# neverendingbooks Posts

Tomorrow
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
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
algebras
. In it, Markus gave a method to compute (at least in
principle) the number of points of the non-commutative Hilbert
scheme
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.

I
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
code
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
URL

http://iBookLieven.local/cgi-bin/Indigo.acgi

but all I got
was a ‘Server Error’. I figured out that the mistake was caused by the
acgi
dispatcher
program. The first time this is run, it asks for your
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
httpd.conf

#BEGIN acgi dispatcher Include
/Library/WebServer/CGI-Executables/dispatcher.app/Contents/acgi.conf\r\
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
Interface
. 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…

I\’ve barely managed to implement the six
great tips for homemade dot mac servers
by Alan
Graham
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
1
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 intellihome.be. 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
then…).

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

—–BEGIN GEEK CODE BLOCK—–
Version:
3.1
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?
——END GEEK CODE BLOCK——

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.

To
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
wrong).
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
you!

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

psync
/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…

Over
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
Monitor
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
typed

/sw/bin/rsync -a -e ssh
iMatrixLieven.local:/Users/lieven/Documents
/Users/lieven/docsLieven

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… 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 matrices). – 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. Tweedledum is a first-generation iMac (233 MHz slot-loading, 192Mb RAM, No Airport) whereas Tweedledee is 2nd-generation (350 MHz front-loading, 192Mb RAM, Airport card). A couple of weeks ago I replaced their original hard-discs (4 Gb resp. 6 Gb) by fat 120 Gb discs and from this weekend they serve as our backup-facility. Tweedledee is connected via Airport to our network and is a fully functional 10.3 computer, everyone has a login on it and is encouraged to dump important files onto it as a secondary copy. Tweedledum. on the other hand, is invisible to the network but forms a one-wire network with Tweedledee (they are connected by a crossed ethernet cable which results in having a self-assigned IP address in the 169.254 range and hence they can see each other; moreover using the Sharing-pane in the System Preferences I allowed Tweedledee to share its internet connection to other computers, connected to it via Ethernet, so Tweedledum can go online to get system-updates when necessary). A house-computer rule is that all important files (which means those you don’t like to loose in a crash) are kept in the Documents folder of your Home-folder on your own computer. At regular intervals I make sure that these folders are synchronized with backup-copies on both Tweedledee & Tweedledum, so at any given time there are at least 3 computers containing the essential files (usually more as everyone has a login at each of the 4 ‘work’-computers and can drop extra copies around, but must clean-up when asked). To synchronise I use the shareware program ExecutiveSync. It is no longer possible to obtain this from its original homepage as they seem to have been taken over and invite you to buy You Sinc instead which costs more than twice what ExecutiveSync costs (19.95$). Fortunately, for now you can
ExecutiveSync running on Tweedledee (you are only allowed to run it on
one computer, you can install it on every computer but then the
synchronizing process is sometimes not possible which is why I came to
the following work-around). In ExecutiveSync you make several
Projects which involve choosing a Local folder and a
Remote folder somewhere on your network which you want to keep in
Sync. In my Home folder on Tweedledee I made several (originally
empty) folders such as docsGitte. Then my ExecutiveSync-project
syncGitte takes docsGitte as the local folder and the
/Users/gitte/Documents-folder on iBookGitte as the remote
folder. The first time you synchronise takes a lot of time (especially
over the wireless network, it may be better to do the first sync via
ethernet) but afterwards it works pleasantly.
Once I
synchronised all the local Documents-folders with the corresponding
folders in my home-folder on Tweedledee, I have another
ExecutiveSync-project BACKUP which takes as the Local-folder my
Home-folder and as the remote folder a folder BACKUP I did create
on Tweedledum. Fortunately, here the synchronising is done over Ethernet
or it would take forever.

Today I
did prepare my lectures for tomorrow for the NOG master-class on
non-commutative geometry. I\’m still doubting whether it is worth TeXing
my handwritten notes. Anyway, here is what I will cover tomorrow :

– Examples of l-algebras (btw. l is an
arbitrary field) : matrix-algebras, group-algebras lG of finite
groups, polynomial algebras, free and tensor-algebras, path algebras
lQ of a finite quiver, coordinaterings O(C) of affine smooth
curves C etc.
– Refresher on homological algebra : free and
projective modules, exact sequences and complexes, Hom and Ext groups
and how to calculate them from projective resolutions, interpretation of
Ext^1 via (non-split) short exact sequences and stuff like that.
– Hochschild cohomology and noncommutative differential forms.
Bimodules and their Hochschild cohomology, standard complex and
connection with differential forms, universal bimodule of derivations
etc.
– Non-commutative manifolds. Interpretation of low degree
Hochschild cohomology, characterization of non-commutative points as
separable l-algebras and examples. Formally smooth algebras
(non-commutative curves) characterised by the lifting property for
square-free extensions and a proof that formally smooth algebras are
hereditary.

Next week I will cover the representation
varieties of formally smooth algebras and the semigroup on their
connected (or irreducible) components.