# Posts Tagged: Klein

• 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 »

• 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, representations

## Klein’s dessins d’enfant and the buckyball

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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 »

• 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 »

• 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

## Galois’ last letter

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“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 »

• 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 »

• featured

## Iguanodon series of simple groups

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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 »

• featured

## neverendingbooks-geometry

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Here a list of saved pdf-files of previous NeverEndingBooks-posts on geometry in reverse chronological order.

• featured

## permutation representations of monodromy groups

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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 »

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## noncommutative curves and their maniflds

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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 loop-structure 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 »

• featured

## the noncommutative manifold of a Riemann surface

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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 »

• stories

## The best rejected proposal ever

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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 »

• web

## simple group of order 2

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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 »

• stories

## non-(commutative) geometry

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Now that my non-geometry post is linked via the comments in this string-coffee-table post which in turn is available through a trackback from the Kontsevich-Soibelman paper it is perhaps useful to add a few links. The little I’ve learned from reading about Connes-style non-commutative geometry is this : if you have a situation where a… Read more »

• web

## Oberwolfach files

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If you go to Oberwolfach and the weather predictions are as good as last weeks, try to bring your mountain-bike 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, off-road along… Read more »

• featured

## the Klein stack

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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 $84(g-1)$ elements) is as large as possible,… Read more »

• featured

## sexing up curves

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Here the story of an idea to construct new examples of non-commutative 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 quasi-free algebras, aka formally smooth algebras) are the \’affine\’ pieces of non-commutative… Read more »