One
cannot fight fashion… Following ones own research interest is a
pretty frustrating activity. Not only does it take forever to get a
paper refereed but then you have to motivate why you do these things
and what their relevance is to other subjects. On the other hand,
following fashion seems to be motivation enough for most…
Sadly, the same begins to apply to teaching. In my Geometry 101 course I
have to give an introduction to graphs&groups&geometry. So,
rather than giving a standard intro to graph-theory I thought it would
be more fun to solve all sorts of classical graph-problems (Konigsberger
bridges, Instant
Insanity, Gas-
water-electricity, and so on…) Sure, these first year
students are (still) very polite, but I get the distinct feeling that
they think “Why on earth should we be interested in these old
problems when there are much more exciting subjects such as fractals,
cryptography or string theory?” Besides, already on the first day
they made it pretty clear that the only puzzle they are interested in is
Sudoku.
Next week I’ll have to introduce groups and I was planning to do
this via the Rubik
cube but I’ve learned my lesson. Instead, I’ll introduce
symmetry by considering micro-
sudoku that is the baby 4×4 version of the regular 9×9
Sudoku. The first thing I’ll do is work out the number of
different solutions to micro-Sudoku. Remember that in regular Sudoku
this number is 6,670,903,752,021,072,936,960 (by a computer search
performed by Bertram
Felgenhauer ). For micro-Sudoku there is an interesting
(but ratther confused) thread on the
Sudoku forum and after a lot of guess-work the consensus seems to be
that there are precisely 288 distinct solutions to micro-Sudoku. In
fact, this is easy to see and uses symmetry. The symmetric group $S_4$
acts on the set of all solutions by permuting the four numbers, so one
may assume that a solution is in the form where the upper-left 2×2
block is 12 and 34 and the lower right 2×2 block consists of the
rows ab and cd. One quickly sees that either this leeds to a
unique solution or so does the situation with the roles of b and c
changed. So in all there are $4! \\times \\frac{1}{2} 4!=24 \\times 12 =
288$ distinct solutions. Next, one can ask for the number of
_essentially_ different solutions. That is, consider the action
of the _Sudoku-symmetry group_ (including things such as
permuting rows and columns, reflections and rotations of the grid). In
normal 9×9 Sudoku this number was computed by Ed Russell
and Frazer Jarvis to be 5,472,730,538 (again,heavily using the
computer). For micro-Sudoku the answer is that there are just 2
essentially different solutions and there is a short nice argument,
given by ‘Nick70′ at the end of the above mentioned thread. Looking a bit closer one verifies easily that the
two Sudoku-group orbits have different sizes. One contains 96 solutions,
the other 192 solutions. It will be interesting to find out how these
calculations will be received in class next week…
Tag: geometry
The third volume in the NeverEndingBooks-series will be written by Geert Van de Weyer and will
be about (double) Poisson structures in the noncommutative world. Volume
4&5 are becoming clearer every day and if you think you have a
project fitting in this series, you can always email to
[info@neverenedingbooks.org][3].
As for the NeverEndingBooks-URL, I will
probably close this blog by the end of the month (at its first
birthday). The main reason is that I found out that it takes several
people to maintain a mathematical blog for some time. So, if you want to
co-author a group-blog on noncommutative algebra and/or noncommutative
geometry, please [email me][5] (or even better, leave a comment here so
that other people may be willing to join in too) and if there is enough
critical mass to go ahead with the plan I will be happy to set up a
group-blog at noncommutative.org.
At
this URL I’ll probably put a frontpage for the book-series we started
and which you can buy at all times via lulu.com/neverendingbooks. It will contain errata- and suggestions-pages for each volume and
details about forthcoming books, links etc… Btw. it would probably
be a good commercial move to delete TheLibrary links sooner, now that
even String Theorists are driven to this site via
Lazariou’s paper On
the non-commutative geometry of topological D-branes
As my
main objective next year will be to write courses (from first year down
to post-doc level) I will set up (again) a Moodle site (mainly in English,
although UA-students will be free to add to it in Dutch). News about it
will be posted eventually at my regular, but forgotten
homepage and perhaps here.
