Tag: geometry

music of the primes

Published March 28, 2006 by lievenlb

Let me
admit it : i was probably wrong in this post to
advise against downloading A walk in the noncommutative
garden by Alain Connes and Matilde Marcolli. After all, it seems
that Alain&Matilde are on the verge of proving the biggest open
problem in mathematics, the Riemann
hypothesis using noncommutative geometry. At least, this is the
impression one gets from reading through The music of the
primes, why an unsolved problem in mathematics matters by Oxford
mathematician Prof.
Marcus du Sautoy… At the moment I’ve only read the first
chapter (_Who wants to be a millionaire?_) and the final two
chapters (_From orderly zeros to quantum chaos_ and _The
missing piece of the jigsaw_) as I assume I’ll be familiar with most
of the material in between (and also, I’m saving these chapters for some
vacation reading). From what I’ve read, I agree most with the final
review at amazon.co.uk

Fascinating
and infuriating, October 5, 2004
Reviewer: pja_jennings
from Southampton, Hants. United Kingdom
This is a book I found
fascinating and infuriating in turns. It is an excellent layman’s
history of number theory with particular reference to prime numbers and
the Riemann zeta function. As such it is well worth the reading.
However I found that there are certain elements, more of style than
anything else, that annoyed me. Most of the results are handed to us
without any proof whatsoever. All right, some of these proofs would be
obviously well beyond the layman, but one is described as being
understandable by the ancient Greeks (who started the whole thing) so
why not include it as a footnote or appendix?
Having established
fairly early on that the points where a mathematical function
“reaches sea level” are known as zeros, why keep reverting
to the sea level analogy? And although the underlying theme throughout
the book is the apparent inextricable link between the zeta function’s
zeros and counting primes, the Riemann hypothesis, I could find no
clear, concise statement of exactly what Riemann said.
Spanning
over 2000 years, from the ancient Greeks to the 21st century, this is a
book I would thoroughly recommend.

Books on Fermat’s last
theorem (and there are some nice ones, such as Alf Van der Poorten’s
Notes on
Fermat’s last theorem) can take Wiles’ solution as their focal
point. Failing a solution, du Sautoy constructs his book around an
April’s Fool email-message by Bombieri in which he claimed that a young
physicist did prove the Riemann hypothesis after hearing a talk by Alain
Connes. Here’s du Sautoy’s account (on page 3)

According
to his email, Bombieri has been beaten to his prize. ‘There are
fantastic developments to Alain Connes’s lecture at IAS last wednesday.’
Bombieri began. Several years previously, the mathematical world had
been set alight by the news that Alain Connes had turned his attention
to trying to crack the Riemann Hypothesis. Connes is one of the
revolutionaries of the subject, a benign Robespierre of mathematics to
Bombieri’s Louis XVI. He is an extraordinary charismatic figure whose
fiery style is far from the image of the staid, awkward mathematician.
He has the drive of a fanatic convinced of his world-view, and his
lectures are mesmerising. Amongst his followers he has almost cult
status. They will happily join him on the mathematical barricades to
defend their hero against any counter-offensive mounted from the ancien
regime’s entrenched positions.

Contrary to physics,
mathematics doesn’t produce many books aimed at a larger public. To a
large extend this is caused by most mathematicians’ unwillingness to
sacrifice precision and technical detail. Hence, most of us would never
be able to come up with something like du Sautoy’s description of Weil’s
work on the zeta function of curves over finite fields (page 295)

It was while exploring some of these related landscapes that
Weil discovered a method that would explain why points at sea level in
them like to be in a straight line. The landscapes where Weil was
successful did not have to do with prime numbers, but held the key to
counting how many solutions an equation such as $y^2=x^3-x$ will have if
you are working on one of Gauss’s clock calculators.

But,
it is far too easy to criticize people who do want to make the effort.
Books such as this one will bring more young people to mathematics than
any high-publicity-technical-paper. To me, the chapter on quantum chaos
was an eye-opener as I hadn’t heard too much about all of this before.
Besides, du Sautoy accompanies this book with an interesting website musicofprimes and several of
his articles for newspapers available from his homepage are
a good read (in case you wonder why the book-cover is full of joggers
with a prime number on their T-shirt, you might have a look at Beckham in his
prime number). The music of the
primes will definitely bring many students to noncommutative
geometry and its possible use to proving the Riemann Hypothesis.

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why nag? (1)

Published March 26, 2006 by lievenlb

Let us
take a hopeless problem, motivate why something like non-commutative
algebraic geometry might help to solve it, and verify whether this
promise is kept.

Suppose we want to know all solutions in invertible
matrices to the braid relation (or Yang-Baxter equation)

$X Y X = Y X Y$

All such solutions (for varying size of matrices)
form an additive Abelian category $\mathbf{rep}~B_3$ , so a big step forward would be to know all its
simple solutions (that is, those whose matrices cannot be brought in
upper triangular block form). A literature check shows that even this
task is far too ambitious. The best result to date is the classification
due to Imre Tuba and
Hans Wenzl of simple solutions of which the matrix size is at most
5.

