
At the moment I’m rereading Siobhan Roberts’ biography of John Horton Conway, Genius at play – the curious mind of John Horton Conway. In fact, I’m also rereading Alexander Masters’ biography of Simon Norton, The genius in my basement – the biography of a happy man. [full_width_image] [/full_width_image] If you’re in for a suggestion, try… Read more »

In several of his talks on #IUTeich, Mochizuki argues that usual scheme theory over $\mathbb{Z}$ is not suited to tackle problems such as the ABCconjecture. The idea appears to be that ABC involves both the additive and multiplicative nature of integers, making rings into ‘2dimensional objects’ (and clearly we use both ‘dimensions’ in the theory… Read more »

Today, Samuel Dehority, Xavier Gonzalez, Neekon Vafa and Roger Van Peski arXived their paper Moonshine for all finite groups. Originally, Moonshine was thought to be connected to the Monster group. McKay and Thompson observed that the first coefficients of the normalized elliptic modular invariant \[ J(\tau) = q^{1} + 196884 q + 21493760 q^2 +… Read more »

Some weeks ago, Robert Kucharczyk and Peter Scholze found a topological realisation of the ‘hopeless’ part of the absolute Galois group $\mathbf{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. That is, they constructed a compact connected space $M_{cyc}$ such that etale covers of it correspond to Galois extensions of the cyclotomic field $\mathbb{Q}_{cyc}$. This gives, at least in theory, a handle on… Read more »

We know embarrassingly little about the symmetries of the roots of all polynomials with rational coefficients, or if you prefer, the absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$. In the title picture the roots of polynomials of degree $\leq 4$ with small coefficients are plotted and coloured by degree: blue=4, cyan=3, red=2, green=1. Sums and products of roots… Read more »

absolute, geometry, number theory, stories
The Log Lady and the Frobenioid of $\mathbb{Z}$
Posted on by lievenlb“Sometimes ideas, like men, jump up and say ‘hello’. They introduce themselves, these ideas, with words. Are they words? These ideas speak so strangely.” “All that we see in this world is based on someone’s ideas. Some ideas are destructive, some are constructive. Some ideas can arrive in the form of a dream. I can… Read more »

Apologies to Joachim Roncin, the guy who invented the slogan “Je suis Charlie”, for this silly abuse of his logo: I had hoped the G+ post below of end december would have been the last I had to say on this (non)issue: (btw. embedded G+post below, not visible in feeds) A quick recap : –… Read more »

math, number theory
$\mathbf{Ext}(\mathbb{Q},\mathbb{Z})$ and the solenoid $\widehat{\mathbb{Q}}$
Posted on by lievenlbNote to self: check Jack Morava’s arXiv notes on a more regular basis! It started with the G+post below by +David Roberts: Suddenly I realised I hadn’t checked out Morava‘s “short preprints with ambitious ideas, but no proofs” lately. A couple of years ago I had a brief email exchange with him on the Habiro… Read more »

The sporadic second Janko group $J_2$ is generated by an element of order two and one of order three and hence is a quotient of the modular group $PSL_2(\mathbb{Z}) = C_2 \ast C_3$. This Janko group has a 100dimensional permutation representation and hence there is an index 100 subgroup $G$ of the modular group such… Read more »

One of the coolest (pure math) facts in Conway’s book ONAG is the explicit construction of the algebraic closure $\overline{\mathbb{F}_2}$ of the field with two elements as the set of all ordinal numbers smaller than $(\omega^{\omega})^{\omega}$ equipped with nimber addition and multiplication. Some time ago we did run a couple of posts on this. In… Read more »

absolute, math, number theory
Quiver Grassmannians and $\mathbb{F}_1$geometry
Posted on by lievenlbReineke’s observation that any projective variety can be realized as a quiver Grassmannian is bad news: we will have to look at special representations and/or dimension vectors if we want the Grassmannian to have desirable properties. Some people still see a silver lining: it can be used to define a larger class of geometric objects… Read more »

The odd Knight of the round table problem asks for a consistent placement of the nth Knight in the row at an odd root of unity, compatible with the two different realizations of the algebraic closure of the field with two elements. The first identifies the multiplicative group of its nonzero elements with the group… Read more »

This is a belated response to a MathOverflow 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 ConnesMarcolli book Noncommutative Geometry, Quantum Fields and Motives can be read as… Read more »

In 1956, Alexander Grothendieck (middle) introduced $\lambda $rings in an algebraicgeometric context to be commutative rings A equipped with a bunch of operations $\lambda^i $ (for all numbers $i \in \mathbb{N}_+ $) satisfying a list of rather obscure identities. From the easier ones, such as $\lambda^0(x)=1, \lambda^1(x)=x, \lambda^n(x+y) = \sum_i \lambda^i(x) \lambda^{ni}(y) $ to those… Read more »

The Knightseating problems asks for a consistent placing of nth Knight at an odd root of unity, compatible with the two different realizations of the algebraic closure of the field with two elements.

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 »

Hendrik Lenstra found an effective procedure to compute the mysterious elements alpha(p) needed to do actual calculations with infinite nimarithmetic.

Conway’s nimarithmetic on ordinal numbers leads to many surprising identities, for example who would have thought that the third power of the first infinite ordinal equals 2…

Surely Georg Cantor’s transfinite ordinal numbers do not have a reallife importance? Well, think again.

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

Some quotes of Andre Weil on the Riemann hypothesis.

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 »

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 »
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