Drorbn - User contributions [en]

10-327/Homework Assignment 8

2010-12-06T05:26:24Z

Jdw:

{{10-327/Navigation}}
===Change in Plans===

Time to rest, a bit. For the lack of fun questions, I've decided to postpone this assignment a bit, and to cancel assignment number 9. So this assignment will only be written on Monday November 29, and will be due on the following Monday, December 6.

===Reading===
'''Read''' sections 35, 43, and 45 in Munkres' textbook (Topology, 2nd edition). Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, '''preread''' section 48 and 49, just to get a feel for the future. For fun alone, read also section 44.

===Doing===
Solve and submit the following problems from Munkres' book:
* Problems 2, 5a on pages 223-224.
* Problems 1, 4, 5 on page 270 (the relevant material will be covered on Thursday December 2).
* Problem 1 on page 280 (same comment).

In addition, solve and submit the following problem:

'''Problem IX.''' Show that every metric space <math>X</math> can be embedded in the cube <math>I^X</math>.

''Hint.'' Given a point <math>x\in X</math>, what real-valued function on <math>X</math> immediately comes to mind?

===Due date===
This assignment is due at the end of class on Wednesday, December 8, 2010.

{{Template:10-327:Dror/Students Divider}}
===Discussion===
* Then how do we count the homework in the final grade? I remember the old rule is to select the best 7 assignments from the total of 9. - Jolin
** The rule was "worst two assignments will not count", see [[10-327/About This Class]]. [[User:Drorbn|Drorbn]] 05:40, 27 November 2010 (EST)

Question.
For question 4. It is to show the intersection is nonempty by creating a Cauchy sequence from these A_n's. But I can't think of a way to prove the other direction. Are we supposed to create a nested sequence of closed sets for every Cauchy sequence? Then how do we ensure closeness?
-Kai

Take the closure of each set, as your set.
-Jdw

10-327/Classnotes for Monday November 22

2010-12-01T01:56:07Z

Jdw:

{{10-327/Navigation}}

See some blackboard shots at {{BBS Link|10_327-101122-143551.jpg}}.

{{10-327/vp|1122}}

{{Template:10-327:Dror/Students Divider}}

Here are some lecture notes..

[http://katlas.math.toronto.edu/drorbn/images/4/44/10-327-l17-p01.jpg Lecture 17 page 1]

[http://katlas.math.toronto.edu/drorbn/images/6/6b/10-327-l17-p02.jpg Lecture 17 page 2]

===Riddle Along===
What is the configuration space of a 3 legged spider

[[10-327/Solution to configuration of a 3 legged spider]]

Here is a lecture notes for today:

[http://katlas.org/drorbn/images/1/1f/10-327_Nov_22_Lec.pdf Lecture Nov 22]

File:10-327-l17-p02.jpg

2010-12-01T01:55:54Z

Jdw:

File:10-327-l17-p01.jpg

2010-12-01T01:55:08Z

Jdw:

10-327/Classnotes for Monday November 29

2010-12-01T01:51:25Z

Jdw:

{{10-327/Navigation}}

See some blackboard shots at {{BBS Link|10_327-101129-142544.jpg}}.

{{10-327/vp|1129}}

{{Template:10-327:Dror/Students Divider}}

Here are some lecture notes..

[http://katlas.math.toronto.edu/drorbn/images/a/a0/10-327-l19-p01.jpg Lecture 19 page 1]

[http://katlas.math.toronto.edu/drorbn/images/5/52/10-327-l19-p02.jpg Lecture 19 page 2]

[http://katlas.math.toronto.edu/drorbn/images/5/5f/10-327-l19-p03.jpg Lecture 19 page 3]

File:10-327-l19-p03.jpg

2010-12-01T01:49:40Z

Jdw:

File:10-327-l19-p02.jpg

2010-12-01T01:48:47Z

Jdw:

File:10-327-l19-p01.jpg

2010-12-01T01:46:04Z

Jdw:

10-327/Homework Assignment 7

2010-11-19T16:24:45Z

Jdw:

{{10-327/Navigation}}
===Reading===
'''Read''' sections <math>\{31,32,33\}</math> in Munkres' textbook (Topology, 2nd edition). Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, '''preread''' sections <math>\emptyset</math>, just to get a feel for the future.

