Tag: geometry

noncommutative@newton

Published September 3, 2006 by lievenlb

At the
moment a Noncommutative
Geometry Programme is being organized at the Newton Institute. This half year
programme started with a workshop on Noncommutative
Geometry and Cyclic Cohomology at the beginning of august. This
week they’ll be running their second workshop Noncommutative
Geometry and Physics: Fundamental Structure of Space and Time
including a speculative evening session The Nature
of Space and Time: An Evening of Speculation where

A
distinguished panel of mathematicians, physicists, theologians and
philosophers will explore the nature of space and time from a personal
perspective. What do science and philosophical theology have to say to
each other about space and time? Is time a continuum? Can the nature of
time be separated from the nature of existence and from the human
condition? There will be short presentations from each panel member
followed by a wide-ranging discussion led by questions from the
audience. This is expected to be a lively event fully accessible to the
wider public.

Perhaps the most interesting workshop,
from a ringtheorist’s point of view, is the closing workshop Trends in
Noncommutative Geometry to be organized in December. Oh, I see that
the closing date for applications has already passed… Still, for
the rest of us, we can follow this programme from the luxury (?!) of our
home using the Newton
Web-Seminar page which includes the slides and a full audio of the
lectures. Last night I sat through Iain Gordon’s talk and there are more talks I intend to upload to
my iPod.

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noncommutative Fourier transform

Published June 23, 2006 by lievenlb

At the
noncommutative algebra program in MSRI 1999/2000, Mikhail Kapranov gave
an intriguing talk Noncommutative neighborhoods and noncommutative Fourier transform
and over the years I’ve watched the video of this talk a number
of times. The first part of the talk is about his work on Noncommutative geometry
based on commutator expansions and as I’ve once worked through it
this part didn’t present problems. On the other hand, I’ve never
understood much from the second part of the talk which claims to relate
these noncommutative formal neighborhoods to _noncommutative Fourier
transforms_. The string coffee table has a post Kapranov
and Getzler on Higher Stuff linking to two recent talks by Kapranov
on noncommutative Fourier transforms at the Streetfest. Marni
Sheppeard made handwritten notes available. I definitely should find the time
to get through them and have another go at the Kapranov-video…

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non-(commutative) geometry

Published June 21, 2006 by lievenlb

Now
that my non-geometry
post is linked via the comments in this
string-coffee-table post which in turn is available through a
trackback from the Kontsevich-Soibelman
paper it is perhaps useful to add a few links.

The little
I’ve learned from reading about Connes-style non-commutative geometry is
this : if you have a situation where a discrete group is acting with a
bad orbit-space (for example, $GL_2(\mathbb{Z})$ acting on the whole
complex-plane, rather than just the upper half plane) you can associate
to this a $C^*$-algebra and study invariants of it and interprete them
as topological information about this bad orbit space. An intruiging
example is the one I mentioned and where the additional noncommutative
points (coming from the orbits on the real axis) seem to contain a lot
of modular information as clarified by work of Manin&Marcolli and
Zagier. Probably the best introduction into Connes-style
non-commutative geometry from this perspective are the Lecture on
Arithmetic Noncommutative Geometry by Matilde Marcolli. To
algebraists : this trick is very similar to looking at the
skew-group algebra $\mathbb{C}[x_1,\ldots,x_n] * G$ if
you want to study the _orbifold_ for a finite group action on affine
space. But as algebraist we have to stick to affine varieties and
polynomials so we can only deal with the case of a finite group,
analysts can be sloppier in their functions, so they can also do
something when the group is infinite.

By the way, the
skew-group algebra idea is also why non-commutative algebraic
geometry enters string-theory via the link with orbifolds. The
easiest (and best understood) example is that of Kleinian singularities.
The best introduction to this idea is via the Representations
of quivers, preprojective algebras and deformations of quotient
singularities notes by Bill Crawley-Boevey.

Artin-style non-commutative geometry aka
non-commutative projective geometry originated from the
work of Artin-Tate-Van den Bergh (in the west) and Odeskii-Feigin (in
the east) to understand Sklyanin algebras associated to elliptic curves
and automorphisms via ‘geometric’ objects such as point- (and
fat-point-) modules, line-modules and the like. An excellent survey
paper on low dimensional non-commutative projective geometry is Non-commutative curves and surfaces by Toby
Stafford and
Michel Van den Bergh. The best introduction is the (also
neverending…) book-project Non-
commutative algebraic geometry by Paul Smith who
maintains a
noncommutative geometry and algebra resource page page (which is
also available from the header).

