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Tag: non-commutative

NOG master class update

Published January 21, 2004 by lievenlb


Yesterday I made a preliminary program for the first two months
of the masterclass non-commutative geometry. It is likely that
the program will still undergo changes as at the moment I included only
the mini-courses given by Bernhard
Keller and Markus Reineke but several other people have
already agreed to come and give a talk. For example, Jacques Alev (Reims),
Tom Lenagan (Edinburgh),
Shahn Majid (London),
Giovanna Carnovale (Padua) among others. And in
may, Fred assures me, Maxim Kontsevich will give a couple of talks.

As for the contents of the two courses I will be
teaching I changed my mind slightly. The course non-commutative
geometry I teach jointly with Markus Reineke and making the program
I realized that I have to teach the full 22 hours before he will start
his mini-course in the week of March 15-19 to explain the few
things he needs, like :

To derive all the
counting of points formulas, I only need from your course:
–
the definition of formally smooth algebras basic properties, like
being
hereditary
– the definition of the component
semigroup
– the fact that dim Hom-dim Ext is constant along
components. This I need
even over finite fields $F_q$, but I
went through your proof in “One quiver”,
and it works. The
key fact is that even over $F_q$, the infinitesimal lifting
property implies smoothness in the sense Dimension of variety =
dimension of
(schematic) tangent space in any $F_q$-valued
point. But I think it’s fine for
the students if you do all
this over C, and I’ll only sketch the (few)
modifications for
algebras over $F_q$.

So my plan is to do all of
this first and leave the (to me) interesting problem of trying to
classify formally smooth algebras birationally to the second
course projects in non-commutative geometry which fits the title
as a lot of things still need to be done. The previous idea to give in
that course applications of non-commutative orders to the resolution of
singularities (in particular of quotient singularities) as very roughly
explained in my three talks on non-commutative geometry@n I now
propose to relegate to the friday afternoon seminar. I’ll be
happy to give more explanations on all this (in particular more
background on central simple algebras and the theory of (maximal)
orders) if other people work through the main part of the paper in the
seminar. In fact, all (other) suggestions for seminar-talks are welcome
: just tell me in person or post a comment to this post.

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Bill Schelter’s Maxima

Published January 20, 2004 by lievenlb

Bill
Schelter was a remarkable man. First, he was a top-class mathematician.
If you allow yourself to be impressed, read his proof of the
Artin-Procesi theorem. Bill was also among the first to take
non-commutative geometry seriously. Together with Mike Artin he
investigated a notion of non-commutative integral extensions and he was
the first to focuss attention to formally smooth algebras (a
suggestion later taken up by a.o. Cuntz-Quillen and Kontsevich) and a
relative version with respect to algebras satisfying all identities of
n x n matrices which (via work of Procesi) led to smooth@n
algebras. To youngsters, he is probably best know as the co-inventor of
Artin-Schelter regular algebras. I still vividly remember an
overly enthusiastic talk by him on the subject in Oberwolfach, sometime
in the late eighties. Secondly, Bill was a genuine Lisp-guru and
a strong proponent of open source software, see for example his
petition against software patents. He maintanind
his own version of Kyoto Common Lisp which developed into Gnu
Common Lisp. A quote on its history :

GCL is
the product of many hands over many years. The original effort was known
as the Kyoto Common Lisp system, written by Taiichi Yuasa and Masami
Hagiya in 1984. In 1987 new work was begun by William Schelter, and that
version of the system was called AKCL (Austin Kyoto Common Lisp). In
1994 AKCL was released as GCL (GNU Common Lisp) under the GNU public
library license. The primary purpose of GCL during that phase of it’s
existence was to support the Maxima computer algebra system, also
maintained by Dr. Schelter. It existed largely as a subproject of
Maxima.

Maxima started as Bill’s version of
Macsyma an MIT-based symbolic computation program to which he
added many routines, one of which was Affine a package that
allowed to do Groebner-like computations in non-commutative
algebras (implementing Bergman’s diamond lemma) and which he
needed to get a grip on 3-dimensional Artin-Schelter regular
algebras. Michel and me convinced Fred to acquire funds to
buy us a work-station (costing at the time 20 to 30 iMacs) and have Bill
flown in from the States with his tape of maxima and let him
port it to our Dec-station. Antwerp was probably for years
the only place in the world (apart from MIT) where one could do
calculations in affine (probably highly illegal at the time).
Still, lots of people benefitted from this, among others Michaela
Vancliff and Kristel Van Rompay in their investigation
of 4-dimensional Artin-Schelter regular algebras associated to an
automorphism of a quadric in three-dimensional projective space.
Yesterday I ran into Bill (alas virtually) by browsing the
crypto-category of Fink. There it was, maxima, Bill’s package! I tried to install it
with the Fink Commander and failed but succeeded from the command line.
So, if you want to have your own version of it type 

sudo fink
install maxima

from the Terminal and it will install without
problems (giving you also a working copy of common lisp). Unfortunately
I do not remember too much of Macsyma or Affine but there is plenty of
documentation on the net. Manuals and user guides can be obtained from
the maxima homepage and the University of Texas
(Bill’s university) maintains an online manual, including a cryptic description of
some Affine-commands. But probably I’ll have to send Michaela an
email asking for some guidance on this… Here, as a tribute to Bill who
died in july 2001 the opening banner 

 iMacLieven:~ lieven$
/sw/bin/maxima Maxima 5.9.0 http://maxima.sourceforge.net 
Distributed under the GNU Public License. 
See the file COPYING. 
Dedicated to the memory of William Schelter. 
This is a development version of Maxima. 
The function bug_report() provides bug reporting information. 
(C1)
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NOG master class

Published January 14, 2004 by lievenlb


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.

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