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Tag: geometry

Grothendieck’s survival talks

The Grothendieck circle is a great resource to find published as well as unpublished texts by Alexander Grothendieck.

One of the text I was unaware of is his Introduction to Functorial Algebraic Geometry, a set of notes written up by Federico Gaeta based on tape-recordings (!) of an 100-hour course given by Grothendieck in Buffalo, NY in the summer of 1973. The Grothendieck-circle page adds this funny one-line comment: “These are not based on prenotes by Grothendieck and to some extent represent Gaeta’s personal understanding of what was taught there.”.

It is a bit strange that this text is listed among Grothendieck’s unpublished texts as Gaeta writes on page 3 : “GROTHENDIECK himself does not assume any responsability for the publication of these notes”. This is just one of many ‘bracketed’ comments by Gaeta which make these notes a great read. On page 5 he adds :

“Today for many collegues, GROTHENDIECK’s Algebraic Geometry looks like one of the most abstract and unapplicable products of current mathematical thought. This prejudice caused har(‘m’ or ‘ess’, unreadable) even before the students of mathematics within the U.S. were worried about the scarcity of academic positions… . If they ever heard GROTHENDIECK deliver one of his survival talks against modern Science, research, technology, etc., … their worries might become unbearable.”

Together with Claude Chevalley and Pierre Cartier, Grothendieck was an editor of “Survivre et Vivre“, the bulletin of the ecological association of the same name which appeared at regular intervals from 1970 to 1973. Scans of all but two of these volumes can be found here. All of this has a strong 60ties feel to it, as does Gaeta’s decription of Grothendieck : “He is a very liberal man and in spite of that he allowed us to use plenty of tape recorders!” (p.5).

On page 11, Gaeta records a little Q&A exchange from one of these legendary ‘survival talks’ by Grothendieck :

Question : We understand your worries about expert knowledge,… by the way, if we try to explain to a layman what algebraic geometry is it seems to me that the title of the old book of ENRIQUES, “Geometrical theory of equations”, is still adequate. What do you think?

GROTHENDIECK : Yes, but your ‘layman’ should know what a sustem of algebraic equations is. This would cost years of study to PLATO.

Question : It should be nice to have a little faith that after two thousand years every good high school graduate can understand what an affine scheme is … What do you think?

GROTHENDIECK : …. ??

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introducing : the n-geometry cafe

It all started with this comment on the noncommutative geometry blog by “gabriel” :

Even though my understanding of noncommutative geometry is limited, there are some aspects that I am able to follow.
I was wondering, since there are so few blogs here, why don’t you guys forge an alliance with neverending books, you blog about noncommutative geometry anyways. That way you have another(n-category cafe) blogspot and gives well informed views(well depending on how well defined a conversational-style blog can be).

The technology to set up a ‘conversational-style blog’, where anyone can either leave twitter-like messages or more substantial posts, is available thanks to the incredible people from Automattic.

For starters, they have the sensational p2 wordpress theme : “blogging at the speed of thought”



A group blog theme for short update messages, inspired by Twitter. Featuring: Hassle-free posting from the front page. Perfect for group blogging, or as a liveblog theme. Dynamic page updates. Threaded comment display on the front page. In-line editing for posts and comments. Live tag suggestion based on previously used tags. A show/hide feature for comments, to keep things tidy. Real-time notifications when a new comment or update is posted. Super-handy keyboard shortcuts.

Next, any lively online community is open for intense debate : “supercharge your community”



Fire up the debate with commenter profiles, reputation scores, and OpenID. With IntenseDebate you’ll tap into a whole new network of sites with avid bloggers and commenters. And that’s just the tip of the iceberg!

And finally, as we want to talk math, both in posts and comments, they provide us with the WP-LaTeX plugin.

All these ingredients make up the n-geometry cafe ((with apologies to the original cafe but I simply couldn’t resist…)) to be found at noncommutative.org (explaining the ‘n’).



Anyone can walk into a Cafe and have his/her say, that’s why you’ll get automatic author-privileges if you register.

Fill in your nick and email (please take your IntenseDebate setting and consider signing up with Gravator.com to get a nice image next to your contributions), invent your own password, show that you’re human by answering the reCapcha question and you’ll get a verification email within minutes ((if you don’t get an email within the hour, please notify me)). This will take you to your admin-page, allowing you to start blogging. For more info, check out the FAQ-pages.