Non-geometry
started with the seminal paper ‘Algebra extensions and
nonsingularity’, J. Amer. Math. Soc. 8 (1995), 251-289 by Joachim
Cuntz and Daniel Quillen but which is not available online. An
online introduction is Noncommutative smooth
spaces by Kontsevich and Rosenberg. Surely, different people have
different motivations to study non-geometry. I assume Cuntz got
interested because inductive limits of separable algebras are quasi-free
(aka formally smooth aka qurves). Kontsevich and Soibelman want to study
morphisms and deformations of $A_{\infty}$-categories as they explain in
their recent
paper. My own motivation to be interested in non-geometry is the
hope that in the next decades one will discover new exciting connections
between finite groups, algebraic curves and arithmetic groups (monstrous
moonshine being the first, and still not entirely understood,
instance of this). Part of the problem is that these three topics seem
to be quite different, yet by taking group-algebras of finite or
arithmetic groups and coordinate rings of affine smooth curves they all
turn out to be quasi-free algebras, so perhaps non-geometry is the
unifying theory behind these seemingly unrelated topics.

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non-geometry

Published June 16, 2006 by lievenlb

Here’s
an appeal to the few people working in Cuntz-Quillen-Kontsevich-whoever
noncommutative geometry (the one where smooth affine varieties
correspond to quasi-free or formally smooth algebras) : let’s rename our
topic and call it non-geometry. I didn’t come up with
this term, I heard in from Maxim Kontsevich in a talk he gave a couple
of years ago in Antwerp. There are some good reasons for this name
change.

The term _non-commutative geometry_ is already taken by
much more popular subjects such as _Connes-style noncommutative
differential geometry_ and _Artin-style noncommutative algebraic
geometry_. Renaming our topic we no longer have to include footnotes
(such as the one in the recent Kontsevich-Soibelman
paper) :

We use “formal” non-commutative geometry
in tensor categories, which is different from the non-commutative
geometry in the sense of Alain Connes.

or to make a
distinction between _noncommutative geometry in the small_ (which is
Artin-style) and _noncommutative geometry in the large_ (which in
non-geometry) as in the Ginzburg notes.

Besides, the stress in _non-commutative geometry_ (both in Connes-
and Artin-style) in on _commutative_. Connes-style might also be called
‘K-theory of $C^*$-algebras’ and they use the topological
information of K-theoretic terms in the commutative case as guidance to
speak about geometrical terms in the nocommutative case. Similarly,
Artin-style might be called ‘graded homological algebra’ and they
use Serre’s homological interpretation of commutative geometry to define
similar concepts for noncommutative algebras. Hence, non-commutative
geometry is that sort of non-geometry which is almost
commutative…

But the main point of naming our subject
non-geometry is to remind us not to rely too heavily on our
(commutative) geometric intuition. For example, we would expect a
manifold to have a fixed dimension. One way to define the dimension is
as the trancendence degree of the functionfield. However, from the work
of Paul Cohn (I learned about it through Aidan Schofield) we know that
quasi-free algebras usually do’nt have a specific function ring of
fractions, rather they have infinitely many good candidates for it and
these candidates may look pretty unrelated. So, at best we can define a
_local dimension_ of a noncommutative manifold at a point, say given by
a simple representation. It follows from the Cunz-Quillen tubular
neighborhood result that the local ring in such a point is of the
form

$M_n(\mathbb{C} \langle \langle z_1,\ldots,z_m \rangle
\rangle) $

(this s a noncommutative version of the classical fact
than the local ring in a point of a d-dimensional manifold is formal
power series $\mathbb{C} [[ z_1,\ldots,z_d ]] $) but in non-geometry both
m (the _local_ dimension) and n (the dimension of the simple
representation) vary from point to point. Still, one can attach to the
quasi-free algebra A a finite amount of data (in fact, a _finite_ quiver
and dimension vector) containing enough information to compute the (n,m)
couples for _all_ simple points (follows from the one quiver to rule them
all paper or see this for more
details).

