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Tag: Connes

mathML versus LaTeXRender

Published March 21, 2007 by lieven

No math
today. If you’re interested in the latest on noncommutative geometry,
head over to the NCG-blog where Alain Connes has a post on
Time.
Still, Alain’s post is a good illustration of what Ill be rambling about
TeX and how to use it in a blog.

If you’re running a math-blog,
sooner or later you want to say something more than new-age speak like
‘points talking to each other’ and get to the essence of it. In short,
you want to talk math and it’s a regrettable fact that math doesnt go
well with ASCII. In everyday life we found a way around this : we all
use TeX to write papers and even email-wise (among mathematicians) we
write plain TeX-commands as this language is more common to us than
English. But, plain TeX and the blogosphere don’t mix well. If you’re
expecting only professional mathematicians to read what you write, you
might as well arXiv your thoughts. Im convinced the majority of people
coming here (for whatever reason) dont speak plain-TeX. Fortunately,
there is technology to display TeX-symbols on a blog. Personally, I was
an early adapter to
LaTeXRender and even today a
fair share of page-views relates to the few
posts I did on
how to get latexrender working on a mac. Some time ago I
switched to mathML and now I’m
regretting I ever did…

Mind you, I’m convinced that mathML is the
‘proper’ way to get TeX to the internet but there are at the moment some
serious drawbacks. For starters, it is highly user-unfriendly. You
simply cannot expect people to switch browsers (as well as installing
extra fonts) just because they come to your site (or you have to be a
pretty arrogant git). Speaking for myself, Im still having (against my
better judgment) Safari as my default browser, so when I come to a site
like the n-category cafe I just
skim the plain-text in between and if (and only if) the topic interests
me tremendously I’ll allow myself to switch to Flock or Firefox to read
the post in detail. I’m convinced most of you have a similar
surfing-attitude. MathML also has serious consequences on the
server-side. If you want to serve mathML you have to emit headers which
expect everything to follow to be purified XHTML. If I ever forget a
closing tag in a post, this is enough to break down NeverEndingBooks to
all Firefox-users. I’ve been writing HTML since the times when the best
browser around was something called NCSA Mosaic so Ive a
pretty lax attitude to end-tags (especially in IMG-tags) and Im just
getting too old to change these bad habbits now… It seems I’m not the
only one. Many developers of WordPress-plugins write bad XHTML-code, so
the last couple of weeks I’ve been spending more time fixing up code
than writing posts. If you want to run a mathML-wordpress site you might
find the following hints helpfull. If you get a ‘yellow screen of
death’ when viewing your site with Firefox, chances are that one of your
plugin-authors missed a closing tag in the HTML-rendering of his/her
plugin. As a rule of thumb : go for the IMG-tags first! I’m sorry to
say, but Latexrender-Steve
is among the XHTML-offenders. (On a marginal note, LaTeXrender also has
its drawbacks : to mathematicians this may seem incredible but what
Latexrender does to get one expression displayed is to TeX an entire
file, get the image from the ps-file turn it into a gif and display it,
so one gets a GIF-folder of enrmous proportions. Hence, use Latexrender
only if you have your own server and dont have to care about memory
constraints. Another disadvantedge was that the GIFs were displayed with
a vertical offset, but this has been solved recently (use the ‘offset
beta’ files in the distribution)). Wrt. to that offset-beta version, use
this latex.php file instead (I
changed the IMG-line). Some plugins may not serve the correct headers
to display mathML. So, if you want to allow readers to have a
printer-friendly version of your mathML-post, get the WP-print plugin BUT
change to this wp-print.php file in order to
send the proper headers. Sometimes there are just forgotten lines/tags
in the code, such as in the [future calendar plugin](http://anthologyoi.com/wordpress/plugins/future-posts-calendar-
plugin.html). So, please use this version
of the future.calendar.php file. And so on, and so on. The joys of
trying to maintain a mathML-based blog… So, no surprise I’m seriously
considering to ditch mathML and change to normal headers soon. One of
the things I like about LaTeXRender is that it can be extended, meaning
that you can get your own definitions and packages loaded whereas with
mathML you’re bound to write iTeX, which Ill never manage. But, again,
mathML will be the correct technology once all major browsers are mathML
capable and the font-problem is resolved. Does anyone know whether
Safari 3 (in Leopard, that is Mac OS 10.5 to the rest of you) will be
mathML-able?