I’m well aware of the obvious dangers of non-moderated sites, but also a strong believer in any Cafe’s self-regulating powers…

If you are interested in noncommutative geometry, and feel like sharing, please try it out.

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Bourbakism & the queen bee syndrome

Probably the smartest move I’ve made after entering math-school was to fall in love with a feminist.

Yeah well, perhaps I’ll expand a bit on this sentence another time. For now, suffice it to say that I did pick up a few words in the process, among them : the queen bee syndrome :

women who have attained senior positions do not use their power to assist struggling young women or to change the system, thereby tacitly validating it.

A recent study by the Max Planck Institute for Human Development asserts that the QBS

likely stems from women at the top who feel threatened by other women and therefore, prefer to surround themselves with men. As a result, these Queen Bees often jeapordize the promotions of other females at their companies.

Radical feminists of the late 70-ties preferred a different ‘explanation’, clearly.

Women who fought their way to the top, they said, were convinced that overcoming all obstacles along the way made them into the strong persons they became. A variant on the ‘what doesn’t kill you, makes you stronger’-mantra, quoi. These queen bees genuinely believed it to be beneficial to the next generation of young women not to offer them any shortcuts on their journey through the glass ceiling.

But, let’s return to mathematics.

By and large, the 45+generation decides about the topics that should be (or shouldn’t be) on the current math-curriculum. They also write most of the text-books and course-notes used, and inevitably, the choices they make have an impact on the new generation of math-students.

Perhaps too little thought is given to the fact that the choices we (yes, I belong to that age group) make, the topics we deem important for new students to master, are heavily influenced by our own experiences.

In the late 60ties, early 70ties, Bourbaki-style mathematics influenced the ‘modern mathematics’ revolution in schools, certainly in Belgium through the influence of George Papy.

In kintergarten, kids learned the basics of set theory. Utensils to draw Venn diagrams were as indispensable as are pocket-calculators today. In secondary school, we had a formal axiomatic approach to geometry, we learned abstract topological spaces and other advanced topics.

Our 45+generation greatly benefitted from all of this when we started doing research. We felt comfortable with the (in retrospect, over)abstraction of the EGAs and SGAs and had little difficulties in using them or generalizing them to noncommutative levels…

Bourbakism made us into stronger mathematicians. Hence, we are convinced that new students should master it if they ever want to do ‘proper’ research.

Perhaps we pay too little attention to the fact that these new students are a lot worse prepared than we were in the old days. Every revolution inevitably provokes a counter-revolution. Secondary school mathematics sank over the last two decades to a debilitating level under the pretense of ‘usability’. Tim Gowers has an interesting Ivory tower post on this.

We may deplore this evolution, we may try to reverse it. But, until we succeed, it may not be fair to freshmen to continue stubbornly as if nothing changed since our good old days.

Perhaps, Bourbakism has become our very own queen bee syndrome…

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Now here’s an idea

Boy, do I feel stupid for having written close to 500 blog-posts hoping (in vain) they might eventually converge into a book project…

Gil Kalai is infinitely smarter. Get a fake gmail account, invent a fictitious character and start COMMENTING and provoking responses. That’s how “Gina” appeared on the scene, cut and pasted her comments (and the replies to them) and turned all of this into a book : “Gina says”, Adventures in the Blogsphere String War.

So, who’s Gina? On page 40 : “35 years of age, Gina is of Greek and Polish descent. Born in the quaint island of Crete, she currently resides in the USA, in quiet and somewhat uneventful Wichita, Kansas. Gina has a B.Sc in Mathematics (from the University of Athens, with Honors), and a Master’s Degree in Psychology (from the University of Florence, with Honors).
Currently in-between jobs (her last job was working with underprivileged children), she has a lot of free time on her hands, which gives her ample opportunities to roam the blogosphere.”

So far, the first 94 pages are there to download, the part of the book consisting of comments left at Peter Woit’s Not Even Wrong. Judging from the table of contents, Gina left further traces at the n-category cafe and Asymptotia.