In fact, one can even extend this to points
corresponding to semi-simple representations in which case one has to
replace the matrix-ring above by a ring Morita equivalent to the
completion of the path algebra of a finite quiver, the _local quiver_ at
the point (which can also be computer from the one-quiver of A. The
local coalgebras of distributions at such points of
Kontsevich&Soibelman are just the dual coalgebras of these local
algebras (in math.RA/0606241 they
merely deal with the n=1 case but no doubt the general case will appear
in the second part of their paper).

The case of the semi-simple
point illustrates another major difference between commutative geometry
and non-geometry, whereas commutative simples only have self-extensions
(so the distribution coalgebra is just the direct sum of all the local
distributions) noncommutative simples usually have plenty of
non-isomorphic simples with which they have extensions, so to get at the
global distribution coalgebra of A one cannot simply add the locals but
have to embed them in more involved coalgebras.

The way to do it
is somewhat concealed in the
third version of my neverending book (the version that most people
found incomprehensible). Here is the idea : construct a huge uncountable
quiver by taking as its vertices the isomorphism classes of all simple
A-representations and with as many arrows between the simple vertices S
and T as the dimension of the ext-group between these simples (and
again, these dimensions follow from the knowledge of the one-quiver of
A). Then, the global coalgebra of distributions of A is the limit over
all cotensor coalgebras corresponding to finite subquivers). Maybe I’ll
revamp this old material in connection with the Kontsevich&Soibelman
paper(s) for the mini-course I’m supposed to give in september.

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a good day at the arxiv

Published June 13, 2006 by lievenlb

The
arXiv is a bit like cable tv : on certain days there seems to be nothing
interesting on, whereas on others it’s hard to decide what to see in
real time and what to record for later. Today was one of the better
days, at least on the arXiv. Pavel Etingof submitted the
notes of a course he gave at ETH in the spring and summer of 2005 Lectures on
Calogero-Moser systems. I always sympathize with people taking time
to explain what they are interested in to non-experts, especially if
they even take more time to write up course notes so that the rest of us
can also benefit from these talks. Besides, it is always more rewarding
to learn a topic from a key-figure such as Etingof, rather than sitting
through talks on this given by people who only embrace a topic as a
career move. However, as I’m no longer that much into Calogero-Moser
stuff I’ve put Pavel’s notes in recording mode as I definitely have to
spend some time getting through that other paper posted today : Notes on A-infinity
algebras, A-infinity categories and non-commutative geometry. I by
Maxim
Kontsevich and Yan
Soibelman. They really come close to things that interest me right
now and although I’m not the greatest coalgebra-fan, they may give me
just enough reasons to bite the bullet. On a different topic : with
plenty of help from Jacques Distler, my
neverending planet
is now also serving MathML, but you need to view it using Firefox and
have all the required fonts
installed.

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writing with gloves on

Published June 8, 2006 by lievenlb

Okay, let’s have it out in the open :

I’m officially diagnosed as being depressed by both PD1 and PD2!

Coming from the two top experience-experts on my mood swings, I’d better take this
seriously. So, do they come up with an explanation for this ‘depression’?

PD1 blames it on the celebrated mid-life-crisis which in her world is merely the generic phrase uttered when a parent does something ‘odd’.

If thePartner wants to spend some time among old friends, or wants to get involved in community work, it’s called ‘mid-life crisis’.

When both of us join a demonstration for the first time in over a decade, it’s MLC etc. etc.

In recent years I heard her say the MLC- phrase often enough referring to her friends’ parents and thePartner but somehow I always got away, until recently…

PD2 blames it on my turning 48 last week, a fact I cannot deny but then, what’s so special about 48? I don’t get it.

Feeble as their explanations may be, they still may have a point. Sure, some losses do affect me. Some recent, some imminent, some unfortunately permanent, some hopefully temporary…

I realise this is a bit cryptic to the uninitiated, but then I’ve given up writing about personal stuff a long time ago (to the dismay of PD2 who would welcome more web-presence when self-googling…).

But wait… Hey, that may be part of the problem :

I’ve given up writing about so many things recently that there’s hardly anything sufficiently interesting left to write about.

In the post-Dutroux scare I did remove all pictures and references to our daughters from my web-pages, for you don’t want to know the weirdos that have a look at it and you definitely do not want to think about what they might do when they obtain my address from the university web-page….

