neverendingbooks

I’ve LaTeXed $48=2 \times 24$ posts into a 114 page booklet Monsters and Moonshine for you to download.

The $24$ ‘Monsters’ posts are (mostly) about finite simple (sporadic) groups : we start with the Scottish solids (hoax?), move on to the 14-15 game groupoid and a new Conway $M_{13}$-sliding game which uses the sporadic Mathieu group $M_{12}$. This Mathieu group appears in musical compositions of Olivier Messiaen and it can be used also to get a winning strategy of ‘mathematical blackjack’. We discuss Galois’ last letter and the simple groups $L_2(5),L_2(7)$ and $L_2(11)$ as well as other Arnold ‘trinities’. We relate these groups to the Klein quartic and the newly discovered ‘buckyball’-curve. Next we investigate the history of the Leech lattice and link to online games based on the Mathieu-groups and Conway’s dotto group. Finally, preparing for moonshine, we discover what the largest sporadic simple group, the Monster-group, sees of the modular group.

The $24$ ‘Moonshine’ posts begin with the history of the Dedekind (or Klein?) tessellation of the upper half plane, useful to determine fundamental domains of subgroups of the modular group $PSL_2(\mathbb{Z})$. We investigate Grothendieck’s theory of ‘dessins d’enfants’ and learn how modular quilts classify the finite index subgroups of the modular group. We find generators of such groups using Farey codes and use those to give a series of simple groups including as special members $L_2(5)$ and the Mathieu-sporadics $M_{12}$ and $M_{24}$ : the ‘iguanodon’-groups. Then we move to McKay-Thompson series and an Easter-day joke pulled by John McKay. Apart from the ‘usual’ monstrous moonshine conjectures (proved by Borcherds) John McKay also observed a strange appearance of $E(8)$ in connection with multiplications of involutions in the Monster-group. We explain Conway’s ‘big picture’ which makes it easy to work with the moonshine groups and use it to describe John Duncan’s solution of the $E(8)$-observation.

I’ll try to improve the internal referencing over the coming weeks/months, include an index and add extra material as we will be studying moonshine for the Mathieu groups as well as a construction of the Monster-group in next semester’s master-seminar. All comments, corrections and suggestions for extra posts are welcome!

If you are interested you can also download two other booklets : The Bourbaki Code (38 pages) containing all Bourbaki-related posts and absolute geometry (63 pages) containing the posts related to the “field with one element” and its connections to (noncommutative) geometry and number theory.

I’ll try to add to the ‘absolute geometry’-booklet the posts from last semester’s master-seminar (which were originally posted at angs@t/angs+) and write some new posts covering the material that so far only exists as prep-notes. The links above will always link to the latest versions of these booklets.

We’ve had three seminar-sessions so far, and the seminar-blog ‘angs+’ contains already 20 posts and counting. As blogging is not a linear activity, I will try to post here at regular intervals to report on the ground we’ve covered in the seminar, providing links to the original angs+ posts.

This year’s goal is to obtain a somewhat definite verdict on the field-with-one-element hype.

In short, the plan is to outline Smirnov’s approach to the ABC-conjecture using geometry over $\mathbb{F}_1$, to describe Borger’s idea for such an $\mathbb{F}_1$-geometry and to test it on elusive objects such as $\mathbb{P}^1_{\mathbb{F}_1} \times_{\mathbb{F}_1} \mathsf{Spec}(\mathbb{Z})$ (relevant in Smirnov’s paper) and $\mathsf{Spec}(\mathbb{Z}) \times_{\mathbb{F}_1} \mathsf{Spec}(\mathbb{Z})$ (relevant to the Riemann hypothesis).

We did start with an historic overview, using recently surfaced material such as the Smirnov letters. Next, we did recall some standard material on the geometry of smooth projective curves over finite fields, their genus leading up to the Hurwitz formula relating the genera in a cover of curves.

Using this formula, a version of the classical ABC-conjecture in number theory can be proved quite easily for curves.

By analogy, Smirnov tried to prove the original ABC-conjecture by viewing $\mathsf{Spec}(\mathbb{Z})$ as a ‘curve’ over $\mathbb{F}_1$. Using the connection between the geometric points of the projective line over the finite field $\mathbb{F}_p$ and roots of unity of order coprime to $p$, we identify $\mathbb{P}^1_{\mathbb{F}_1}$ with the set of all roots of unity together with $\{ [0],[\infty] \}$. Next, we describe the schematic points of the ‘curve’ $\mathsf{Spec}(\mathbb{Z})$ and explain why one should take as the degree of the ‘point’ $(p)$ (for a prime number $p$) the non-sensical value $log(p)$.

To me, the fun starts with Smirnov’s proposal to associate to any rational number $q = \tfrac{a}{b} \in \mathbb{Q} – \{ \pm 1 \}$ a cover of curves

$q~:~\mathsf{Spec}(\mathbb{Z}) \rightarrow \mathbb{P}^1_{\mathbb{F}_1}$

by mapping primes dividing $a$ to $[0]$, primes dividing $b$ to $[\infty]$, sending the real valuation to $[0]$ or $[\infty]$ depending onw whether or not $b > a$ and finally sending a prime $p$ not involved in $a$ or $b$ to $[n]$ where $n$ is the order of the unit $\overline{a}.\overline{b}^{-1}$ in the finite cyclic group $\mathbb{F}_p^*$. Somewhat surprisingly, it does follow from Zsigmondy’s theorem that this is indeed a finite cover for most values of $q$. A noteworthy exception being the map for $q=2$ (which fails to be a cover at $[6]$) and of which Pieter Belmans did draw this beautiful graph

True believers in $\mathbb{F}_1$ might conclude from this graph that there should only be finitely many Mersenne primes… Further, the full ABC-conjecture would follow from a natural version of the Hurwitz formula for such covers.