Having read the first 20 odd pages in full and skimmed the rest, two remarks : (1) it shouldn’t be too difficult to borrow this idea and make a much better book out of it and (2) it raises the question about copyrights on blog-comments…

If the noncommutative geometry blog could be persuaded to awake from its present dormant state, I’d love to get some discussions started, masquerading as AG. Or, given the fact that I’ll use the summer-break to re-educate myself as an n-categorist, the guys running the cafe are hereby warned…

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Pollock your own noncommutative space

I really like Matilde Marcolli’s idea to use some of Jackson Pollock’s paintings as metaphors for noncommutative spaces. In her talk she used this painting



and refered to it (as did I in my post) as : Jackson Pollock “Untitled N.3”. Before someone writes a post ‘The Pollock noncommutative space hoax’ (similar to my own post) let me point out that I am well aware of the controversy surrounding this painting.

This painting is among 32 works recently discovered and initially attributed to Pollock.
In fact, I’ve already told part of the story in Doodles worth millions (or not)? (thanks to PD1). The story involves the people on the right : from left to right, Jackson Pollock, his wife Lee Krasner, Mercedes Matter and her son Alex Matter.

Alex Matter, whose father, Herbert, and mother, Mercedes, were artists and friends of Jackson Pollock, discovered after his mother died a group of small drip paintings in a storage locker in Wainscott, N.Y. which he believed to be authentic Pollocks.

Read the post mentioned above if you want to know how mathematics screwed up his plan, or much better, reed the article Anatomy of the Jackson Pollock controversy by Stephen Litt.

So, perhaps the painting above was not the smartest choice, but we could take any other genuine Pollock ‘drip-painting’, a technique he taught himself towards the end of 1946 to make an image by splashing, pouring, sloshing colors onto the canvas. Typically, such a painting consists of blops of paint, connected via thin drip-lines.

What does this have to do with noncommutative geometry? Well, consider the blops as ‘points’. In commutative geometry, distinct points cannot share tangent information ((technically : a commutative semi-local ring splits as the direct sum of local rings and this does no longer hold for a noncommutative semi-local ring)). In the noncommutative world though, they can!, or if you want to phrase it like this, noncommutative points ‘can talk to each other’. And, that’s what we cherish in those drip-lines.

But then, if two points share common tangent informations, they must be awfully close to each other… so one might imagine these Pollock-lines to be strings holding these points together. Hence, it would make more sense to consider the ‘Pollock-quotient-painting’, that is, the space one gets after dividing out the relation ‘connected by drip-lines’ ((my guess is that Matilde thinks of the lines as the action of a group on the points giving a topological horrible quotient space, and thats precisely where noncommutative geometry shines)).

For this reason, my own mental picture of a genuinely noncommutative space ((that is, the variety corresponding to a huge noncommutative algebra such as free algebras, group algebras of arithmetic groups or fundamental groups)) looks more like the picture below



The colored blops you see are really sets of points which you might view as, say, a FacebookGroup ((technically, think of them as the connected components of isomorphism classes of finite dimensional simple representations of your favorite noncommutative algebra)). Some chatter may occur between two distinct FacebookGroups, the more chatter the thicker the connection depicted ((technically, the size of the connection is the dimension of the ext-group between generic simples in the components)). Now, there are some tiny isolated spots (say blue ones in the upper right-hand quadrant). These should really be looked at as remote clusters of noncommutative points (sharing no (tangent) information whatsoever with the blops in the foregound). If we would zoom into them beyond the Planck scale (if I’m allowed to say a bollock-word in a Pollock-post) they might reveal again a whole universe similar to the interconnected blops upfront.

The picture was produced using the fabulous Pollock engine. Just use your mouse to draw and click to change colors in order to produce your very own noncommutative space!

For the mathematicians still around, this may sound like a lot of Pollock-bollocks but can be made precise. See my note Noncommutative geometry and dual coalgebras for a very terse reading. Now that coalgebras are gaining popularity, I really should write a more readable account of it, including some fanshi-wanshi examples…

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Views of noncommutative spaces

The general public expects pictures from geometers, even from non-commutative geometers. Hence, it is important for researchers in this topic to make an attempt to convey the mental picture they have of their favourite noncommutative space, … somehow. Two examples :



This picture was created by Shahn Majid. It appears on his visions of noncommutative geometry page as well as in an extremely readable Plus-magazine article on Quantum geometry, written by Marianne Freiberger, explaining Shahn’s ideas. For more information on this, read Shahn’s SpaceTime blog.