Surely a valid point. So, away with all writing about personal stuff.

Then, more recently (and I hope at least some of you noticed it…) I’ve imposed a ban on critical postings about people or events going on in noncommutative algebra/geometry. The reason behind this decision is personal, so if I didnt tell you in private you’ll never find it here.

Speaking about this with Paul Smith at the last Oberwolfach, he had an hilarious reply.

“I wouldn’t say you were critical. I’d say you are sometimes pretty intense and I love it, as long as I’m not on the receiving end…”

But see, that’s just the problem. Mathematicians are so vane that there is always someone who feels to be on the receiving end!

Let’s say, hypothetically speaking, that I write a somewhat critical post about the ongoing cluster-algebra hype, we all know some people who will not like it. Ditto about (again hypothetically…) symplectic-reflection algebras, ditto about etc. etc.

Compare this with the entertaining about-life-or-death fights going on in physics-blogs. If you don’t know what I’m talking about and want to have a good laugh, have a go at the comments to this Not Even Wrong Post.

Possibly, I should come to terms with the fact that blogging is an activity which will never be tolerated by the autism-enriched environment of mathematicians and that I should just give it up.

Or, perhaps, I should regain my writing-freedom and blog about whatever I feel strongly about at that particular moment in time (and remember, I do suffer from violent mood-swings so these opinions may change overnight…), be it critical or if you want ‘intense’, and hope that not too many will think they are on the receiving end…

I realize that I will sometimes be accused of ‘jealousy’, sometimes of being ‘frustrated’. But, let’s face it : bottling up one’s frustrations, that’s precisely the thing that leads to a genuine depression…

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sage

Published May 24, 2006 by lievenlb

SAGE
(which stands for ‘Software for Algebra and Geometry
Experimentation’) includes and offers an interface to GAP, Singular,
Maxima and even PARI as
well as an interface to other packages such as Maple, Magma and
Mathematica (see here
for a full list of its features). More importantly, Sage offers a binary
for both PPC and Intel-Macs! I did check this out and it runs without
problems, in fact, after this initial check I installed from the sources
on my MacBook Pro and after one hour of compiling I did have working
(though not full) versions of GAP, Maxima and Singular. At first I
was a bit worried that only small subsets of the three systems were
installed, but it is quite easy to extend your Sage with additional
packages. From the Unix-prompt do a sage -optional
and you will get a list of all (additional) packages you have already
installed and those available for installation. SAGE is pretty well
documented with tutorials and reference manual to be found here. Even if you do not want to learn (yet) the Sage-commands but just
want to continue using the programs under its hood, this is pretty easy.
For example, to get to Maxima, you only have to type
!maxima from the sage-prompt to open up a maxima-session
(and similarly for Gap and Singular).

Bill
Schelter’s Affine-package is not included, but you can load and install
it from the maxima-prompt by load("affine.lisp"); but some
commands such as ‘fast_central_elements’ do not seem to
work as expected (or maybe I forgot the drill over the years, I’ll try
it out again).

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hold on to those PPC macs

Published May 22, 2006 by lievenlb

On my
return from O a brand new 15inch MacBook Pro lie waiting in
my office. By that evening I had wrecked the system to the extend that I
could no longer login and had to reinstall from scratch… I was
about to trow it away but tried it out for a few more days and
eventually began to understand it a bit. In short : the new Intel Macs
promise to be really good hardware, unfortunately some essential
software lags behind, so if you want a stress-free Mac-life… hold
on to your PPC mac a few months longer. If you are impatient and want to
learn some of the pitfalls, read on… I’m ashamed to admit this
but the first thing I did on my new machine was to create a WindowsXP
partition… BootCamp does what it
promises to do and is extremely easy to use once you can start it. The
installation guide does tell that you jave to update your systems
software and firmware, but that’s what you do anyway after a new
install, right? Wrong! You update the software but _not_ the
firmware and it took me some time to come to this simple conclusion. How
to check whether your firmware is up to date? Go under the apple to
‘About this Mac’, click on ‘More Info’ and look at your
‘ Boot ROM Version:’ if it says MBP11.0055.B03 you’re ok, if not
you have to install the newest firmware which is a slightly terrifying experience
with soundsignals included, but works fine. Once this is done, you can
start BootCamp and have a Windows partition in no time. At a certain
moment you have to decide on possible partition-formats for the Windows
part, I choose the ‘Fat’ option to be able to swap files across
the partitions. Next, what does a mathematician wants from a
computer? To run LaTeX! I’ve installed LaTeX on more Macs than I
remember so I continued on automatic pilot, getting Gerben Wierda’s i-Installer, startd it up and
… my machine froze! Nothing, not even a ‘Force Quit’, was
possible any more. Today, there is a clear warning message as the
i-Installer page (i don’t recall seeing it there last week, but then it
is a recent problem. Things broke down on May 11th when I was still in
O)