For fixed matrix size n, finding solutions in $\mathbf{rep}~B_3$ is the same as solving a system of $n^2$ cubic
polynomial relations in $2n^2$
unknowns, which quickly becomes a daunting task. Algebraic geometry
tells us that all solutions, say $\mathbf{rep}_n~B_3$ form an affine closed subvariety of $n^2$ -dimensional affine space. If we assume that $\mathbf{rep}_n~B_3$ is a smooth variety (that is, a manifold) and
if we know one solution explicitly, then we can use the tangent space in
this point to linearize the problem and to get at all solutions in a
neighborhood.

So, here is an idea : assume that $\mathbf{rep}~B_3$ itself would be a non-commutative manifold, then
we might linearize our problem by considering tangent spaces and obtain
new solutions out of already known ones. But, what is a non-commutative
manifold? Well, by the above we at least require that for all integers n
the commutative variety $\mathbf{rep}_n~B_3$ is a commutative manifold.

But, there
is still some redundancy in our problem : if $(X,Y)$ is a
solution, then so is any conjugated pair $(g^{-1}Xg,g^{-1}Yg)$ where $g \in GL_n$ is a basechange matrix. In categorical terms, we are only
interested in isomorphism classes of solutions. Again, if we fix the
size n of matrix-solutions, we consider the affine variety $\mathbf{rep}_n~B_3$ as a variety with a $GL_n$ -action
and we like to classify the orbits of simple solutions. If $\mathbf{rep}_n~B_3$ is a manifold then the theory of Luna slices
provides a method, both to linearize the problem as well as to reduce
its complexity. Instead of the tangent space we consider the normal
space N to the $GL_n$ -orbit
(in a suitable solution). On this affine space, the stabilizer subgroup
$GL(\alpha)$ acts and there is a natural one-to-one
correspondence between $GL_n$ -orbits
in $\mathbf{rep}_n~B_3$ and $GL(\alpha)$ -orbits in the normal space N (at least in a
neighborhood of the solution).

So, here is a refinement of the
idea : we would like to view $\mathbf{rep}~B_3$ as a non-commutative manifold with a group action
given by the notion of isomorphism. Then, in order to get new isoclasses
of solutions from a constructed one we want to reduce the size of our
problem by considering a linearization (the normal space to the orbit)
and on it an easier isomorphism problem.

However, we immediately
encounter a problem : calculating ranks of Jacobians we discover that
already $\mathbf{rep}_2~B_3$ is not a smooth variety so there is not a
chance in the world that $\mathbf{rep}~B_3$ might be a useful non-commutative manifold.
Still, if $(X,Y)$ is a
solution to the braid relation, then the matrix $(XYX)^2$
commutes with both X and Y.

If $(X,Y)$ is a
simple solution, this means that after performing a basechange, $C=(XYX)^2$ becomes a scalar matrix, say $\lambda^6 1_n$ . But then, $(X_1,Y_1) = (\lambda^{-1}X,\lambda^{-1}Y)$ is a solution to

$XYX = YXY , (XYX)^2 = 1$

and all such solutions form a
non-commutative closed subvariety, say $\mathbf{rep}~\Gamma$ of $\mathbf{rep}~B_3$ and if we know all (isomorphism classes of)
simple solutions in $\mathbf{rep}~\Gamma$ we have solved our problem as we just have to
bring in the additional scalar $\lambda \in \mathbb{C}^*$ .

Here we strike gold : $\mathbf{rep}~\Gamma$ is indeed a non-commutative manifold. This can
be seen by identifying $\Gamma$
with one of the most famous discrete infinite groups in mathematics :
the modular group $PSL_2(\mathbb{Z})$ . The modular group acts by Mobius
transformations on the upper half plane and this action can be used to
write $PSL_2(\mathbb{Z})$ as the free group product $\mathbb{Z}_2 \ast \mathbb{Z}_3$ . Finally, using
classical representation theory of finite groups it follows that indeed
all $\mathbf{rep}_n~\Gamma$ are commutative manifolds (possibly having
many connected components)! So, let us try to linearize this problem by
looking at its non-commutative tangent space, if we can figure out what
this might be.

Here is another idea (or rather a dogma) : in the
world of non-commutative manifolds, the role of affine spaces is played
by $\mathbf{rep}~Q$ the representations of finite quivers Q. A quiver
is just on oriented graph and a representation of it assigns to each
vertex a finite dimensional vector space and to each arrow a linear map
between the vertex-vector spaces. The notion of isomorphism in $\mathbf{rep}~Q$ is of course induced by base change actions in all
of these vertex-vector spaces. (to be continued)

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MyLife@300dpi

Published March 20, 2006 by lievenlb

Three years ago I did spend three weeks next to my Canonscan, painstakingly scanning all individual pages of every preprint I ever wrote. Next, I converted every page to PDF, resized it (in order to control the size) and bundled them into PDF-files. A typical preprint would take me roughly three quarters of an hour and the final result was mediocre. For example, here a blown-up sample from the original 1992 ‘Moduli
spaces of right ideals of the Weyl algebra’ -preprint, resulting in a 1.7Mb PDF-file

Recentlty, the department bought a Ricoh-copier which makes scanning a lot more fun. To scan a preprint at 300dpi and convert it into a single PDF-file takes under a minute (actually, downloading the file using a web-interface takes longer…). For this particular preprint, the resulting PDF-file took up 1.2Mb and looks a lot nicer