===Doing===
Solve and submit the following problems from Munkres' book:
* Problem 1 on page 199.
* Problem 1 on page 205.
* Problems 1, 4, 5, 8, 9 on pages 212-213.

===Due date===
This assignment is due at the end of class on Thursday, November 25, 2010.

{{Template:10-327:Dror/Students Divider}}

Question: In problem 1 p205, is asks us to show that any closed subspace of a normal space is also normal. Do we really need the condition that the subspace be closed? - Jdw

10-327/Homework Assignment 7

2010-11-19T16:24:29Z

Jdw:

10-327/Homework Assignment 6

2010-11-16T04:26:17Z

Jdw:

{{10-327/Navigation}}
===Reading===
'''Read''' sections 37-38 in Munkres' textbook (Topology, 2nd edition). Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, '''preread''' sections 30-33, just to get a feel for the future.

===Doing===
Solve and submit the following problems.

'''Problem 1.''' Problem 1 on page 235 of Munkres' book.

'''Problem 2.''' Show that <math>\{0,1\}^{\mathbb N}</math> is homeomorphic to the cantor set <math>C</math>.

'''Problem 3.''' Show that any function <math>f\colon{\mathbb N}\to I^A</math> from the integers into a "cube" <math>I^A=[0,1]^A</math> has a unique continuous extension to <math>\beta{\mathbb N}</math>.

'''Problem 4.''' Use the fact that there is a countable dense subset within <math>I^I</math> to show that the cardinality of <math>\beta{\mathbb N}</math> is greater than or equal to the cardinality of <math>I^I</math>.

'''Problem 5.''' Show that the cardinality of <math>\beta{\mathbb N}</math> is also less than or equal to the cardinality of <math>I^I</math>, and therefore it is equal to the cardinality of <math>I^I</math>.

'''Problem 6.''' Show that if <math>\mu\in\beta{\mathbb N}\backslash{\mathbb N}</math> and if <math>\mbox{Lim}_\mu</math> is the corresponding generalized limit, and if <math>b</math> is a bounded sequence and <math>f\colon{\mathbb R}\to{\mathbb R}</math> is a continuous function, then <math>\mbox{Lim}_\mu f(b_k) = f(\mbox{Lim}_\mu b_k)</math>.

'''Problem 7.''' Show that there is no super-limit function <math>\mbox{SuperLim}</math> defined on bounded sequences of reals with values in the reals which has the following 4 properties:
# <math>\mbox{SuperLim}(a_k)=\lim a_k</math>, if the sequence <math>a_k</math> is convergent.
# <math>\mbox{SuperLim}(a_k+b_k) = \mbox{SuperLim}(a_k) + \mbox{SuperLim}(b_k)</math>.
# <math>\mbox{SuperLim}(a_k\cdot b_k) = \mbox{SuperLim}(a_k) \cdot \mbox{SuperLim}(b_k)</math>.
# <math>\mbox{SuperLim}(a'_k) = \mbox{SuperLim}(a_k)</math>, where <math>a'</math> is <math>a</math> "shifted once": <math>a'_k=a_{k+1}</math>.

===Due date===
This assignment is due at the end of class on Thursday, November 18, 2010.

{{Template:10-327:Dror/Students Divider}}

Hi Dror, how do we prove some set has smaller cardinality to another set? I mean what do you mean by that?
Why in problem 5 it says that the cardinality of a set is less or equal to another set so that the cardinality are the same?
[[User:Xwbdsb|Xwbdsb]] 00:25, 13 November 2010 (EST)
*One way to do this is that if you have a map <math>f(x):A \to B</math> such that f(x) is 1-1, then <math>A</math> has cardinality less than or equal to <math>B</math>. I haven't worked on this question yet, so it may be a red herring, but in general I find the technique useful. [[User:Johnfleming|John]]

