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Tag: geometry

writing

A long
time ago Don Passman
told me the simple “secret” for writing books : “Get up and,
before you do anything else, try to write 2 or 3 pages. If you do this
every day, by the end of the year you’ll have a pretty thick book.”

Probably the best advice ever for those who need to get a thesis or book
finished. I’ve managed to live by this rule for several months in a
row (the first half of 2000 leading to version 2 and the winter of 2001-2002
resulting in version 3) and I can recommend it to
anyone in need for some (self)dicipline. It feels just like training,
hard in the beginning but after a couple of weeks you’re addicted.
Also the pitfalls are similar. On certain days you have so much energy
that it is easy to write 10 or more pages (or in the revision process,
to revise 30 or more pages). Don’t do it! Tomorrow you will be
exhausted and you will not be able to do a single page but you will
convince yourself that it is not needed as you did more than enough the
day before. And you’ll feel and say the same thing the day after, and
the next day! and before you realize it you’ll be way behind
schedule. So, rule 1 : do 2 pages mimimun, 3 or 4 if possible but never
more than 5!

Another useful bit of advice comes from
Lewis Caroll’s ‘Through the looking glass’
in which the Red
King says

Start at the beginning, then continue until
you reach the end. Then stop.

Too many bookprojects
never get past the planning stages. It is much more fun to dream up the
perfect book than it is to write the first paragraph. Also, when the
writing on chapter X goes slow, it is tempting to begin with chapter X+1
or any other chapter that seems like more fun, and before you know
you’ll end up with a complete mess (and believe me, I know what I’m
talking about here).

Armed with these two guiding rules I began
the new year writing version pi of my book. (Oh, a marginal note : some
people seem to think that I set up ‘NeverEndingBooks’ to get my
book published. It may surely be the case that I’ll get _a_
book published there, but _the_ book I promised already a long
time ago to the EMS-publishing
house
! So, if you have an interesting bookproject for
‘NeverEndingBooks’ please contact us.) Anyway, the writing goes
slow! I’m already far behind schedule. So far I produced just over 20
pages! Part of the problem is that I want the book to be self-contained
and from past experiences with our ‘masterclass non-commutative
geometry’ I know that this means including a lot of elementary
material (it seems that sudents are eager on entering a masterclass on
non-commutative geometry without knowing the basics of either
non-commutative algebra or algebraic geometry). So. I started out with
believe it or not the definition of matrix-multiplication! But the book
has a pretty steap learning curve, by page 3 I’m already using
Grassmannians to classify left ideals in matrix-algebras! But I was
surprised how long it took me to come up with my own proofs of all this
‘trivial’ material. But the main problem is : lack of motivation.
I’m no longer convinced that one has to write technical books to aid
the younger generation. They are already far too technical!Perhaps it
would be far better to write books helping to develop creativity? But
how? And why are there so few of such books around. In fact, I know of
only one book trying to achieve this : An Invitation to General
Algebra and Universal Constructions
By George Bergman. His chapter 0
‘about the course and these notes’ comes very close to how I would
like to teach masterclass courses or how I’d love to write books if
only I’d know how. Perhaps, over the next couple of weeks, I’ll use
this weblog again to write up a micro-course on noncommutative geometry,
some people tell me they begin to miss the mathematics on this
site.

