Drorbn - User contributions [en]

Talk:06-1350

2007-06-13T13:36:10Z

134.58.33.227:

Guys,

I thought it might be nice to establish a lively discussion board on which we can discuss, develop and clarify any, or all, aspects of material pertaining to the class! I'll try to get things started with something that I had on my mind.

From lectures, we've opted to get our teeth stuck into things from the outset, with the point being that a thorough discussion of the elementary concepts would eat up a substancial amount of time, from which the only outcome would be a rigorous confirmation of what we all already understood knot, and equivalence of knots, to mean.

Being eager, I thought I'd jot down a couple of sentences (having just finished the post, I can say that this is was a gross underestimate of length) indicating what I think the pedantic issues at stake to be, and was hoping that perhaps we could iron out a couple of things I find to be confusing.

As I see it, the first, and main, point that might warrent a discussion, is on the correspondence between smooth and peicewise linear embeddings of the circle in three space.

[Recall from the first lecture that we would like a knot to be an equivalence class of embeddings, where the embeddings can be taken to be peicewise linear or smooth and the equivalence relation is determined by ambient isotopy; namely f and g are said to be ambient isotopic embeddings if and only if we have a smooth/peicewise linear (depending on whether we're talking about smooth or peicewise linear embeddings) map H:S^3x[0,1]->S^3 subject to H(_,t) being a smooth/peicewise linear homeomorphism with H(_,0) the identity and H(f(S^1),1)=g(S^1).]

The first question is: what is the correspondence between the two theories? Clearly any peicewise linear knot may be smoothed at the corners, and any smooth embedding has a linear approximation. However, if we try to talk about isotopies of such knots we're talking about something living in S^3x[0,1]xS^3. I think smooth manifolds, of which the graph of H is one, can be triangulated, though I don't have a good reference for this. Further, if the peicewise linear isotopies could be "smoothed" then we could conclude that thinking of knots as being peicewise linear or smooth is really the same for our purposes. Does anyone know about this or have any good references?

The second question (which is really dependant on the first question) I had in mind was on the connection between representations of knots in the form of quadravalent planar diagrams up to Reidmeister moves, and knots up to ambient isotopy.

Firstly, well defined parallel projections of knots exist

[infact there is a lovely visual argument that show that almost all parallel projections are "nice" in the sence that the inverse image of points of the diagram have at most two preimages and such points are isolated... equate the projections with points on the sphere and observe that two linear segments of the knot project onto each other only on arcs of the sphere, and further that projections having preimage sets of cardinality greater or equal to two are isolated or again lie on an arc. Thus the measure of such a set is zero.]

Now ambient isotopies of peicewise linear isotopies of knots can be broken down into moves where you can either replace one segement of a knot with the other two edges of a triangle containing the fixed segment as an edge as long as the spanning triangle doesn't intersect the knot anywhere else and vica versa (a picture would be very handy here, but I don't really know how to use any such feature yet), and consequently we see the Reidemister moves of planar diagrams all correspond to isotopies. The other direction is Reidemeister's famous theorem (the equivalence that we assumed axiomatically) and is quite a complicated argument. The question that arises in my mind is that if the smooth and peicewise linear theories are not equivalent [see question one], then it would seem that Reidemeister's proof may not carry over!

I'd love to have this put straight in my mind, and hope that someone can help me via a reference or authorative confirmation.

Sorry this was long winded, confused and above all practically irrelevant to the course!

Fionntan

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06-240/About This Class

2007-06-13T13:06:32Z

134.58.33.227: /* Crucial Information */

{{06-240/Navigation}}

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===Abstract===
Taken from the Faculty of Arts and Science [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Calendar]:
<blockquote>
A theoretical approach to: vector spaces over arbitrary fields including <math>{\mathbb C}</math>, <math>{\mathbb Z}_p</math>. Subspaces, bases and dimension. Linear transformations, matrices, change of basis, similarity, determinants. Polynomials over a field (including unique factorization, resultants). Eigenvalues, eigenvectors, characteristic polynomial, diagonalization. Minimal polynomial, Cayley-Hamilton theorem.
</blockquote>

*Prerequisite: MCB4U, MGA4U
*Co-requisite: MAT157Y1

[[Image:Friedberg_Insel_Spence_Cover.jpg|right|200px]]
===Text Book(s)===
Our main text book will be ''Linear Algebra'' (fourth edition) by Friedberg, Insel and Spence, ISBN 0-13-008451-4; it is a required reading. Note that the eratta for various printings of the 4th edition of the text is available from the publisher here at http://www.math.ilstu.edu/linalg/errata.html.

I am told that Schaum’s ''Outline of Linear Algebra'', ISBN 0-07-136200-2, may contain useful examples; it is not a required reading.