Also I am wondering what is the super-limit because it wasn't done in class and it is not covered in the book too??[[User:Xwbdsb|Xwbdsb]] 00:36, 13 November 2010 (EST)
I remember that when proving the homeomorphism in question 2 there is another intricate argument for continuity that is
not the standard definition in our topology class? Sorry Dror I am sort of lost of what to do for this assignment...[[User:Xwbdsb|Xwbdsb]] 00:36, 13 November 2010 (EST)

Also what is the generalized limit? I search for this idea in the book but I didn't find anything. I think in the book before we understand Stone-Cech compatification theorem we need to understand what a regular space is and also we need to understand one-point compactification? Isn't Stone-Cech compatification just a special way to compatify the some topological space so that the continuous function with uniquely be extended to the compatification? [[User:Xwbdsb|Xwbdsb]] 00:58, 13 November 2010 (EST)

* I actually genuinely enjoyed writing this assignment, thinking that I've asked a lovely collection of questions that will challenge you in the positive sense of the word - make you scratch your heads, and struggle a bit, and fight a bit to understand what is going on and eventually get some satisfaction and gain some appreciation of the intricacy and beauty of the subject matter as you understand and solve these problems. So please challenge yourself and think and struggle a bit more. [[User:Drorbn|Drorbn]] 10:32, 13 November 2010 (EST)

* I have a question concerning Problem 7. Perhaps I am misunderstanding it, but if we just define <math>SuperLim(a_k)=0</math> for all sequences the 3 properties seem to hold, maybe that <math>SuperLim(a_k)=Lim(a_k)</math> if the sequence does converge should be added? Or maybe just ruling out constant functions is suffient... -[[User:Johnfleming|John]]
** Thanks! You are absolutely right. The question has been modified. [[User:Drorbn|Drorbn]] 15:23, 13 November 2010 (EST)

----

Hi Dror, I understand you want to make it challenging but I think it is also a bit unfair for some students like me. I never learned super-limit function. It is not done in your lecture or discussed in the book. So solving this question totally relies on other background knowledge. I don't have such strong background compared to some other students and I didn't even have a serious lecture on cardinality. I am just hoping since you are asking these kind of questions could you possibly talk about it in class? Trust me Dror it is not that I am not working hard or not trying to struggle but my background is not even enough for me to understand the question. For example I don't know what super-limit function is. I couldn't find it in Munkres book I couldn't find it by googling? Are you talking about limsup?

And what is the generalized definition of limit? I cannot find the definition anywhere.[[User:Xwbdsb|Xwbdsb]] 10:01, 14 November 2010 (EST)

* In the question about super-limits you are asked to prove that "super-limits", objects with the given properties, do not exist. It is not surprising they are not in the literature. Cardinalities are covered in MAT246, which is a pre-requisite to this class, I believe. I'll say more about generalized limits in the coming class tomorrow. [[User:Drorbn|Drorbn]] 10:37, 14 November 2010 (EST)
** Munkres Chapter 1 is all about set theory. In particular sections 6,7 and 9 have discussions about cardinality, this may be a good resource if you are having difficulties with the ideas. - [[User:Johnfleming|John]]

----

*A quick question (and I hope it isn't stupid...), but I'm not sure what "A" is when you talk about the cube <math>I^A</math>. What product of the unit interval are you talking about? --[[User:Wesley|Wesley]] 12:15, 14 November 2010 (EST)
** <math>A</math> is any set whatsoever. So the "cube" <math>I^A</math> is finite dimensional if <math>A</math> is finite, but it is infinite dimensional if <math>A</math> is infinite. [[User:Drorbn|Drorbn]] 13:28, 14 November 2010 (EST)

**Is it true that for question 6 the function f has to be bounded? Since even though b is a bounded sequence but f(b_k) might not be bounded so that the generalized limit is not defined. given that f is continuous. suppose f=1/x and b_k = 1/k. [[User:Xwbdsb|Xwbdsb]] 22:12, 15 November 2010 (EST)

* In problem 1 part c, the errata to Munkres says that should be Hausdorff.

