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Tag: noncommutative

MyLife@300dpi

Published March 20, 2006 by lievenlb

Three years ago I did spend three weeks next to my Canonscan, painstakingly scanning all individual pages of every preprint I ever wrote. Next, I converted every page to PDF, resized it (in order to control the size) and bundled them into PDF-files. A typical preprint would take me roughly three quarters of an hour and the final result was mediocre. For example, here a blown-up sample from the original 1992 ‘Moduli
spaces of right ideals of the Weyl algebra’ -preprint, resulting in a 1.7Mb PDF-file

Recentlty, the department bought a Ricoh-copier which makes scanning a lot more fun. To scan a preprint at 300dpi and convert it into a single PDF-file takes under a minute (actually, downloading the file using a web-interface takes longer…). For this particular preprint, the resulting PDF-file took up 1.2Mb and looks a lot nicer

Still, 1.2Mb is a huge file but converting it to a DjVu-file (DjVu=deja vu) using the handy Any2DjVu Service gives us a mere 236Kb file which comes a lot closer to the filesize of a PDFLaTeX-file and the output is still very legible

So, I decided to rescan my entire life at 300dpi and convert it into DjVu. Next, I got the MOPP-package (MOPP = My Online Publications Page) working using the instructions from this page and some obvious MacOSX-modifications (if I can do it, so can you but perhaps I’ll write up the details in another post, just to remind myself). You can see the result at my homepage. I’ll update the latter one regularly (there are still some preprints missing, as are all my courses etc. and cross-references) and only afterwards I’ll update my homepage again. So far there is 250Mb to download (including all versions of the noncommutative geometry@n book, including the published ones…) so this should keep you busy for a while…

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noncommutative topology (4)

Published February 9, 2006 by lievenlb

For a
qurve (aka formally smooth algebra) A a *block* is a (possibly infinite
dimensional over the basefield) left A-module X such that its
endomorphism algebra $D = End_A(X)$ is a division algebra and X
(considered as a right D-module) is finite dimensional over D. If a
block X is finite dimensional over the basefield, we call it a *brick*
(aka a *Schur representation*). We want to endow the set of all blocks
with a topology and look at the induced topology on the subset of
bricks. It is an old result due to Claus Ringel
that there is a natural one-to-one correspondence between blocks of A
and algebra epimorphisms (in the categorical sense meaning that identify
equality of morphisms to another algebra) $A \rightarrow M_n(D) =
End_D(X_D)$. This result is important as it allows us to define a
partial order on teh set of all A-blocks via the notion of
*specialization*. If X and Y are two A-blocks with corresponding
epimorphisms $A \rightarrow M_n(D),~A \rightarrow M_m(E)$ we say that Y
is a specialization of X and we denote $X \leq Y$ provided there is an
epimorphism $A \rightarrow B$ making the diagram below commute

$\xymatrix{& M_n(D) \\\ A \ar[ru] \ar[r] \ar[rd] & B \ar[u]^i
\ar[d]^p \\\ & M_m(E)} $

where i is an inclusion and p is a
onto. This partial ordering was studied by Paul Cohn, George Bergman and
Aidan Schofield who use
the partial order to define the _closed subsets_ of blocks to be
those closed under specialization.

There are two important
constructions of A-blocks for a qurve A. One is Aidan’s construction of
a universal localization wrt. a *Sylvester rank function* (and which
should be of use in noncommutative rationality problems), the other
comes from invariant theory and is related to Markus Reineke’s monoid in
the special case when A is the path algebra of a quiver. Let X be a
GL(n)-closed irreducible subvariety of an irreducible component of
n-dimensional A-representations such that X contains a brick (and hence
a Zariski open subset of bricks), then taking PGL(n)-equivariant maps
from X to $M_n(\mathbb{C})$ determines a block (by inverting all central
elements). Now, take a *sensible* topology on the set of all A-bricks.
I would go for defining as the open wrt. a block X, the set of all
A-bricks which become simples after extending by the epimorphism
determined by a block Y such that $Y \leq X$. (note that this seems to
be different from the topology coming from the partial ordering…).
Still, wrt. this topology one can then again define a *noncommutative
topology* on the Abelian category $\mathbf{rep}~A$ of all finite
dimensional A-representations
but this time using filtrations with successive quotients being bricks
rather than simples.