Still, 1.2Mb is a huge file but converting it to a DjVu-file (DjVu=deja vu) using the handy Any2DjVu Service gives us a mere 236Kb file which comes a lot closer to the filesize of a PDFLaTeX-file and the output is still very legible

So, I decided to rescan my entire life at 300dpi and convert it into DjVu. Next, I got the MOPP-package (MOPP = My Online Publications Page) working using the instructions from this page and some obvious MacOSX-modifications (if I can do it, so can you but perhaps I’ll write up the details in another post, just to remind myself). You can see the result at my homepage. I’ll update the latter one regularly (there are still some preprints missing, as are all my courses etc. and cross-references) and only afterwards I’ll update my homepage again. So far there is 250Mb to download (including all versions of the noncommutative geometry@n book, including the published ones…) so this should keep you busy for a while…

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get your brain subscribed to

Published February 8, 2006 by lievenlb

In the
‘subscribe
to my brain’ post I promised to blog on how-to get your own

button up and running on your homepage. It seems rather unlikely
that I’ll ever keep that promise if I don’t do it right away. So, here
we go for a quick tour :

step 1 : set up a rudimentary
FoaF-file : read the FoaF post if
you dont know what it’s all about. The easiest way to get a simple
FoaF-file of your own is to go to the FoaF-a-matic
webpage and fill in the details you feel like broadcasting over the
web, crucial is your name and email information (for later use) but
clearly the more details you fill out and the more Friends you add the
more useful your file becomes. Click on the ‘foaf-me’ button and
copy the content created. Observe that there is no sign of my email
adress, it is encrypted in the _mbox_sha1sum_ data. Give this
file a name like _foaf.rdf_ or _myname.rdf_ and put it on
your webserver to make it accessible. Also copy your
_mbox_sha1sum_ info for later smushing.

step 2 : subscribe to online services and modify your
online-life accordingly : probably you have already a few of
these accounts, but if not, take a free subscription just for fun and
(hopefully) later usage to the following sites :

del.icio.us a social bookmarks manager
citeUlike a service to
organise your academic papers
connotea a reference management
service for scientists
bloglines a web-based personal news
aggregator
43things a
‘What do you want to do with your life?’ service
audioscrobbler a database that
tracks listening habits and does wonderful things with statistics
backpackit a ‘be better organized’ service (Update october 2017 : Tom Howard emails: “I thought I’d reach out because we’ve just updated our guide which reviews the best alternatives to Backpack. Here’s the link”
flickr an online photo management and
sharing application
technorati a Google-for-weblogs
upcoming a social event
calendar
webjay a playlist
community

So far, I’m addicted to del.icio.us and use
citeUlike but hardly any of the others (but I may come back to this
later). The great thing about these services is that you get more
value-information back if you feed more into the system. For example, if
you use del.icio.us as your ‘public’ bookmarks-file you get to
know how many other people have bookmarked the same site and you can
access their full bookmarks which often is a far more sensible way to
get at the information you are after than mindless Googling. So, whereas
I was at first a bit opposed to the exhibisionist-character of these
services (after all, anyone with web-access can have a look at
‘your’ info), I’ve learned that the ‘social’ feature of
these services can be beneficial to get the right information I want.
Hence, the hardest part is not to get an account with these services but
to adopt your surfing behavior in such a way that you maximize this
added value. And, as I mentioned before, I’m doing badly myself but hope
that things will improve…

step 3 : turn these
accounts into an OPML file : Knowing the URL of your foaf-file
and sha1-info (step 1) and your online accounts, go to the FOAF Online Account
Description Generator and feed it with your data. You will then get
another foaf-file back (save the source in a file such as
_accounts.rdf_ and put it on your webserver). Read the Lost Boy’s
posts Subscribe to my
brain and foaf:
OnlineAccount Generator for more background info. Then, use the SubscribeToMyBrain-
form to get an OPML-file out of the account.rdf file and your sha1.
Save the source as _mybrain.opml_.

step 4 :
add/delete information you want : The above method uses generic
schemes to deduce relevant RSS-data from an account name, which works
for some services, but doesn’t for all. So, if you happen to know the
URL of RSS-feeds for one of these services, you can always add it
manually to the OPML-file (or delete data you don’t want to
publish…). My own attitude is to make all public web-data
available and to leave it to the subscriber to unsubscribe those parts
of my brain (s)he is not interested in. I know there are people whoo are
mainly interested to find out whether I put another paper online, would
tolerate some weblog-posts but have no interest in my musical tast,
whereas there are others who would like me to post more on 43things,
flickr or upcoming and don’t give a damn about my mathematics…
Apart from these online subscriptions, it is also a good idea to include
additional RSS-feeds you produce, such as those of your weblog or use my
Perl
script to have your own arXiv-feeds.

step 5 : make
your ‘subscribe to my brain’-button : Now, put the
OPML-file on your webserver, put the button

on your
homepage and link it to the file. Also, add information on your site,
similar to the one I gave in my own
subscription post so that your readers know what to do when do want
to subscribe to (parts of) your brain. Finally, (and optionally though
I’d wellcome it) send me an email with your URL so that I can subscribe
(next time you’re in Antwerp I’ll buy you a beer) and for the first few
who do so and are working in noncommutative geometry and/or
noncommutative algebra, I’ll send a copy of a neverending book. Mind
you, this doesn’t apply to local people, I’m already subscribed to their
brain on a daily basis…

