Handy, if you want to (re)populate your RSS-aggregator with interesting mathematical blogs, is their graphical presentation of (nearly) all math-blogs ordered by type : group blogs, individual researchers, teachers and educators, journalistic writers, communities, institutions and microblogging (twitter). Links to the last 7 posts are given so you can easily determine whether that particular blog is of interest to you.

The three people behind the project, Felix Breuer, Frederik von Heymann and Peter Krautzberger, welcome you to send them links to (micro)blogs they’ve missed. Surely, there must be a lot more mathematicians with a twitter-account than the few ones listed so far…

Even more convenient is their list of latest posts from their collection, ordered by date. I’ve put that page in my Bookmarks Bar the moment I discovered it! It would be nice, if they could provide an RSS-feed of this list, so that people could place it in their sidebar, replacing old-fashioned and useless blogrolls. The site does provide two feeds, but they are completely useless as they click through to empty pages…

While we’re on the topic of math-blogging, the results of the ‘What should we write about next?’-poll that ran the previous two days on the entry page. Of all people visiting that page, 2.6% left suggestions.

The vast majority (67%) wants more posts on noncommutative geometry. Most of you are craving for introductions (and motivation) accessible to undergraduates (as ‘it’s hard to find quality, updated information on this’). In particular, you want posts giving applications in mathematics (especially number theory), or explaining relationships between different approaches. One person knew exactly how I should go about to achieve the hoped-for accessibility : “As a rule, I’d take what you think would be just right for undergrads, and then trim it down a little more.”

Others want rather specialized posts, such as on ‘connection and parallel transport in noncommutative geometry’ or on ‘trees (per J-L. Loday, M. Aguiar, Connes/Kreimer renormalization (aka Butcher group)), or something completely other tree-related’.

Fortunately, some of you told me it was fine to write about ‘combinatorial games and cool nim stuff, finite simple groups, mathematical history, number theory, arithmetic geometry’, pushed me to go for ‘anything monstrous and moonshiney’ (as if I would know the secrets of the ‘connection between the Mathieu group M24 and the elliptic genus of K3’…) or wrote that ‘various algebraic geometry related posts are always welcome: posts like Mumford’s treasure map‘.

Over the years NeB survived three hardware-upgrades of ‘the Matrix’ (the webserver hosting it), more themes than I care to remember, and a couple of dramatic closure announcements…

But then we’re still here, soldiering on, still uncertain whether there’s a point to it, but grateful for tiny tokens of appreciation.

Such as this morning’s story: Chandan deemed it necessary to correct two spelling mistakes in a 2 year old Fun-math post on Weil and the Riemann hypothesis (also reposted on Neb here). Often there’s a story behind such sudden comments, and a quick check of MathOverflow revealed this answer and the comments following it.

I thank Ed Dean for linking to the Fun-post, Chandan for correcting the misspellings and Georges for the kind words. I agree with Georges that a cut&copy of a blogpost-quoted text does not require a link to that post (though it is always much appreciated). It is rewarding to see such old posts getting a second chance…

Above the Google Analytics graph of the visitors coming here via a mobile device (at most 5 on a good day…). Anticipating much more iPads around after tonights presents-session I’ve made NeB more accessible for iPods, iPhones, iPads and other mobile devices.

The first time you get here via your Mac-device of choice you’ll be given the option of saving NeB as an App. It has its own icon (lowest row middle, also the favicon of NeB) and flashy start-up screen.

Of course, the whole point trying to make NeB more readable for Mobile users you get an overview of the latest posts together with links to categories and tags and the number of comments. Sliding through you can read the post, optimized for the device.

I do hope you will use the two buttons at the end of each post, the first to share or save it and the second to leave a comment.

I wish you all a lot of mathematical (and other) fun in 2011 :: lieven.

• Mathematics somewhat less high-brow than MathO.
• Physics still in the beta-phase (see below)
• Theoretical computer science
• TeX for TeX and LaTeX-lovers
• iPad 4 edu for those who want to use their iPad in the classroom
• etc. etc.