===Wiki===
The class web site is a wiki, as in [http://www.wikipedia.org Wikipedia] - meaning that anyone can and is welcome to edit almost anything and in particular, students can post notes, comments, pictures, whatever. Some rules, though -
* This wiki is a part of my ([[User:Drorbn|Dror's]]) academic web page. All postings on it must be class-related (or related to one of the other projects I'm involved with).
* If there's no specific reason for your edit to be anonymous, please log in and don't have it anonymous.
* Criticism is fine, but no insults or foul language, please.
* I ([[User:Drorbn|Dror]]) will allow myself to exercise editorial control, when necessary.
* The titles of all pages related to this class should begin with "06-240/" or with "06-240-", just like the title of this page.
Some further editing help is available at [[Help:Contents]].

===Marking Scheme===
There will be one term test (25% of the total grade) and a final exam (50%), as well as about 10 homework assignments (25%).

====The Term Test====
The term test will take place in class on Tuesday October 24th. A student who misses the term test without providing a valid reason (for example, a doctor’s note) within one week of the test will receive a mark of 0 on the term test. There will be no make-up term test. If a student misses the term test for a valid reason, the weight of the problem sets will increase to 35% and the weight of the final exam to 65%.

====Homework====
Assignments will be posted on the course web page and distributed in class (usually on Thursdays) on the weeks shown in the class timeline. They will be due a week later at the tutorials and they will be (at least partially) marked by the TAs. All students (including those who join the course late) will receive a mark of 0 on each assignment not handed in; though in computing the homework grade, your worst two assignments will not count. I encourage you to discuss the assignments with other students or even browse the web, so long as you do at least some of the thinking on your own and you write up your own solutions. Remember that cheating is always possible and may increase your homework grade a bit. But it will hurt your exam grades a lot more.

===Good Deeds===
Students will be able to earn up to 25 "good deeds" points throughout the year for doing services to the class as a whole. There is no pre-set system for awarding these points, but the following will definitely count:
* Drawing a beautiful picture to illustrate a point discussed in class and posting it on this site.
* Taking class notes in nice handwriting, scanning them and posting them here.
* Typing up or formatting somebody else's class notes, correcting them or expanding them in any way.
* Writing an essay on expanding on anything mentioned in class and posting it here; correcting or expanding somebody else's article.
* Doing anything on our [[06-240/To do]] list.
* Any other service to the class as a whole.

Good deed points will count towards your final grade! If you got <math>n</math> of those, they are solidly your and the formula for the final grade below will only be applied to the remaining <math>100-n</math> points. So if you got 25 good deed points (say) and your final grade is 80, I will report your grade as <math>25+80(100-25)/100=85</math>. Yet you can get an overall 100 even without doing a single good deed.

'''Important.''' For your good deeds to count, you '''must''' do them under your own name. So you must set up an account for yourself on this wiki and you must use it whenever you edit something. I will periodically check [[Special:Recentchanges|Recent changes]] to assign good deeds credits.

===Class Photo===
To help me learn your names, I will take a class photo on Thursday of the third week of classes. I will post the picture on the class' web site and you will be ''required'' to send me an email and identify yourself in the picture or to identify yourself on the [[06-240/Class Photo|Class Photo]] page of this wiki.

===How to Succeed in this Class===
* '''Keep up!''' Don't fall behind on reading, listening, and doing assignments! University goes at a different pace than high school. New material is covered once and just once. There will be no going over the same thing again and again - if you fall behind, you stay behind. Unless you are an Einstein, there is ''no way'' to do well in this class merely by attending lectures - you '''must''' think about the material more than 3 or 5 hours a week if you want it to sink in. And if you are planning on not attending lectures, well, think again. Most people find it very hard to pace their own studies without a human contact; if you'll try, you are likely to discover the hard way that you belong to the majority.
* If in high school you were the best in your class in math, now remember that everybody around you was the same. You may find that what was enough then simply doesn't cut it any more. Try to catch that early in the year!
* Math is about '''understanding''', not about memorizing. To understand is to internalize; it is to come to the point where whatever the professor does on the blackboard or whatever is printed in the books becomes '''yours'''; it is to come to the point where you appreciate why everything is done the way it is done, what does it mean, what are the reasons and motivations and what is it all good for. Don't settle for less!
* Keep asking yourself questions; many of them will be answered in class, but not all. Remember the old Chinese proverb:

<div style="text-align: center; font-size: 18px">'''"Teachers open the door, but you must enter by yourself"'''</div>

The saddest that can happen to you in this class is if you won't notice the door being opened.

06-240

2007-06-13T12:54:17Z

134.58.33.227:

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