Two lecture series on absolute geometry

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 “absolute point” $\mathbf{Spec}(\mathbb{F}_1)$.

Starting with the paper by Bertrand Toen and Michel Vaquie, Under $\mathbf{Spec}(\mathbb{Z})$, topos theory entered this topic.

First there was the proposal by Jim Borger to view $\lambda$-rings as $\mathbb{F}_1$-algebras. More recently, Alain Connes and Katia Consani introduced the arithmetic site.

Now, there are lectures series on these two approaches, one by Yuri I. Manin, the other by Alain Connes.


Yuri I. Manin in Ghent

On Tuesday, February 3rd, Yuri I. Manin will give the inaugural lectures of the new $\mathbb{F}_1$-seminars at Ghent University, organised by Koen Thas.

Coffee will be served from 13.00 till 14.00 at the Department of Mathematics, Ghent University, Krijgslaan 281, Building S22 and from 14.00 till 16.30 there will be lectures in the Emmy Noether lecture room, Building S25:

14:00 – 14:25: Introduction (by K. Thas)
14:30 – 15:20: Lecture 1 (by Yu. I. Manin)
15:30 – 16:20: Lecture 2 (by Yu. I. Manin)

Recent work of Manin related to $\mathbb{F}_1$ includes:

Local zeta factors and geometries under $\mathbf{Spec}(\mathbb{Z})$

Numbers as functions

Alain Connes on the Arithmetic Site

Until the beginning of march, Alain Connes will lecture every thursday afternoon from 14.00 till 17.30, in Salle 5 – Marcelin Berthelot at he College de France on The Arithmetic Site (hat tip Isar Stubbe).

Here’s a two minute excerpt, from a longer interview with Connes, on the arithmetic site, together with an attempt to provide subtitles:



And,in this example, we saw the wonderful notion of a topos, developed by Grothendieck.

It was sufficient for me to open SGA4, a book written at the beginning of the 60ties or the late fifties.

It was sufficient for me to open SGA4 to see that all the things that I needed were there, say, how to construct a cohomology on this site, how to develop things, how to see that the category of sheaves of Abelian groups is an Abelian category, having sufficient injective objects, and so on … all those things were there.

This is really remarkable, because what does it mean?

It means that the average mathematician says: “topos = a generalised topological space and I will never need to use such things. Well, there is the etale cohomology and I can use it to make sense of simply connected spaces and, bon, there’s the chrystaline cohomology, which is already a bit more complicated, but I will never need it, so I can safely ignore it.”

And (s)he puts the notion of a topos in a certain category of things which are generalisations of things, developed only to be generalisations…

But in fact, reality is completely different!

In our work with Katia Consani we saw not only that there is this epicyclic topos, but in fact, this epicyclic topos lies over a site, which we call the arithmetic site, which itself is of a delirious simplicity.

It relies only on the natural numbers, viewed multiplicatively.

That is, one takes a small category consisting of just one object, having this monoid as its endomorphisms, and one considers the corresponding topos.

This appears well … infantile, but nevertheless, this object conceils many wonderful things.

And we would have never discovered those things, if we hadn’t had the general notion of what a topos is, of what a point of a topos is, in terms of flat functors, etc. etc.



I will try to report here on Manin’s lectures in Ghent. If someone is able to attend Connes’ lectures in Paris, I’d love to receive updates!

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 inspiring conjectures. He considers the ring


where $\mathbb{Z}$ are the integers, the $\xi_i$ are the “odd” variables anti-commuting among themselves and commuting with the “even” variables $x_j$. To this ring, Manin wants to associate a geometric object of dimension $1+m+n$ where $1$ refers to the “arithmetic dimension”, $m$ to the ordinary geometric dimensions $(x_1,\ldots,x_m)$ and $n$ to the new “odd dimensions” represented by the coordinates $(\xi_1,\ldots,\xi_n)$. Manin writes :

“Before the advent of ringed spaces in the fifties it would have been difficult to say precisely what me mean when we speak about this geometric object. Nowadays we simply define it as an “affine superscheme”, an object of the category of topological spaces locally ringed by a sheaf of $\mathbb{Z}_2$-graded supercommutative rings.”

Here’s my own image (based on Mumford’s depiction of $\mathsf{Spec}(\mathbb{Z}[x])$) of what Manin calls the three-space-2000, whose plain $x$-axis is supplemented by the set of primes and by the “black arrow”, corresponding to the odd dimension.

Manin speculates : “The message of the picture is intended to be the following metaphysics underlying certain recent developments in geometry: all three types of geometric dimensions are on an equal footing”.

