
In preparing for next year’s ‘seminar noncommutative geometry’ I’ve converted about 30 posts to LaTeX, centering loosely around the topics students have asked me to cover : noncommutative geometry, the absolute point (aka the field with one element), and their relation to the Riemann hypothesis. The idea being to edit these posts thoroughly, add much… Read more »

Last time we discovered that the mental picture to view prime numbers as knots in $S^3$ was first dreamed up by David Mumford. Today, we’ll focus on where and when this happened. 3. When did Mazur write his unpublished preprint? According to his own website, Barry Mazur did write the paper Remarks on the Alexander… Read more »

In an interview with readers of the Guardian, December 3rd 2010, Julian Assange made a somewhat surprising comparison between WikiLeaks and Bourbaki, sorry, The Bourbaki (sic) : “I originally tried hard for the organisation to have no face, because I wanted egos to play no part in our activities. This followed the tradition of the… Read more »

One of the more surprising analogies around is that prime numbers can be viewed as knots in the 3sphere $S^3$. The motivation behind it is that the (etale) fundamental group of $\pmb{spec}(\mathbb{Z}/(p))$ is equal to (the completion) of the fundamental group of a circle $S^1$ and that the embedding $\pmb{spec}(\mathbb{Z}/(p)) \subset \pmb{spec}(\mathbb{Z})$ embeds this circle… Read more »

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 RSSaggregator with interesting mathematical blogs, is their graphical presentation of (nearly) all mathblogs ordered by type : group blogs, individual researchers, teachers and educators, journalistic writers,… Read more »

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 »

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 rediscovered 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 »

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 organistincharge at the SaintGermaindesPrés church in 1939,… Read more »

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 »

The algebraic fundamental group of a scheme gives the MazurKapranovReznikov dictionary between primes in number fields and knots in 3manifolds.

To Gavin Wraiht a mathematical phantom is a “nonexistent entity which ought to be there but apparently is not; but nevertheless obtrudes its effects so convincingly that one is forced to concede a broader notion of existence”. Mathematics’ history is filled with phantoms getting the kiss of life. Nobody will deny the ancient Greek were… 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 »

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 »

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

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 »

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 »

The one thing that makes it hard for an outsider to get through a cryptopaper is their shared passion for using nonsensical abbreviations. ECSTR stands for “Efficient Compact Subgroup Trace Representation” and we are fortunate that Arjen Lenstra and Eric Verheul shortened it in their paper The XTR public key system to just XTR. As… 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 »

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 »

Even if you don’t know the formal definition of a profinte group, you know at least one example which explains the concept : the Galois group of the algebraic numbers $Gal = Gal(\overline{\mathbb{Q}}/\mathbb{Q}) $ aka the absolute Galois group. By definition it is the group of all $\mathbb{Q} $isomorphisms of the algebraic closure $\overline{\mathbb{Q}} $…. 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 »

For what it is worth, Ive computed some more terms in the iguanodon series. Here they are $L_2(7),M_{12},A_{16},M_{24},A_{28},A_{40},A_{48},A_{60},A_{68},A_{88},A_{96},A_{120},A_{132},A_{148},A_{164},A_{196},\ldots $ By construction, the nth iguanodon group $Ig_n $ (corresponding to the nth Farey sequence) is a subgroup of the alternating group on its (half)legs. Hence to prove that all remaining iguanodons are alternating groups boils down… Read more »

Here the details of the iguanodon series. Start with the Farey sequence $F(n) $of order n which is the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size. Here are the first eight Fareys F(1) =… Read more »

Conway’s puzzle M(13) involves the 13 points and 13 lines of $\mathbb{P}^2(\mathbb{F}_3) $. On all but one point numbered counters are placed holding the numbers 1,…,12 and a move involves interchanging one counter and the ‘hole’ (the unique point having no counter) and interchanging the counters on the two other points of the line determined… Read more »

In older NeverEndingBooksposts (and here) proofs were given that the modular group $\Gamma = PSL_2(\mathbb{Z}) $ is the group free product $C_2 \ast C_3 $, so let’s just skim over details here. First one observes that $\Gamma $ is generated by (the images of) the invertible 2×2 matrices $U= \begin{bmatrix} 0 & 1 \\\ 1… Read more »
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