ISEF ’08 Winner: Knot Theory Mathematics

August 20, 2008 at 1:57 pm | Posted in History and News | 2 Comments
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Sana Raoof, 17 of Jericho High School in Jericho, New York, is one of the winners of Intel Foundation Scientist Award from Intel International Science & Engineering Fair (ISEF) on May 16, 2008. (from the Winner Announcement and News Release). And will attend Harvard University this fall.

The vedio shows us how the intelligent girl introduce her research, “Computation of the Alexander-Conway polynomial on the Chord Diagrams of Singular Knots” to ISEF.  The lucky girl motivated by something from MIT, maybe in some graduate summer program, which provided her a great introduction of knot theory and lots of paper for this subject. She focus on computation of the δ-function that can distinguish if those on Chord Diagrams.

Congratulation to the shiny star…She won the prize to the royal road!!

P.s.

• Want to know more about “Chord Digrams“, you may also see the article of Greg in The Everything Seminar to discuss with them.
• You can also click “knot theory and biochemistry” by Jesse Johnson to study the subject more.
• Gee didn’t think the result application of protein or DNA in biochemistry can be helpful… We all know that the identified methods of biochemistry are quite differenent from the knot computation. Knot model allowed knots to be contracted, distorted and twisted infinitely. That’s not the same for DNA nor protein. The force energies between slight masses of protein or DNA need more detailed discussion or computation. It’s more more important than to compute the free knot polynomial of the model. … Just my little experience in the program of Institue of Bio-Chemistry to learn something about protein in Academia sinica, 2008.

Bala-bala Lie algebras

August 16, 2008 at 1:40 am | Posted in Math Learning | 1 Comment
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Introduction to Lie algebras and Representation Theory, translated by Jui-Ji Chang

Gee was so happy to met Jyun-Ao , a student of  Shun-Jen, remind me to pick my memory about Lie algebras…  It had been several years since I was struggle for knowing what’s the key math touch my mind. Lie algebras was really a unexpected beauty for me. The history of me to touch Lie algebras maybe an accident(or not)…

• Beginning, Wen Long Lin who taught Applied mathematics in department of physics shows me two books of Lie algebras in 2001. Which are:(A)Howard Gergi. Lie Algeras in Partical Physics(1999). ISBN:0-738-20233-9. (B)Robert . Chahn. Semi-Simle Lie Algebras & their Representations.BenJamin/cummings(1984). ISBN:0-805-31600-0. …He said “Department of Mathematics may be better than Department of Physics if you want to learn these.”… Did not agree with him that time because I never saw or learned Lie algebras in Department of Mathematics till met him.
• Found what “root system”is of the paper on the webpage I saw in 2004 : Valentina Golubeva:Hamiltonians of the Calogeto-Sutherland Type Models Associated to the Root Systems and Corresponding Fock Spaces….For understanding this paper and others I did not mention here, I was been kicked out the door of Roger. It became a penal pain pricked my heart that hard to be cured in my memory…
• Audit a course of Chia-Hsin Liu to know root systems, He used the Humphreys’ book… There is no feelings happened in my mind in his class, thus gave the course up immediately…Audit his class just one time…For knowing that course did open for those  interested in ideal and rings.
• The story after baby lie groups(class of Chun Chung in 2004) and lie group(class of Ong, Ping-Zen) attract me, especially for the class of Chun Chung, Lie algebras just like the drum in heaven that drumed me…
• Met I-Hsun in 2004. Wanted to go to his Differential Geometry course to listen more things about Lie algebras but fixed in the Lie algebras course of Shun-Jen for one year.
• ??—to present.

m…m… let me talk about translation works of mathematics.  After doing “My oops translate“, I felt a little better. But offer such volunteer effort really not a work or plan that should be do. Anyway I get something I like to do from doing that… It’s a long time not touch math, this way can cure my life a moment…Good translation is not a easy work, especially for mathematics. Take my translating “18.238 Geometry and Quantum Field Theory, Fall 2002” for example, I never know what I type…Just a machine kala~ka-la.  Keep the questions in your stomach or you cannot finish the document.

Borrowed the book,「張瑞吉 譯。李代數與表現理論之導引。國立編譯館主編。台北：黎明文化(民70)。」which is translated from the famous book: “Jumes E. Humphreys. Introduction to Lie Algebras and Representation Theory, GTM9. Springer-Verlag New York,1972.” ,from Taitung County library.  Prof. Rui-Ji Chang translate the book very well. Didn’t know if the teachers in the university before like translate. Gee regard them are all tips on tops during those days. They wrote, edited and translated excellent books by themselves…seems a little different than present day. Anyway,Each period must exist its great events.Back to translate…

Because of the translation mistake always happened for general translation. Gee still like read the original version than the second. Even for this good book translated by Prof. Chang…

Example.(p.1)

J. E. Humphreys wrote…

Definition. A vector space L over a field F, with an operation L × L→L, dented (x,y) →[xy]and called the bracket or commutator of x and y, is called a Lie algebra over F if the following axioms are satisfied:

(L1) The bracket operation is bilinear.
(L2) [xx]=0 for all x in L
(L3)[x[yz]]+[y[xz]]+[z[xy]]=0 (x,y,z∈L)

Rui-Ji Chang translated

(L1) 括弧運算為雙線性。
(L2)[xx]=0 對所有 x ∈ L。
(L3)[x[yz]]+[y[xz]]+[z[xy]]=0 (x,y,z∈L)

You catch the points here? That’s why I like to read the original book than the translated one, even though the Chinese words are so cute for me. Each translated book like this must have the tiny mistakes like these~ They will confuse me a lot to read the book (From these, you may understand that “to read a book well” is such a difficult hobby for me><).