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A for aggregates

Let us
begin with a simple enough question : what are the points of a
non-commutative variety? Anyone? Probably you\’d say something like :
standard algebra-geometry yoga tells us that we should associate to a
non-commutative algebra $A$ on object, say $X_A$ and an arbitrary
variety is then build from \’gluing\’ such things together. Ok, but what
is $X_A$? Commutative tradition whispers $X_A=\mathbf{spec}~A$ the
[prime spectrum][1] of $A$, that is, the set of all twosided prime
ideals $P$ (that is, if $aAb \subset P$ then either $a \in P$ or $b \in
P$) and \’points\’ of $\mathbf{spec}~A$ would then correspond to
_maximal_ twosided ideals. The good news is that in this set-up, the
point-set comes equipped with a natural topology, the [Zariski
topology][2]. The bad news is that the prime spectrum is rarely
functorial in the noncommutative world. That is, if $\phi~:~A
\rightarrow B$ is an algebra morphism then $\phi^{-1}(P)$ for $P \in
\mathbf{spec}~B$ is not always a prime ideal of $A$. For example, take
$\phi$ the inclusion map $\begin{bmatrix} C[x] & C[x] \\ (x) & C[x]
\end{bmatrix} \subset \begin{bmatrix} C[x] & C[x] \\ C[x] & C[x]
\end{bmatrix}$ and $P$ the prime ideal $\begin{bmatrix} (x) & (x) \\ (x)
& (x) \end{bmatrix}$ then $P Cap \begin{bmatrix} C[x] & C[x] \\ (x) &
C[x] \end{bmatrix} = P$ but the corresponding quotient is
$\begin{bmatrix} C & C \\ 0 & C \end{bmatrix}$ which is not a prime
algebra so $\phi^{-1}(P)$ is not a prime ideal of the smaller algebra.
Failing this, let us take for $X_A$ something which obviously is
functorial and worry about topologies later. Take $X_A = \mathbf{rep}~A$
the set of all finite dimensional representations of $A$, that is
$\mathbf{rep}~A = \bigsqcup_n \mathbf{rep}_n~A$ where $\mathbf{rep}_n~A
= \{ Chi~:~A \rightarrow M_n(C)~\}$ with $Chi$ an algebra morphism. Now,
for any algebra morphism $\phi~:~A \rightarrow B$ there is an obvious
map $\mathbf{rep}~B \rightarrow \mathbf{rep}~A$ sending $Chi \mapsto Chi
Circ \phi$. Alernatively, $\mathbf{rep}_n~A$ is the set of all
$n$-dimensional left $A$-modules $M_{Chi} = C^n_{Chi}$ with $a.m =
Chi(m)m$. As such, $\mathbf{rep}~A$ is not merely a set but a
$C$-_category_, that is, all objects are $C$-vectorspaces and all
morphisms $Hom(M,N)$ are $C$-vectorspaces (the left $A$-module
morphisms). Moreover, it is an _additive_ category, that is if
$Chi,\psi$ are representations then we also have a direct sum
representation $Chi \oplus \psi$ defined by $a \mapsto \begin{bmatrix}
Chi(a) & 0 \\ 0 & \psi(a) \end{bmatrix}$. Returning at the task at
hand let us declare a _non-commutative variety_ $X$ to be (1) _an
additive_ $C$-_category_ which \’locally\’ looks like $\mathbf{rep}~A$
for some non-commutative algebra $A$ (even if we do not know at the
momemt what we mean by locally as we do not have defined a topology,
yet). Let is call objects of teh category $X$ the \’points\’ of our
variety and $X$ being additive allows us to speak of _indecomposable_
points (that is, those objects that cannot be written as a direct sum of
non-zero objects). By the local description of $X$ an indecomposable
point corresponds to an indecomposable representation of a
non-commutative algebra and as such has a local endomorphism algebra
(that is, all non-invertible endomorphisms form a twosided ideal). But
if we have this property for all indecomposable points,our category $X$
will be a Krull-Schmidt category so it is natural to impose also the
condition (2) : every point of $X$ can be decomposed uniquely into a
finite direct sum of indecomposable points. Further, as the space of
left $A$-module morphisms between two finite dimensional modules is
clearly finite dimensional we have also the following strong finiteness
condition (3) : For all points $x,y \in X$ the space of morphisms
$Hom(x,y)$ is a finite dimensional $C$-vectorspace. In their book
[Representations of finite-dimensional algebras][3], Peter Gabriel and
Andrei V. Roiter call an additive category such that all endomorphism
algebras of indecomposable objects are local algebras and such that all
morphism spaces are finite dimensional an _aggregate_. So, we have a
first tentative answer to our question **the points of a
non-commutative variety are the objects of an aggregate** Clearly, as
$\mathbf{rep}~A$ has stronger properties like being an _Abelian
category_ (that is, morphisms allow kernels and cokernels) it might also
be natural to replace \’aggregate\’ by \’Abelian Krull-Schmidt category
with finite dimensional homs\’ but if Mr. Abelian Category himself finds
the generalization to aggregates useful I\’m not going to argue about
this. Are all aggregates of the form $\mathbf{rep}~A$ or are there
other interesting examples? A motivating commutative example is : the
category of all coherent modules $Coh(Y)$ on a _projective_ variety $Y$
form an aggegate giving us a mental picture of what we might expect of a
non-commutative variety. Clearly, the above tentative answer cannot be
the full story as we haven\’t included the topological condition of
being locally of the form $\mathbf{rep}~A$ yet, but we will do that in
the next episode _B for Bricks_. [1]:
http://planetmath.org/encyclopedia/PrimeSpectrum.html [2]:
http://planetmath.org/encyclopedia/ZariskiTopology.html [3]:

1/ref=sr_1_8_1/026-3923724-4530018

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TheLibrary (version 2)

Just in time for 2005 : a pretty good approx of what I had in mind
with TheLibrary.
The major new feature is one-page view. That
is, if you click on a bookmark or search-page link you will get a page
(as in the screenshot below) consisting of three frames. The left-bottom
frame contains the pdf file of _just_ the requested page, hence
your brwoser no longer has to download the full pdf-file to get at the
wanted page which speeds up the process. The downside is that you can no
longer scroll in neighbouring pages. To compensate for this there are
_previous page_ and _next page_ buttons in the top frame
as well as a link to the index and search page of the document.

An added bonus of this set-up is that the author of a document can
control what readers can do with these pdf-pages. For example, the pages
of 3 talks on noncommutative geometry@n
admit all features (such as content-copy, merging pages, printing etc.)
whence a determined reader can reconstruct the full pdf-document if
he/she so desires. On the other hand, the pages of version 2 can only be printed at a low
resolution and those of version 3 do not even permit this.

The bottom-right frame of the pages allow the reader
to read (and post) marginal notes wrt. the content of the document-text
(such as : extra references, errors, suggestions etc.). As always,
comments are great; obnoxious comments get deleted. Deal!
Once
again, if you like your courses and or books (on a subject from either
non-commutative geometry or non-commutative algebra) to be included in
TheLibrary email.
All scripts are adapted from the original
scripts from pdf
hacks
.

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FAQ

This is the first version of a set of general frequently asked
questions relating to _NeverEndingBooks_. An extended and updated
version of it is at all times available from the sidebar under the
heading 'FAQ'. If you have a question and/or suggestion for
these FAQs, please email and it will be answered/used in an updated
version.

What is NeverEndingBooks ?
neverendingbooks.org is a
non-profit PublishingHouse specializing in courses and books on
non-commutative algebra and/or non-commutative geometry. Our authors
have a set of notes on a subject in na&g and hope to turn it into a
book one fine day. We offer them help throughout this process, from
secure on-screen viewing & commenting of their successive versions
to the production of a genuine hardcopy version distributed worldwide.
At all times we aim to keep the costs of our books minimal and the
royalties & copyrights for our authors maximal.
NeverEndingBooks' coordinates are :

NeverEndingBooks.org

c/o Lieven Le Bruyn
Department Mathematics UA
Middelheimlaan 1
B-2020 Antwerp, Belgium

URL : www.neverendingbooks.org
ISBN-prefix : 90-809390

Why NeverEndingBooks
?

We all complain about exaggerated prices of
mathematical books from certain publishers, poor quality of editing and
refereeing offered as well as far too stringent book-contracts. Rather
than lamenting about this, NeverEndingBooks gives itself one year to
learn (and report) about the many aspects of the book-production cyclus
and to explore whether an alternative exists. If at the end of this year
we will have produced at least one book this experiment will be
considered a success. If, however, we find out that it is an impossible
task, we will explain where it all went wrong and why it is better to
stick to an established PublishingHouse and accept its terms.