“Opening a StackExchange site is damn hard. First you have to find at least 60 people interested in the site. Then, when this limit is reached, a large amount of people (in the hundreds, but it really depends on the reputation of each participant) must commit and promise to create momentum for the site, adding questions and answers. When this amount is reached, the site is open and stays in closed beta for seven days. During this time, the committers have to enrich the site so that the public beta (which starts after the first seven days) gets enough hits and participants to show a self-sustained community.” (quote from ForTheScience’s StackExchange sites proliferation, this post also contains a list of StackExchange-projects in almost every corner of Life)

The site keeping you up to date with StackExchange proposals and their progress is area51. Perhaps, you want to commit to some of these proposals

• High Energy Physics
• Theoretical Physics
• Mathematica
• Math Errata Database
• Blogging and the Blogosphere
• Cryptography

or simply browse around area51 until you find the ideal community for you to belong to…

• Spam filter : normally between 10 and 20 spam-comments are trapped. They are removed automatically every 14 days so I only look at them when someone complains a valid comment didn’t get through. This moment, there are 1007 spam-comments held (already 1020 by the time I post this).
• Akismet-plugin : for those who don’t run a blog, Akismet is an extremely useful WordPress-plugin getting rid of most spam-comments (before entering the spam-filter). I must admit I don’t check Akismet-stats regularly but today the message reads : ‘Akismet has protected your site from 104,145 spam comments already’ (which is about 10% of all visits…).
• Mail-notifications : this week my email-account is bombarded with spam-comments to approve. Usually there are 1 to 2 such comments a week, now about 20 a day. Part of the problem was that the Akismet-servers were obstructed earlier this week, but the problem persists.
• Today’s stats show more than 4000 hits, 500 uniques. About 3 times average.

Do we all really need to get caught in the cross-fire between WikiLeaks- and government-hackers?

I have no intention to post my take on WikiLeaks, simply because it won’t be original. As to Assange’s problems with justice : until further evidence comes to light I’m with Naomi Wolf on this (thanks Kea for the link).

If NeB goes off-line for a while, you know why…

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Courageous because as the talk progresses, she gives more and more examples from her own experiences, thereby breaking the taboo surrounding the topic of bipolar mood disorder among scientists. Interesting because she raises a couple of valid points, well worth repeating.

We didn’t can see it coming

We are always baffled when someone we know commits suicide, especially if that person is extremely successful in his/her work. ‘(S)he was so full of activity!’, ‘We did not see it coming!’ etc. etc.

Matilde argues that if a person suffers from bipolar mood disorder (from mild forms to full-blown manic-depression), a condition quite common among scientists and certainly mathematicians, we can see it coming, if we look for the proper signals!

We, active scientists, are pretty good at hiding a down-period. We have collected an arsenal of tricks not to send off signals when we feel depressed, simply because it’s not considered cool behavior. On the other hand, in our manic phases, we are quite transparent because we like to show off our activity and creativity!

Matilde tells us to watch out for people behaving orders-of-magnitude out of their normal-mode behavior. Say, someone who normally posts one or two papers a year on the arXiv, suddenly posting 5 papers in one month. Or, someone going rarely to a conference, now spending a summer flying from one conference to the next. Or, someone not blogging for months, suddenly flooding you with new posts…

As scientists we are good at spotting such order-of-magnitude-out-behavior. So we can detect friends and colleagues going through a manic-phase and hence should always take such a person serious (and try to offer help) when they send out signals of distress.

Mood disorder, a Faustian bargain

The Faust legend :
“Despite his scholarly eminence, Faust is bored and disappointed. He decides to call on the Devil for further knowledge and magic powers with which to indulge all the pleasures of the world. In response, the Devil’s representative Mephistopheles appears. He makes a bargain with Faust: Mephistopheles will serve Faust with his magic powers for a term of years, but at the end of the term, the Devil will claim Faust’s soul and Faust will be eternally damned.”

Mathematicians suffering from mood disorder seldom see their condition as a menace, but rather as an advantage. They know they do their best and most creative work in short spells of intense activity during their manic phase and take the down-phase merely as a side effect. We fear that if we seek treatment, we may as well loose our creativity.

That is, like Faust, we indulge the pleasures of our magic powers during a manic-phase, knowing only too well that the devilish depression-phase may one day claim our life or mental sanity…

NeverEndingBooks didn’t make it to the list, and I can live with that. For reasons only relevant to myself, posting has slowed down over the last year and the most recent post dates back from february!

More puzzling to me was the fact that F-un mathematics got in place 26! OnlineDegree had this to say about F-un Math : “Any students studying math must bookmark this blog, which provides readers with a broad selection of undergraduate and graduate concerns, quotes, research, webcasts, and much, much more.” Well, personally I wouldn’t bother to bookmark this site as prospects for upcoming posts are virtually inexistent…

As I am privy to both sites’ admin-pages, let me explain my confusion by comparing their monthly hits. Here’s the full F-un history

After a flurry of activity in the fall of 2008, both posting and attendance rates dropped, and presently the site gets roughly 50 hits-a-day. Compare this to the (partial) NeB history

The whopping 45000 visits in january 2008 were (i think) deserved at the time as there was then a new post almost every other day. On the other hand, the green bars to the right are a mystery to me. It appears one is rewarded for not posting at all…

The only explanation I can offer is that perhaps more and more people are recovering from the late 2008-depression and do again enjoy reading blog-posts. Google then helps blogs having a larger archive (500 NeB-posts compared to about 20 genuine Fun-posts) to attract a larger audience, even though the blog is dormant.