10-327/Classnotes for Thursday November 4

2010-11-06T17:37:06Z

Jdw:

{{10-327/Navigation}}

See some blackboard shots at {{BBS Link|10_327-101104-142342.jpg}}.

{{Template:10-327:Dror/Students Divider}}

Question: Regarding the proof of the Lebesque Number Lemma, let T(x) be the function we were working with in the proof. I am confused with how we reached the conclusion that if d(x,y)<E, then T(y)>= T(x) - E. I know that it was said that this is just an application of the triangle inequality, but I am having a bit of trouble seeing that. Hopefully someone can make this point a bit clearer for me. Thanks! Jason.

User talk:Jdw

2010-11-02T01:31:40Z

Jdw:

Jdw - thanks for updating [[template:10-327/Navigation]]. Though after modifying it, there's no need to re-write all the pages that include it - if a page is cached and you want to purge its cached version, you load it once with <tt>&action=purge</tt> appended to the URL. For example, you would load

http://katlas.math.toronto.edu/drorbn/index.php?title=10-327&action=purge

instead of just

http://katlas.math.toronto.edu/drorbn/index.php?title=10-327

(My apologies if I mis-understood what you were trying to do).

Best,

Dror.

Dror - Thank you, that was what I was trying to do. Jdw

== Linking. ==

Jdw -

There's no point to uploading images if they are not linked anywhere.

[[User:Drorbn|Drorbn]] 21:04, 1 November 2010 (EDT)

Sorry, I lost track of what I was doing. Fixed now. Jdw

10-327/Classnotes for Monday November 1

2010-11-02T01:29:34Z

Jdw:

{{10-327/Navigation}}

See some blackboard shots at {{BBS Link|10_327-101101-142717.jpg}}.

{{10-327/vp|1101}}

{{Template:10-327:Dror/Students Divider}}

Here are some lecture notes..

[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:10-327l12p01.jpg Lecture 12 page 1]

[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:10-327l12p02.jpg Lecture 12 page 2]

[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:10-327l12p03.jpg Lecture 12 page 3]

10-327/Classnotes for Monday October 25

2010-11-02T01:29:19Z

Jdw:

{{10-327/Navigation}}

See some blackboard shots at {{BBS Link|10_327-101025-142133.jpg}}.

{{10-327/vp|1025}}

{{Template:10-327:Dror/Students Divider}}

Here are some lecture notes..

[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:10-327l11p01.jpg Lecture 11 page 1]

[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:10-327l11p02.jpg Lecture 11 page 2]

[http://katlas.math.toronto.edu/drorbn/index.php?title=Image:10-327l11p03.jpg Lecture 11 page 3]

*Everybody good luck for the midterm! Cheers~~-Kai[[User:Xwbdsb|Xwbdsb]] 23:52, 27 October 2010 (EDT)

10-327/Classnotes for Monday November 1

2010-11-02T01:27:35Z

Jdw:

File:10-327l11p03.jpg

2010-11-01T19:25:28Z

Jdw:

File:10-327l11p02.jpg

2010-11-01T19:24:41Z

Jdw:

File:10-327l11p01.jpg

2010-11-01T19:23:40Z

Jdw:

File:10-327l12p03.jpg

2010-11-01T19:23:08Z

Jdw:

File:10-327l12p02.jpg

2010-11-01T19:21:52Z

Jdw:

File:10-327l12p01.jpg

2010-11-01T19:21:45Z

Jdw:

10-327/Classnotes for Thursday October 21

2010-10-21T20:29:57Z

Jdw:

{{10-327/Navigation}}

{{Template:10-327:Dror/Students Divider}}

Here are some lecture notes..

