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Borcherds’ monster papers

Published March 27, 2004 by lievenlb


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
spaces

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

Published March 23, 2004 by lievenlb

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
it

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

Published March 22, 2004 by lievenlb

Tomorrow
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
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.

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