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Borcherds’ monster papers

Yesterday morning I thought that I could use some discussions I had a
week before with Markus Reineke to begin to make sense of one
sentence in Kontsevich’ Arbeitstagung talk Non-commutative smooth
spaces :

It seems plausible that Borcherds’ infinite rank
algebras with Monstrous symmetry can be realized inside Hall-Ringel
algebras for some small smooth noncommutative

However, as I’m running on a 68K RAM-memory, I
didn’t recall the fine details of all connections between the monster,
moonshine, vertex algebras and the like. Fortunately, there is the vast
amount of knowledge buried in the arXiv and a quick search on Borcherds gave me a
list of 17 papers. Among
these there are some delightful short (3 to 8 pages) expository papers
that gave me a quick recap on things I once must have read but forgot.
Moreover, Richard Borcherds has the gift of writing at the same time
readable and informative papers. If you want to get to the essence of
things in 15 minutes I can recommend What
is a vertex algebra?
(“The answer to the question in the title is
that a vertex algebra is really a sort of commutative ring.”), What
is moonshine?
(“At the time he discovered these relations, several
people thought it so unlikely that there could be a relation between the
monster and the elliptic modular function that they politely told McKay
that he was talking nonsense.”) and What
is the monster?
(“3. It is the automorphism group of the monster
vertex algebra. (This is probably the best answer.)”). Borcherds
maintains also his homepage on which I found a few more (longer)
expository papers : Problems in moonshine and Automorphic forms and Lie algebras. After these
preliminaries it was time for the real goodies such as The
fake monster formal group
, Quantum vertex algebras and the like.
After a day of enjoyable reading I think I’m again ‘a point’
wrt. vertex algebras. Unfortunately, I completely forgot what all this
could have to do with Kontsevich’ remark…

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chicken of the VNC

If I
ever get our home automation system configured I’ll use my (partly
broken) old iBook as my Indigo-server (or my MisterHouse-server when I brush up my
Perl-knowledge). It should then run quietly put away somewhere and I
don’t want to take it out every time I want to add another routine to
the program.
Fortunately there is a way to do this by turning
the iBook into a VNC-server, where VNC stands for
Virtual Network Computer. Here is how RealVNC describes

VNC (Virtual Network Computing) software makes it
possible to view and fully-interact with one computer from any other
computer or mobile device anywhere on the Internet. VNC software is
cross-platform, allowing remote control between different types of
computer. For ultimate simplicity, there is even a Java viewer, so that
any desktop can be controlled remotely from within a browser without
having to install software.

But can all this be done under
Mac OS X without too much hassle? The first step is to download
OSXvnc and install it on the iBook. Some of the
sourceforge-sites do not seem to have this package, but fortunately some
still do. Installation is no problem and when you fire OSXvnc up
you have to fill in a password which you need later to connect to your
OSXvnc-server (the iBook). Most other options one can leave at their
default values but in the Startup-pane it is useful to click on
the Configure Startup Item button. When all this is done, press
the Start button to launch the VNC-server.
Next step is
to go to the computer you want to use to control the VNC-server (an iMac
in my case). On it one needs to install the Chicken of the VNC software which makes the iMac
into a VNC-client. Fire it up and fill out the Host (the name of
your OSXvnc-server, iBookLieven.local in my case) and the
Password (the one of the OSXvnc-server program), press the
Connect button and the screen of your VNC-server will appear
which you can control with your mouse as if you were actually working on
the thing. Very handy as I managed to break the touch-control on my
iBook when installing a new hard-drive and I need the only USB-port to
connect to the X10-network…

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projects in noncommutative geometry

