# Posts Tagged: modular

• groups, math

## Monsters and Moonshine : a booklet

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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… Read more »

• stories

## the Reddit (after)effect

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Sunday january 2nd around 18hr NeB-stats went crazy. Referrals clarified that the post ‘What is the knot associated to a prime?’ was picked up at Reddit/math and remained nr.1 for about a day. Now, the dust has settled, so let’s learn from the experience. A Reddit-mention is to a blog what doping is to a… 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 »

• stories

## Who discovered the Leech lattice?

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The Leech lattice was, according to wikipedia, ‘originally discovered by Ernst Witt in 1940, but he did not publish his discovery’ and it ‘was later re-discovered in 1965 by John Leech’. However, there is very little evidence to support this claim. The facts What is certain is that John Leech discovered in 1965 an amazingly… Read more »

• noncommutative, number theory

## Langlands versus Connes

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This is a belated response to a Math-Overflow exchange between Thomas Riepe and Chandan Singh Dalawat asking for a possible connection between Connes’ noncommutative geometry approach to the Riemann hypothesis and the Langlands program. Here’s the punchline : a large chunk of the Connes-Marcolli book Noncommutative Geometry, Quantum Fields and Motives can be read as… Read more »

• stories

## Seriously now, where was the Bourbaki wedding?

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A few days before Halloween, Norbert Dufourcq (who died december 17th 1990…), sent me a comment, containing lots of useful information, hinting I did get it wrong about the church of the Bourbali wedding in the previous post. Norbert Dufourcq, an organist and student of Andre Machall, the organist-in-charge at the Saint-Germain-des-Prés church in 1939,… Read more »

• stories

## Bourbakism & the queen bee syndrome

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Probably the smartest move I’ve made after entering math-school was to fall in love with a feminist. Yeah well, perhaps I’ll expand a bit on this sentence another time. For now, suffice it to say that I did pick up a few words in the process, among them : the queen bee syndrome : women… 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 »

• featured

## Mumford’s treasure map

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In the series “Brave new geometries” we give an introduction to ‘strange’ but exciting new ideas. We start with Grothendieck’s scheme-revolution, go on with Soule’s geometry over the field with one element, Mazur’s arithmetic topology, Grothendieck’s anabelian geometry, Connes’ noncommutative geometry etc.

• geometry, noncommutative

## noncommutative F_un geometry (2)

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We use Kontsevich’s idea of thin varieties to define complexified varieties over F\_un.

• absolute, geometry, noncommutative

## noncommutative F_un geometry (1)

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We propose to extend the Connes-Consani definition to noncommuntative F_un varieties.

• featured

## what does the monster see?

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The Monster is the largest of the 26 sporadic simple groups and has order 808 017 424 794 512 875 886 459 904 961 710 757 005 754 368 000 000 000 = 2^46 3^20 5^9 7^6 11^2 13^3 17 19 23 29 31 41 47 59 71. It is not so much the size… Read more »

• geometry, groups, number theory

## the buckyball curve

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We are after the geometric trinity corresponding to the trinity of exceptional Galois groups The surfaces on the right have the corresponding group on the left as their group of automorphisms. But, there is a lot more group-theoretic info hidden in the geometry. Before we sketch the $L_2(11)$ case, let us recall the simpler… Read more »

• geometry, groups

## the buckyball symmetries

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The buckyball is without doubt the hottest mahematical object at the moment (at least in Europe). Recall that the buckyball (middle) is a mixed form of two Platonic solids the Icosahedron on the left and the Dodecahedron on the right. For those of you who don’t know anything about football, it is that other ball-game,… 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 »

• geometry, groups, math, number theory

## Arnold’s trinities

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Referring to the triple of exceptional Galois groups $L_2(5),L_2(7),L_2(11)$ and its connection to the Platonic solids I wrote : “It sure seems that surprises often come in triples…”. Briefly I considered replacing triples by trinities, but then, I didnt want to sound too mystic… David Corfield of the n-category cafe and a dialogue 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 »

• 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 »

• featured

## Farey symbols of sporadic groups

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John Conway once wrote : There are almost as many different constructions of $M_{24}$ as there have been mathematicians interested in that most remarkable of all finite groups. In the inguanodon post Ive added yet another construction of the Mathieu groups $M_{12}$ and $M_{24}$ starting from (half of) the Farey sequences and… Read more »

• featured

## Quiver-superpotentials

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It’s been a while, so let’s include a recap : a (transitive) permutation representation of the modular group $\Gamma = PSL_2(\mathbb{Z})$ is determined by the conjugacy class of a cofinite subgroup $\Lambda \subset \Gamma$, or equivalently, to a dessin d’enfant. We have introduced a quiver (aka an oriented graph) which comes from a… Read more »