 0.5 of manifold on the last page. Think about why this is equivalent to ours.
10/1	Chapter 4 full and Ch. 5, p. 27-28
10/8	Chapter 5	See Chapter 1 of these notesfor a nice resource on Lie groups.
10/10	MIDTERM 1
10/15	Chapter 6	Don't forget to read the starred exercises for Ch. 5: you can use them as theorems.
10/29	Chapter 7
10/31	Chapter 8
11/5	CHAPTER 2, Section 1
11/7	Chapter 2, Section 2
11/12	Chapter 2, Section 3
11/14	Chapter 2, Section 4

Old homework sets.

HW1, HW2, HW3, HW4 HW5, HW6, HW7, HW8, HW9, HW10

Old resources

A scan of a few homework pages from G-P.
An expanded exposition of class proof of theorem about differentials and linear approximation.
Solutions to Quiz #1.
Midterm 1 Prep problems and solutions.