I’ll start with the course Projects in non-commutative geometry
in our masterclass. The idea of this course (and its companion
Projects in non-commutative algebra run by Fred Van Oystaeyen) is
that students should make a small (original if possible) work, that may
eventually lead to a publication.
At this moment the students
have seen the following : definition and examples of quasi-free algebras
(aka formally smooth algebras, non-commutative curves), their
representation varieties, their connected component semigroup and the
Euler-form on it. Last week, Markus Reineke used all this in his mini-course
Rational points of varieties associated to quasi-free
. In it, Markus gave a method to compute (at least in
principle) the number of points of the non-commutative Hilbert
and the varieties of simple representations over a
finite field. Here, in principle means that Markus demands a lot of
knowledge in advance : the number of points of all connected components
of all representation schemes of the algebra as well as of its scalar
extensions over finite field extensions, together with the action of the
Galois group on them … Sadly, I do not know too many examples were all
this information is known (apart from path algebras of quivers).
Therefore, it seems like a good idea to run through Markus’
calculations in some specific examples were I think one can get all this
: free products of semi-simple algebras. The motivating examples
being the groupalgebra of the (projective) modular group
PSL(2,Z) = Z(2) * Z(3) and the free matrix-products $M(n,F_q) *
M(m,F_q)$. I will explain how one begins to compute things in these
examples and will also explain how to get the One
quiver to rule them all
in these cases. It would be interesting to
compare the calculations we will find with those corresponding to the
path algebra of this one quiver.
As Markus set the good
example of writing out his notes and posting them, I will try to do the
same for my previous two sessions on quasi-free algebras over the next
couple of weeks.

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bought a couple of X10-building blocks : a tranceiver, an appliance- and
a lamp-module, a computer-interface and a motion detector and started
playing using the Indigo help-page. All modules worked immediately
and getting them under Indigo‘s control was also no problem.
Clearly it is fun to control a living room lamp and the coffee maker
from your computer but it gets even better when you program actions.
With Indigo you can let your home automation system react to
incoming emails. For example, if it is a rainy workday and I want to
have a cup of coffee when I bicycle home I can just send an email with
subject “Make coffee”. Indigo checks at home my email every
two-minutes and when it scans this subject-title it will send a signal
to the coffee maker to turn on (assuming I filled it with water and
coffee beforehand, otherwise it may result in a fire…). One can also
program it the other direction. For example, with Indigo I can
program things so that when the motion detector detects movement from
opening the front door, I can ask to send an email to work (or to a
mobile but as I am not using these things this is no option) with
message “Someone just walked in…”.
Getting the
motion-detector (MS13 for the experts) working was so far the second
hardest thing to do. I couldnt work out how to give it a home&unit
but I found a readable manual page which made everything work. I have to
remember to change the other default options of the detector.
The hardest thing to solve was to get the Indigo Web Interface working. Following the
instructions on this page to the letter I thought that I could control
my X10-stuff from any other computer (assuming Indigo is running
on my iBook) by accessing the


but all I got
was a ‘Server Error’. I figured out that the mistake was caused by the
program. The first time this is run, it asks for your
admin password to write some extra lines to the httpd.conf
Apache configuration file, but for some strange reason it didn’t want
to accept my password… Changing permissions on httpd.conf and
even creating a genuine root-account didn’t help so I was stuck
for a while. But then I found the Mac OS X
hack #91
which not only explains the use of the dispatcher tool but
also explains what it adds to the httpd.conf-file. So, I just
copied the following lines manually at the end of

#BEGIN acgi dispatcher Include
nEND acgi dispatcher

did restart the Apache webserved by a

sudo apachectl graceful

after which the acgi dispatcher
tool started up without problems and I got a working Indigo Web
. I must remember to put both Indigo and the
dispatcher into my StartUp items.
The Web Interface is
very basic compared to other house automation programs such as MisterHouse
which makes up for its sexist name by being open source! It is entirely
written in Perl
but as I am only halfway through the Learning Perl book at the moment, this will have
to wait a bit longer…

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home automation, the next project??