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a noncommutative topology 2

Published February 1, 2006 by lievenlb

A *qurve*
is an affine algebra such that $~\Omega^1~A$ is a projective
$~A~$-bimodule. Alternatively, it is an affine algebra allowing lifts of
algebra morphisms through nilpotent ideals and as such it is the ‘right’
noncommutative generalization of Grothendieck’s smoothness criterium.
Examples of qurves include : semi-simple algebras, coordinate rings of
affine smooth curves, hereditary orders over curves, group algebras of
virtually free groups, path algebras of quivers etc.

Hence, qurves
behave a lot like curves and as such one might hope to obtain one day a
‘birational’ classification of them, if we only knew what we mean
by this. Whereas the etale classification of them is understood (see for
example One quiver to
rule them all or Qurves and quivers )
we don’t know what the Zariski topology of a qurve might be.

Usually, one assigns to a qurve $~A~$ the Abelian category of all its
finite dimensional representations $\mathbf{rep}~A$ and we would like to
equip this set with a topology of sorts. Because $~A~$ is a qurve, its
scheme of n-dimensional representations $\mathbf{rep}_n~A$ is a smooth
affine variety for each n, so clearly $\mathbf{rep}~A$ being the disjoint
union of these acquires a trivial but nice commutative topology.
However, we would like open sets to hit several of the components
$\mathbf{rep}_n~A$ thereby ‘connecting’ them to form a noncommutative
topological space associated to $~A~$.

In a noncommutative topology on
rep A I proposed a way to do this and though the main idea remains a
good one, I’ll ammend the construction next time. Whereas we don’t know
of a topology on the whole of $\mathbf{rep}~A$, there is an obvious
ordinary topology on the subset $\mathbf{simp}~A$ of all simple finite
dimensional representations, namely the induced topology of the Zariski
topology on $~\mathbf{spec}~A$, the prime spectrum of twosided prime ideals
of $~A~$. As in commutative algebraic geometry the closed subsets of the
prime spectrum consist of all prime ideals containing a given twosided
ideal. A typical open subset of the induced topology on $\mathbf{simp}~A$
hits many of the components $\mathbf{rep}_n~A$, but how can we extend it to
a topology on the whole of the category $\mathbf{rep}~A$ ?

Every
finite dimensional representation has (usually several) Jordan-Holder
filtrations with simple successive quotients, so a natural idea is to
use these filtrations to extend the topology on the simples to all
representations by restricting the top (or bottom) of the Jordan-Holder
sequence. Let W be the set of all words w such as $U_1U_2 \ldots U_k$
where each $U_i$ is an open subset of $\mathbf{simp}~A$. We can now define
the *left basic open set* $\mathcal{O}_w^l$ consisting of all finite
dimensional representations M having a Jordan-Holder sequence such that
the i-th simple factor (counted from the bottom) belongs to $U_i$.
(Similarly, we can define a *right basic open set* by counting from the
top or a *symmetric basic open set* by merely requiring that the simples
appear in order in the sequence). One final technical (but important)
detail is that we should really consider equivalence classes of left
basic opens. If w and w’ are two words we will denote by $\mathbf{rep}(w
\cup w’)$ the set of all finite dimensional representations having a
Jordan-Holder filtration with enough simple factors to have one for each
letter in w and w’. We then define $\mathcal{O}^l_w \equiv
\mathcal{O}^l_{w’}$ iff $\mathcal{O}^l_w \cap \mathbf{rep}(w \cup w’) =
\mathcal{O}^l_{w’} \cap \mathbf{rep}(w \cup w’)$. Equivalence classes of
these left basic opens form a partially ordered set (induced by
set-theoretic inclusion) with a unique minimal element 0 (the empty set
corresponding to the empty word) and a uunique maximal element 1 (the
set $\mathbf{rep}~A$ corresponding to the letter $w=\mathbf{simp}~A$).
Set-theoretic union induces an operation $\vee$ and the operation
$~\wedge$ is induced by concatenation of words, that is,
$\mathcal{O}^l_w \wedge \mathcal{O}^l_{w’} \equiv \mathcal{O}^l_{ww’}$.
This then defines a **left noncommutative topology** on $\mathbf{rep}~A$ in
the sense of Van Oystaeyen (see [part
1](http://www.neverendingbooks.org/index.php/noncommutative-topology-1 $
). To be precise, it satisfies the axioms in the left and middle column
of the following picture and
similarly, the right basic opens give a right noncommutative topology
(satisfying the axioms of the middle and right columns) whereas the
symmetric opens satisfy all axioms giving the basis of a noncommutative
topology. Even for very simple finite dimensional qurves such as
$\begin{bmatrix} \mathbb{C} & \mathbb{C} \\ 0 & \mathbb{C}
\end{bmatrix}$ this defines a properly noncommutative topology on the
Abelian category of all finite dimensional representations which
obviously respect isomorphisms so is really a noncommutative topology on
the orbits. Still, while this may give a satisfactory local definition,
in gluing qurves together one would like to relax simple representations
to *Schurian* representations. This can be done but one has to replace
the topology coming from the Zariski topology on the prime spectrum by
the partial ordering on the *bricks* of the qurve, but that will have to
wait until next time…