Probably, by the addition “2000” Manin meant that by the year 2000 we would as easily switch between these three types of dimensions as we were able to draw arithmetic schemes in the mid-80ties. Quod non.

Twelve years into the new millenium we are only able to decode fragments of this. We know that symmetric algebras and exterior algebras (that is the “even” versus the “odd” dimensions) are related by Koszul duality, and that the precise relationship between the arithmetic axis and the geometric axis is the holy grail of geometry over the field with one element.

For aficionados of $\mathbb{F}_1$ there’s this gem by Manin to contemplate :

“Does there exist a group, mixing the arithmetic dimension with the (even) geometric ones?”

Way back in 1984 Manin conjectured : “There is no such group naively, but a ‘category of representations of this group’ may well exist. There may exist also certain correspondence rings (or their representations) between $\mathsf{Spec}(\mathbb{Z})$ and $x$.”

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 of its importance, rather than giving a visual illustration of the varieties over $\mathbb{F}_1$ the approach proposes.

Some time ago, we’ve had a couple of posts trying to depict noncommutative varieties, first the illustrations used by Shahn Majid and Matilde Marcolli, and next my own mental picture of it.

In this post, we’ll try to do something similar for affine varieties over the absolute point. To simplify things drastically, I’ll divide the islands in the Lopez Pena-Lorscheid map of $\mathbb{F}_1$ land in two subsets : the former approaches (all but the $\Lambda$-schemes) and the current approach (the $\Lambda$-scheme approach due to James Borger).

The former approaches : Francis Bacon “The Pope” (1953)

The general consensus here was that in going from $\mathbb{Z}$ to $\mathbb{F}_1$ one looses the additive structure and retains only the multiplicative one. Hence, ‘commutative algebras’ over $\mathbb{F}_1$ are (commutative) monoids, and mimicking Grothendieck’s functor of points approach to algebraic geometry, a scheme over $\mathbb{F}_1$ would then correspond to a functor

$h_Z~:~\mathbf{monoids} \longrightarrow \mathbf{sets}$

Such functors are described largely by combinatorial data (see for example the recent blueprint-paper by Oliver Lorscheid), and, if the story would stop here, any Rothko painting could be used as illustration.

Most of the former approaches add something though (buzzwords include ‘Arakelov’, ‘completion at $\infty$’, ‘real place’ etc.) in order to connect the virtual geometric object over $\mathbb{F}_1$ with existing real, complex or integral schemes. For example, one can make the virtual object visible via an evaluation map $h_Z \rightarrow h_X$ which is a natural transformation, where $X$ is a complex variety with its usual functor of points $h_X$ and to connect both we associate to a monoid $M$ its complex monoid-algebra $\mathbb{C} M$. An integral scheme $Y$ can then be said to be ‘defined over $\mathbb{F}_1$’, if $h_Z$ becomes a subfunctor of its usual functor of points $h_Y$ (again, assigning to a monoid its integral monoid algebra $\mathbb{Z} M$) and $Y$ is the ‘best’ integral scheme approximation of the complex evaluation map.

To illustrate this, consider the painting Study after Velázquez’s Portrait of Pope Innocent X by Francis Bacon (right-hand painting above) which is a distorded version of the left-hand painting Portrait of Innocent X by Diego Velázquez.

Here, Velázquez’ painting plays the role of the complex variety which makes the combinatorial gadget $h_Z$ visible, and, Bacon’s painting depicts the integral scheme, build up from this combinatorial data, which approximates the evaluation map best.

All of the former approaches more or less give the same very small list of integral schemes defined over $\mathbb{F}_1$, none of them motivically interesting.

The current approach : Jackson Pollock “No. 8” (1949)

An entirely different approach was proposed by James Borger in $\Lambda$-rings and the field with one element. He proposes another definition for commutative $\mathbb{F}_1$-algebras, namely $\lambda$-rings (in the sense of Grothendieck’s Riemann-Roch) and he argues that the $\lambda$-ring structure (which amounts in the sensible cases to a family of endomorphisms of the integral ring lifting the Frobenius morphisms) can be viewed as descent data from $\mathbb{Z}$ to $\mathbb{F}_1$.

The list of integral schemes of finite type with a $\lambda$-structure coincides roughly with the list of integral schemes defined over $\mathbb{F}_1$ in the other approaches, but Borger’s theory really shines in that it proposes long sought for mystery-objects such as $\mathbf{spec}(\mathbb{Z}) \times_{\mathbf{spec}(\mathbb{F}_1)} \mathbf{spec}(\mathbb{Z})$. If one accepts Borger’s premise, then this object should be the geometric object corresponding to the Witt-ring $W(\mathbb{Z})$. Recall that the role of Witt-rings in $\mathbb{F}_1$-geometry was anticipated by Manin in Cyclotomy and analytic geometry over $\mathbb{F}_1$.

