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%\url{http://drorbn.net/AKT-17}\newline
\bf\href{\myurl}{Dror Bar-Natan}:
\href{\myurl/classes/}{Classes}:
\href{\myurl/classes/\#1617}{2017}:
\href{http://drorbn.net/?title=AKT-17}{MAT 1350}:
\newline Algebraic Knot Theory --- Poly-Time Computations:
}
\hfill\text{\huge\bf About This Class}
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{\tt\small Dear All,}\hfill\text{\bf Shameless Marketing.}
{\tt\small\frenchspacing This email is to shamelessly market to you the graduate class I will be teaching this semester, "MAT 1350 Algebraic Knot Theory - Polynomial Time Computations".
Over the last couple of years it emerged that there exists a class of very strong knot invariants that can be computed in polynomial-time (better than the reigning champs, which are all exponential time) - meaning that in principle these invariants can be computed even for very very large knots. Furthermore, there is interesting and promising topology behind this class of invariants, and some novel algebra, especially around Lie algebras and their universal enveloping algebras. There are also many directions to explore. [Yet unfortunately, at least for now, some of this material is *hard*.]
My class will be an introduction to the topic, by means of pushing it further. If we work hard during the semester, we may, just may, be able to push things on from sl(2) to sl(3), thus turning the invariant much stronger (even if a bit less computable). It will be truly wonderful if we succeed. Though even if we fail, we will learn a great deal about knots and tangles and virtual knots and virtual tangles and expansions and Lie algebras and Lie bialgebras and about sophisticated computations using Mathematica.
The prerequisites are mathematical maturity and no fear of computers, total comfort with linear algebra: vector spaces, duals, quotients, tensor products, etc., and some appreciation of Lie algebras.
%The class will meet on Tuesdays 11-1 and Fridays 11-12 at Bahen 6180. More details at \url{http://drorbn.net/AcademicPensieve/Classes/17-1350-AKT/About.pdf}.
Sincerely, Dror Bar-Natan.}
\columnbreak\null\hfill{\bf Web page @ \url{http://drorbn.net/AKT-17}:}
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{\bf Classes.} Tuesdays 11-1 and Fridays 11-12 at Bahen 6180. There will also be a ``HW meeting'', covering no new material, on Fridays at 6:10PM at or near my office.
{\bf Instructor.} Dror Bar-Natan, \href{mailto:drorbn@math.toronto.edu}{drorbn@\linebreak[0]math.\linebreak[0]toronto.\linebreak[0]edu}, \href{\myurl}{http:\linebreak[0]//\linebreak[0]www.\linebreak[0]math.\linebreak[0]toronto.\linebreak[0]edu/\linebreak[0]$\sim$drorbn/}, Bahen 6178, 416-946-5438. Office hours: See website, or by appointment, or by opportunity.
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{\bf Agenda.} Group-discover and group-implement the strongest ever truly computable knot invariant. Along the way, learn some of the why (topology!), what (Lie theory!), and how (Mathematica!). Leave behind a complete documentation trail.
Alternatively, ``understand everything in \url{http://drorbn.net/dbnvp/GWU-1612.php} (video there, handout attached), and beat it''.
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{\bf Plan.} Yes, I mean it. I plan to do deep and real and significant research as a part of this class; the sketch I already know, and the details and the implementation we will work out. We will split our time at a roughly 1:1 ratio between background material (Tuesdays and later Fridays), and shortest and most direct\&brute path to discovery (Fridays and later Tuesdays).
{\bf Warning.} By the nature of these plans, this will be a hard class, and likely messy, and possibly we will fail to achieve the main goal. Yet even then we will learn a great deal.
{\bf Optimistic tentative ``Background Material'' Plan.} Course introduction (hour 1). Knots, Reidemeister moves and the Jones polynomial (h2). Tangles and a faster Jones program (h3). The Alexander polynomial as a determinant and using $\Gamma$-calculus (h4-5). Seifert surfaces and genus, ribbon knots and ``algebraic knot theory'' (h6-7). Finite type invariants and expansions (h8-9). The relationship with metrized Lie algebras and PBW (h10-11). The variants $v$, $w$, $bv$, and $rv$, and their expansions (h12-14). Lie bialgebras and solvable approximation (h15-16). Odds and ends (h17-18).
