
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 When did Mazur write his unpublished preprint?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 »

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 »

In view or recents events & comments, some changes have been made or will be made shortly : categories : Sanitized the plethora of wordpresscategories to which posts belong. At the moment there are just 5 categories : ‘stories’ and ‘web’ (for all posts with low mathcontent) and three categories ‘level1’, ‘level2’ and ‘level3’, loosely… Read more »

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 »

Sometimes a MathOverflow question gets deleted before I can post a reply… Yesterday (NewYear) PD1&2 were visiting, so I merely bookmarked the What is the knot associated to a prime?topic, promising myself to reply to it this morning, only to find out that the page no longer exists. From what I recall, the OP interpreted… Read more »

A few MathOverflow threads I bookmarked in 2010 for various reasons. Honest answer : Applications of algebraic geometry over a field with one element. James Borger’s answer : “I’m confident that the answer to the original question is no. There are hardly any theorems at all in the subject, much less ones with external applications!… 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 »

Over the weekend I read The artist and the mathematician (subtitle : The story of Nicolas Bourbaki, the genius mathematician who never existed) by Amir D. Aczel. Whereas the central character of the book should be Bourbaki, it focusses more on two of Bourbaki’s most colorful members, André Weil and Alexander Grothendieck, and the many… Read more »

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

Manin proposes the idea of projecting spec(Z[x]) not only onto spec(Z), but also to a geometric axis by considering the integers as an algebra over the field with one element.

In the series “Brave new geometries” we give an introduction to ‘strange’ but exciting new ideas. We start with Grothendieck’s schemerevolution, go on with Soule’s geometry over the field with one element, Mazur’s arithmetic topology, Grothendieck’s anabelian geometry, Connes’ noncommutative geometry etc.

Before we can even attempt to describe the adelic description of the BostConnes Hecke algebra and its symmetries, we’d probably better recall the construction and properties of adeles and ideles. Let’s start with the padic numbers $\hat{\mathbb{Z}}_p $ and its field of fractions $\hat{\mathbb{Q}}_p $. For p a prime number we can look at the… 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 »

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 »

Last time we introduced the tangent quiver $\vec{t}~A $ of an affine algebra A to be a quiver on the isoclasses of simple finite dimensional representations. When $A=\mathbb{C}[X] $ is the coordinate ring of an affine variety, these vertices are just the points of the variety $X $ and this set has the extra structure… Read more »

Here pdffiles of older NeverEndingBooksposts on geometry. For more recent posts go here.

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

Yesterday, Yuri Manin and Matilde Marcolli arXived their paper Modular shadows and the LevyMellin infinityadic transform which is a followup of their previous paper Continued fractions, modular symbols, and noncommutative geometry. They motivate the title of the recent paper by : In [MaMar2](http://www.arxiv.org/abs/hepth/0201036), these and similar results were put in connection with the so called… 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 »

Sociologists are a constant source of enlightenment as CNN keeps reminding Kids who are turned off by math often say they don’t enjoy it, they aren’t good at it and they see little point in it. Who knew that could be a formula for success? The nations with the best scores have the least happy,… Read more »

The CategoryCafe ran an interesting post The history of ncategories claiming that “mathematicians’ histories are largely ‘Royalroadtome’ accounts” To my mind a key difference is the historians’ emphasis in their histories that things could have turned out very differently, while the mathematicians tend to tell a story where we learn how the present has emerged… Read more »

For a qurve (aka formally smooth algebra) A a *block* is a (possibly infinite dimensional over the basefield) left Amodule X such that its endomorphism algebra $D = End_A(X)$ is a division algebra and X (considered as a right Dmodule) is finite dimensional over D. If a block X is finite dimensional over the basefield,… Read more »

For finite dimensional hereditary algebras, one can describe its noncommutative topology (as developed in part 2) explicitly, using results of Markus Reineke in The monoid of families of quiver representations. Consider a concrete example, say $A = \begin{bmatrix} \mathbb{C} & V \\ 0 & \mathbb{C} \end{bmatrix}$ where $V$ is an ndimensional complex vectorspace, or equivalently,… Read more »

A *qurve* is an affine algebra such that $~\Omega^1~A$ is a projective $~A~$bimodule. Alternatively, it is an affine algebra allowing lifts of algebra morphisms through nilpotent ideals and as such it is the ‘right’ noncommutative generalization of Grothendieck’s smoothness criterium. Examples of qurves include : semisimple algebras, coordinate rings of affine smooth curves, hereditary orders… Read more »
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