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noncommutative topology (1)

Published January 29, 2006 by lievenlb

A couple of days ago Ars Mathematica had a post Cuntz on noncommutative topology pointing to a (new, for me) paper by Joachim Cuntz

A couple of years ago, the Notices of the AMS featured an article on noncommutative geometry a la Connes: Quantum Spaces and Their Noncommutative Topology by Joachim Cuntz. The hallmark of this approach is the heavy reliance on K theory. The first few pages of the article are fairly elementary (and full of intriguing pictures), before the K theory takes over.

A few comments are in order. To begin, the paper is **not** really about noncommutative geometry a la Connes, but rather about noncommutative geometry a la Cuntz&Quillen (based on quasi-free algebras) or, equivalently, a la Kontsevich (formally smooth algebras) or if I may be so bold a la moi (qurves).

About the **intruiging pictures** : it seems to be a recent trend in noncommutative geometry research papers to include meaningless pictures to lure the attention of the reader. But, unlike aberrations such as the recent pastiche by Alain Connes and Mathilde Marcolli A Walk in the Noncommutative Garden, Cuntz is honest about their true meaning

I am indebted to my sons, Nicolas and Michael,
for the illustrations to the examples above. Since
these pictures have no technical meaning, they
are only meant to provide a kind of suggestive
visualization of the corresponding quantum spaces.

As one of these pictures made it to the cover of the **Notices** an explanation was included by the cover-editor

About the Cover :

The image on this month’s cover arose from
Joachim Cuntz’s effort to render into visible art
his own internal vision of a noncommutative
torus, an object otherwise quite abstract. His
original idea was then implemented by his son
Michael in a program written in Pascal. More
explicitly, he says that the construction started
out with a triangle in a square, then translated
the triangle by integers times a unit along a line
with irrational slope; plotted the images thus
obtained in a periodic manner; and stopped
just before the figure started to seem cluttered.
Many mathematicians carry around inside
their heads mental images of the abstractions
they work with, and manipulate these objects
somehow in conformity with their mental imagery. They probably also make aesthetic judgements of the value of their work according to
the visual qualities of the images. These presumably common phenomena remain a rarely
explored domain in either art or psychology.

—Bill Casselman(covers@ams.org)

There can be no technical meaning to the pictures as in the Connes and Cuntz&Quillen approach there is only a noncommutative algebra and _not_ an underlying geometric space, so there is no topology, let alone a noncommutative topology. Of course, I do understand why Cuntz&others name it as such. They view the noncommutative algebra as the ring of functions on some virtual noncommutative space and they compute topological invariants (such as K-groups) of the algebras and interprete them as information about the noncommutative topology of these virtual and unspecified spaces.

Still, it is perfectly possible to associate to a qurve (aka quasi-free algebra or formally smooth algebra) a genuine noncommutative topological space. In this series of posts I’ll explain the little I know of the history of this topic, the thing I posted about it a couple of years ago, why I abandoned the project and the changes I made to it since and the applications I have in mind, both to new problems (such as the birational_classification of qurves) as well as classical problems (such as rationality problems for $PGL_n $ quotient spaces).

Although others have tried to define noncommutative topologies before, I learned about them from Fred Van Oystaeyen. Fred spend the better part of his career constructing structure sheaves associated to noncommutative algebras, mainly to prime Noetherian algebras (the algebras of preference for the majority of non-commutative algebraists). So, suppose you have an ordinary (meaning, the usual commutative definition) topological space X associated to this algebra R, he wants to define an algebra of sections on every open subset $X(\sigma) $ by taking a suitable localization of the algebra $Q_{\sigma}(R) $. This localization is taken with respect to a suitable filter of left ideals $\mathcal{L}(\sigma) $ of R and is defined to be the subalgebra of the classiocal quotient ring $Q(R) $ (which exists because $R$ is prime Noetherian in which case it is a simple Artinian algebra)

$Q_{\sigma}(R) = { q \in Q(R)~|~\exists L \in \mathcal{L}(\sigma)~:~L q \subset R } $

(so these localizations are generalizations of the usual Ore-type rings of fractions). But now we come to an essential point : if we want to glue this rings of sections together on an intersection $X(\sigma) \cap X(\tau) $ we want to do this by ‘localizing further’. However, there are two ways to do this, either considering $~Q_{\sigma}(Q_{\tau}(R)) $ or considering $Q_{\tau}(Q_{\sigma}(R)) $ and these two algebras are only the same if we impose fairly heavy restrictions on the filters (or on the algebra) such as being compatible.

As this gluing property is essential to get a sheaf of noncommutative algebras we seem to get stuck in the general (non compatible) case. Fred’s way out was to make a distinction between the intersection $X_{\sigma} \cap X_{\tau} $ (on which he put the former ring as its ring of sections) and the intersection $X_{\tau} \cap X_{\sigma} $ (on which he puts the latter one). So, the crucial new ingredient in a noncommutative topology is that the order of intersections of opens matter !!!