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get your brain subscribed to

Published February 8, 2006 by lievenlb

In the
‘subscribe
to my brain’ post I promised to blog on how-to get your own

button up and running on your homepage. It seems rather unlikely
that I’ll ever keep that promise if I don’t do it right away. So, here
we go for a quick tour :

step 1 : set up a rudimentary
FoaF-file : read the FoaF post if
you dont know what it’s all about. The easiest way to get a simple
FoaF-file of your own is to go to the FoaF-a-matic
webpage and fill in the details you feel like broadcasting over the
web, crucial is your name and email information (for later use) but
clearly the more details you fill out and the more Friends you add the
more useful your file becomes. Click on the ‘foaf-me’ button and
copy the content created. Observe that there is no sign of my email
adress, it is encrypted in the _mbox_sha1sum_ data. Give this
file a name like _foaf.rdf_ or _myname.rdf_ and put it on
your webserver to make it accessible. Also copy your
_mbox_sha1sum_ info for later smushing.

step 2 : subscribe to online services and modify your
online-life accordingly : probably you have already a few of
these accounts, but if not, take a free subscription just for fun and
(hopefully) later usage to the following sites :

  • del.icio.us a social bookmarks manager
  • citeUlike a service to
    organise your academic papers
  • connotea a reference management
    service for scientists
  • bloglines a web-based personal news
    aggregator
  • 43things a
    ‘What do you want to do with your life?’ service
  • audioscrobbler a database that
    tracks listening habits and does wonderful things with statistics
  • backpackit a ‘be better organized’ service (Update october 2017 : Tom Howard emails: “I thought I’d reach out because we’ve just updated our guide which reviews the best alternatives to Backpack. Here’s the link”
  • flickr an online photo management and
    sharing application
  • technorati a Google-for-weblogs
  • upcoming a social event
    calendar
  • webjay a playlist
    community

So far, I’m addicted to del.icio.us and use
citeUlike but hardly any of the others (but I may come back to this
later). The great thing about these services is that you get more
value-information back if you feed more into the system. For example, if
you use del.icio.us as your ‘public’ bookmarks-file you get to
know how many other people have bookmarked the same site and you can
access their full bookmarks which often is a far more sensible way to
get at the information you are after than mindless Googling. So, whereas
I was at first a bit opposed to the exhibisionist-character of these
services (after all, anyone with web-access can have a look at
‘your’ info), I’ve learned that the ‘social’ feature of
these services can be beneficial to get the right information I want.
Hence, the hardest part is not to get an account with these services but
to adopt your surfing behavior in such a way that you maximize this
added value. And, as I mentioned before, I’m doing badly myself but hope
that things will improve…

step 3 : turn these
accounts into an OPML file : Knowing the URL of your foaf-file
and sha1-info (step 1) and your online accounts, go to the FOAF Online Account
Description Generator and feed it with your data. You will then get
another foaf-file back (save the source in a file such as
_accounts.rdf_ and put it on your webserver). Read the Lost Boy’s
posts Subscribe to my
brain and foaf:
OnlineAccount Generator for more background info. Then, use the SubscribeToMyBrain-
form to get an OPML-file out of the account.rdf file and your sha1.
Save the source as _mybrain.opml_.

step 4 :
add/delete information you want : The above method uses generic
schemes to deduce relevant RSS-data from an account name, which works
for some services, but doesn’t for all. So, if you happen to know the
URL of RSS-feeds for one of these services, you can always add it
manually to the OPML-file (or delete data you don’t want to
publish…). My own attitude is to make all public web-data
available and to leave it to the subscriber to unsubscribe those parts
of my brain (s)he is not interested in. I know there are people whoo are
mainly interested to find out whether I put another paper online, would
tolerate some weblog-posts but have no interest in my musical tast,
whereas there are others who would like me to post more on 43things,
flickr or upcoming and don’t give a damn about my mathematics…
Apart from these online subscriptions, it is also a good idea to include
additional RSS-feeds you produce, such as those of your weblog or use my
Perl
script to have your own arXiv-feeds.

step 5 : make
your ‘subscribe to my brain’-button : Now, put the
OPML-file on your webserver, put the button

on your
homepage and link it to the file. Also, add information on your site,
similar to the one I gave in my own
subscription post so that your readers know what to do when do want
to subscribe to (parts of) your brain. Finally, (and optionally though
I’d wellcome it) send me an email with your URL so that I can subscribe
(next time you’re in Antwerp I’ll buy you a beer) and for the first few
who do so and are working in noncommutative geometry and/or
noncommutative algebra, I’ll send a copy of a neverending book. Mind
you, this doesn’t apply to local people, I’m already subscribed to their
brain on a daily basis…

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