Once again, if you are interested
to contribute at unregular intervals to a noncommutative group-blog,
please leave a comment!
[3]: mailto: info@neverendingbooks.org
[5]: mailto: lieven.lebruyn@ua.ac.be
A quick reply to some of the comments
to the lulu/neverendingbooks post.
_Are they also
responsible for the graphical design in your books ?_ No! In fact it
was one of the more pleasant experiences of the last couple of weeks to
develop our own format, LaTeX-style and covers. The usual gang had their
say in all of this but it is only fair to say that Jan did most of the work. We developed
the cover-concept (that is, macro shots of games in duotones and placing
of titles etc) by trial and error. Jan is responsible for the
photo-shoot, I did choose the shots to be used and did the initial
coloring and placing of titles and left the final tweaking to Jan, who
did some lay-out work before. We, at least, are happy with the
result… As mentioned before, the LaTeX-style sheets were made
using the
memoir package.
_Who is responsible for trying to sell
the book, you or them?_ I dont think we are doing great efforts to
try to sell the books, yet. Up to now, you can only get to the
book-sites via this blog or via my homepage. Lulu claims that they will only make
money if we do… and as this is clearly sales-talk (they make money
on every book they print) it involves no (or a very small) financial
risk on our part. Anyone who wants to have a copy of one of our books,
orders them at Lulu, they print it and ship it to you. But beware! They
have several shipping options and for most of them it costs you more to
get them shipped than to buy the books… In fact, that was the main
reason why we didnt put the URL online before we had two volumes out.
The reason being that if you buy for over 25dollars you can have them
shipped via their “SuperSaver” option, that is, shipping is free (but
probably slow). But, based on my own experience it works well (I ordered
a few copies of book 1 via SuperSaver and another one via their
InternationalShipment and got the free SuperSaver package a day before
the costly other shipment…). Our real investment is that we have
bought ISBN-numbers for the books (at a price of 35dollars/book) and
hope to earn this back from a small royalty we get from each book (the
Lulu-rule is that they get 25% of any royalties you set). Even though we
are not entirely happy with the distribution process we opted for this
series for an unusual book-format making it handy and fun to use (the
square 7.5 x 7.5 inch format is very pleasing to read and the
coil-binding makes it extremely handy to lay flat on the table). So we
view this series as a student-edition of the books and we keep them as
cheap as possible. At a later stage it may happen that we will also have
a library-edition of the books which will have global distribution
(meaning that you can order them via Amazon or your local bookshop). For
this to work, you have not so much freedom in your book-format and can
only have regular binding. Besides, buying such a global-ISBN is more
costly and will make this edition (a lot) more expensive. But, as you
can see from the picture, the books get printed and shipped and look
VERY nice. In fact, of the few copies I ordered, I had to hand out
already two because some people just liked the feel and touch of it. I
think, people will only gradually be willing to buy their own copy when
(1) they have glanced through a copy at some meeting or seminar and (2)
if more volumes come out and they have a greater choice in bying 2
volumes to get free shipping. On this last issue : already three people
have expressed interest in writing a book in our series. My own hunch is
that the next book out will have to do with Poisson noncommutative
geometry and will have a macro shot of a war-game on its cover (authors
can give suggestions for which games they want on their cover), curious
how this will work out…
_How many have you sold so
far?_ Well, not enough so far to get our ISBN-investment back…
But, once again, I think it will take some time for people to trust the
series enough to buy a volume or two. In the first week we made the URL
available we sold 16 books, so if you want to increase our sales-index
please do by going to this
page for the first volume and to this page for the second
volume. But perhaps it is easier to bookmark the lulu/neverendingbooks if
you want the latest news on the series. I”ll keep you posted on our
sales via this page. If you buy a book and like the result, please tell
others about it (or even better, let them see and feel the copy.