Of course, this is just the germ of an idea. He then went on to properly define what a noncommutative topology (and even more generally a noncommutative Grothendieck topology) should be by using this localization-example as guidance. I will not state the precise definition here (as I will have to change it slightly later on) but early version of it can be found in the Antwerp Ph.D. thesis by Luc Willaert (1995) and in Fred’s book Algebraic geometry for associative algebras.

Although _qurves_ are decidedly non-Noetherian (apart from trivial cases), one can use Fred’s idea to associate a noncommutative topological space to a qurve as I will explain next time. The quick and impatient may already sneak at my old note a non-commutative topology on rep A but please bear in mind that I changed my mind since on several issues…

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upgrade to wp 2.0

Published January 24, 2006 by lievenlb

All
activity on this site this week (apart from changing the theme) was done
behind the scenes. Finally, _neverendingbooks_ is upgraded to WordPress 2.0.
It is a straightforward well-explained procedure but somehow I decided
to try this out in between a WorkShop and a
Ph.D. defense. As a consequence I had to reclone twice…
Some of the Plugins‘ functionality
didn’t survive the upgrade. In particular, the anti-spam plugin BotCheck doesn’t work any longer (one could fill out any code and
still get a reply posted) as I found out sunday-morning when I was
greeted with about 20 spam-replies… Fortunately, WP 2.0 comes
bundled with its own anti-spam plugin Akismet but one needs a WordPress.com API key which
meens signing up to a WordPress-account (free). When Akismet is
activated, it really bans all spam (it even shows how many spam-messages
it found, 30 over two days…), the only problem being that it seems
to de-activate itself at random… The new theme is called Kiwi which is a lot more
compact than the default neverending(sic) page. But there is a (heavy
some will say) price to pay : only summaries of posts are on the
front-page and the font is (too some will say) small. Still, Kiwi has
some nice extra features : the Featured Post Plugin which
allows to re-cycle changing selected old posts to the right of the
banner. Another changing part is the _Elsewhere_ list (second row
to the right) where one can display any feed. At the moment (but I may
change this as the del.icio.us site
seems to be having some problems) all _del.icio.us_ links tagged
noncommutative are shown (if the site is up…). It
appears that apart from Graham
Leuschke nobody has a del.icio.us account or doesn’t use the
noncommutative tag. So, if you want to change this site a bit every day,
you know what to do. Speaking of tags, several new
_categories_ were created so that posts now get multiples tags,
describing better their (intended) content. Something I learned by
tagging papers at citeUlike. Btw.
you are still invited to join the
NoncommutativeGeometry Group over there… Clearly, re-tagging
every individual post was a painstaking experience. A WordPress 2.0
feature I like is the ability to write _pages_ (as opposed to
Posts) which are kept alive in the sidebar and therefore resemble
‘stickies’ (in WP parlace ‘they live outside of the usual
timeline’). At the moment there is just one test-page NAGworldMAP
on which you can see that geocoding was added to
this site via the Geo
Plugin (allowing to add geographic data to posts) and the instant google world map Plugin plotting these data on a Google Map. At the moment you can see
the distance I have to cycle to get to the university, but I have plans
to do something more substantial with this feature soon, so please
familiarize yourself with dragging and zooming the map (for US-citizens,
European countries often do not put geographic data in the public
domain, so there is a limit to the zoom-factor and I use the
‘satellite’-view rather than any of the other two).

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citeUlike

Published January 13, 2006 by lievenlb

Thanks
to Andrei Sobolevskii for his comment
pointing me to a wonderful initiative : CiteULike.

What is CiteULike?
CiteULike is a
free service to help academics to share, store, and organise the
academic papers they are reading. When you see a paper on the web that
interests you, you can click one button and have it added to your
personal library. CiteULike automatically extracts the citation details,
so there’s no need to type them in yourself. It all works from
within your web browser. There’s no need to install any special
software.
Because your library is stored on the server, you
can access it from any computer. You can share you library with others,
and find out who is reading the same papers as you. In turn, this can
help you discover literature which is relevant to your field but you may
not have known about.
When it comes to writing up your
results in a paper, you can export your library to either BibTeX or
Endnote to build it in to your bibliography. CiteULike has a flexible
filing system, so you actually stand a chance of being able to find that
article that you stored a few months ago when you need
it.

