If you happen to have a couple of hours to kill, you might have a look at the
arXiv trackback policy debate over at Jacques Distler’s blog Musings. But before you dive into this it is perhaps useful to glance at what went before. Distler did pester (his wording, not mine) the arXiv to add trackbacks from certain weblogs to hep-th postings (i’m not aware of math-papers having trackbacks). So far so good, the more information about a paper the better i’d say, but it seems that not all weblogs’ trackbacks are allowed… A small commitee has the power to divide hep-th people into ‘crackpots’ or ‘active researchers’
(mother nature may very well decide to add all stringtheorists from the second category to the first in a couple of years… but, i’m digressing) and accordingly censor specific blogs and frustrate their authors, Peter Woit’s blog Not Even Wrong being the main victim. The whole trackback-policy is yet another futile academics power-game. Futile because there is an obvious way around it : type into Technorati either the arXiv-number or title or author and you will get all (!) weblog postings mentioning the paper (Technorati even has a slider if you only want to read postings with ‘authority’ rather than all). Perhaps one of the more tech-abled stringtheorists should spend an afternoon to write a
bookmarklet to perform this trick from any arXiv abstract page…
neverendingbooks Posts
Once
upon a time, not so long before the video-games era, people needed the
command-line and knowledge of esoteric commands like _examine_,
_look_, _take_, _drop_, _go south_ etc. to
get into the mysterious worlds of dungeons &
dragons. If you have nostalgia to the heroic times of text-based
adventure games (nowadays called IF for _interactive fiction_),
there is a short message : get Inform(ed)! Here’s a
slightly longer message for those who have a mac running OSX and want to
know the quickest way to get to a screen like 
and start
playing Christminster (or another of 300 IF-games) (if you’re on a
different system, things will be just as simple but you’ll have to find
it out yourself starting from the Inform-Z
machine page). step 1 : Get a
copy of an inform installation and expand it to get an
Inform-folder and place this in your Home-folder. step 2
: Go in the Finder to Inform/Games/MyGame1 and double click on
the _MyGame1.command_ file. A Terminal window will open and exit
and you will see that a new file appeared in the Folder :
_MyGame1.z5_. Double click it and a warning message will appear
that this is the first time you will open _Zoom_, tell it’s ok
and Zoom will launch and you can play your first (though primitive)
Inform game! step 3 : If you want to play other
games (such as Christminster), go to the Z-
code archive and pick one of the 346 games. For example, click on
the minster.z5 link and the file will download to your
Desktop. Place it in the Inform/Games folder (not necessary) double
click it and you should see the above wellcoming message. That’s it,
start playing. step 4 : If you don’t know how to
play such games, there are excellent tutorials
available on the Inform site.
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.