Hopefully you will get it back…)
If you recognize where this picture was
taken, you will know that I\’m back from France. If you look closer you
will see two bikes, my own Bulls mountainbike
in front and Stijn\’s
lightweight bike behind.
If you see the relative position of the
saddles, you will know that Stijn is at least 20 cm taller. Let me add
that he is also at least 20 yrs. younger and 20 kgs. stronger and it
will be clear that I had a hard (but fun) time trying to follow him
uphill. Btw. this picture (and the next dozen or so) was taken by Jan and I\’ll try to add the next
days a couple of shots he likes more.
Since then I\’ve been
writing up a paper which I hope will be ready to put online by
september. It\’s all about using non-commutative geometry to construct
representations of arithmetic groups, a bit like the Granada Notes but with a dash of
Double Poisson
Algebras to it.
A positive outcome of this short break is
a renewed interest in the NeverEndingBooks project, but more on this
later. For now, let me just add that Raf
decided to feed my noncommutative geometry@n (version 2)
to a printing on demand publisher. So, if you want a perfect bound
paperback version of it (for 12 Euro approx.) you\’d better email him at once (at the
moment he will order just 5 copies).
Here some
possible uses of Markdown and the
HumaneText Service.
As an example, let us take the
noncommutative geometry & algebra page maintained by Paul Smith.
If you copy the source of this page to BBEdit and use the
html2txt.py script in the #! menu (see
this post)
you get a nicely readable Markdown-file which strips the page of all its
layout and which is easy to modify, for example to include author and
URL at the start, remove some additional empty lines, make relative URLs
absolute and so on.
Applying the Markdown.pl
script to it one gets a nice RetroCool version
of the page. For starters, this gives a way to make your own collection
of websites you like in a uniform layout (of course, later on you can
add your own CSS to them).
More important is that the
Markdown-version (see here for
the text-file) is extremely readable and allows to _mine_ all
links easily (as you can see all links contained in the HTML-page are
referenced together at the end of the file). So, this is a quick way to
collect homepage- and email-links from link-pages.
Btw. there
are different ways to include links in a markdown text, for example I
like to write it immediately after the reference, so doing a Markdown.pl
followed by a html2txt.py doesn’t have to reproduce your original file
and fortunately you will always end up with a file having all links
referenced at the end. So, this procedure allows you to have uniformity
in a collection of markdown-files.
Equally important for me (for
later use in an intelligent database using DevonThink ) is that the Markdown file is the best way to safe the
HTML file in the database (as a RTF file) while maintaining readability
(which is important when DevonThink returns snippets of
information).
I’m always
extremely slow to pick up a trend (let alone a hype), in mathematics as
well as in real life. It took me over a year to know of the existence of
_blogs_ and to realize that they were a much easier way to
maintain a webpage than manually modifying HTML-pages. But, eventually I
sometimes get there, usually with the help of the mac-dev-center. So, once again,
I read their gettings things done with your mac article long after it was
posted and completely unaware of the Getting Things Done (or GTD) hype.
At first, it just
sounds as one of those boring managament-nonsense-peptalk things (and
probably that is precisely what it generically is). Or what do you think
about the following resume from Getting
started with ‘Getting things done’ :
- identify all the
stuff in your life that isnÕt in the right place (close all open
loops) - get rid of the stuff that isnÕt yours or you donÕt
need right now - create a right place that you trust and that
supports your working style and values - put your stuff in the
right place, consistently - do your stuff in a way that honors
your time, your energy, and the context of any given moment - iterate and refactor mercilessly
But in fact there is
also some interesting material around at the 43 folders website which bring this
management-talk closer to home such as the How does a
nerd hack GTD? post.
Also of interest are his findings after
a year working with the GTD setup. These are contained in three posts :
A Year
of Getting Things Done: Part 1, The Good Stuff, followed by A Year of
Getting Things Done: Part 2, The Stuff I Wish I Were Better At to
end with A Year of
Getting Things Done: Part 3, The Future of GTD?. If these three
postings don’t get you intrigued, nothing else will.
