General information
| Schedule |
Monday, Wednesday, & Friday, 9:00 - 9:50 |
| Location | Building 380 (Math), room 380-W
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| Professor |
Brian Munson (munson at math dot stanford dot edu)
Office: 382-H (2nd floor, Math building 380)
Office Phone: 3-7829
Office Hours: Monday 1-2:30, Wednesday 2-3:30, or by appointment.
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Course
Assistant |
Andres Angel (andresangel at stanford dot edu)
Office: 380-T
Office hours: Tuesday 3-4 and Wednesday 11-12
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| Textbook |
Differential Topology, by Victor Guillemin and Alan Pollack. Unfortunately, you can buy six or seven new paperbacks for the price of this book, and be at least that many times more entertained. But this is a good book. Since there's only one edition, you should try to buy a used copy. I put a copy on reserve in the math library, but you will have a hard time getting by without one of your very own. |
Course
content | We will cover the first three chapters of Guillemin and Pollack. Please download a copy the syllabus for a more detailed description and other course policies not covered here. The homework section of this page will also give you a better idea of the topics, since it lists the relevant reading next to each assignment. Please read BEFORE coming to class. I do, and so should you. Ultimately I want class to feel like a discussion, even if I'm the one doing most of the talking. |
| Prerequisites |
The prerequisites are Math 115 or Math 171 (the latter is preferred but not strictly necessary). This means you should already know how to read and write proofs, you should know the linear algebra from Math 51, and you should know something of the theory of calculus. A little basic point-set topology is useful too (open, closed, compact sets). A good portion of this class is devoted to rigorously developing intuitive geometric ideas, so keep that in mind. |
Announcements & Dates
- Important Dates
- April 23rd: Add deadline
- Evening of May 15th: Midterm Exam 5-7 PM, in 420-048 (building 420 is next door to the math department, and houses the psychology department. Room 048 is alsmost certainly in the basement).
- April 30th: Drop deadline
- June 12th: Final Exam due no later than 12:00 PM in my office
- The Midterm will cover everything we have covered so far up through and including Chapter 2 section 4. The rough format is this: I will ask you to state some definitions and some theorems, and I will ask you to prove various statements, which may include theorems from the book and/or homework exercises verbatim, but will certainly include new stuff. There are six problems, most of which might have several parts. You will have two hours to complete the exam. Mostly you should think of it like a homework assignment where you don't have your book with you.
- Advice: Memorize the definitions and statements of theorems. Definitions are the italicized terms in the text. Some of them appear in the exercises. You should know those too, unless it's part of a homework problem I did not ask you to do. You probably already have most of the deinitions memorized because we've used them so often, but it's a good idea to make sure you know them by heart, because you'll need to use them on the exam. Memorize the statements of theorems, especially those theorems with special names (for example, The Inverse Function Theorem, Sard's Theorem, The Brouwer fixed point Theorem, The Transversality Theorem, etc.). The reason for memorizing the statements of theorems is the same as that for the definitions: you will need to use them and so you need to know what they say in a precise way. Other preparation for the exam: you should look at your homework problems, your notes from class, and you should read the text. Chances are good that ideas I presented in class or asked you to work on for homework will come up on the exam, and that is because I think those ideas are important. Hence your homework and especially your class notes are a great way to filter the information from the textbook into a more manageable amount of information from which to study.
- The Final Exam will be take-home and is due at noon on June 12th in my office.
Homework
Homework assignments are due in class one week from the day they are assigned, unless otherwise noted. Late homework is not accepted. Typically this will be a Friday. Homework will be assigned on a weekley basis, and you will always know at least a week ahead of time what the assignment is. I will make you aware in class when a new homeowrk has been posted, but ultimately you are responsible for keeping up to date by frequenting this web page. There are about ten problems per assignment, with two exceptions, one of which is notable: the second assignment has twelve problems, owing to the importance of this set of concepts (inverse function theorem and transversality).