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stalking the Riemann hypothesis

Published March 12, 2007 by lieven

There
seems to be a neverending (sic) stream of books and posts on the
Riemann hypothesis. A while ago I
wrote about du Sautoy’s The music of primes and over a snow-sparse
skiing holiday I read Stalking the Riemann Hypothesis by Daniel N. Rockmore.
Here’s the blurb

Like a hunter who sees ‘a bit of blood’
on the trail, that’s how Princeton mathematician Peter Sarnak describes
the feeling of chasing an idea that seems to have a chance of success.
If this is so, then the jungle of abstractions that is mathematics is
full of frenzied hunters these days. They are out stalking big game: the
resolution of ‘The Riemann Hypothesis’, seems to be in their sights. The
Riemann Hypothesis is about the prime numbers, the fundamental numerical
elements. Stated in 1859 by Professor Bernhard Riemann, it proposes a
simple law which Riemann believed a ‘very likely’ explanation for the
way in which the primes are distributed among the whole numbers,
indivisible stars scattered without end throughout a boundless numerical
universe. Just eight years later, at the tender age of thirty-nine
Riemann would be dead from tuberculosis, cheated of the opportunity to
settle his conjecture. For over a century, the Riemann Hypothesis has
stumped the greatest of mathematical minds, but these days frustration
has begun to give way to excitement. This unassuming comment is
revealing astounding connections among nuclear physics, chaos and number
theory, creating a frenzy of intellectual excitement amplified by the
recent promise of a one million dollar bountry. The story of the quest
to settle the Riemann Hypothesis is one of scientific exploration. It is
peopled with solitary hermits and gregarious cheerleaders, cool
calculators and wild-eyed visionaries, Nobel Prize-winners and Fields
Medalists. To delve into the Riemann Hypothesis is to gain a window into
the world of modern mathematics and the nature of mathematics research.
Stalking the Riemann Hypothesis will open wide this window so that all
may gaze through it in amazement.

Personally, I prefer
this book over du Sautoy’s. Ok, the first few chapters are a bit pompous
but the latter half gives a (much) better idea of the ‘quantum chaos’
connection to the RH. At the Arcadian Functor, there was the post
Riemann rumbling on
pointing to the book Dr, Riemann’s zeros by Karl Sabbagh.

From
what Kea wrote I understand it also involves quantum chaos. Im not sure
whether I’ll bother to buy this one though, as one reviewer wrote

I stopped reading this rather fast: it had errors in it,
and while a lovely story for the non-mathematician, for anyone who knows
and loves mathematics (and who else really does buy these books?) it’s
really rather frustrating that, after a few chapters, you’re still not
much clearer on what Reimann’s Hypothesis really is.
Not worth the
money: try The Music of the Primes (utterly brilliant) instead. This
book simply cannot begin to compete.

The last line did it
for me, but then “Des gouts et des couleurs, on ne dispute pas”.
Speaking of which, over at Noncommutative geometry there was a post
by Alain Connes on his approach to the Riemann Hypothesis Le reve mathematique which
some found

A masterpiece of
mathematical blogging, a post by Alain Connes in Noncommutative
Geometry. Strongly recommended.

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noncommutative geometry : a medieval science?

Published February 28, 2007 by lieven

According to a science article in the New York Times, archeologists have discovered “signs of advanced math” in medieval mosaics. An example of a quasi-crystalline Penrose pattern was found at the Darb-i Imam shrine in Isfahan, Iran.

“A new study shows that the Islamic pattern-making process, far more intricate than the laying of oneโ€šร„รดs bathroom floor, appears to have involved an advanced math of quasi crystals, which was not understood by modern scientists until three decades ago. Two years ago, Peter J. Lu, a doctoral student in physics at Harvard University, was transfixed by the geometric pattern on a wall in Uzbekistan. It reminded him of what mathematicians call quasi-crystalline designs. These were demonstrated in the early 1970s by Roger Penrose, a mathematician and cosmologist at the University of Oxford. Mr. Lu set about examining pictures of other tile mosaics from Afghanistan, Iran, Iraq and Turkey, working with Paul J. Steinhardt, a Princeton cosmologist who is an authority on quasi crystals and had been Mr. Lu’s undergraduate adviser. ”

Penrose tilings are one of the motivating examples of Alain Connes’ book as there is a $C^* $-algebra associated to it. In fact, the algebra is AF ( a limit of semi-simple finite dimensional algebras) so is even a formally smooth algebra in Kontsevichian noncommutative geometry (it is remarkable how quickly one gets used to silly terminology…). However, the Penrose algebra is simple, so rather useless from the point of view of finite dimensional representations… Still, Connesian noncommutative geometry may be a recent incarnation of the medieval Tehran program (pun intended). Thanks to easwaran for the link (via Technorati).

Added, March 1 : I haven’t looked at the Connes-Marcolli paper A walk in the noncommutative garden for a while but now that I do, I see that they mentioned the above already at the end of their section on Tilings (page 32). They also include clearer pictures.


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