[http://katlas.math.toronto.edu/drorbn/images/8/86/10-327-lec11p01.jpg Lecture 11 page 1]

[http://katlas.math.toronto.edu/drorbn/images/0/09/10-327-lec11p02.jpg Lecture 11 page 2]

[http://katlas.math.toronto.edu/drorbn/images/f/f1/10-327-lec11p03.jpg Lecture 11 page 3]

[http://katlas.math.toronto.edu/drorbn/images/1/15/10-327-lec11p04.jpg Lecture 11 page 4]

[http://katlas.math.toronto.edu/drorbn/images/c/c4/10-327-lec11p05.jpg Lecture 11 page 5]

[http://katlas.math.toronto.edu/drorbn/images/3/39/10-327-lec11p06.jpg Lecture 11 page 6]

[http://katlas.math.toronto.edu/drorbn/images/e/e5/10-327-lec11p07.jpg Lecture 11 page 7]

10-327/Navigation

2010-10-21T20:27:48Z

Jdw:

<noinclude>Back to [[10-327]]. </noinclude>
{| border="1px" cellpadding="1" cellspacing="0" width="100%" style="font-size: small; align: left"
|- align=left
!#
!Week of...
!Notes and Links
|- align=left
|align=center|1
|Sep 13
|[[10-327/About This Class|About This Class]], [[10-327/Classnotes for Monday September 13|Monday]] - Continuity and open sets, [[10-327/Classnotes for Thursday September 16|Thursday]] - topologies, continuity, bases.
|- align=left
|align=center|2
|Sep 20
|[[10-327/Classnotes for Monday September 20|Monday]] - More on bases, [[10-327/Classnotes for Thursday September 24|Thursdsay]] - Products, Subspaces, Closed sets, [[10-327/Homework Assignment 1|HW1]]
|- align=left
|align=center|3
|Sep 27
|[[10-327/Classnotes for Monday September 27|Monday]] - the Cantor set, closures, [[10-327/Classnotes for Thursday September 30|Thursday]], [[10-327/Class Photo|Class Photo]], [[10-327/Homework Assignment 2|HW2]]
|- align=left
|align=center|4
|Oct 4
|[[10-327/Classnotes for Monday October 4|Monday]] - the axiom of choice and infinite product spaces, [[10-327/Classnotes for Thursday October 7|Thursday]] - the box and the product topologies, metric spaces, [[10-327/Homework Assignment 3|HW3]]
|- align=left
|align=center|5
|Oct 11
|Monday is Thanksgiving. [[10-327/Classnotes for Thursday October 14|Thursday]] - metric spaces, sequencial closures, various products. Final exam's date announced on Friday.
|- align=left
|align=center|6
|Oct 18
|[[10-327/Classnotes for Monday October 18|Monday]] - connectedness in <math>{\mathbb R}</math>, [[10-327/Classnotes for Thursday October 21|Thursday]], [[10-327/Homework Assignment 4|HW4]]
|- align=left
|align=center|7
|Oct 25
|Term test on Thursday, HW5
|- align=left
|align=center|8
|Nov 1
|Wednesday is the last date to drop this course, HW6
|- align=left
|align=center|9
|Nov 8
|Monday-Tuesday is Fall Break
|- align=left
|align=center|10
|Nov 15
|HW7
|- align=left
|align=center|11
|Nov 22
|HW8
|- align=left
|align=center|12
|Nov 29
|HW9
|- align=left
|align=center|13
|Dec 6
|Classes end Tuesday
|- align=left
|align=center|F
|?
|
|- align=left
|colspan=3 align=center|[[10-327/Register of Good Deeds|Register of Good Deeds]] / [[10-327/To Do|To Do List]]
|- align=left
|colspan=3 align=center|[[Image:10-327-ClassPhoto.jpg|310px]] [[10-327/Class Photo|Add your name / see who's in!]]
|- align=left
|colspan=3 align=center|[[Image:10-327-Splash.png|310px]] See {{Home Link|Talks/Fields-0911/|Hilbert's 13th}}
|}

File:10-327-lec11p07.jpg

2010-10-21T20:27:06Z

Jdw:

File:10-327-lec11p06.jpg

2010-10-21T20:27:02Z

Jdw:

File:10-327-lec11p05.jpg

2010-10-21T20:26:58Z

Jdw:

File:10-327-lec11p04.jpg

2010-10-21T20:26:52Z

Jdw:

File:10-327-lec11p03.jpg

2010-10-21T20:26:48Z

Jdw:

File:10-327-lec11p02.jpg

2010-10-21T20:26:42Z

Jdw:

File:10-327-lec11p01.jpg

2010-10-21T20:26:37Z

Jdw:

10-327/Homework Assignment 4

2010-10-19T01:06:17Z

Jdw:

{{10-327/Navigation}}

===Reading===
Read sections 23 through 25 in Munkres' textbook (Topology, 2nd edition). Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, '''preread''' sections 26 through 27, just to get a feel for the future.