There will be no posted solutions to the homework. I don't think this is a valuable use of the person's time writing them (myself or Andres), and while you might get what you want out of it (a solution!), I don't think you get what you need out of it. We are more than happy to help you solve these problems while you're working on them, and to discuss them in detail after you've handed them in.
How to read this portion of the page: The first column tells you the problems to be done, the second when they are due, and the third the appropriate sections of the book. The third column, Relevant Reading, also tells you what sections of the book we will cover in class on a given day. For instance, on Wednesday April 5th we will cover the first section of the first chapter, and on Friday the second section of the first chapter, and so on. You will turn in your homework on these sections (listed in the left-hand column) on Friday of the following week (in this case April 14th), giving you a weekend plus a full week of classes to get problems done.
|
Assignment |
Due date |
Relevant Reading |
| #1 |
Chapter 1:
- Section 1, pp. 5-7: 7, 12, 17
- Section 2, pp.11-13: 4, 9, 11
| Due: Friday, April 14th
|
- Monday, April 3rd: No class
- Wednesday: Section 1
- Friday: Section 2
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| #2 |
Chapter 1:
- Section 3, pp.18-19: 2, 4, 9
- Section 4, pp.25-27: 1, 2, 7, 11
- Section 5, pp.32-33: 4, 5, 7, 9, 10
| Due: Friday, April 21st
|
- Monday, April 10th: Section 3
- Wednesday: Section 4
- Friday: Section 5
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| #3 |
Chapter 1:
- Section 6, pp.37-39: 2, 3, 4, 5, 7, 10
- Section 7, pp.45-48: 6 (see number 6 of section 1.6 for a definition of simply connected)
- Section 8, pp.54-56: 6, 7, 8
| Due: Friday, April 28th
|
- Monday, April 17th: Section 6
- Wednesday: Section 7 (through page 40; we will talk about Morse functions later in the quarter). You should read Appendix 1 if you have not seen Sard's theorem before, especially if you are not familiar with the concept of "measure zero". I will not discuss this in detail, since its proof, like that of the inverse function theorem, belongs in a course on analysis.
- Friday: Section 8 (read through page 51. We will skip partitions of unity for the moment.)
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| #4 |
Chapter 2:
- Section 1, pp.62-64: 3, 7, 8, 10
- Section 2, pp.66-67: 3, 5, 7
| Due: Friday, May 5th
|
- Monday, April 24th: Embeddings
- Wednesday: Section 1
- Friday: Section 2
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| #5 |
Chapter 2:
- Section 3, pp.74-77: 5, 7, 12, 18, 19
- Section 4, pp.82-85: 1, 2, 4, 5, 7
| Due: Friday, May 12th
|
- Monday, May 1st: Section 3
- Wednesday: Section 3
- Friday: Sections 4
|
| #6 |
Chapter 2:
- Section 4, pp.82-85: 6, 8, 10, 13
- Section 5, pp.87: 4, 6, 7, 8, 9, 10, 11 (this isn't as long as it looks: all the exercises in Section 5 have hints)
| Due: Friday, May 19th
|
- Monday, May 8th: Section 4
- Wednesday: Sections 4, 5
- Friday: Section 5
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| #7 |
Chapter 2 & Linking Number:
| Due: Friday, May 26th
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- Monday, May 15th: Review for exam
- Wednesday: Chapter 2, Section 6
- Friday: Applications of Borsuk-Ulam and The Linking Number
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| #8 |
Chapter 3:
- Section 2, pp. 103-107 4, 6, 13, 18, 19
- Section 3, pp.116-119: 2, 8, 11
| Due: Friday, June 2nd
|
- Monday, May 22nd: Section 2
- Wednesday: Sections 2, 3
- Friday: Section 3
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"#9" |
Chapter 3:
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- Monday, May 29th: Memorial Day Holiday, NO CLASS
- Wednesday: Fundamental Theorem of Algebra
- Friday: More degree theory
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