I\’ve barely managed to implement the six
great tips for homemade dot mac servers
by Alan
or he is already off on a new project : Home Automation
with Mac OS X
. I thought that home automation only could be
installed in new, highly wired, houses but I was wrong. In part
Alan Graham gives an overview why you might consider home
automation, gives the set-up in his house and outlines the hardware
necessary to do it. Clearly, most of his hardware is American but even
in Belgium it is not difficult to find vendors, for example One can either control the
X10-machinery by remote control or via computer. For Macintosh Alan
Graham suggests to use the indigo program, of which one can download a fully
functional version for a 30 day tryout. The only piece I could not find
(yet) in Belgium was the PowerLink USB device but there is a serial-port
alternative available which seems to work just fine using a USB to
Serial cable (which are fairly expensive). In part 2 Alan Graham explains the basics of X10 technology and how you can
install all the hardware. In part 3 and later he promises to explain the
software part of things (if he hasnt started a new project by

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my temporary geek code

haven’t mastered by far all nuances of this fine device yet, but here
it is : a first approximation to my geek

GM d- s: a+ C+ UB+ P+ L+ E- W++ N o? K- w– O? M+ V? PS+
PE- Y+ PGP t 5?
X- R- tv+ b+++ DI D? G e++++ h—- r+++ z?

No doubt I’ll post a
revision in a few weeks. If you do not feel like compiling your own
geek-code but rather want to find out what all this gibberish means,
there is the geek code decoder page to assist you.

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ASCII math

a large extent mathematics has to do with elaborate typography. Many
youngsters have been attracted over the centuries to maths because they
wanted to understand the meaning of these beautiful pages filled with
integrals, partial derivatives and other bizarre hieroglyphs. But now we
have come to the point that this obsession for symbols is working
against mathematics…
Have you ever wondered why there are so
few mathematics-pages on the net compared to computer-science pages
(apart from the fact that a lot more exiting things are happening in
web-technology these days than in mathematics), why forums dedicated to
math-problems never get off the ground (apart from boring housework
sites) or why it is so seldom that you discuss serious math with
colleagues or students via emails (apart from the fact that more and
more mathematicians seem to turn off their sharing mode) ???
One of the reasons might be that our default way of writing and
communicating math (LaTeX) is incompatible with either HTML or email
(and for those of you who think that LaTeX2HTML or
tth or similar programs offer an alternative, just
try to make an attractive looking website with them and prove me
If we want mathematics to survive and flourish (and
whether you like it or not that may depend heavily on its
web-visibility) it is high time to develop some
ascii-math, that is, a way to write mathematical formulas in
plain typewriter symbols. This cannot be totally impossible as
programming languages are capable of defining a large number of
complicated objects with ascii and for those of you who discard the idea
on beauty-reasons, I never found a piece of code in a computer
book particularly ugly.
Of course I realise that not too many
people will be willing to make this paradigm-shift right now, but can we
at least ask of people introducing new symbols to add as an appendix to
their paper a suggestion for the transition to ascii-symbols for
those who value the net and/or sharing more than they do. Thank


once more : synchronizing

Carbon Copy Cloner is a tool to make a full backup
of your hard-disk on an external firewire disk or iPod. Here’s
how it sells itself

Have you ever wanted a simple, complete,
bootable backup of your hard drive? Have you ever wanted to upgrade to
a larger hard drive with minimal hassle and without reinstalling your
OS and all of your applications? Have you ever wanted to move your
entire Mac OS X installation to a new computer? Then CCC is the tool
for you! CCC makes these tasks simple by harnessing the Unix power
built into Mac OS X. In addition to the features that CCC has provided
in the past, version 2 offers synchronization of the source and target
as well as scheduled backup tasks.