{\bf Optimistic tentative ``Direct\&Brute Path'' Plan.} The Lie algebras $sl_2$, $sl_3$, $\frakg_0$ and $\frakg_1$, universal enveloping algebras and low degree computations (hours 1-2). Algebras, Yang-Baxter elements, and invariants (h3). Ordering symbols and commutation relations for $\frakg_0$ (h4-5). The $\frakg_0$ invariant (h6). The {\greektext L'ogos} and $\frakg_1$ computations (h7-8). Morse knots and the $\frakg_1$ invariant (h9). $\frakg_0$ and $\frakg_1$ as approximations of $sl_2$, approximating $sl_3$ (h10). The $sl_3^0$ invariant (h10-11). The $sl_3^1$ invariant, fame, and glory (h12-18).
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{\bf Prerequisites.} Mathematical maturity and no fear of computers. Total comfort with linear algebra: vector spaces, duals, quotients, tensor products, etc. Some appreciation of Lie algebras.
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{\bf Mathematica.} Almost everything we say, we will immediately implement using Mathematica\footnote{{\bf Q.} Can I use language $X$ instead of Mathemtica? {\bf A.} Theoretically, you could and it would be a wonderful contribution if you did. In practice you'd be taking something very difficult and turning it into nearly impossible. So the honest answer is ``No''.}. If you are seriously taking this class, you {\bf must} have a working copy on your computer. If you don't already have one, ``Mathematica Student Edition'' (web-search that) is CDN\$150. Save the receipt! I will make every effort to refund this expense for students officially taking this class.
{\bf Class Photo.} To help me learn your names, I will take a class photo on Wednesday of the third week of classes. I will post the picture on the class' web site and you will be encouraged to identify yourself on the Class Photo page (\url{http://drorbn.net/?title=AKT-17/Class_Photo}) of the class' wiki.
{\bf Videos and Wiki.} We will videotape all classes and the course's web site will be centered around these videos. I have set up a system (see below) that allows anyone signed-up to index and annotate these videos on a wiki, and allows for the inclusion and linking of other pages and further material to this wiki.
Anyone signed-up can, is welcome and is encouraged to edit and add to the class' web site. In particular, students can post video annotations, notes, comments, pictures, solution to open problems, whatever. Some rules, though ---
$\bullet$ This wiki is a part of my (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).
$\bullet$ I (Dror) will allow myself to exercise editorial control, when necessary.
$\bullet$ The titles of all pages related to this class should contain and preferably begin with the string ``{\tt AKT-17}'', just like the title the classes' main page.
$\bullet$ For most {\tt AKT-17} pages, it is a good idea to put a line containing only the string \verb${{AKT-17/Navigation}}$ at the top of the page. This template inserts the class' ``navigation panel'' at the top right of the page.
$\bullet$ To edit the navigation panel itself, click on the word ``Navigation'' on the upper right of the panel. Use caution! Such edits affect many other pages!
$\bullet$ Neatness matters! Material that is posted in an appealing manner will be read more, and thus will be more useful.
$\bullet$ Some further editing help is available at \url{http://drorbn.net/?title=Help:Contents}.
{\bf Wiki Sign-Up.} Email me with you full name, email address and preferred userid if you need an account on the class wiki.
{\bf Homework} assignments will usually be jointly written, usually on the Friday HW meetings, usually they will be assigned on Mondays, and usually be due on the following Monday. There will be about 11 assignments; your HW mark will be the average of your best 6 assignments. Late assignments will be marked down by 1\% per day.
{\bf Student Presentations.} During class I will occasionally suggest topics for student presentations. Typically these will involve reading some research papers and lecturing to class about them a week or two after the end of formal classes.
{\bf Final Exam.} If people will so prefer, there will also be a Final Examination.
{\bf Good deeds.} You will be able to earn ``good deed'' points throughout the term. You may earn up to 90 good deed points for removing one of the major roadblocks we will encounter, up to 80 good deed points for writing a book-quality open-source and copyleft exposition of a significant and deep portion of this class. More easily, for lively participation in and markup of the class wiki, you may receive up to about 40 good deed points.
{\bf Semi-Final grade.} The higher of 70\% HW and 30\% Presentation Mark or Final Examination, or 20\% HW and 80\% Presentation Mark or Final Examination.
{\bf Final Grade.} If you earn $0\leq\gamma\leq 90$ good deed points during the term, and your semi-final grade is $\sigma$, your final grade will be $\gamma+(100-\gamma)\sigma/100$. This can be $100$ even with $\gamma=0$, yet with $\gamma=90$, it will be $95$ even with $\sigma=50$.
{\bf Dror's Open Notebook} for this class is at \url{http://drorbn.net/AcademicPensieve/Classes/17-1350-AKT/}. Use at your own risk.
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