# Tag: geometry

Yesterday I made reservations for lecture rooms to run the
master class on non-commutative geometry sponsored by the ESF-NOG project. We have a lecture room on
monday- and wednesday afternoon and friday the whole day which should be
enough. I will run two courses in the program : non-commutative
geometry
and projects in non-commutative geometry both 30
hours. I hope that Raf Bocklandt will do most of the work on the
Geometric invariant theory course so that my contribution to it
can be minimal. Here are the first ideas of topics I want to cover in my
courses. As always, all suggestions are wellcome (just add a
comment).

non-commutative geometry : As
I am running this course jointly with Markus Reineke and as Markus will give a
mini-course on his work on non-commutative Hilbert schemes, I will explain
the theory of formally smooth algebras. I will cover most of the
paper by Joachim Cuntz and Daniel Quillen “Algebra extensions and
nonsingularity”, Journal of AMS, v.8, no. 2, 1995, 251?289. Further,
I’ll do the first section of the paper by Alexander Rosenberg and Maxim Kontsevich,
Noncommutative smooth spaces“. Then, I will
explain some of my own work including the “One
quiver to rule them all
” paper and my recent attempts to classify
all formally smooth algebras up to non-commutative birational
equivalence. When dealing with the last topic I will explain some of Aidan Schofield‘s paper
Birational classification of moduli spaces of representations of quivers“.

projects in
non-commutative geometry
: This is one of the two courses (the other
being “projects in non-commutative algebra” run by Fred Van Oystaeyen)
for which the students have to write a paper so I will take as the topic
of my talks the application of non-commutative geometry (in particular
the theory of orders in central simple algebras) to the resolution of
commutative singularities and ask the students to carry out the detailed
analysis for one of the following important classes of examples :
quantum groups at roots of unity, deformed preprojective algebras or
symplectic reflexion algebras. I will explain in much more detail three talks I gave on the subject last fall in
Luminy. But I will begin with more background material on central simple
algebras and orders.

The
game of sprouts is a two-person game invented by John Conway and Michael Paterson in 1967 (for some
historical comments visit the encyclopedia). You just need pen and paper to
play it. Here are the rules : Two players, Left and Right, alternate
moves until no more moves are possible. In the normal game, the last
person to move is the winner. In misere play, the last person to move is
the loser. The starting position is some number of small circles called
“spots”. A move consists of drawing a new spot g and then drawing two
lines, in the loose sense, each terminating at one end at spot g and at
the other end at some other spot. (The two lines can go to different
spots or the same spot, subject to the following conditions.) The lines
drawn cannot touch or cross any line or spot along the way. Also, no
more than three lines can terminate at any spot. A spot with three lines
attached is said to be “dead”, since it cannot facilitate any further
action.

You can play sprouts online using this Java applet.
There is also an ongoing discussion about sprouts on the geometry math forum. Probably the most complete
information can be found at the world game
of sprouts association
. The analysis of the game involves some nice
topology (the Euler number) and as the options for Left and Right are
the same at each position it is an impartial game and the outcome
depends on counting arguments. There is also a (joke) variation on the
game called Brussels sprouts (although some people seem to miss the point
entirely).

Some years ago I invented some variations
on sprouts making it into a partizan game (that is, at a given
position, Left and Right have different legal moves). Here are the rules
:

dots. Left is allowed to connect two White dots or a White and bLue dot
or two bLue dots and must draw an additional Red dot on the connecting
line. Right is allowed to connect two White dots, a Red and a White dot
or two Red dots and must draw an additional bLue dot on the connecting
line.

White dots. Left is allowed to connect two White dots or a White and
bLue dot or two bLue dots and must draw an additional bLue dot on the
connecting line. Right is allowed to connect two White dots, a Red and a
White dot or two Red dots and must draw an additional Red dot on the
connecting line.

Although the rules look pretty
similar, the analysis of these two games in entirely different. On
february 11th I’ll give a talk on this as an example in
Combinatorial Game Theory. I will show that Cold Antwerp Sprouts
is very similar to the game of COL, whereas Hot Antwerp Sprouts resembles SNORT.

A
longer term project is to get the web-server www.matrix.ua.ac.be integrated in our home-network
as an external WebDAV-server (similar to the .Mac-service
offered by Apple). But as this server runs all information about the
master-class on non-comutative geometry connecting to it via HTTP to use
combinations will be send without encryption. Hence the natural question
whether this server can be set up to run SSL (Secure Sockets
Layer) such that one can connect via HTTPS and all exchanged information
will be encrypted. As the server is an Apache it comes down to get
mod-ssl running. A Google on mod_ssl OS X gives the
ADC-document Using mod-ssl on Mac OS X which seems to be just
instructions of using the openssl command. However, the
end-result is rather weak : it only makes the localhost running
HTTPS, that is, one can connect to your own computer safely… which is
pretty ridiculous (other computers in the same network cannot even
connect safely).