(to be continued)

F_un Mathematics

Hardly a ‘new’ blog, but one that is getting a new life! On its old homepage you’ll find a diagonal banner stating ‘This site has moved’ and clicking on it will guide you to its new location : cage.ugent.be/~kthas/Fun.

From now on, this site will be hosted at the University of Ghent and maintained by Koen Thas. So, please update your bookmarks and point your RSS-aggregator to the new feed.

Everyone interested in contributing to this blog dedicated to the mathematics of the field with one element should contact Koen by email.

angst

Though I may occasionally (cross)post at F_un mathematics, my own blog-life will center round a new blog to accompany the master-course ‘seminar noncommutative geometry’ I’m running at Antwerp University this semester. Its URL is noncommutative.org and it is called :

Here, angs is short for Antwerp Noncommutative Geometry Seminar and the additions @t resp. + are there to indicate we will experiment a bit trying to find useful interactions between the IRL seminar, its blog and social media such as twitter and Google+.

The seminar (and blog) are scheduled to start in earnest september 30th, but I may post some prep-notes already. This semester the seminar will try to decode Smirnov’s old idea to prove the ABC-conjecture in number theory via geometry over the field with one element and connect it with new ideas such as Borger’s $\mathbb{F}_1$-geometry using $\lambda$-rings and noncommutative ideas proposed by Connes, Consani and Marcolli.

Again, anyone willing to contribute actively is invited to send me an email or to comment on ‘angst’, tweet about it using the hashtag #angs (all such tweets will appear on the frontpage) or share its posts on Google+.

Noncommutative Arithmetic Geometry Media Library

Via the noncommutative geometry blog a new initiative maintained by Alain Connes and Katia Consani was announced : the Noncommutative Arithmetic Geometry Media Library.

This site is dedicated to maintain articles, videos, and news about meetings and activities related to noncommutative arithmetic geometry. The website is still `under construction’ and the plan is to gradually add more videos (also from past conferences and meetings), as well as papers and slides.

In the early days of math-blogging, one was happy to get LaTeXRender working. Some years later, the majority of math-blogs were using the, more user-friendly, wp-latex plugin to turn LaTeX-code into png-images. Today, everyone uses MathJax which works with modern CSS and web fonts instead of equation images, so equations scale with surrounding text at all zoom levels.

However, MathJax has one downside : it doesn’t parse in ePub-readers. Peter Krautzberger wrote a post Epub and mathematics in which he suggested two methods to turn MathJax into ePub, but after dozens of experiments I still fail to reproduce these.

No doubt, someone will soon come up with a working alternative, but for the impatient here’s a quick but dirty method to turn your MathJax powered wordpress post into ePub :

the tools

• download and install the ePub export plugin. It automatically creates an ePub file when a post or page is published or updated. The ePubs are stored in the uploads directory (to be found in the wp-contents directory).
• download and install the wp-latex plugin. MathJax uses the normal \$ tex-delimeters whereas wp-latex requires \$latex, so this plugin doesn’t interfere with the default use of MathJax.
• download the wp2latex python script. It converts a standard LaTeX file into a format that is ready to be copied into WordPress.

the routine

• Edit the post you want to convert to ePub. Copy the contents of the post box to a file say post1.tex and save this in the same directory containing the latex2wp.py script.
• In Terminal go to that directory and type the command ‘python latex2wp.py post1.tex’. It will produce a new file post1.html in the same directory.
• Copy the contents of post1.html into the post box of your WordPress-post and press the update button. This time the TeX-commands in your post will be rendered using wp-latex and the ePub export-plugin will have created an ePub-version of it.
• Locate this newly created ePub file in the relevant wp-contents/uploads/ folder (file has a number.epub name) and, if wanted, change its name into something easier to recognize and copy it somewhere outside the uploads directory. This will be your desired ePub-version of the post.
• Replace the contents of the post box of your WordPress-post with the contents of the post1.tex file and hit the ‘Update’ button, to restore your original post (powered by MathJax).
• Email your ePub-file to your iPad and open it with iBooks. Not quite as nice as MathJax-parsed TeX but a lot better than reading unparsed TeX-commands.

Note to students following this year’s ‘seminar noncommutative geometry’ : the seminar starts friday september 30th at 13h in room G 0.16.

However, if you have another good reason to be in Ghent on thursday september 22nd, consider attending the inaugural lecture of Koen Thas at 17h in auditorium Emmy Noether, campus De Sterre, Krijgslaan 281, 9000 Gent.

Koen’s lecture has one of the longest titles i’ve seen : “De lange weg – een verhaal over wiskundige problemen die denkers al eeuwenlang teisteren, zonderlingen die in afgelegen berghutten de existentie van de duivel willen aantonen, en een mythisch object dat niet bestaat, maar waar we toch naar zoeken” (“The long road – a story on mathematical problems torturing scientists for centuries, weirdos trying to prove the existence of the Devil in desolated mountain-huts and the search for a mythical object that doesn’t exist”).

Knowing Koen a bit I’d say it will be on the Riemann hypothesis, Grothendieck’s theory of motives and the field with one element. A sneak preview of our upcoming seminar, quoi?

More information on the event and to register see here.