This painting is Jackson Pollock‘s “Untitled N.3”. It depicts the way Matilde Marcolli imagines a noncommutative space. It is taken from her slides of her talk for a general audience Mathematicians look at particle physics.

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The Scottish solids hoax

A truly good math-story gets spread rather than scrutinized. And a good story it was : more than a millenium before Plato, the Neolithic Scottish Math Society classified the five regular solids : tetrahedron, cube, octahedron, dodecahedron and icosahedron. And, we had solid evidence to support this claim : the NSMS mass-produced stone replicas of their finds and about 400 of them were excavated, most of them in Aberdeenshire.

Six years ago, Michael Atiyah and Paul Sutcliffe arXived their paper Polyhedra in physics, chemistry and geometry, in which they wrote :

Although they are termed Platonic solids there is
convincing evidence that they were known to the Neolithic people of Scotland at least a
thousand years before Plato, as demonstrated by the stone models pictured in fig. 1 which
date from this period and are kept in the Ashmolean Museum in Oxford.

Fig. 1 is the picture below, which has been copied in numerous blog-posts (including my own scottish solids-post) and virtually every talk on regular polyhedra.



From left to right, stone-ball models of the cube, tetrahedron, dodecahedron, icosahedron and octahedron, in which ‘knobs’ correspond to ‘faces’ of the regular polyhedron, as best seen in the central dodecahedral ball.

But then … where’s the icosahedron? The fourth ball sure looks like one but only because someone added ribbons, connecting the centers of the different knobs. If this ribbon-figure is an icosahedron, the ball itself should be another dodecahedron and the ribbons illustrate the fact that icosa- and dodeca-hedron are dual polyhedra. Similarly for the last ball, if the ribbon-figure is an octahedron, the ball itself should be another cube, having exactly 6 knobs.
Who did adorn these artifacts with ribbons, thereby multiplying the number of ‘found’ regular solids by two (the tetrahedron is self-dual)?

The picture appears on page 98 of the book Sacred Geometry (first published in 1979) by Robert Lawlor. He attributes the NSMS-idea to the book Time Stands Still: New Light on Megalithic Science (also published in 1979) by Keith Critchlow. Lawlor writes

The five regular polyhedra or
Platonic solids were known and worked with
well before Plato’s time. Keith Critchlow in
his book Time Stands Still presents convincing
evidence that they were known to the Neolithic peoples of Britain at least 1000 years
before Plato. This is founded on the existence
of a number of sphericalfstones kept in the
Ashmolean Museum at Oxford. Of a size one
can carry in the hand, these stones were carved
into the precise geometric spherical versions of
the cube, tetrahedron, octahedron, icosahedron
and dodecahedron, as well as some additional
compound and semi-regular solids, such as the
cube-octahedron and the icosidodecahedron.
Critchlow says, ‘What we have are objects
clearly indicative of a degree of mathematical
ability so far denied to Neolithic man by any
archaeologist or mathematical historian’. He
speculates on the possible relationship of these
objects to the building of the great astronomical stone circles of the same epoch in Britain:
‘The study of the heavens is, after all, a
spherical activity, needing an understanding of
spherical coordinates. If the Neolithic inhabitants of Scotland had constructed Maes Howe
before the pyramids were built by the ancient
Egyptians, why could they not be studying the
laws of three-dimensional coordinates? Is it not
more than a coincidence that Plato as well as
Ptolemy, Kepler and Al-Kindi attributed
cosmic significance to these figures?’

As Lawlor and Critchlow lean towards mysticism, their claims should not be taken for granted. So, let’s have a look at these famous stones kept in the Ashmolean Museum. The Ashmolean has a page dedicated to their Stone Balls, including the following picture (the Critchlow/Lawlor picture below, for comparison)



The Ashmolean stone balls are from left to right the artifacts with catalogue numbers :

  • Stone ball with 7 knobs from Marnoch, Banff (AN1927.2728)
  • Stone ball with 6 knobs and isosceles triangles between, from Fyvie, Aberdeenshire (AN1927.2731)
  • Stone ball with 6 knobs and isosceles triangles between, from near Aberdeen (AN1927.2730)
  • Stone ball with 4 knobs from Auchterless, Aberdeenshire (AN1927.2729)
  • Stone ball with 14 knobs from Aberdeen (AN1927.2727)

Ashmolean’s AN 1927.2729 may very well be the tetrahedron and AN 1927.2727 may be used to forge the ‘icosahedron’ (though it has 14 rather than 12 knobs), but the other stones sure look different. In particular, none of the Ashmolean stones has exactly 12 knobs in order to be a dodecahedron.