So, is
there something like _GMD : Getting Mathematics Done_? Clearly, I
don’t mean getting theorems proved, that’s a thing of a few seconds of
inspiration and months to fill in the gaps. But, perhaps all this GTD
and the software mentioned can be of some help to manage the
everyday-workflow of mathematicians, such as checking the arXiv and the
web, maintaining an email-, pdf- and BiBTeX-database, drafting papers,
books and courses etc.
In the next few weeks I’ll try out some
of the tricks. Probably another way to state this is the question “which
Apps will survive Tiger?” Now that it is official that Tiger (that is, Mac
10.4 to non-apple eaters) will be released by the end of the month it is
time to rethink which of the tools I really like to keep and which is
just useless garbage I picked up along the road. For example, around
this time last year I had a Perl
phase and bought half a meter or so of O’Reilly Perl-books. And yes
I did write a few simple scripts, some useful such as my own arXiv RSS-feeds,
some not so useful as a web-spider I wrote to check on changes in the
list of hamepages of people working in non-commutative algebra and
geometry. A year later I realize I’ll never become a Perl Monk. So from now on I want to
make my computer-life as useful and easy as possible, relying on wizards
to provide me with cool software to use and help me enjoy mathematics
even more. I’ll keep you posted how my GMD-adventure goes.
I
expect to be writing a lot in the coming months. To start, after having
given the course once I noticed that I included a lot of new material
during the talks (mainly concerning the component coalgebra and some
extras on non-commutative differential forms and symplectic forms) so
I\’d better update the Granada notes
soon as they will also be the basis of the master course I\’ll start
next week. Besides, I have to revise the Qurves and
Quivers-paper and to start drafting the new bachelor courses for
next academic year (a course on representation theory of finite groups,
another on Riemann surfaces and an upgrade of the geometry-101 course).
So, I\’d better try to optimize my LaTeX-workflow and learn
something about the pdfsync package.
Here is what it is supposed to do :
pdfsync is
an acronym for synchronization between a pdf file and the TeX or so
source file used in the production process. As TeX system is not a
WYSIWYG editor, you cannot modify the output directly, instead, you must
edit a source file then run the production process. The pdfsync helps
you finding what part of the output corresponds to what line of the
source file, and conversely what line of the source file corresponds to
a location of a given page in the ouput. This feature is achieved with
the help of an auxiliary file: foo.pdfsync corresponding to a foo.pdf.
All you have to do is to put the pdfsync.sty file
in the directory _~/Library/texmf/tex/latex/pdfsync.sty_ and to
include the pdfsync-package in the preamble of the LaTeX-document. Under
my default iTex-front-end TeXShop it
works well to go from a spot in the PDF-file to the corresponding place
in the source-code, but in the other direction it only shows the
appropriate page rather than indicate the precise place with a red dot
as it does in the alternative front-end iTeXMac.
