
Sunday january 2nd around 18hr NeBstats 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 Redditmention is to a blog what doping is to a… Read more »

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 »

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 grouptheoretic info hidden in the geometry. Before we sketch the $L_2(11) $ case, let us recall the simpler… Read more »

We saw that the icosahedron can be constructed from the alternating group $A_5 $ by considering the elements of a conjugacy class of order 5 elements as the vertices and edges between two vertices if their product is still in the conjugacy class. This description is so nice that one would like to have a… Read more »

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 ballgame,… Read more »

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 ncategory cafe and a dialogue on… 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 »

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 »

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 »

Here a list of saved pdffiles of previous NeverEndingBooksposts on geometry in reverse chronological order.

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 »

Last time we have seen that the noncommutative manifold of a Riemann surface can be viewed as that Riemann surface together with a loop in each point. The extra loopstructure tells us that all finite dimensional representations of the coordinate ring can be found by separating over points and those living at just one point… Read more »

The natural habitat of this lesson is a bit further down the course, but it was called into existence by a comment/question by Kea I don’t yet quite see where the nc manifolds are, but I guess that’s coming. As I’m enjoying telling about all sorts of sources of finite dimensional representations of $SL_2(\mathbb{Z}) $… 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 »

The Klein Four Group is an a capella group from the maths department of Northwestern. Below a link to one of their songs (grabbed from P.P. Cook’s Tangent Space ). Finite Simple Group (of order two) A Klein Four original by Matt Salomone The path of love is never smoothBut mine’s continuous for youYou’re the… Read more »

Now that my nongeometry post is linked via the comments in this stringcoffeetable post which in turn is available through a trackback from the KontsevichSoibelman paper it is perhaps useful to add a few links. The little I’ve learned from reading about Connesstyle noncommutative geometry is this : if you have a situation where a… Read more »

If you go to Oberwolfach and the weather predictions are as good as last weeks, try to bring your mountainbike along! Here is a nice 1hr30 to 2hrs tour : from the institute to Walke (height 300m), follow the road north to Rankach and at the Romanes Hof turn left to Hackerhof. Next, offroad along… Read more »

Klein’s quartic $X$is the smooth plane projective curve defined by $x^3y+y^3z+z^3x=0$ and is one of the most remarkable mathematical objects around. For example, it is a Hurwitz curve meaning that the finite group of symmetries (when the genus is at least two this group can have at most $84(g1)$ elements) is as large as possible,… Read more »

Here the story of an idea to construct new examples of noncommutative compact manifolds, the computational difficulties one runs into and, when they are solved, the white noise one gets. But, perhaps, someone else can spot a gem among all gibberish… [Qurves](http://www.neverendingbooks.org/toolkit/pdffile.php?pdf=/TheLibrary/papers/qaq.pdf) (aka quasifree algebras, aka formally smooth algebras) are the \’affine\’ pieces of noncommutative… Read more »
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