But this still doesn’t explain why FunMath made it to the top 50-list and NeB did not. Perhaps the fault is entirely mine and a consequence of a bad choice of blog-title. ‘NeverEndingBooks’ does not ring like a math-blog, does it?

Still, I’m not going to change the title into something more math-related. NeverEndingBooks will be around for some time (unless my hard-disk breaks down). On the other hand, I plan to start something entirely new and learn from the mistakes I made over the past 6 years. Regulars of this blog will have a pretty good idea of the intended launch date, not?

Until then, my online activity will be limited to tweets.

To most mathematicians, a good LaTeX-frontend (such as TeXShop for Mac-users) is the crucial tool to get the work done. We use it to draft ideas, write papers and courses, or even to take notes during lectures.

However, after six years of blogging, my own LaTeX-routine became rusty. I rarely open a new tex-document, and when I do, I’d rather copy-paste the long preamble from an old file than to start from scratch with a minimal list of packages and definitions needed for the job at hand. The few times I put a paper on the arXiv, the resulting text resembles a blog-post more than a mathematical paper, here’s an example.

As I desperately need to get some math-writing done, I need to pull myself away from the lure of an ever-open WordPress admin browser-screen and reacquaint myself with the far more efficient LaTeX-environment.

Perhaps even my blogging will benefit from the change. Whereas I used to keep on adding to most of my tex-files in order to keep them up-to-date, I rarely edit a blog-post after hitting the ’publish’ button. If I really want to turn some of my better posts into a book, I need them in a format suitable for neverending polishing, without annoying the many RSS-feed aggregators out there.

Who better than Terry Tao to teach me a more proficient way of blogging? A few days ago, Terry announced he will soon have his 5th (!!) book out, after three years of blogging…

How does he manage to do this? Well, as far as I know, Terry blogs in LaTeX and then uses a python-script called LaTeX2WP ’a program that converts a LaTeX file into something that is ready to be cut and pasted into WordPress. This way, you can write, and preview, your post in LaTeX, then run LaTeX2WP, and post into WordPress whatever comes out.’ More importantly, one retains a pure-tex-file of the post on which one can keep on editing to get it into a (book)-publishable form, eventually.

Nice, but one can do even better, as Eric from Curious Reasoning worked out. He suggests to install two useful python-packages : WordPressLib “with this library you can control remotely a WordPress installation. Use of library is very simple, you can write a small scripts or full applications that allows you to automate publishing of articles on your blog/site powered by WordPress” and plasTeX “plasTeX is a LaTeX document processing framework written entirely in Python. It currently comes bundled with an XHTML renderer (including multiple themes), as well as a way to simply dump the document to a generic form of XML”. Installation is easy : download and extract the files somewhere, go there and issue a **sudo python setup.py install** to add the packages to your python.

Finally, get Eric’s own wplatex package and install it as explained there. WpLaTeX has all the features of LaTeX2WP and much more : one can add titles, tags and categories automatically and publish the post from the command-line without ever having to enter the taboo WordPress-admin page! Here’s what I’ve written by now in TeXShop

I’ve added the screenshot and the script will know where to find it online for the blog-version as well as on my hard-disk for the tex-version. Very handy is the iftex … fi versus ifblog … fi alternative which allows you to add pure HTML to get the desired effect, when needed. Remains only to go into Terminal and issue the command

wplpost -x https://lievenlebruyn.github.io/neverendingbooks/xmlrpc.php ReturnToLatex.tex

(if your blog is on WordPress.com it even suffices to give its name, rather than this work-around for stand-alone wordpress blogs). The script asks for my username and password and will convert the tex-file and post it automatic.

Remember from last time that we identified Olivier Messiaen as the ‘Monsieur Modulo’ playing the musical organ at the Bourbaki wedding. This was based on the fact that his “modes à transposition limitée” are really about epimorphisms between modulo rings Z/12Z→Z/3Z and Z/12Z→Z/4Z.

However, Messiaen had more serious mathematical tricks up his sleeve. In two of his compositions he did discover (or at least used) one of the smaller sporadic groups, the Mathieu group $M_{12} $ of order 95040 on which we have based a whole series of Mathieu games two and a half years ago.

Messiaen’s ‘Ile de fey 2’ composition for piano (part of Quatre études de rythme (“Four studies in rhythm”), piano (1949–50)) is based on two concurrent permutations. The first is shown below, with the underlying motive rotational permutation shown.