A major
drawback for me is that pdfsync doesn\’t live in harmony with my
favorite package for drawing commutative diagrams diagrams.sty. For example, the 75 pages of the current
version of the Granada notes become blown-up to 96 pages because each
commutative diagram explodes to nearly page size! So I will also have to
translate everything to xymatrix&#
8230;
Here is
the construction of this normal space or chart 
. The sub-semigroup of 
(all
dimension vectors of Q) consisting of those vectors 
satisfying the numerical condition 
is generated by six dimension vectors,
namely those of the 6 non-isomorphic one-dimensional solutions in 
![S_1 = \xymatrix@=.4cm{ & & & & \vtx{1} \\ \vtx{1} \ar[rrrru]^1
\ar[rrrrd] \ar[rrrrddd] & & & & \\ & & & & \vtx{0} \\ \vtx{0}
\ar[rrrruuu] \ar[rrrru] \ar[rrrrd] & & & & \\ & & & & \vtx{0}} \qquad
S_2 = \xymatrix@=.4cm{ & & & & \vtx{0} \\ \vtx{0} \ar[rrrru] \ar[rrrrd]
\ar[rrrrddd] & & & & \\& & & & \vtx{1} \\\vtx{1} \ar[rrrruuu]
\ar[rrrru]^1 \ar[rrrrd] & & & & \\ & & & & \vtx{0}}](http://www.math.ua.ac.be/~lebruyn/latexrender/pictures/
a59961f7695f6a329d40347aa9f16498.gif "S_1 = \xymatrix@=.4cm{ & &
& & \vtx{1} \\ \vtx{1} \ar[rrrru]^1 \ar[rrrrd] \ar[rrrrddd] & & & & \\ &
& & & \vtx{0} \\ \vtx{0} \ar[rrrruuu] \ar[rrrru] \ar[rrrrd] & & & & \\ &
& & & \vtx{0}} \qquad S_2 = \xymatrix@=.4cm{ & & & & \vtx{0} \\ \vtx{0}
\ar[rrrru] \ar[rrrrd] \ar[rrrrddd] & & & & \\& & & & \vtx{1} \\\vtx{1}
\ar[rrrruuu] \ar[rrrru]^1 \ar[rrrrd] & & & & \\ & & & & \vtx{0}}")
![S_3 = \xymatrix@=.4cm{ & & & & \vtx{0} \\ \vtx{1} \ar[rrrru]
\ar[rrrrd] \ar[rrrrddd]^1 & & & & \\ & & & & \vtx{0} \\ \vtx{0}
\ar[rrrruuu] \ar[rrrru] \ar[rrrrd] & & & & \\ & & & & \vtx{1}} \qquad
S_4 = \xymatrix@=.4cm{ & & & & \vtx{1} \\ \vtx{0} \ar[rrrru] \ar[rrrrd]
\ar[rrrrddd] & & & & \\ & & & & \vtx{0} \\ \vtx{1} \ar[rrrruuu]^1
\ar[rrrru] \ar[rrrrd] & & & & \\ & & & & \vtx{0}}](http://www.math.ua.ac.be/~lebruyn/latexrender/pictures/
2bf710496d35295d2ae9b7a3322d0fab.gif "S_3 = \xymatrix@=.4cm{ & &
& & \vtx{0} \\ \vtx{1} \ar[rrrru] \ar[rrrrd] \ar[rrrrddd]^1 & & & & \\ &
& & & \vtx{0} \\ \vtx{0} \ar[rrrruuu] \ar[rrrru] \ar[rrrrd] & & & & \\ &
& & & \vtx{1}} \qquad S_4 = \xymatrix@=.4cm{ & & & & \vtx{1} \\ \vtx{0}
\ar[rrrru] \ar[rrrrd] \ar[rrrrddd] & & & & \\ & & & & \vtx{0} \\ \vtx{1}
\ar[rrrruuu]^1 \ar[rrrru] \ar[rrrrd] & & & & \\ & & & & \vtx{0}}")
![S_5 = \xymatrix@=.4cm{ & & & & \vtx{0} \\ \vtx{1} \ar[rrrru]
\ar[rrrrd]^1 \ar[rrrrddd] & & & & \\ & & & & \vtx{1} \\ \vtx{0}
\ar[rrrruuu] \ar[rrrru] \ar[rrrrd] & & & & \\ & & & & \vtx{0}} \qquad
S_6 = \xymatrix@=.4cm{ & & & & \vtx{0} \\ \vtx{0} \ar[rrrru] \ar[rrrrd]
\ar[rrrrddd] & & & & \\ & & & & \vtx{0} \\ \vtx{1} \ar[rrrruuu]
\ar[rrrru] \ar[rrrrd]^1 & & & & \\ & & & & \vtx{1}}](http://www.math.ua.ac.be/~lebruyn/latexrender/pictures/
59fe70dac8703efdbb3f8abb19188d9e.gif "S_5 = \xymatrix@=.4cm{ & &
& & \vtx{0} \\ \vtx{1} \ar[rrrru] \ar[rrrrd]^1 \ar[rrrrddd] & & & & \\ &
& & & \vtx{1} \\ \vtx{0} \ar[rrrruuu] \ar[rrrru] \ar[rrrrd] & & & & \\ &
& & & \vtx{0}} \qquad S_6 = \xymatrix@=.4cm{ & & & & \vtx{0} \\ \vtx{0}