Perhaps the Ashmolean has a larger collection of Scottish balls and today’s selection is different from the one in 1979? Well, if you have the patience to check all 9 pages of the Scottish Ball Catalogue by Dorothy Marshall (the reference-text when it comes to these balls) you will see that the Ashmolean has exactly those 5 balls and no others!

The sad lesson to be learned is : whether the Critchlow/Lawlor balls are falsifications or fabrications, they most certainly are NOT the Ashmolean stone balls as they claim!

Clearly this does not mean that no neolithic scott could have discovered some regular polyhedra by accident. They made an enormous amount of these stone balls, with knobs ranging from 3 up to no less than 135! All I claim is that this ball-carving thing was more an artistic endeavor, rather than a mathematical one.

There are a number of musea having a much larger collection of these stone balls. The Hunterian Museum has a collection of 29 and some nice online pages on them, including 3D animation. But then again, none of their balls can be a dodecahedron or icosahedron (according to the stone-ball-catalogue).

In fact, more than half of the 400+ preserved artifacts have 6 knobs. The catalogue tells that there are only 8 possible candidates for a Scottish dodecahedron (below their catalogue numbers, indicating for the knowledgeable which museum owns them and where they were found)

  • NMA AS 103 : Aberdeenshire
  • AS 109 : Aberdeenshire
  • AS 116 : Aberdeenshire (prob)
  • AUM 159/9 : Lambhill Farm, Fyvie, Aberdeenshire
  • Dundee : Dyce, Aberdeenshire
  • GAGM 55.96 : Aberdeenshire
  • Montrose = Cast NMA AS 26 : Freelands, Glasterlaw, Angus
  • Peterhead : Aberdeenshire

The case for a Scottish icosahedron looks even worse. Only two balls have exactly 20 knobs

  • NMA AS 110 : Aberdeenshire
  • GAGM 92 106.1. : Countesswells, Aberdeenshire

Here NMA stands for the National Museum of Antiquities of Scotland in Edinburg (today, it is called ‘National Museums Scotland’) and
GAGM for the Glasgow Art Gallery and Museum. If you happen to be in either of these cities shortly, please have a look and let me know if one of them really is an icosahedron!

UPDATE (April 1st)

Victoria White, Curator of Archaeology at the
Kelvingrove Art Gallery and Museum, confirms that the Countesswells carved stone ball (1892.106.l) has indeed 20 knobs. She gave this additional information :

The artefact came to Glasgow Museums in the late nineteenth century as part of the John Rae collection. John Rae was an avid collector of prehistoric antiquities from the Aberdeenshire area of Scotland. Unfortunately, the ball was not accompanied with any additional information regarding its archaeological context when it was donated to Glasgow Museums. The carved stone ball is currently on display in the ‘Raiders of the Lost Art’ exhibition.

Dr. Alison Sheridan, Head of Early Prehistory, Archaeology Department, National Museums Scotland makes the valid point that new balls have been discovered after the publication of the catalogue, but adds :

Although several balls have turned up since Dorothy Marshall wrote her synthesis, none has 20 knobs, so you can rely on Dorothy’s list.

She has strong reservations against a mathematical interpretation of the balls :

Please also note that the mathematical interpretation of these Late Neolithic objects fails to take into account their archaeological background, and fails to explain why so many do not have the requisite number of knobs! It’s a classic case of people sticking on an interpretation in a state of ignorance. A great shame when so much is known about Late Neolithic archaeology.

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Geometry of the Okubo algebra

Last week, Melanie Raczek gave a talk entitled ‘Cubic forms and Okubo product’ in our Artseminar, based on her paper On ternary cubic forms that determine central simple algebras of degree 3.

I had never heard of this strange non-associative product on 8-dimensional space, but I guess it is an instance of synchronicity that now the Okubo algebra seems to pop-up everywhere.