For being an oops mathematic courses translator…Keep learning is an important things. Even though the problems also a lot, and lots problems like the upper statement…

“Just Go!” give myself the words!! I will be a good translator like Rui-Ji.

Equations Editor in WordPress

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To Tell the truth, there are not only amazing persons, but also good tools here that support a good  environment to learn mathematics in WordPress.

Yes! WordPress can include $\LaTeX$ code in the post. See the announcement “Can I put Math or Equations in my Posts? ” of FAQ. You can type the mathematical formula on the blog. Just as the article show us :
So, taking these two definitions for example,

Definition3.6 The Jones polynomial V(L) of an oriented link L is the Laurent polynomial in t1/2, with integer coefficients, defined by

$V(L)=((-A))^{-3(D)})\langle{D}\rangle)_{t^{1/2}=A^{-2}}\in{Z[t^{-1/2},t^{1/2}]}$

where D is any oriented diagram for L .

Definition6.6  The rth Alexander ideal of an oriented link L is the rth elementary ideal of the Z[t-1,t] module H1[X;Z]. The rth Alexander polynomial of L is a generator of the smallest principal ideal of Z[t-1,t] that contains the rth Alexander polynomial. The first Alexander polynomial is the called the Alexander polynomial and is written ΔL(t)

(Definition Resourse: W.B. Raymond Lickorish. An Introduction to Knot Theory, GTM175. New York:Springer-Verlag (1997). p. 26,55.)

Using this useful Latex code… expressing and discussing mathematic are never difficulity problems here. You can also try it. But if you haven’t known what Latex would be, you may also click”LaTeX–A document preparation system” to keep close and to attach it.

p.s. As for the two definitions in the post, Gee still not catch their true meanings in my mind after typing them. If anybody can explain them more clearly…Please and please tell me.

Knot knock

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Do you know how hard to draw the knot picture?!…
You must try it!!
Look at the eight crossings knot table…

• First, You should know how to tie a knot, thus you can know the upper segment and lower part!
• Second, Upper segment is a real black line, and the lower part must anticipate an empty.
※You should keep all basic knot form deeply in your brain.
• Third, Draw it with your pen. 3-dimension memory and 2-dimension drawing. Keep practice, practice and practice or you cannot draw it as well.(Actually speaking I still cannot do it well now)
• Fourth, Reverse what you draw…to show the different positive picture and negative picture like.

This is What I did before…A never succeed work><

After going to Taipei, I open on the book: W.B. Raymond Lickorish. An Introduction to Knot Theory, GTM;175. Springer-Verlag New York(1997).To turn on my memory of Knot…It was Chun Chung tell me the good one. Seems a hard reading book for a undergraduate guy…I always confuse that Why I can easily understand what knot is after Chun Chung easily explained it, but cannot understand what the book show us. Alas~O if there I can take part in a course of he to tell me what the knot is…Not just a book.

Take something for example, look at the definition1.1 to 1.4, and compare with knot table to Eight Crossing(table one). To make the definition clearly had been a hard work for me…

Definition1.1: A link L of m components is a subset of S3 or of R3, that consists of m disjoint, piecewise linear, simple closed curves. A link of one component is a Knot.

Definition1.2: Links L1 and L2 in S3 are equivalent if there is an orientation-preserving piecewise linear homeomorphism H: S3 → S3 such that h(L1)=(L2)

Note Definition: A knot is said to be the unknot if it bounds an embedded piecewise linear discs in S3.

Definition1.3: A knot K is a prime knot if it is not the unknot, and K=K1+K2 implies that K1 or K2 is the unknot.

Definition1.4 Suppose that L is a two-component oriented link with components L1 and L2. The linking number lk(L,L) of L1 and L2 is half the sum of the sign, in a diagram for L, of the crossings at which one strand is from L1 and the other is from L2

My learning problems always stubs my brain..

Here: Who knows the “link”?! In mathworld shows that ” Formally, a link is one or more disjointly embedded circles in three space. More informally, a link is an assembly of knots with mutual entaglements”(http://mathworld.wolfram.com/Link.html). The definitions of knot and link are much easier on the page with picture(http://library.thinkquest.org/12295/)..of course, I understand it now, but it quite hard for me to state the definition. Cause I found at least 3 definition of Knot or link…

And: Is there anybody can show me the knot that been bounded an embedded piecewise linear discs in S3 (in definition1.2) ?….It sounds like noodles in a big pan. God~There must no people read this book like me…@@

OK~take it easy: Who knows the number of Prime knots after see the definition? We can distinguish the prime knot from definion1.3.. also can know the number of Prime knots not equal to the prime number from the knot table. As for the definition1.4…How does the “half the sum of the sign” work?…It just says that something related in homology theory. How can I catch the meanings of it from such few words?!

After looking over these, how you think about to know other important invariant-related topics like Alexander polynomial, Jones polynomial and so on in knot theory? It should be another hard work to overcome.  Of course, you can see what the stupid student’s story about learning math, especially in complex geometry , Lie algebra, knots and those with Quantization on the blog from now on.  Thus for the important part of knot,Alexander polynomial and Jones polynomial, you will see what the  learning problem I met.