But, Witt-rings and their associated Witt-spaces are huge objects, so one needs to extend arithmetic geometry drastically to include such ‘integral schemes of infinite type’. Borger has made a couple of steps in this direction in The basic geometry of Witt vectors, II: Spaces.

To depict these new infinite dimensional geometric objects I’ve chosen for Jackson Pollock‘s painting No. 8. It is no coincidence that Pollock-paintings also appeared in the depiction of noncommutative spaces. In fact, Matilde Marcolli has made the connection between $\lambda$-rings and noncommutative geometry in Cyclotomy and endomotives by showing that the Bost-Connes endomotives are universal for $\lambda$-rings.

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 3-dimensional simply connected manifold which, after Perelman, has to be $S^3$. For more see the what is the knot associated to a prime?-post.

In recent months new evidence has come to light allowing us to settle the genesis of this marvelous idea.

1. The former consensus

Until now, the generally accepted view (see for example the ‘Mazur-dictionary-post’ or Morishita’s expository paper) was that the analogy between knots and primes was first pointed out by Barry Mazur in the middle of the 1960’s when preparing for his lectures at the Summer Conference on Algebraic Geometry, at Bowdoin, in 1966. The lecture notes where later published in 1973 in the Annales of the ENS as ‘Notes on etale cohomology of number fields’.

For further use in this series of posts, please note the acknowledgement at the bottom of the first page, reproduced below : “It gives me pleasure to thank J.-P. Serre for his vigorous editing and his suggestions and corrections, which led to this revised version.”

Independently, Yuri I. Manin spotted the same analogy at around the same time. However, this point of view was quickly forgotten in favor of the more classical one of viewing number fields as analogous to algebraic function fields of one variable. Subsequently, in the mid 1990’s Mikhail Kapranov and Alexander Reznikov took up the analogy between number fields and 3-manifolds again, and called the resulting study arithmetic topology.

2. The new evidence

On december 13th 2010, David Feldman posted a MathOverflow-question Mazur’s unpublished manuscript on primes and knots?. He wrote : “The story of the analogy between knots and primes, which now has a literature, started with an unpublished note by Barry Mazur. I’m not absolutely sure this is the one I mean, but in his paper, Analogies between group actions on 3-manifolds and number fields, Adam Sikora cites B. Mazur, Remarks on the Alexander polynomial, unpublished notes.

Two months later, on february 15th David Feldman suddenly found the missing preprint in his mail-box and made it available. The preprint is now also available from Barry Mazur’s website. Mazur adds the following comment :

“In 1963 or 1964 I wrote an article Remarks on the Alexander Polynomial [PDF] about the analogy between knots in the three-dimensional sphere and prime numbers (and, correspondingly, the relationship between the Alexander polynomial and Iwasawa Theory). I distributed some copies of my article but never published it, and I misplaced my own copy. In subsequent years I have had many requests for my article and would often try to search through my files to find it, but never did. A few weeks ago Minh-Tri Do asked me for my article, and when I said I had none, he very kindly went on the web and magically found a scanned copy of it. I’m extremely grateful to Minh-Tri Do for his efforts (and many thanks, too, to David Feldman who provided the lead).”

The opening paragraph of this unpublished preprint contains a major surprise!

Mazur points to David Mumford as the originator of the ‘primes-are-knots’ idea : “Mumford has suggested a most elegant model as a geometric interpretation of the above situation : $\pmb{spec}(\mathbb{Z}/p\mathbb{Z})$ is like a one-dimensional knot in $\pmb{spec}(\mathbb{Z})$ which is like a simply connected three-manifold.”

In a later post we will show that one can even pinpoint the time and place when and where this analogy was first dreamed-up to within a few days and a couple of miles.

For the impatient among you, have a sneak preview of the cradle of birth of the primes=knots idea…

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, we will include here Javier’s and Oliver’s insights on the colored pieces below in their map : CC=Connes-Consani, Generalized torified schemes=Lopez Pena-Lorscheid, Generalized schemes with 0=Durov and, this time, $\Lambda$=Manin-Marcolli.