===Doing===
Solve and submit problems 1-3 and 8-10 Munkres' book, pages 157-158.

===Due date===
This assignment is due at the end of class on Monday, October 25, 2010.

===Suggestions for Good Deeds===
Annotate our Monday videos (starting with {{10-327/vp|0927}}) in a manner similar to (say) {{dbnvp link|AKT-090910-1|AKT-090910-1}}, and/or add links to the blackboard shots, in a manner similar to {{dbnvp link|Alekseev-1006-1|Alekseev-1006-1}}. Also, make ''constructive'' suggestions to me, {{Dror}} and / or the videographer, Qian (Sindy) Li, on how to improve the videos and / or the software used to display them. Note that "constructive" means also, "something that can be implemented relatively easily in the real world, given limited resources".

{{Template:10-327:Dror/Students Divider}}

===Questions===
1)Hi, quick question. I am wondering if the term test will cover the material on this assignment, or only the material before the assignment. Thanks! Jason.

10-327/Classnotes for Monday October 18

2010-10-18T19:18:51Z

Jdw:

{{10-327/Navigation}}

{{Template:10-327:Dror/Students Divider}}

Here are some lecture notes..

[http://katlas.math.toronto.edu/drorbn/images/f/f8/10-327-lec10p01.pdf Lecture 10 page 1]

[http://katlas.math.toronto.edu/drorbn/images/6/6a/10-327-lec10p02.pdf Lecture 10 page 2]

[http://katlas.math.toronto.edu/drorbn/images/3/35/10-327-lec10p03.pdf Lecture 10 page 3]

[http://katlas.math.toronto.edu/drorbn/images/9/98/10-327-lec10p04.pdf Lecture 10 page 4]

10-327/Navigation

2010-10-18T19:17:22Z

Jdw:

<noinclude>Back to [[10-327]]. </noinclude>
{| border="1px" cellpadding="1" cellspacing="0" width="100%" style="font-size: small; align: left"
|- align=center
|colspan=3 style="color: red"|'''Important! The due date for HW3 has been postponed to Monday October 18.'''
|- align=left
!#
!Week of...
!Notes and Links
|- align=left
|align=center|1
|Sep 13
|[[10-327/About This Class|About This Class]], [[10-327/Classnotes for Monday September 13|Monday]] - Continuity and open sets, [[10-327/Classnotes for Thursday September 16|Thursday]] - topologies, continuity, bases.
|- align=left
|align=center|2
|Sep 20
|[[10-327/Classnotes for Monday September 20|Monday]] - More on bases, [[10-327/Classnotes for Thursday September 24|Thursdsay]] - Products, Subspaces, Closed sets, [[10-327/Homework Assignment 1|HW1]]
|- align=left
|align=center|3
|Sep 27
|[[10-327/Classnotes for Monday September 27|Monday]] - the Cantor set, closures, [[10-327/Classnotes for Thursday September 30|Thursday]], [[10-327/Class Photo|Class Photo]], [[10-327/Homework Assignment 2|HW2]]
|- align=left
|align=center|4
|Oct 4
|[[10-327/Classnotes for Monday October 4|Monday]] - the axiom of choice and infinite product spaces, [[10-327/Classnotes for Thursday October 7|Thursday]] - the box and the product topologies, metric spaces, [[10-327/Homework Assignment 3|HW3]]
|- align=left
|align=center|5
|Oct 11
|Monday is Thanksgiving. [[10-327/Classnotes for Thursday October 14|Thursday]] - metric spaces, sequencial closures, various products. Final exam's date announced on Friday.
|- align=left
|align=center|6
|Oct 18
|[[10-327/Homework Assignment 4|HW4]], [[10-327/Classnotes for Monday October 18|Monday]]
|- align=left
|align=center|7
|Oct 25
|Term test on Thursday, HW5
|- align=left
|align=center|8
|Nov 1
|Wednesday is the last date to drop this course, HW6
|- align=left
|align=center|9
|Nov 8
|Monday-Tuesday is Fall Break
|- align=left
|align=center|10
|Nov 15
|HW7
|- align=left
|align=center|11
|Nov 22
|HW8
|- align=left
|align=center|12
|Nov 29
|HW9
|- align=left
|align=center|13
|Dec 6
|Classes end Tuesday
|- align=left
|align=center|F
|?
|
|- align=left
|colspan=3 align=center|[[10-327/Register of Good Deeds|Register of Good Deeds]] / [[10-327/To Do|To Do List]]
|- align=left
|colspan=3 align=center|[[Image:10-327-ClassPhoto.jpg|310px]] [[10-327/Class Photo|Add your name / see who's in!]]
|- align=left
|colspan=3 align=center|[[Image:10-327-Splash.png|310px]] See {{Home Link|Talks/Fields-0911/|Hilbert's 13th}}
|}

