Posts Tagged: moonshine

• rants, stories

what have quivers done to students?

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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 opening-lines to address me and, inevitably, the most disastrous teacher-student-conversation-ever followed (also on my part, i’m… Read more »

• web

mathblogging and poll-results

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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 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,… Read more »

• stories

So, who did discover the Leech lattice?

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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 »

• web

bloomsday, again

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Bloomsday has a tradition of bringing drastic changes to this blog. Two years ago, it signaled a bloomsday-ending 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 math-blog can only… Read more »

• groups

E(8) from moonshine groups

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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 vertex-valency coincides with the valency of the corresponding group? Duncan describes a subset of 9 moonshine groups for which the valencies… Read more »

• groups

looking for the moonshine picture

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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 »

• groups, number theory

Conway’s big picture

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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 »

• groups

the monster graph and McKay’s observation

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While the verdict on a neolithic Scottish icosahedron is still open, let us recall Kostant’s group-theoretic construction of the icosahedron from its rotation-symmetry 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 »

• stories

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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 weeks-max, this year. So, last… Read more »

• web

the future of this blog

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Some weeks ago Peter Woit of Not Even Wrong and Bee of Backreaction had a video-chat 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 »

• groups, math

Arnold’s trinities version 2.0

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Arnold has written a follow-up to the paper mentioned last time called “Polymathematics : is mathematics a single science or a set of arts?” (or here for a (huge) PDF-conversion). 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 »

• groups, noncommutative

Monstrous frustrations

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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 »

• web

bloomsday 2 : BistroMath

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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 »

• groups, stories

Dedekind or Klein ?

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The black&white psychedelic picture on the left of a tessellation of the hyperbolic upper-halfplane, 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) 70-76. But is this correct terminology? Nobody else uses it apparently. So, let’s try… Read more »

• rants, stories

wankers

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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 »

• stories

the secret revealed…

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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 track-record 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 »

• stories

Monstrous Easter Egg Race

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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 »

• featured

the McKay-Thompson series

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

• stories

Finding Moonshine

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On friday, I did spot in my regular Antwerp-bookshop 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 »

• stories

censured post : bloggers’ block

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Below an up-till-now hidden post, written november last year, trying to explain the long blog-silence at neverendingbooks during october-november 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 »

• stories

sobering-up

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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 »

• featured

Hexagonal Moonshine (3)

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Hexagons keep on popping up in the representation theory of the modular group and its close associates. We have seen before that singularities in 2-dimensional 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 »

• featured

Hexagonal Moonshine (2)

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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 »

• featured

Hexagonal Moonshine (1)

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Over at the Arcadian Functor, Kea is continuing her series of blog posts on M-theory (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 »

• stories

The miracle of 163

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On page 227 of Symmetry and the Monster, Mark Ronan tells the story of Conway and Norton computing the number of independent _mini j-functions_ (McKay-Thompson 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 »