0708-1300/Homework Assignment 3: Difference between revisions

Revision as of 09:55, 18 October 2007

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Fall Semester
1	Sep 10	About, Tue, Thu
2	Sep 17	Tue, HW1, Thu
3	Sep 24	Tue, Photo, Thu
4	Oct 1	Questionnaire, Tue, HW2, Thu
5	Oct 8	Thanksgiving, Tue, Thu
6	Oct 15	Tue, HW3, Thu
7	Oct 22	Tue, Thu
8	Oct 29	Tue, HW4, Thu, Hilbert sphere
9	Nov 5	Tue,Thu, TE1
10	Nov 12	Tue, ~~Thu~~
11	Nov 19	Tue, Thu, HW5
12	Nov 26	Tue, Thu
13	Dec 3	Tue, Thu, HW6
Spring Semester
14	Jan 7	Tue, Thu, HW7
15	Jan 14	Tue, Thu
16	Jan 21	Tue, Thu, HW8
17	Jan 28	Tue, Thu
18	Feb 4	Tue
19	Feb 11	TE2, Tue, HW9, Thu, Feb 17: last chance to drop class
R	Feb 18	Reading week
20	Feb 25	Tue, Thu, HW10
21	Mar 3	Tue, Thu
22	Mar 10	Tue, Thu, HW11
23	Mar 17	Tue, Thu
24	Mar 24	Tue, HW12, Thu
25	Mar 31	Referendum,Tue, Thu
26	Apr 7	Tue, Thu
R	Apr 14	Office hours
R	Apr 21	Office hours
F	Apr 28	Office hours, Final (Fri, May 2)
Register of Good Deeds
Errata to Bredon's Book

In Preparation

The information below is preliminary and cannot be trusted! (v)

Read sections 8-10 of chapter II of Bredon's book three times:

First time as if you were reading a novel - quickly and without too much attention to detail, just to learn what the main keywords and concepts and goals are.
Second time like you were studying for an exam on the subject - slowly and not skipping anything, verifying every little detail.
And then a third time, again at a quicker pace, to remind yourself of the bigger picture all those little details are there to paint.

Also, read section 12 of chapter I of Bredon's book, but you can be a little less careful here.

Solve the following problems from Bredon's book, but submit only the solutions of the problems marked with an "S":

problems	on page(s)
1, 2, 3, S4, S5, S6	82
1, 2, S3, S4, 5	86

This assignment is due in class on Thursday November 1, 2007.

Trace the proof of the Whitney embedding theorem to find an embedding of the two dimensional real projective plane, ${\mathbb {R} }{\mathbb {P} }^{2}=S^{2}/(p=-p)$ , inside ${\mathbb {R} }^{5}$ . Do not do anything explicitly; just convince yourself that indeed you can find a small atlas (how small?), use it to embed ${\mathbb {R} }{\mathbb {P} }^{2}$ in some large ${\mathbb {R} }^{N}$ (how large?), and figure out how many times you will need to use Sard's theorem before you're down to the target, ${\mathbb {R} }^{5}$ .
Now see if you can come up with some cleverer way of viewing ${\mathbb {R} }{\mathbb {P} }^{2}$ , that will allow you to explicitly embed it in ${\mathbb {R} }^{5}$ .

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 ==Just for Fun==
+* Trace the proof of the Whitney embedding theorem to find an embedding of the two dimensional real projective plane, <math>{\mathbb R}{\mathbb P}^2=S^2/(p=-p)</math>, inside <math>{\mathbb R}^5</math>. Do not do anything explicitly; just convince yourself that indeed you can find a small atlas (how small?), use it to embed <math>{\mathbb R}{\mathbb P}^2</math> in some large <math>{\mathbb R}^N</math> (how large?), and figure out how many times you will need to use Sard's theorem before you're down to the target, <math>{\mathbb R}^5</math>.
+* Now see if you can come up with some cleverer way of viewing <math>{\mathbb R}{\mathbb P}^2</math>, that will allow you to explicitly embed it in <math>{\mathbb R}^5</math>.