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TheLibrary (demo)

It is far from finished but you can already visit a demo-version of
TheLibrary which I hope will one day be a useful collection of
online courses and books on non-commutative algebra & geometry. At
the moment it just contains a few of my own things but I do hope that
others will find the format interesting enough to allow me to include
their courses and/or books. So, please try this demo out! But before you
do, make sure that you have a good webbrowser-plugin to view
PDF-documents from within your browser (rather than having to download
the files). If you are using Macintosh 10.3 or better there is a very
nice plugin freely
available whch you only have to drag into your _/Library/Internet
Plug-Ins/_-folder to get it working (after restarting Safari).
If you click on the title you will get a page with hyper-links to all
bookmarks of the pdf-file (for example, if you have used the hyperref package to
(La)TeX your file, you get these bookmarks for free). If you only have a
PDF-file you can always include the required bookmarks using Acrobat.
No doubt the most useful feature (at this moment) of the set-up is
that all files are fully searchable for keywords.
For example, if
you are at the page of my 3 talks on noncommutative
geometry@n
-course and fill out “Azumaya” in the Search
Document-field you will get a screen like the one below

That is, you wlll get all occurrences of 'Azumaya' in
the document together with some of the context as well as page- or
section-links nearby that you can click to get to the paragraph you are
looking for. In the weeks to come I hope to extend the usability of
_TheLibrary_ by offering a one-page view, modular security
enhancements, a commenting feature as well as a popularity count. But,
as always, this may take longer than I want…
If you think
that the present set-up might already be of interest to readers of your
courses or books and if you have a good PDF-file of it available
(including bookmarks) then email and we will try to include your
material!

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From Galois to NOG


Evariste Galois (1811-1832) must rank pretty high on the all-time
list of moving last words. Galois was mortally wounded in a duel he
fought with Perscheux d\’Herbinville on May 30th 1832, the reason for
the duel not being clear but certainly linked to a girl called
Stephanie, whose name appears several times as a marginal note in
Galois\’ manuscripts (see illustration). When he died in the arms of his
younger brother Alfred he reportedly said “Ne pleure pas, j\’ai besoin
de tout mon courage pour mourir ‚àö‚Ć 20 ans”. In this series I\’ll
start with a pretty concrete problem in Galois theory and explain its
elegant solution by Aidan Schofield and Michel Van den Bergh.
Next, I\’ll rephrase the problem in non-commutative geometry lingo,
generalise it to absurd levels and finally I\’ll introduce a coalgebra
(yes, a co-algebra…) that explains it all. But, it will take some time
to get there. Start with your favourite basefield $k$ of
characteristic zero (take $k = \mathbb{Q}$ if you have no strong
preference of your own). Take three elements $a,b,c$ none of which
squares, then what conditions (if any) must be imposed on $a,b,c$ and $n
\in \mathbb{N}$ to construct a central simple algebra $\Sigma$ of
dimension $n^2$ over the function field of an algebraic $k$-variety such
that the three quadratic fieldextensions $k\sqrt{a}, k\sqrt{b}$ and
$k\sqrt{c}$ embed into $\Sigma$? Aidan and Michel show in \’Division
algebra coproducts of index $n$\’ (Trans. Amer. Math. Soc. 341 (1994),
505-517) that the only condition needed is that $n$ is an even number.
In fact, they work a lot harder to prove that one can even take $\Sigma$
to be a division algebra. They start with the algebra free
product
$A = k\sqrt{a} \ast k\sqrt{b} \ast k\sqrt{c}$ which is a pretty
monstrous algebra. Take three letters $x,y,z$ and consider all
non-commutative words in $x,y$ and $z$ without repetition (that is, no
two consecutive $x,y$ or $z$\’s). These words form a $k$-basis for $A$
and the multiplication is induced by concatenation of words subject to
the simplifying relations $x.x=a,y.y=b$ and $z.z=c$.