\ar[rrrru] \ar[rrrrd] \ar[rrrrddd] & & & & \\ & & & & \vtx{0} \\ \vtx{1}
\ar[rrrruuu] \ar[rrrru] \ar[rrrrd]^1 & & & & \\ & & & & \vtx{1}}")
In
particular, in any component 
containing an open subset of
representations corresponding to solutions in we have a particular semi-simple solution
and in
particular 
. The normal space
to the 
-orbit of M in can be identified with the representation
space 
where 
and Q is the quiver of the following
form
![\xymatrix{ &
\vtx{g_1} \ar@/^/[ld]^{C_{16}} \ar@/^/[rd]^{C_{12}} & \\ \vtx{g_6}
\ar@/^/[ru]^{C_{61}} \ar@/^/[d]^{C_{65}} & & \vtx{g_2}
\ar@/^/[lu]^{C_{21}} \ar@/^/[d]^{C_{23}} \\ \vtx{g_5}
\ar@/^/[u]^{C_{56}} \ar@/^/[rd]^{C_{54}} & & \vtx{g_3}
\ar@/^/[u]^{C_{32}} \ar@/^/[ld]^{C_{34}} \\ & \vtx{g_4}
\ar@/^/[lu]^{C_{45}} \ar@/^/[ru]^{C_{43}} & }](http://www.math.ua.ac.be/~lebruyn/latexrender/pictures/
ec5bfddac46a6eed7dbbd791918b9aed.gif "\xymatrix{ & \vtx{g_1}
\ar@/^/[ld]^{C_{16}} \ar@/^/[rd]^{C_{12}} & \\ \vtx{g_6}
\ar@/^/[ru]^{C_{61}} \ar@/^/[d]^{C_{65}} & & \vtx{g_2}
\ar@/^/[lu]^{C_{21}} \ar@/^/[d]^{C_{23}} \\ \vtx{g_5}
\ar@/^/[u]^{C_{56}} \ar@/^/[rd]^{C_{54}} & & \vtx{g_3}
\ar@/^/[u]^{C_{32}} \ar@/^/[ld]^{C_{34}} \\ & \vtx{g_4}
\ar@/^/[lu]^{C_{45}} \ar@/^/[ru]^{C_{43}} & }")
and we can
even identify how the small matrices 
fit
into the 
block-decomposition of the base-change matrix B
Hence, it makes sense
to call Q the non-commutative normal space to the isomorphism problem in
. Moreover, under this correspondence simple
representations of Q (for which both the dimension vectors and
distinguishing characters are known explicitly) correspond to simple
solutions in .
Having completed our promised
approach via non-commutative geometry to the classification problem of
solutions to the braid relation, it is time to collect what we have
learned. Let with 
, then for every
non-zero scalar 
the matrices
give a solution of size
n to the braid relation. Moreover, such a solution can be simple only if
the following numerical relations are satisfied
where indices are viewed
modulo 6. In fact, if these conditions are satisfied then a sufficiently
general representation of Q does determine a simple solution in 
and conversely, any sufficiently general simple n
size solution of the braid relation can be conjugated to one of the
above form. Here, by sufficiently general we mean a Zariski open (hence
dense) subset.
That is, for all integers n we have constructed
nearly all (meaning a dense subset) simple solutions to the braid
relation. As to the classification problem, if we have representants of
simple 
-dimensional representations of the quiver Q, then the corresponding
solutions 
of
the braid relation represent different orbits (up to finite overlap
coming from the fact that our linearizations only give an analytic
isomorphism, or in algebraic terms, an etale map). Such representants
can be constructed for low dimensional .
Finally, our approach also indicates why the classification of
braid-relation solutions of size 
is
easier : from size 6 on there are new classes of simple
Q-representations given by going round the whole six-cycle!