WARNING: i-Installer on Mac OS X 10.4.6 may trigger
the Mac OS X 10.4.6 bug that partially freezes your system. May 2006:
i-Installer did work perfectly on Mac OS X 10.4.3, the version of Tiger
that was shipped with the Developer Transition Kit. When the first intel
machines were sold by Apple, these contained 10.4.4 and on that system,
i-Installer experiences troubles because of problems deep inside Apple’s
Frameworks. The only way I could solve this was to make i-Installer a
PowerPC-only application again and ask for Apple’s help to determine
where the problem was. So far, this has been s slow process without any
noticeable results. The PowerPC-only version worked fine until Apple
released 10.4.6 and especially the latest upgrades (Security Upgrade
2006-003 and maybe QuickTime). As I am writing this (May 21) a
completely updated Mac OS X 10.4.6 on intel will partially freeze in
various circumstances, triggered by various applications (MatLab,
i-Installer, etc.). Sadly, the just released MacBook (successor of the
iBook) is shipping with this broken version of the OS. Hence, there is
now no i-Installer that reliably works on intel machines with recent OS
versions and even worse, i-Installer may trigger a nasty bug in recent
Mac OS X intel versions.

Scary isn’t it? You have a
brand new expensive machine but cannot typeset a single paper…
Fortunately, the TeXShop
page not only mentions the problem, but also a workaround

On May 11, 2006, Apple provided security updates for Mac OS
X. These updates broke i-Installer on Intel (it continues to work on
PowerPC). If you have an Intel Mac and you have installed this update,
you must use the MacTeX install package until this problem is fixed.
Once TeX is installed, it works fine.

The first
assertion is true : installing the MacTeX package gives you a working
TeX-installation, with TeXShop, Excalibur, BibTeX and i-Installer coming
for free. But don’t think the i-Installer problem has been solved, I
tried it out and voila another ice-age… So far so good but
sometimes we like to compute things, don’t we? Like some commutative
algebra or algebraic geometry things via Singular? I remembered to
install this via the Fink
project but already their news-items are not very promissing

A preliminary version of Fink for the Intel architecture is
now ready. No binary packages are available, and things are still rough
around the edges, but it should be usable if you are patient! To
install it, you need to install the XCode compiler and SDK packages (at
minimum). Then you need to get the file fink-0.24.14.tar.gz from the
Sourceforge file release page for Fink, expand the file, and run the
command ./bootstrap.sh . At the end of the bootstrap process, run fink
selfupdate and you’ll get the currently available packages. At last
check, there were about 1750 packages in the “stable” tree,
but about 150 of those did not build. When things are truly stable,
another annoucement will be made here.

The normal
FinkCommander didn’t work either but then I found a version which does
at Charles K. C. Lo’s
Homepage. I verified it by having the fink-TeTeX package installed
(which works!) and then I wanted to do a Singular-install… Things
seemed to start off well (once you change the freferences to install
also unstable packages) but then the installation procedure halted with
the message

Failed: phase compiling: singular-3.0.1-1013
failed Before reporting any errors, please run “fink
selfupdate” and try again. If you continue to have issues, please
check to see if the FAQ on fink’s website solves the problem. If not,
ask on the fink-users or fink-beginners mailing lists. As a last
resort, you can try e-mailing the maintainer directly: Michael
Brickenstein bricken at mathematik.uni-kl.de Note that many
fink package maintainers do not (yet) have access to OS X on Intel
hardware, so you may have better luck on the mailing lists.