Next, they look
at the affine $k$-varieties $\mathbf{rep}(n) A$ of $n$-dimensional
$k$-representations of $A$ and their irreducible components. In the
parlance of $\mathbf{geometry@n}$, these irreducible components correspond
to the minimal primes of the level $n$-approximation algebra $\int(n) A$.
Aidan and Michel worry a bit about reducedness of these components but
nowadays we know that $A$ is an example of a non-commutative manifold (a
la Cuntz-Quillen or Kontsevich-Rosenberg) and hence all representation
varieties $\mathbf{rep}n A$ are smooth varieties (whence reduced) though
they may have several connected components. To determine the number of
irreducible (which in this case, is the same as connected) components
they use _Galois descent
, that is, they consider the algebra $A
\otimes_k \overline{k}$ where $\overline{k}$ is the algebraic closure of
$k$. The algebra $A \otimes_k \overline{k}$ is the group-algebra of the
group free product $\mathbb{Z}/2\mathbb{Z} \ast \mathbb{Z}/2\mathbb{Z}
\ast \mathbb{Z}/2\mathbb{Z}$. (to be continued…) A digression : I
cannot resist the temptation to mention the tetrahedral snake problem
in relation to such groups. If one would have started with $4$ quadratic
fieldextensions one would get the free product $G =
\mathbb{Z}/2\mathbb{Z} \ast \mathbb{Z}/2\mathbb{Z} \ast
\mathbb{Z}/2\mathbb{Z} \ast \mathbb{Z}/2\mathbb{Z}$. Take a supply of
tetrahedra and glue them together along common faces so that any
tertrahedron is glued to maximum two others. In this way one forms a
tetrahedral-snake and the problem asks whether it is possible to make
such a snake having the property that the orientation of the
\’tail-tetrahedron\’ in $\mathbb{R}^3$ is exactly the same as the
orientation of the \’head-tetrahedron\’. This is not possible and the
proof of it uses the fact that there are no non-trivial relations
between the four generators $x,y,z,u$ of $\mathbb{Z}/2\mathbb{Z} \ast
\mathbb{Z}/2\mathbb{Z} \ast \mathbb{Z}/2\mathbb{Z} \ast
\mathbb{Z}/2\mathbb{Z}$ which correspond to reflections wrt. a face of
the tetrahedron (in fact, there are no relations between these
reflections other than each has order two, so the subgroup generated by
these four reflections is the group $G$). More details can be found in
Stan Wagon\’s excellent book The Banach-tarski paradox, p.68-71.

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neverendingbooks.org

There is a nice, cosy 2nd hand book shop Never Ending
Books
located at 308 Hibiscus Highway, Orewa Beach (New Zealand).
Further, someone registered the domain-name www.neverendingbooks.com and
isn't doing a thing with it at this moment. And that's about it!

As this site will be a strictly non-profit set-up, it made sense
to register the domain-name www.neverendingbooks.org
instead. Partly because many of you seem to find www.matrix.ua.ac.be way too
difficult to remember (judging from the number of times people end up
here Googling _lieven le bruyn_). Unfortunately, registering the
domain-name is the only of three urgent goals I set myself that actually
panned out so far (the other two, _getting a prefix_ and
_partnering up_, won't mean much to you and I'll explain
them later when (if) they work out).
Over the next couple of
weeks it will become gradually clear what this site is all about.
I've worked out things (in theory) over several sleepless nights,
but making them happen will require a lot of extra work.
Oh, you
don't believe I did think some things through? Have a look at the
new header-picture. Recognize those eyes? If you do, you will agree that
this choice was almost forced upon me as I wanted to capture at the same
time the _non-commutative-algebra_, the _non-commutative
geometry_ as well as the _neverending_ aspect of this
site…

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quintominal dodecahedra


A _quintomino_ is a regular pentagon having all its sides
colored by five different colours. Two quintominoes are the same if they
can be transformed into each other by a symmetry of the pentagon (that
is, a cyclic rotation or a flip of the two faces). It is easy to see
that there are exactly 12 different quintominoes. On the other hand,
there are also exactly 12 pentagonal faces of a dodecahedron
whence the puzzling question whether the 12 quintominoes can be joined
together (colours mathching) into a dodecahedron.
According to
the Dictionnaire de
mathematiques recreatives
this can be done and John Conway found 3
basic solutions in 1959. These 3 solutions can be found from the
diagrams below, taken from page 921 of the reprinted Winning Ways for your Mathematical
Plays (volume 4)
where they are called _the_ three
quintominal dodecahedra giving the impression that there are just 3
solutions to the puzzle (up to symmetries of the dodecahedron). Here are
the 3 Conway solutions