File:10-327-lec10p04.pdf

2010-10-18T19:16:16Z

Jdw:

File:10-327-lec10p03.pdf

2010-10-18T19:15:20Z

Jdw:

File:10-327-lec10p02.pdf

2010-10-18T19:14:44Z

Jdw:

File:10-327-lec10p01.pdf

2010-10-18T19:14:35Z

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10-327/Classnotes for Thursday October 14

2010-10-15T00:15:25Z

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{{10-327/Navigation}}

See some blackboard shots at {{BBS Link|10_327-101014-142707.jpg}}.

{{Template:10-327:Dror/Students Divider}}

Here are some lecture notes..

[http://katlas.math.toronto.edu/drorbn/images/5/51/10-327-lec09p01.jpg Lecture 9 page 1]

[http://katlas.math.toronto.edu/drorbn/images/7/76/10-327-lec09p02.jpg Lecture 9 page 2]

[http://katlas.math.toronto.edu/drorbn/images/e/e3/10-327-lec09p03.jpg Lecture 9 page 3]

[http://katlas.math.toronto.edu/drorbn/images/9/95/10-327-lec09p04.jpg Lecture 9 page 4]

[http://katlas.math.toronto.edu/drorbn/images/8/84/10-327-lec09p05.jpg Lecture 9 page 5]

[http://katlas.math.toronto.edu/drorbn/images/9/97/10-327-lec09p06.jpg Lecture 9 page 6]

File:10-327-lec09p06.jpg

2010-10-15T00:13:29Z

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File:10-327-lec09p05.jpg

2010-10-15T00:11:51Z

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File:10-327-lec09p04.jpg

2010-10-15T00:09:07Z

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File:10-327-lec09p03.jpg

2010-10-15T00:07:28Z

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File:10-327-lec09p02.jpg

2010-10-15T00:06:23Z

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File:10-327-lec09p01.jpg

2010-10-15T00:05:10Z

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10-327/Classnotes for Thursday October 7

2010-10-07T20:23:20Z

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{{10-327/Navigation}}

{{Template:10-327:Dror/Students Divider}}

Here are some lecture notes..

[http://katlas.math.toronto.edu/drorbn/images/0/03/10-327-lec08p01.jpg Lecture 7 page 1]

[http://katlas.math.toronto.edu/drorbn/images/6/67/10-327-lec08p02.jpg Lecture 7 page 2]

[http://katlas.math.toronto.edu/drorbn/images/3/30/10-327-lec08p03.jpg Lecture 7 page 3]

[http://katlas.math.toronto.edu/drorbn/images/5/5b/10-327-lec08p04.jpg Lecture 7 page 4]

File:10-327-lec08p04.jpg

2010-10-07T20:23:02Z

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File:10-327-lec08p03.jpg

2010-10-07T20:22:09Z

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File:10-327-lec08p02.jpg

2010-10-07T20:21:46Z

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File:10-327-lec08p01.jpg

2010-10-07T20:21:39Z

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