So, back to the Google-list on which
one link raises my interest Configuring mod-ssl on Mac OS X which looks like
the previous link but has one essential difference : the page is written
by Marc Liyanage. If you ever tried to get PHP and/or MySQL
running under OS X you will have noticed that his pages are by far the
most reliable on the subject, hence maybe he has also something
interesting to say on mod-ssl. However, the bottom line of the
document is not very promising :

You
should now be able to access the content with https://127.0.0.1 from
the same machine.

which is again the
localhost. So perhaps it is just impossible to run mod-ssl
without having an X-server. Anyway, let us try out his procedure.
Begin by issuing the following commands in the Terminal

sudo -s cd /etc/httpd mkdir ssl chmod 700 ssl cd
ssl gzip -c --best /var/log/system.log > random.dat openssl rand
-rand file:random.dat 0

Next, we need a server certificate. If you
want to do it properly you need a certificate from a certification
authority
such as Thawte but this costs at least \$200 a year which I
am not willing to pay. The alternative is to use a self-signed
certificate
which will force the browser to display an error-message
but if the user dismisses it all traffic exchanged with the server will
still be encrypted which is just what I want. So, type the command

openssl req -keyout privkey-2001.pem -newkey rsa:1024
-nodes -x509 -days 365 -out cert-2001.pem

(all on one line).
You will be asked a couple of questions (the only important one is the
Common Name (eg, YOUR name). Here you should take care to enter
the host name of your web server exactly as it will be used later in the
common name field. In my test-case, if I want to get my server
used by other computers in the network this name will be
imaclieven.local. (note the trailing .). Now issue the following
commands

chmod 600 privkey-2001.pem chown root
privkey-2001.pem apxs -e -a -n ssl /usr/libexec/httpd/libssl.so


which will activate the SSL-module (if at a later state you want
to de-activate it you have to change -a by -A in the last command).
Finally, we have to change the /etc/httpd/httpd.conf file so
first save a backup-version and then add the following lines at the end
of the file :

(IfModule mod-ssl.c)     Listen 80
Listen 443     SSLCertificateFile /etc/httpd/ssl/cert-2001.pem
SSLCertificateKeyFile /etc/httpd/ssl/privkey-2001.pem
SSLRandomSeed startup builtin     SSLRandomSeed connect builtin
(VirtualHost -default- :443)         SSLEngine on
(/VirtualHost) (/IfModule)

Observe that round brackets ()
should be replaced by <>. Finally, we do

apachectl
stop apachectl start

and we are done! Going to another computer
in the network and typing in Safari https://imaclieven.local./
will result in an error message

Just click Continue and you will have a secure connection
to the server. Thanks Marc Liyanage!

11th) Whereas the above allows one to make a HTTPS connection it is not
enough for my intended purposes. In order to get a secure connection to
a WebDAV server, this server must have the mod-auth-digest module
running which seems to be impossible for the standard Apache server of
10.3. You need an X-server to have this facility. So I think I have to
scale down my ambitions a bit.

So far I
found it rather easy to post one or more messages a day as I was
installing a lot of software or trying to get things working and was
merely logging my progress for future reference. These notes are useful
to me but probably not to the rest of the world. Another thing I noticed
is that I’m using this blog sometimes as a replacement for my
Bookmarks, merely listing interesting web-pages without too much
personal comments. I will continue to post both install-logs and
bookmark-logs but in addition I want to write (say weekly) a lengthier
post on a specific topic with more background, more details (such as
screenshots) and more personal comments. We will see how this works out
in the coming weeks…

Another thing that slightly
worried me is that people visiting my homepage and clicking on to my
blog may expect entirely different things there. But this cant be
helped, I’m sitting on an OSX-cloud at the moment but no doubt this
will change quickly. Beginning of february I have to give a talk on
Combinatorial Game Theory and soon afterwards the
Non-commutative Geometry Master Class starts in which I’m giving
a couple of courses, so mathematics will become more dominant in this
blog from next month on…

On a
blog-tech matter : I found a quite good editor pMpost
which is meant to write pMachine-blogs offline and upload them by one
click. It also synchronizes categories etc. on login. Further, it has a
spelling-checker but the thing I really like about it is that you can
save texts as a draft and continue at a later time (sadly, it remember
the date/hour when you start your post so when you finally submit it it
will be posted at the starting- rather than the posting-day. Still,
there is nothing that copy/paste cannot solve. I hope to use this
facility when (read if) I’m going for a more in-depth post. Another
matter that I will address to as quickly as possible (probably over the
weekend) is teh layout of this site. The main annoying thing is that the
text doesnt resize when you increase/decrease window width. So I will
address this matter first and probably leave a personal layout and
color-scheme to later. Fortunately, I did find a good site containg a
lot of CSS templates for pMachine weblogs. Another site I’ll have to
investigate over the weekend is pMtemplates. But don’t expect too much from the
layout-side, I still have other projects to worry about : SSL, WebDAV,
streaming iTunes, getting on Ethernet-DVD player to work and so
on.