
A few years ago a student entered my office asking suggestions for his master thesis. “I’m open to any topic as long as it has nothing to do with those silly quivers!” At that time not the best of openinglines to address me and, inevitably, the most disastrous teacherstudentconversationever followed (also on my part, i’m… Read more »

Mathblogging.org is a recent initiative and may well become the default starting place to check on the status of the mathematical blogosphere. Handy, if you want to (re)populate your RSSaggregator with interesting mathematical blogs, is their graphical presentation of (nearly) all mathblogs ordered by type : group blogs, individual researchers, teachers and educators, journalistic writers,… Read more »

For the better part of the 30ties, Ernst Witt (1) did hang out with the rest of the ‘Noetherknaben’, the group of young mathematicians around Emmy Noether (3) in Gottingen. In 1934 Witt became Helmut Hasse‘s assistent in Gottingen, where he qualified as a university lecturer in 1936. By 1938 he has made enough of… Read more »

Bloomsday has a tradition of bringing drastic changes to this blog. Two years ago, it signaled a bloomsdayending to the original neverendingbooks, giving birth (at least for a couple of months) to MoonshineMath. Last year, the bloomsday 2 post was the first of several ‘conceptual’ blog proposals, voicing my conviction that a mathblog can only… Read more »

Are the valencies of the 171 moonshine groups are compatible, that is, can one construct a (disconnected) graph on the 171 vertices such that in every vertex (determined by a moonshine group G) the vertexvalency coincides with the valency of the corresponding group? Duncan describes a subset of 9 moonshine groups for which the valencies… Read more »

We have seen that Conway’s big picture helps us to determine all arithmetic subgroups of $PSL_2(\mathbb{R}) $ commensurable with the modular group $PSL_2(\mathbb{Z}) $, including all groups of monstrous moonshine. As there are exactly 171 such moonshine groups, they are determined by a finite subgraph of Conway’s picture and we call the minimal such subgraph… Read more »

Conway and Norton showed that there are exactly 171 moonshine functions and associated two arithmetic subgroups to them. We want a tool to describe these and here’s where Conway’s big picture comes in very handy. All moonshine groups are arithmetic groups, that is, they are commensurable with the modular group. Conway’s idea is to view… Read more »

While the verdict on a neolithic Scottish icosahedron is still open, let us recall Kostant’s grouptheoretic construction of the icosahedron from its rotationsymmetry group $A_5 $. The alternating group $A_5 $ has two conjugacy classes of order 5 elements, both consisting of exactly 12 elements. Fix one of these conjugacy classes, say $C $ and… Read more »

Over the last month a pile of books grew in our living room to impressive heights, intended to be packed for our usual 3+week vacation to the south of France. From the outset it was clear that ‘circumstances’ (see title for hint) forced us to slim it down to 2 weeksmax, this year. So, last… Read more »

Some weeks ago Peter Woit of Not Even Wrong and Bee of Backreaction had a videochat on all sorts of things (see the links above to see the whole clip) including the nine minute passage below on ‘the future of (science) blogs’. click here to see the video The crucial point being that blogging takes… Read more »

Arnold has written a followup to the paper mentioned last time called “Polymathematics : is mathematics a single science or a set of arts?” (or here for a (huge) PDFconversion). On page 8 of that paper is a nice summary of his 25 trinities : I learned of this newer paper from a comment by… Read more »

Thanks for clicking through… I guess. If nothing else, it shows that just as much as the stock market is fueled by greed, mathematical reasearch is driven by frustration (or the pleasure gained from knowing others to be frustrated). I did spend the better part of the day doing a lengthy, if not laborious, calculation,… Read more »

Exactly one year ago this blog was briefly renamed MoonshineMath. The concept being that it would focus on the mathematics surrounding the monster group & moonshine. Well, I got as far as the Mathieu groups… After a couple of months, I changed the name back to neverendingbooks because I needed the freedom to post on… Read more »

The black&white psychedelic picture on the left of a tessellation of the hyperbolic upperhalfplane, was called the Dedekind tessellation in this post, following the reference given by John Stillwell in his excellent paper Modular Miracles, The American Mathematical Monthly, 108 (2001) 7076. But is this correct terminology? Nobody else uses it apparently. So, let’s try… Read more »

You may not have noticed, but I’m in a foul mood for weeks now because of comments and reactions to the last line of the post on Finding Moonshine. I wrote Du Sautoy is a softy! I’d throw such students out of the window… and got everyone against me for this (first floor) defenestration threat…… Read more »

Often, one can appreciate the answer to a problem only after having spend some time trying to solve it, and having failed … pathetically. When someone with a trackrecord of coming up with surprising mathematical tidbits like John McKay sends me a mystery message claiming to contain “The secret of Monstrous Moonshine and the universe”,… Read more »

Here’s a sweet Easter egg for you to crack : a mysterious message from none other than the discoverer of Monstrous Moonshine himself… From: mckayj@Math.Princeton.EDU Date: Mon 10 Mar 2008 07:51:16 GMT+01:00 To: lieven.lebruyn@ua.ac.be The secret of Monstrous Moonshine and the universe. Let j(q) = 1/q + 744 + sum( c[k]*q^k,k>=1) be the Fourier expansion… Read more »

Monstrous moonshine was born (sometime in 1978) the moment John McKay realized that the linear term in the jfunction $j(q) = \frac{1}{q} + 744 + 196884 q + 21493760 q^2 + 864229970 q^3 + \ldots $ is surprisingly close to the dimension of the smallest nontrivial irreducible representation of the monster group, which is 196883…. Read more »

On friday, I did spot in my regular Antwerpbookshop Finding Moonshine by Marcus du Sautoy and must have uttered a tiny curse because, at once, everyone near me was staring at me… To make matters worse, I took the book from the shelf, quickly glanced through it and began shaking my head more and more,… Read more »

Below an uptillnow hidden post, written november last year, trying to explain the long blogsilence at neverendingbooks during octobernovember 2007… A couple of months ago a publisher approached me, out of the blue, to consider writing a book about mathematics for the general audience (in Dutch (?!)). Okay, I brought this on myself hinting at… Read more »

Kea’s post reminded me to have a look at my search terms (the things people type into search engines to get redirected here). Quite a sobering experience… Via Google Analytics I learn that 49,51% of traffic comes from Search Engines (compared to 26,17% from Referring Sites and 24,32% from direct hits) so I should take… Read more »

Hexagons keep on popping up in the representation theory of the modular group and its close associates. We have seen before that singularities in 2dimensional representation varieties of the three string braid group $B_3 $ are ‘clanned together’ in hexagons and last time Ive mentioned (in passing) that the representation theory of the modular group… Read more »

Delving into finite dimensional representations of the modular group $\Gamma = PSL_2(\mathbb{Z}) $ it is perhaps not too surprising to discover numerical connections with modular functions. Here, one such strange observation and a possible explanation. Using the _fact_ that the modular group $\Gamma = PSL_2(\mathbb{Z}) $ is the free group product $C_2 \ast C_3 $… Read more »

Over at the Arcadian Functor, Kea is continuing her series of blog posts on Mtheory (the M is supposed to mean either Monad or Motif). A recurrent motif in them is the hexagon and now I notice hexagons popping up everywhere. I will explain some of these observations here in detail, hoping that someone, more… Read more »

On page 227 of Symmetry and the Monster, Mark Ronan tells the story of Conway and Norton computing the number of independent _mini jfunctions_ (McKayThompson series) arising from the Moonshine module. There are 194 distinct characters of the monster (btw. see the background picture for the first page of the character table as given in… Read more »
Close