So, maybe I should just donate my MacBook Pro to the
Fink-project? A similar problem with installing Maxima… I didn’t
even try out GAP via Fink but went for a niversal Unix-installation for
GAP and this WORKED! even with all packages and tables and the whole I
dont know what. Thank you, GAPpers, perhaps all algebraists on Intel
Macs should shift to GroupTheory? But hey! My Intel-Mac does have a
WindowsXP partition… So, I did a binary Windows install of
Singular and Maxima and both work without problems. Still, it is a
strange situation. Fortunately, I did resolve these issues but that will
have to wait until tomorrow…

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noncommutative geometry master class

Published April 19, 2006 by lievenlb

Here is
the program of the NOG-masterclass 2006 : april 2006

Jacques Alev (Univ. Reims)
“Deformations of Poisson algebras”. : 18-21 april
Geert Van de Weyer
(FWO-Antwerp) “Noncommutative Poisson geometry” : 25-28
april

may 2006

Fred Van Oystaeyen (Univ. Antwerp)
“Dequantization” : 2-5 May
Raf
Bocklandt (FWO-Antwerp) “Geometric invariant theory” :
16-19 May
Bruce Westbury
(Univ. Warwick) “Magic square and magic triangle” : 30
May-2 June

june 2006

Anthony Joseph
(Weizmann Inst.) “Invariants for biparabolic actions and the
geometry of their coadjoint orbits” : 5-9 june
Michel Brion
(Univ. Grenoble) TBA : 12-17 june
Claus Michael
Ringel (Univ. Bielefeld) “Tilted algebras and cluster tilted
algebras” : 15-20 june
Karin Erdmann (Univ.
Oxford) TBA : 19-23 june

Lecture rooms and latest changes
can be consulted here.

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why mathematicians can’t write

Published March 29, 2006 by lievenlb

The Music of the
Primes will attract many young people to noncommutative geometry a
la Connes. It would be great if someone would spend a year trying to
write a similar pamphlet in favour of noncommutative _algebraic_
geometry, but as I mentioned before chances are not very high as most
mathematicians are unwilling to sacrifice precision and technical detail
for popular success. Still, perhaps we should reconsider this position.
A fine illustration why most mathematicians cannot write books for a
bigger audience is to be found in the preface to the book “The
problems of mathematics” (out of print or at least out of
amazon.com) by the Warwick mathematician Ian Stewart.
Below I quote a fraction from his ‘An interview with a
mathematician…’

(I)nterviewer : … So,
Mathematician : what delights do you have in store for us?
(M)athematician : I thought I’d say a bit about how you can get a TOP
but non-DIFF 4-manifold by surgery on the Kummer surface. You see,
there’s this fascinating cohomology intersection form related to the
exceptional Lie algebra $E_8$, and…
(I) : That’s
fascinating.
(M) : Thank you.
(I) : Is all that
gobbledegook really significant?
(M) : Of course! It’s one of the
most important discoveries of the last decade!
(I) : Can you
explain it in words ordinary mortals can understand?
(M) : Look,
buster, if ordinary mortals could understand it, you wouldn’t need
mathematicians to do the job for you, right?
(I) : I don’t want
the technical details. Just a general feeling for what’s going on.
(M) : You can’t get a feeling for what’s going on without
understanding the technical details.
(I) : Why not?
(M) :
Well, you just can’t.
(I) : Physicists seem to manage.
(M)
: But they work with things from everyday experience…
(I) :
Sure. ‘How gluon antiscreening affects the colour charge of a
quark.’ ‘Conduction bands in Gallium Arsenide.’ Trip over
‘em all the time on the way to work, don’t you?
(M) : Yes,
but…
(I) : I’m sure that the physicists find all the
technical details just as fascinating as you do. But they don’t let them
intrude so much.
(M) : But how can I explain things properly if I
don’t give the details?
(I) : How can anyone else
understand them if you do?
(M) : But if I skip the fine
points, some of the things I say won’t be completely true! How can I
talk about manifolds without mentioning that the theorems only work if
the manifolds are finite-dimensional paracompact Hausdorff with empty
boundary?
(I) : Lie a bit.
(M) : Oh, but I couldn’t do
that!
(I) : Why not? Everybody else does.
(M) : But, I
must tell the truth!
(I) : Sure. But you might be prepared to
bend it a little, if it helps people understand what you’re doing.
(M) : Well…

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