Durov : romanticism

In his 568 page long Ph.D. thesis New Approach to Arakelov Geometry Nikolai Durov introduces a vast generalization of classical algebraic geometry in which both Arakelov geometry and a more exotic geometry over $\mathbb{F}_1$ fit naturally. Because there were great hopes and expectations it would lead to a big extension of algebraic geometry, Javier and Oliver associate this approach to romantism. From wikipedia : “The modern sense of a romantic character may be expressed in Byronic ideals of a gifted, perhaps misunderstood loner, creatively following the dictates of his inspiration rather than the standard ways of contemporary society.”

Manin and Marcolli : impressionism

Yuri I. Manin in Cyclotomy and analytic geometry over $\mathbb{F}_1$ and Matilde Marcolli in Cyclotomy and endomotives develop a theory of analytic geometry over $\mathbb{F}_1$ based on analytic functions ‘leaking out of roots of unity’. Javier and Oliver depict such functions as ‘thin, but visible brush strokes at roots of 1’ and therefore associate this approach to impressionism. Frow wikipedia : ‘Characteristics of Impressionist paintings include: relatively small, thin, yet visible brush strokes; open composition; emphasis on accurate depiction of light in its changing qualities (often accentuating the effects of the passage of time); common, ordinary subject matter; the inclusion of movement as a crucial element of human perception and experience; and unusual visual angles.’

Connes and Consani : cubism

In On the notion of geometry over $\mathbb{F}_1$ Alain Connes and Katia Consani develop their extension of Soule’s approach. A while ago I’ve done a couple of posts on this here, here and here. Javier and Oliver associate this approach to cubism (a.o. Pablo Picasso and Georges Braque) because of the weird juxtapositions of the simple monoidal pieces in this approach.

Lopez-Pena and Lorscheid : deconstructivism

Torified varieties and schemes were introduced by Javier Lopez-Pena and Oliver Lorscheid in Torified varieties and their geometries over $\mathbb{F}_1$ to get lots of examples of varieties over the absolute point in the sense of both Soule and Connes-Consani. Because they were fragmenting schemes into their “fundamental pieces” they associate their approach to deconstructivism.

Another time I’ll sketch my own arty-farty take on all this.

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 the search for the absolute point $\pmb{spec}(\mathbb{F}_1)$ originates from the observation that $\pmb{spec}(\mathbb{Z})$, the set of all prime numbers together with $0$, is too large to serve as the terminal object in Grothendieck’s theory of commutative schemes. The last couple of years have seen a booming industry of proposals, to the extent that Javier Lopez Pena and Oliver Lorscheid decided they had to draw a map of $\mathbb{F}_1$-land.

Manin only discusses the colored proposals (TV=Toen-Vaquie, M=Deitmar, S=Soule and $\Lambda$=Borger) and compares them to these art-history trends.

Toen and Vaquie : Abstract Expressionism

In Under $\pmb{spec}(\mathbb{Z})$ Bertrand Toen and Michel Vaquie argue that geometry over $\mathbb{F}_1$ is a special case of algebraic geometry over a symmetric monoidal category, taking the simplest example namely sets and direct products. Probably because of its richness and abstract nature, Manin associates this approach to Abstract Expressionism (a.o. Karel Appel, Jackson Pollock, Mark Rothko, Willem de Kooning).

Deitmar : Minimalism

Because monoids are the ‘commutative algebras’ in sets with direct products, an equivalent proposal is that of Anton Deitmar in Schemes over $\mathbb{F}_1$ in which the basic affine building blocks are spectra of monoids, topological spaces whose points are submonoids satisfying a primeness property. Because Deitmar himself calls this approach a ‘minimalistic’ one it is only natural to associate to it Minimalism where the work is stripped down to its most fundamental features. Prominent artists associated with this movement include Donald Judd, John McLaughlin, Agnes Martin, Dan Flavin, Robert Morris, Anne Truitt, and Frank Stella.

Soule : Critical Realism

in Les varietes sur le corps a un element Christophe Soule defines varieties over $\mathbb{F}_1$ to be specific schemes $X$ over $\mathbb{Z}$ together with a form of ‘descent data’ as well as an additional $\mathbb{C}$-algebra, morally the algebra of functions on the real place. Because of this Manin associates to it Critical Realism in philosophy. There are also ‘realism’ movements in art such as American Realism (o.a. Edward Hopper and John Sloan).

Borger : Futurism

James Borger’s paper Lambda-rings and the field with one element offers a totally new conception of the descent data from $\mathbb{Z}$ to $\mathbb{F}_1$, namely that of a $\lambda$-ring in the sense of Grothendieck. Because Manin expects this approach to lead to progress in the field, he connects it to Futurism, an artistic and social movement that originated in Italy in the early 20th century.