I didn’t try it out yet
but was interested in the final sentence and scrolling down the page I
discovered that the synchronisation is done using Dan Kogai’s psync program, which does not seem to work under
10.3 but has on the page a patch to this. Rather than using the
psync-page to install it, one can use the unoffical psync for Panther dmg-file from the
Carbon Copy Cloner-page. It installs without a problem and to
learn how to use it, there is a manual page. Here is what I do when I want to
synchronize my Documents-folder on iMacLieven to the
backup-machine tweedledee over the Airport-network

/Network/iMacLieven/lieven/Documents /Users/lieven/docsLieven

Watching the packet-flow on the Activity Monitor it seems to be
slightly quicker than the rsync tool. But most of all : it seems
to do a much better job. When I compared the end-result of the
synchronising session with rsync to that of psync I was
surprised to find a 20 Mb difference (on an original .5 Gb Folder) in
psync‘s favour! But even psync seems to have dropped 0.6
Mb in the process…

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nothing beats the command line

the last couple of days I’ve been experimenting a bit with different
backup methods. To begin, I did try out ExecutiveSync and its
successor You Syncronize but they are very, very
slow. Not only did the first synchronizing of a 0.5 Gb Folder between
two computers over our Airport-network took over 2.5 hrs, but also on
subsequent syncs the checking of the database seems to last forever.

So I turned to the fink project
again and did find two interesting packages : wget . GNU Wget is a free network utility to
retrieve files from the World Wide Web using HTTP and FTP, so one way
to backup a folder would be to put it in the Sites folder and
mirror it over the network using wget. I did’t check this out in
great details (did a small test to see it working but I assume it will
be slow for large folders). The other one is rsync It uses the “rsync algorithm” which
provides a very fast method for remote files into sync. It does this by
sending just the differences in the files across the link, without
requiring that both sets of files are present at one of the ends of the
link beforehand. This seems to be precisely what I wanted to do and
after a google for ‘rsync OS X’ I arrived at the RsyncX package which is an implementation of rsync
with HFS support and configuration through a command line (Terminal) or
graphical user interface. I downloaded this package and the GUI seems to
be placed in the Applications/Utilities and tried it out by
filling out the Source and Local Folders and pressing the synchronize
button. Not much progress was reported but the Activity Monitor
showed that it was using up all of the CPU so I was patient for over an
hour and then looked for the Network Activity in the Activity
and virtually no packets were going in or out, so I killed
RsyncX. I am sure I did something wrong but rather than trying to
get it working, I tried the command-line rsync-command I
downloaded from Fink. After a few false attempts I

/sw/bin/rsync -a -e ssh

and suddenly the packets were flying
happily over the network at 250 Kb/sec, so it took me only half an hour
to get a first synchronization done and subsequent changes are added in
no time! Afterwards I discovered that rsync is included in the
standard OS X Developers Tools as RsyncX seems to have replaced
it to rsync_orig and installed a new (quite large) rsync
in /usr/bin. Maybe my problems with RsyncX were caused
because I have /sw/bin earlier in my $PATH than
/usr/bin but verifying this will have to await another day. For
the moment, I’m happy to have a quick syncronizing tool available and
Real Madrid is playing on the TV…

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noncommutative geometry 2

Again I
spend the whole morning preparing my talks for tomorrow in the master
class. Here is an outline of what I will cover :
– examples of
noncommutative points and curves. Grothendieck’s characterization of
commutative regular algebras by the lifting property and a proof that
this lifting property in the category alg of all l-algebras is
equivalent to being a noncommutative curve (using the construction of a
generic square-zero extension).
– definition of the affine
scheme rep(n,A) of all n-dimensional representations (as always,
l is still arbitrary) and a proof that these schemes are smooth
using the universal property of k(rep(n,A)) (via generic
– whereas rep(n,A) is smooth it is in general
a disjoint union of its irreducible components and one can use the
sum-map to define a semigroup structure on these components when
l is algebraically closed. I’ll give some examples of this
semigroup and outline how the construction can be extended over
arbitrary basefields (via a cocommutative coalgebra).

definition of the Euler-form on rep A, all finite dimensional
representations. Outline of the main steps involved in showing that the
Euler-form defines a bilinear form on the connected component semigroup
when l is algebraically closed (using Jordan-Holder sequences and
upper-semicontinuity results).

After tomorrow’s
lectures I hope you are prepared for the mini-course by Markus Reineke on non-commutative Hilbert schemes
next week.

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