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 object over something ‘deeper’, the… Read more →

# Tag Archive for Manin

# Manin’s three-space-2000

Almost three decades ago, Yuri Manin submitted the paper “New dimensions in geometry” to the 25th Arbeitstagung, Bonn 1984. It is published in its proceedings, Springer Lecture Notes in Mathematics 1111, 59-101 and there’s a review of the paper available online in the Bulletin of the AMS written by Daniel Burns. In the introduction Manin makes some highly speculative but… Read more →

# Art and the absolute point (3)

Previously, we have recalled comparisons between approaches to define a geometry over the absolute point and art-historical 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 the approach or an appreciation… Read more →

# Who dreamed up the primes=knots analogy?

One of the more surprising analogies around is that prime numbers can be viewed as knots in the 3-sphere $S^3$. 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 as a knot in a… Read more →

# Art and the absolute point (2)

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 Lopez-Pena wrote that he and Oliver Lorscheid briefly contemplated the idea of extending Manin’s artsy-dictionary to all approaches they did draw on their Map of $\mathbb{F}_1$-land. So this time,… Read more →

# Art and the absolute point

In his paper Cyclotomy and analytic geometry over $\mathbb{F}_1$ Yuri I. Manin sketches and compares four approaches to the definition of a geometry over $\mathbb{F}_1$, the elusive field with one element. He writes : “Preparing a colloquium talk in Paris, I have succumbed to the temptation to associate them with some dominant trends in the history of art.” Remember that… Read more →

# Langlands versus Connes

This is a belated response to a Math-Overflow 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 Connes-Marcolli book Noncommutative Geometry, Quantum Fields and Motives can be read as an exploration of the noncommutative… Read more →

# best of 2008 (1) : wiskundemeisjes

A feeble attempt to translate the Marcolli-post by the ‘wiskundemeisjes’. Read more →

# Manin’s geometric axis

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. Read more →

# Mumford’s treasure map

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

# F_un hype resulting in new blog

At the Max-Planck 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 Lopez-Pena and Bram Mesland will organize a weekly “F_un Study Seminar” starting next tuesday. Over at Noncommutative Geometry there is an Update on the field with one… Read more →

# F_un with Manin

Amidst all LHC-noise, Yuri I. Manin arXived today an interesting paper Cyclotomy and analytic geometry over $\mathbb{F}_1 $. The paper gives a nice survey of the existent literature and focusses on the crucial role of roots of unity in the algebraic geometry over the non-existent field with one element $\mathbb{F}_1 $ (in French called ‘F-un’). I have tried to do… Read more →

# Arnold’s trinities

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 infinity (and perhaps other blogs… Read more →

# bloomsday 2 : BistroMath

Exactly one year ago this blog was briefly renamed MoonshineMath. The concept being that it would focus on the mathematics surrounding the monster group & moonshine. Well, I got as far as the Mathieu groups… After a couple of months, I changed the name back to neverendingbooks because I needed the freedom to post on any topic I wanted. I… Read more →

# The F_un folklore

All esoteric subjects have their own secret (sacred) texts. If you opened the Da Vinci Code (or even better, the original The Holy blood and the Holy grail) you will known about a mysterious collection of documents, known as the “Dossiers secrets“, deposited in the Bibliothèque nationale de France on 27 April 1967, which is rumoured to contain the mysteries… Read more →