If all this seems too abstract, here is an excellent practical
introduction (also suggested by Andrei). This text focusses on
articles from AnthroSource but if you’re a mathematician, do the
same things when you are at the abstract page of a paper on the arXiv or a paper description from MathSciNet. The really nice
thing is that you virtually have to do no typing at all (apart from the
tags you want to add to classify the paper where you want it or, if you
want, to add a note about the paper). Another exciting feature
is that you can upload your personal copy of the paper. A typical
situation : most of us can get the PDF-file of a published paper at work
(because the university has a contract with the publisher) but not at
home, on the road or on vacation. So, while at work, download the PDF,
upload it as your personal copy to citeUlike and you can read that paper
wherever you have internet access! But there is more : you can
export the BibTeX-data of your whole library and use it in your next
paper, every library has its separate RSS-feed so you can feed it to a
news-aggregator (or to bloglines) to find out whether someone with
similar interests added a new paper to his/her library, you can create
Groups that is collections of Libraries of people interested in the same
topic, so that others can help you finding stuff of value (and again,
such Group-libraries have there own RSS-feed so….), all libraries
have all tags used by the Library-owner in a graphical format, the
larger the tag-text the more it is used in the Library, so just by
looking at the right-sidebar you get a good idea what the person’s
interests are, etc. etc. etc. I’m just two days into
citeUlike and there will be tons of features I still have to discover
and I’ll report on this later. At the moment I just added a few
papers to my Library but I will extend this drasticly in the weeks
ahead. If you want to check on my progress here is lieven’s Library
or the citeIlike link in the header of this blog (between the
‘about me’ and the ’search’ link) and I hope
that many of you will add similar buttons on your homepages.
Finally, if you are interested in Noncommutative algebraic and/or
differential geometry, I’ve set up a Group-Library
NoncommutativeGeometry. At the moment it’s just identical to
my own Library, but please register to citeUlike, set up your own
Library and if you’re into NOG join this group!

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Latexrender and dvonn boards

Published January 9, 2006 by lievenlb

In order
to blog a bit about Dvonn-strategy, I made myself a simple Dvonn
LaTeX-template which works very well on paper but which gets mutilated
by Latexrender, for example the first situation of the looks
like

$~\xymatrix@=.3cm @!C @R=.7cm{ & & \Black{2} \connS & &
\bull{d}{5} \conn & & \bull{e}{5} \conn & & \bull{f}{5} \conn & &
\bull{g}{5} \conn & & \bull{h}{5} \conn & & \SWhite \connS & & \SWhite
\connS & & \SWhite \conneS & & \\ & \bull{b}{4} \conn & & \SBlack
\connS & & \Black{6} \connS & & \bull{e}{4} \conn& & \bull{f}{4} \conn &
& \bull{g}{4} \conn & & \bull{h}{4} \conn & & \SWhite \connS & &
\SWhite \connS & & \SWhite \conneS & \\ \SBlack \connbeginS & &
\SBlack \connS & & \BDvonn{7} \connS & & \bull{d}{3} \conn & & \SBlack
\connS & & \BDvonn{6} \connS & & \bull{g}{3} \conn & & \bull{h}{3}
\conn & & \Dvonn \connS & & \SWhite \connS & & \SWhite \connendS \\ &
\Black{5} \connbeginS & & \bull{b}{2} \conn & & \SBlack \connS & &
\bull{d}{2} \conn & & \bull{e}{2} \conn & & \bull{f}{2} \conn & &
\bull{g}{2} \conn & & \bull{h}{2} \conn & & \SWhite \connS & & \SWhite
\connendS & \\ & & \bull{a}{1} \con & & \bull{b}{1} \con & & \Black{5}
\conS & & \bull{d}{1} \con & & \bull{e}{1} \con & & \bull{f}{1} \con & &
\bull{g}{1} \con & & \bull{h}{1} \con & & \White{2} & &} $

The
reason behind this unwanted clipping is that Latexrender uses
**convert** to take the relevant part of a ps-page containing only the
TeXed formula on an empty page by performing clipping and then converts
it into a GIF-file (or any other format you desire). The obvious way
round this is to enlarge my template by adding two additional rows and
columns and putting visible nonsense there (such as dots) to enlarge the
relevant part so that no clipping is done of essential info. But then
(1) the picture generated becomes even larger than that above and (2) I
don’t want you to see the extra nonsensical dots… The essential line
in the **class.latexrender.php** file is

$command = $this->_convert_path." -density ".$this->_formula_density. " -trim -transparent \"#FFFFFF\" ".$this->_tmp_filename.".ps ". $this->_tmp_filename.".".$this->_image_format;

So
I needed to delve into the [manual pages for the convert command](http://amath.colorado.edu/computing/software/man/convert.html)
of the ImageMagick-package. To my surprise, the *-trim* option (which I
thought to adjust somewhat by adding parameters) doesn’t exist! Still, I
got around my second problem using the *crop* option and around the
first by using the very useful *geometry* option. The latter is also
useful if you find that the size of the output of Latexrender is not
compatible with the size of your regular text. Of course you can amend
this somewhat by using the *extarticle* documentclass (as suggested) but
if you want to further adjust it, use for example

-geometry 86%

to size the output to exactly 86% (or whatever you need).
So, whenever I want to do some Dvonn-blogging from now on I’ll change my
class.latexrender.php file as follows