This gives the permutation (1,7,10,2,6,4,5,9,11,12)(3,8). A second concurrent permutation is based on the permutation (1,6,9,2,7,3,5,4,8,10,11) and both of them generate the Mathieu group $M_{12} $. This can be seen by realizing the two permutations as the rotational permutations

and identifying them with the Mongean shuffles generating $M_{12} $. See for example, Dave Benson’s book “Music: A Mathematical Offering”, freely available online.

Clearly, Messiaen doesn’t use all of its 95040 permutations in his piece! Here’s how it sounds. The piece starts 2 minutes into the clip.

The second piece is “Les Yeux dans les Roues” (The Eyes in the Wheels), sixth piece from the “Livre d’Orgue” (1950/51).

According to Hauptwerk, the piece consists of a melody/theme in the pedal, accompanied by two fast-paced homorhythmic lines in the manuals. The pedal presents a sons-durées theme which is repeated six times, in different permutations. Initially it is presented in its natural form. Afterwards, it is presented alternatively picking notes from each end of the original form. Similar transformations are applied each time until the sixth, which is the retrograde of the first. The entire twelve-tone analysis (pitch only, not rhythm) of the pedal is shown below:

That is we get the following five permutations which again generate Mathieu 12 :

• a=(2,3,5,9,8,10,6,11,4,7,12)
• b=(1,2,4,8,9,7,11,3,6,12)(5,10)=e*a
• c=(1,12,11,9,5,4,6,2,10,7)(3,8)=e*d
• d=(1,11,10,8,4,5,3,7,2,9,6)
• e=(1,12)(2,11)(3,10)(4,9)(5,8)(6,7)

Here’s the piece performed on organ :

Considering the permutations $X=d.a^{-1} $ and $Y=(a.d^2.a.d^3)^{-1} $ one obtains canonical generators of $M_{12} $, that is, generators satisfying the defining equations of this sporadic group

$X^2=Y^3=(XY)^{11}=[X,Y]^6=(XYXYXY^{-1})^6=1 $

I leave you to work out the corresponding dessin d’enfant tonight after a couple of glasses of champagne! It sure has a nice form. Once again, a better 2010!

My nomination for the all-time highest quality discussion ever held in a blog comment section goes to the comments on this posting at Secret Blogging Seminar, where several of the best (relatively)-young algebraic geometers in the business discuss the foundations of the subject and how it should be taught.

I follow far too few comment-sections to make such a definite statement, but found the contributions by James Borger and David Ben-Zvi of exceptional high quality. They made a case for using Grothendieck’s ‘functor of points’ approach in teaching algebraic geometry instead of the ‘usual’ approach via prime spectra and their structure sheaves.

The text below was written on december 15th of last year, but never posted. As far as I recall it was meant to be part two of the ‘Brave New Geometries’-series starting with the Mumford’s treasure map post. Anyway, it may perhaps serve someone unfamiliar with Grothendieck’s functorial approach to make the first few timid steps in that directions.

Allyn Jackson’s beautiful account of Grothendieck’s life “Comme Appele du Neant, part II” (the first part of the paper can be found here) contains this gem :

“One striking characteristic of Grothendieck’s
mode of thinking is that it seemed to rely so little
on examples. This can be seen in the legend of the
so-called “Grothendieck prime”.

In a mathematical
conversation, someone suggested to Grothendieck
that they should consider a particular prime number.
“You mean an actual number?” Grothendieck
asked. The other person replied, yes, an actual
prime number. Grothendieck suggested, “All right,
take 57.”

But Grothendieck must have known that 57 is not
prime, right? Absolutely not, said David Mumford
of Brown University. “He doesn’t think concretely.””

We have seen before how Mumford’s doodles allow us to depict all ‘points’ of the affine scheme $\mathbf{spec}(\mathbb{Z}[x]) $, that is, all prime ideals of the integral polynomial ring $\mathbb{Z}[x] $.
Perhaps not too surprising, in view of the above story, Alexander Grothendieck pushed the view that one should consider all ideals, rather than just the primes. He achieved this by associating the ‘functor of points’ to an affine scheme.

