
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 »

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

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 »

“Ne pleure pas, Alfred ! J’ai besoin de tout mon courage pour mourir à vingt ans!” We all remember the last words of Evariste Galois to his brother Alfred. Lesser known are the mathematical results contained in his last letter, written to his friend Auguste Chevalier, on the eve of his fatal duel. Here the… Read more »

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 »

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 »

We have associated to a subgroup of the modular group $PSL_2(\mathbb{Z}) $ a quiver (that is, an oriented graph). For example, one verifies that the fundamental domain of the subgroup $\Gamma_0(2) $ (an index 3 subgroup) is depicted on the right by the region between the thick lines with the identification of edges as indicated…. Read more »

In a couple of days I’ll be blogging for 4 years… and I’m in the process of resurrecting about 300 posts from a databasedump made in june. For example here’s my first post ever which is rather naive. This conversion program may last for a couple of weeks and I apologize for all unwanted pingbacks… Read more »

Here I will go over the last post at a more leisurely pace, focussing on a couple of far more trivial examples. Here’s the goal : we want to assign a quiversuperpotential to any subgroup of finite index of the modular group. So fix such a subgroup $\Gamma’ $ of the modular group $\Gamma=PSL_2(\mathbb{Z}) $… Read more »

Yesterday, Jan Stienstra gave a talk at theARTS entitled “Quivers, superpotentials and Dimer Models”. He started off by telling that the talk was based on a paper he put on the arXiv Hypergeometric Systems in two Variables, Quivers, Dimers and Dessins d’Enfants but that he was not going to say a thing about dessins but… Read more »

Last time (possibly with help from the survival guide) we have seen that the universal map from the modular group $\Gamma = PSL_2(\mathbb{Z}) $ to its profinite completion $\hat{\Gamma} = \underset{\leftarrow}{lim}~PSL_2(\mathbb{Z})/N $ (limit over all finite index normal subgroups $N $) gives an embedding of the sets of (continuous) simple finite dimensional representations $\mathbf{simp}_c~\hat{\Gamma} \subset… Read more »

This is how my attention was drawn to what I have since termed anabelian algebraic geometry, whose starting point was exactly a study (limited for the moment to characteristic zero) of the action of absolute Galois groups (particularly the groups $Gal(\overline{K}/K) $, where K is an extension of finite type of the prime field) on… Read more »

Bruce Westbury has a page on recent work on series of Lie groups including exceptional groups. Moreover, he did put his slides of a recent talk (probably at MPI) online. Probably, someone considered a similar problem for simple groups. Are there natural constructions leading to a series of finite simple groups including some sporadic groups… Read more »

Today we will link modular quilts (via their associated cuboid tree diagrams) to special hyperbolic polygons. The above drawing gives the hyperbolic polygon (the gray boundary) associated to the M(24) tree diagram (the black interior graph). In general, the correspondence goes as follows. Recall that a cuboid tree diagram is a tree such that all… Read more »

Conjugacy classes of finite index subgroups of the modular group $\Gamma = PSL_2(\mathbb{Z}) $ are determined by a combinatorial gadget : a modular quilt. By this we mean a finite connected graph drawn on a Riemann surface such that its vertices are either black or white. Moreover, every edge in the graph connects a black… Read more »

Here a collection of pdffiles of NeverEndingBooksposts on groups, in reverse chronological order.

After a lengthy springbreak, let us continue with our course on noncommutative geometry and $SL_2(\mathbb{Z}) $representations. Last time, we have explained Grothendiecks mantra that all algebraic curves defined over number fields are contained in the profinite compactification $\widehat{SL_2(\mathbb{Z})} = \underset{\leftarrow}{lim}~SL_2(\mathbb{Z})/N $ of the modular group $SL_2(\mathbb{Z}) $ and in the knowledge of a certain subgroup… Read more »

Last time we saw that a curve defined over $\overline{\mathbb{Q}} $ gives rise to a permutation representation of $PSL_2(\mathbb{Z}) $ or one of its subgroups $\Gamma_0(2) $ (of index 2) or $\Gamma(2) $ (of index 6). As the corresponding monodromy group is finite, this representation factors through a normal subgroup of finite index, so it… Read more »

Today we will explain how curves defined over $\overline{\mathbb{Q}} $ determine permutation representations of the carthographic groups. We have seen that any smooth projective curve $C $ (a Riemann surface) defined over the algebraic closure $\overline{\mathbb{Q}} $ of the rationals, defines a _Belyi map_ $\xymatrix{C \ar[rr]^{\pi} & & \mathbb{P}^1} $ which is only ramified over… Read more »

Just as cartographers like Mercator drew maps of the then known world, we draw dessins d ‘enfants to depict the associated algebraic curve defined over $\overline{\mathbb{Q}} $. In order to see that such a dessin d’enfant determines a permutation representation of one of Grothendieck’s cartographic groups, $SL_2(\mathbb{Z}), \Gamma_0(2) $ or $\Gamma(2) $ we need to… Read more »

The Oscar in the category The Best Rejected Research Proposal in Mathematics (ever) goes to … Alexander Grothendieck for his proposal Esquisse d’un Programme, Grothendieck\’s research program from 1983, written as part of his application for a position at the CNRS, the French equivalent of the NSF. An English translation is available. Here is one… Read more »

Even a virtual course needs an opening line, so here it is : Take your favourite $SL_2(\mathbb{Z}) $representation Here is mine : the permutation presentation of the Mathieu group(s). Emile Leonard Mathieu is remembered especially for his discovery (in 1861 and 1873) of five sporadic simple groups named after him, the Mathieu groups $M_{11},M_{12},M_{22},M_{23} $… Read more »
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