One projects the dodecahedron down from the top face which is
supposed to be the quintomino where the five colours red (1), blue (2),
yellow (3), green (4) and black(5) are ordered to form the quintomino of
type A=12345. Using the other quintomino-codes it is then easy to work
out how the quintominoes fit together to form a coloured dodecahedron.

In preparing to explain this puzzle for my geometry-101 course I
spend a couple of hours working out a possible method to prove that
these are indeed the only three solutions. The method is simple : take
one or two of the bottom pentagons and fill then with mathching
quintominoes, then these more or less fix all the other sides and
usually one quickly runs into a contradiction.
However, along the
way I found one case (see top picture) which seems to be a _new_
quintominal dodecahedron. It can't be one of the three Conway-types
as the central quintomino is of type F. Possibly I did something wrong
(but what?) or there are just more solutions and Conway stopped after
finding the first three of them…
Update (with help from
Michel Van den Bergh
) Here is an elegant way to construct
'new' solutions from existing ones, take a permutation $\\sigma
\\in S_5$ permuting the five colours and look on the resulting colored
dodecahedron (which again is a solution) for the (new) face of type A
and project from it to get a new diagram. Probably the correct statement
of the quintominal-dodecahedron-problem is : find all solutions up to
symmetries of the dodecahedron _and_ permutations of the colours.
Likely, the 3 Conway solutions represent the different orbits under this
larger group action. Remains the problem : to which orbit belongs the
top picture??

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changes

Tomorrow
I’ll give my last class of the semester (year?) so it is about time to
think about things to do (such as preparing the courses for the
“master program on noncommutative geometry”) and changes to make to
this weblog (now that it passed the 25000 mark it is time for something
different). In the sidebar I’ve added a little poll to let you guess
what changes 2005 will bring to this blog (if I find the time over
Christmas to implement it). In short, @matrix will
become the portal of a little company I’ll start up (seems
_the_ thing to do now). Here are some possible names/goals. Which
one will it be? Vote and find out after Christmas.

WebMathNess is a Web-service company helping lazy
mathematicians to set up their website and make it LaTeXRender savvy
(free restyling every 6 months).

iHomeEntertaining is a
Tech-company helping Mac-families to get most out of their valuable
computers focussing on Audio-Photo-Video streaming along their Airport-network.

SnortGipfGames is a Game-company focussing on the
mathematical side of the Gipf project
games
by distributing Snort-versions of them.

NeverendingBooks is a Publishing-company specializing
in neverending mathematical course- and book-projects offering their
hopeless authors print on demand and eprint services.

QuiverMerch is a Merchandising-company specializing in
quivers. For example, T-shirts with the tame quiver classification,
Calogero-Moser coffee mugs, Lego-boxes to construct local quivers
etc.

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Scottish solids

John McKay
pointed me to a few interesting links on ‘Platonic’ solids and monstrous
moonshine. If you thought that the ancient Greek discovered the five
Platonic solids, think again! They may have been the first to give a
correct proof of the classification but the regular solids were already
known in 2000BC as some
neolithic stone artifacts
discovered in Scotland show. These
Scottish solids can be visited at the Ashmolean Museum in Oxford. McKay
also points to the paper Polyhedra in physics,
chemistry and geometry
by Michael Atiyah and Paul Sutcliffe. He also
found my posts on a talk I gave on monstrous moonshine for 2nd year students earlier this year and
mentionted a few errors and updates. As these posts are on my old weblog
I’ll repost and update them here soon. For now you can already hear and
see a talk given by John McKay himself 196884=1+196883, a monstrous tale at the Fields Institute.

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