Consider an arbitrary affine integral scheme $X $ with coordinate ring $\mathbb{Z}[X] = \mathbb{Z}[t_1,\ldots,t_n]/(f_1,\ldots,f_k) $, then any ringmorphism
$\phi~:~\mathbb{Z}[t_1,\ldots,t_n]/(f_1,\ldots,f_k) \rightarrow R $
is determined by an n-tuple of elements $~(r_1,\ldots,r_n) = (\phi(t_1),\ldots,\phi(t_n)) $ from $R $ which must satisfy the polynomial relations $f_i(r_1,\ldots,r_n)=0 $. Thus, Grothendieck argued, one can consider $~(r_1,\ldots,r_n) $ an an ‘$R $-point’ of $X $ and all such tuples form a set $h_X(R) $ called the set of $R $-points of $X $. But then we have a functor

$h_X~:~\mathbf{commutative rings} \rightarrow \mathbf{sets} \qquad R \mapsto h_X(R)=Rings(\mathbb{Z}[t_1,\ldots,t_n]/(f_1,\ldots,f_k),R) $

So, what is this mysterious functor in the special case of interest to us, that is when $X = \mathbf{spec}(\mathbb{Z}[x]) $?
Well, in that case there are no relations to be satisfied so any ringmorphism $\mathbb{Z}[x] \rightarrow R $ is fully determined by the image of $x $ which can be any element $r \in R $. That is, $Ring(\mathbb{Z}[x],R) = R $ and therefore Grothendieck’s functor of points
$h_{\mathbf{spec}(\mathbb{Z}[x]} $ is nothing but the forgetful functor.

But, surely the forgetful functor cannot give us interesting extra information on Mumford’s drawing?
Well, have a look at the slightly extended drawing below :

What are these ‘smudgy’ lines and ‘spiky’ points? Well, before we come to those let us consider the easier case of identifying the $R $-points in case $R $ is a domain. Then, for any $r \in R $, the inverse image of the zero prime ideal of $R $ under the ringmap $\phi_r~:~\mathbb{Z}[x] \rightarrow R $ must be a prime ideal of $\mathbb{Z}[x] $, that is, something visible in Mumford’s drawing. Let’s consider a few easy cases :

For starters, what are the $\mathbb{Z} $-points of $\mathbf{spec}(\mathbb{Z}[x]) $? Any natural number $n \in \mathbb{Z} $ determines the surjective ringmorphism $\phi_n~:~\mathbb{Z}[x] \rightarrow \mathbb{Z} $ identifying $\mathbb{Z} $ with the quotient $\mathbb{Z}[x]/(x-n) $, identifying the ‘arithmetic line’ $\mathbf{spec}(\mathbb{Z}) = { (2),(3),(5),\ldots,(p),\ldots, (0) } $ with the horizontal line in $\mathbf{spec}(\mathbb{Z}[x]) $ corresponding to the principal ideal $~(x-n) $ (such as the indicated line $~(x) $).

When $\mathbb{Q} $ are the rational numbers, then $\lambda = \frac{m}{n} $ with $m,n $ coprime integers, in which case we have $\phi_{\lambda}^{-1}(0) = (nx-m) $, hence we get again an horizontal line in $\mathbf{spec}(\mathbb{Z}[x]) $. For $ \overline{\mathbb{Q}} $, the algebraic closure of $\mathbb{Q} $ we have for any $\lambda $ that $\phi_{\lambda}^{-1}(0) = (f(x)) $ where $f(x) $ is a minimal integral polynomial for which $\lambda $ is a root.
But what happens when $K = \mathbb{C} $ and $\lambda $ is a trancendental number? Well, in that case the ringmorphism $\phi_{\lambda}~:~\mathbb{Z}[x] \rightarrow \mathbb{C} $ is injective and therefore $\phi_{\lambda}^{-1}(0) = (0) $ so we get the whole arithmetic plane!

In the case of a finite field $\mathbb{F}_{p^n} $ we have seen that there are ‘fat’ points in the arithmetic plane, corresponding to maximal ideals $~(p,f(x)) $ (with $f(x) $ a polynomial of degree $n $ which remains irreducible over $\mathbb{F}_p $), having $\mathbb{F}_{p^n} $ as their residue field. But these are not the only $\mathbb{F}_{p^n} $-points. For, take any element $\lambda \in \mathbb{F}_{p^n} $, then the map $\phi_{\lambda} $ takes $\mathbb{Z}[x] $ to the subfield of $\mathbb{F}_{p^n} $ generated by $\lambda $. That is, the $\mathbb{F}_{p^n} $-points of $\mathbf{spec}(\mathbb{Z}[x]) $ consists of all fat points with residue field $\mathbb{F}_{p^n} $, together with slightly slimmer points having as their residue field $\mathbb{F}_{p^m} $ where $m $ is a divisor of $n $. In all, there are precisely $p^n $ (that is, the number of elements of $\mathbb{F}_{p^n} $) such points, as could be expected.

Things become quickly more interesting when we consider $R $-points for rings containing nilpotent elements.