
Absolute geometry is the attempt to develop algebraic geometry over the elusive field with one element $\mathbb{F}_1$. The idea being that the set of all prime numbers is just too large for $\mathbf{Spec}(\mathbb{Z})$ to be a terminal object (as it is in the category of schemes). So, one wants to view $\mathbf{Spec}(\mathbb{Z})$ as a geometric… Read more »

Previously, we have recalled comparisons between approaches to define a geometry over the absolute point and arthistorical movements, first those due to Yuri I. Manin, subsequently some extra ones due to Javier Lopez Pena and Oliver Lorscheid. In these comparisons, the art trend appears to have been chosen more to illustrate a key feature of… Read more »

Penrose tilings are aperiodic tilings of the plane, made from 2 sort of tiles : kites and darts. It is well known (see for example the standard textbook tilings and patterns section 10.5) that one can describe a Penrose tiling around a given point in the plane as an infinite sequence of 0’s and 1’s,… Read more »

Last time we did recall Manin’s comparisons between some approaches to geometry over the absolute point $\pmb{spec}(\mathbb{F}_1)$ and trends in the history of art. In the comments to that post, Javier LopezPena wrote that he and Oliver Lorscheid briefly contemplated the idea of extending Manin’s artsydictionary to all approaches they did draw on their Map… 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 »

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 »

noncommutative, web
Jason & David, the Ninja warriors of noncommutative geometry
Posted on by lievenlbSocialMention gives a rather accurate picture of the webbuzz on a specific topic. For this reason I check it irregularly to know what’s going on in noncommutative geometry, at least webwise. Yesterday, I noticed two new kids on the block : Jason and David. Their blogs have (so far ) 44 resp. 27 posts, this… Read more »

No christmas or newyears family party without heated discussions. Often on quite silly topics. For example, which late 19thcentury bookcharacter turned out to be most influential in the 20th century? Dracula, from the 1897 novel by Irish author Bram Stoker or Sir Arthur Conan Doyle’s Sherlock Holmes who made his first appearance in 1887? Well,… 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 »

Here’s a tiny problem illustrating our limited knowledge of finite fields : “Imagine an infinite queue of Knights ${ K_1,K_2,K_3,\ldots } $, waiting to be seated at the unitcircular table. The master of ceremony (that is, you) must give Knights $K_a $ and $K_b $ a place at an odd root of unity, say $\omega_a… Read more »

Today, Alain Connes and Caterina Consani arXived their new paper Schemes over $ \mathbb{F}_1$ and zeta functions. It is a followup to their paper On the notion of geometry over $ \mathbb{F}_1$, which I’ve tried to explain in a series of posts starting here. As Javier noted already last week when they updated their first… Read more »

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

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.

We use Kontsevich’s idea of thin varieties to define complexified varieties over F\_un.

Some quotes of Andre Weil on the Riemann hypothesis.

We propose to extend the ConnesConsani definition to noncommuntative F_un varieties.

Some links to posts on Soule’s algebraic geometry over the field with one element.

A quick recap of last time. We are trying to make sense of affine varieties over the elusive field with one element $\mathbb{F}_1 $, which by Grothendieck’s schemephilosophy should determine a functor $\mathbf{nano}(N)~:~\mathbf{abelian} \rightarrow \mathbf{sets} \qquad A \mapsto N(A) $ from finite Abelian groups to sets, typically giving pretty small sets $N(A) $. Using the… Read more »

Last time we have seen how an affine $\mathbb{C} $algebra R gives us a maxifunctor (because the associated sets are typically huge) $\mathbf{maxi}(R)~:~\mathbf{abelian} \rightarrow \mathbf{sets} \qquad A \mapsto Hom_{\mathbb{C}alg}(R, \mathbb{C} A) $ Substantially smaller sets are produced from finitely generated $\mathbb{Z} $algebras S (therefore called minifunctors) $\mathbf{mini}(S)~:~\mathbf{abelian} \rightarrow \mathbf{sets} \qquad A \mapsto Hom_{\mathbb{Z}alg}(S, \mathbb{Z} A)… Read more »

At the MaxPlanck Institute in Bonn Yuri Manin gave a talk about the field of one element, $\mathbb{F}_1 $ earlier this week entitled “Algebraic and analytic geometry over the field F_1”. Moreover, Javier LopezPena and Bram Mesland will organize a weekly “F_un Study Seminar” starting next tuesday. Over at Noncommutative Geometry there is an Update… Read more »

A couple of weeks ago, Alain Connes and Katia Consani arXived their paper “On the notion of geometry over $\mathbb{F}_1 $”. Their subtle definition is phrased entirely in Grothendieck‘s schemetheoretic language of representable functors and may be somewhat hard to get through if you only had a few years of mathematics. I’ll try to give… Read more »

There are only a handful of human activities where one goes to extraordinary lengths to keep a dream alive, in spite of overwhelming evidence : religion, theoretical physics, supporting the Belgian football team and … mathematics. In recent years several people spend a lot of energy looking for properties of an elusive object : the… Read more »

I’ve always thought of Alain Connes as the unchallengeable worldchampion opaque mathematical writing, but then again, I was proven wrong. Alain’s writings are crystal clear compared to the monstrosity the AMS released to the world : In search of the Riemann zeros – Strings, fractal membranes and noncommutative spacetimes by Michel L. Lapidus. Here’s a… Read more »

On friday, I did spot in my regular Antwerpbookshop Finding Moonshine by Marcus du Sautoy and must have uttered a tiny curse because, at once, everyone near me was staring at me… To make matters worse, I took the book from the shelf, quickly glanced through it and began shaking my head more and more,… Read more »

Terry Tao is reworking some of his better blogposts into a book, to be published by the AMS (here’s a preliminary version of the book “What’s New?”) After some thought, I decided not to transcribe all of my posts from last year (there are 93 of them!), but instead to restrict attention to those articles… Read more »
Close