$command = $this->_convert_path." -crop 0x0-10% -crop 0x0+10% -density ".$this->_formula_density. " -geometry 80% -transparent \"#FFFFFF\" ".$this->_tmp_filename.".ps ". $this->_tmp_filename.".".$this->_image_format;

which
produces the output

$\xymatrix@=.3cm @R=.7cm{.& & & & & & & & & &
& & & \\ & & & \Black{2} \connS & & \bull{d}{5} \conn & & \bull{e}{5}
\conn & & \bull{f}{5} \conn & & \bull{g}{5} \conn & & \bull{h}{5} \conn
& & \SWhite \connS & & \SWhite \connS & & \SWhite \conneS & & & \\ & &
\bull{b}{4} \conn & & \SBlack \connS & & \Black{6} \connS & &
\bull{e}{4} \conn& & \bull{f}{4} \conn & & \bull{g}{4} \conn & &
\bull{h}{4} \conn & & \SWhite \connS & & \SWhite \connS & & \SWhite
\conneS & & \\ & \SBlack \connbeginS & & \SBlack \connS & &
\BDvonn{7} \connS & & \bull{d}{3} \conn & & \SBlack \connS & &
\BDvonn{6} \connS & & \bull{g}{3} \conn & & \bull{h}{3} \conn & &
\Dvonn \connS & & \SWhite \connS & & \SWhite \connendS & . \\ & &
\Black{5} \connbeginS & & \bull{b}{2} \conn & & \SBlack \connS & &
\bull{d}{2} \conn & & \bull{e}{2} \conn & & \bull{f}{2} \conn & &
\bull{g}{2} \conn & & \bull{h}{2} \conn & & \SWhite \connS & & \SWhite
\connendS & & \\ & & & \bull{a}{1} \con & & \bull{b}{1} \con & &
\Black{5} \conS & & \bull{d}{1} \con & & \bull{e}{1} \con & &
\bull{f}{1} \con & & \bull{g}{1} \con & & \bull{h}{1} \con & & \White{2}
& & & \\ . & & & & & & & & & & & & & } $

which (I hope) you will
find slightly better…

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Alain Connes on everything

Published January 6, 2006 by lievenlb

A few
days ago, Ars Mathematica wrote :

Alain Connes and Mathilde Marcolli have posted a
new survey paper on Arxiv A walk in the
noncommutative garden. There are many contenders for the title of
noncommutative geometry, but Connes‚Äô flavor is the most
successful.

Be that as it may, do
not print this 106 page long paper! Browse through it
if you have to, be dazzled by it if you are so inclined, but I doubt it
is the eye-opener you were looking for if you gave up on reading
Connes’ book Noncommutative
Geometry…. Besides, there is much better
_Tehran-material_ on Connes to be found on the web : An interview
with Alain Connes, still 45 pages long but by all means : print it
out, read it in full and enjoy! Perhaps it may contain a lesson or two
for you. To wet your appetite a few quotes

It is
important that different approaches be developed and that one
doesn‚Äôt try to merge them too fast. For instance in noncommutative
geometry my approach is not the only one, there are other approaches
and it‚Äôs quite important that for these approaches there is no
social pressure to be the same so that they can develop
independently. It‚Äôs too early to judge the situation for instance
in quantum gravity. The only thing I resent in string theory is that
they put in the mind of people that it is the only theory that can
give the answer or they are very close to the answer. That I resent.
For people who have enough background it is fine since they know all
the problems that block the road like the cosmological constant, the
supersymmetry breaking, etc etc…but if you take people who are
beginners in physics programs and brainwash them from the very start
it is really not fair. Young physicists should be completely free,
but it is very hard with the actual system.

And here for some (moderate) Michael Douglas bashing :

Physicists tend to shift often and work on the
last fad. I cannot complain because at some point around 98 that fad was
NCG after my paper with Douglas and Schwarz. But after a while when
I saw Michael Douglas and asked him if he had thought more about
these problems the answer was no because it was no longer the last
fad and he wanted to work on something else. In mathematics one
sometimes works for several years on a problem but these young
physicists have a very different type of working habit. The unit of
time in mathematics is about 10 years. A paper in mathematics which is
10 years old is still a recent paper. In physics it is 3 months. So
I find it very difficult to cope with constant
zapping.

To the suggestion that he is the
prophet (remember, it is a Tehran-interview) of noncommutative geometry
he replies

It is flattering but I don‚Äôt think
it is a good thing. In fact we are all human beings and it is a
wrong idea to put a blind trust in a single person and believe in
that person whatever happens. To give you an example I can tell you
a story that happened to me. I went to Chicago in 1996, and gave a
talk in the physics department. A well known physicist was there and
he left the room before the talk was over. I didn‚Äôt meet this
physicist for two years and then, two years later, I gave the same
talk in the Dirac Forum in Rutherford laboratory near Oxford. This
time the same physicist was attending, looking very open and convinced
and when he gave his talk later he mentioned my talk quite
positively. This was quite amazing because it was the same talk and
I had not forgotten his previous reaction. So on the way back to
Oxford, I was sitting next to him in the bus, and asked him openly
how can it be that you attended the same talk in Chicago and you
left before the end and now you really liked it. The guy was not a
beginner and was in his forties, his answer was ‚ÄúWitten was seen
reading your book in the library in Princeton‚Äù! So I don‚Äôt want
to play that role of a prophet preventing people from thinking on
their own and ruling the sub ject, ranking people and all that. I
care a lot for ideas and about NCG because I love it as a branch of
mathematics but I don‚Äôt want my name to be associated with it as a
prophet.

and as if that was not convincing
enough, he continues

Well, the point is that what
matters are the ideas and they belong to nobody. To declare that
some persons are on top of the ladder and can judge and rank the
others is just nonsense mostly produced by the sociology (in fact by the
system of recommendation letters). I don‚Äôt want that to be true in
NCG. I want freedom, I welcome heretics.

But please, read it all for yourself and draw your own conclusions.

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