Yesterday, there was the post the Okubo algebra by John Baez at the n-cafe, telling that Susumu Okubo discovered his algebra while investigating quarks.

I don’t know a thing about the physics, but over the last days I’ve been trying to understand some of the miraculous geometry associated to the Okubo algebra. So, let’s start out by defining the ‘algebra’.

Consider the associative algebra of all 3×3 complex matrices $M_3(\mathbb{C}) $ with the usual matrix-multiplication. In this algebra there is the 8-dimensional subspace of trace zero matrices, usually called the Lie algebra $\mathfrak{sl}_3 $. However, we will not use the Lie-bracket, only matrix-multiplication. Typical elements of $\mathfrak{sl}_3 $ will be written as $X,Y,Z,… $ and their entries will be denoted as

$X = \begin{bmatrix} x_0 & x_1 & x_2 \\ x_3 & x_4 & x_5 \\ x_6 & x_7 & -x_0-x_4 \end{bmatrix} $

For any two elements $X,Y \in \mathfrak{sl}_3 $ one defines their Okubo-product to be the 3×3 matrix

$X \ast Y = \frac{1}{1-\omega}(Y.X-\omega X.Y) – \frac{1}{3}Tr(X.Y) 1_3 $

where $\omega $ is a primitive 3-rd root of unity and $1_3 $ is the identity matrix. Written out in the entries of X and Y this operation looks horribly complicated

$X \ast Y = \frac{1}{1-\omega} \begin{bmatrix} p_{11} & p_{12} & p_{13} \\ p_{21} & p_{22} & p_{23} \\ p_{31} & p_{32} & -p_{11}-p_{22} \end{bmatrix} $

with

[tex]\begin{eqalign} \\ p_{11} &= (1-\omega)x_0y_0+x_3y_1+x_6y_2-\omega(x_1y_3+x_2y_6)-\frac{1}{3}T \\ p_{12} &= x_1y_0+x_4y_1+x_7y_2-\omega(x_0y_1+x_1y_4+x_2y_7) \\ p_{13} &= x_2y_0+x_5y_1-x_0y_2-x_4y_2-\omega(x_0y_2+x_1y_5-x_2y_0-x_2y_4) \\
p_{21} &= x_0y_3+x_3y_4+x_6y_5 – \omega(x_3y_0+x_4y_3+x_5y_6) \\
p_{22} &= (1-\omega)x_4y_4+x_1y_3+x_7y_5 – \omega(x_3y_1+x_5y_7) – \frac{1}{3}T \\
p_{23} &= x_2y_3+x_5y_4-x_0y_5-x_4y_5-\omega(x_3y_2+x_4y_5-x_5y_0-x_5y_4) \\
p_{31} &= x_0y_6+x_3y_7-x_6y_0-x_6y_4-\omega(x_6y_0+x_7y_3-x_0y_6-x_4y_6) \\
p_{32} &= x_1y_6+x_4y_7-x_7y_0-x_7y_4 – \omega(x_6y_1+x_7y_4-x_0y_7-x_4y_7) \\
T &= 2x_0y_0+2x_4y_4+x_1y_3+x_2y_6+x_3y_1+x_5y_7+x_6y_2+x_7y_5+x_0y_4+x_4y_0
\end{eqalign}[/tex]

The crucial remark to make is that $X \ast Y $ is again a trace zero matrix. That is, we have defined a new operation on $\mathfrak{sl}_3 $.

$\mathfrak{sl}_3 \times \mathfrak{sl}_3 \rightarrow \mathfrak{sl}_3~\qquad~\qquad~(X,Y) \mapsto X \ast Y $

This Okubo-product is neither a Lie-bracket, nor an associative multiplication. In fact, it is a lot ‘less associative’ than that other 8-dimensional algebra, the octonions. The only noteworthy identity it has is that $X \ast (Y \ast X) = (X \ast Y) \ast X $. So, why should we be interested in this horrible algebra?

Well, let us consider the subset of $\mathfrak{sl}_3 $ consisting of those matrices X satusfying $Tr(X^2)=0 $. That is, with the above notation, all matrices X such that

$x_0^2+x_4^2+x_1x_3+x_2x_6+x_5x_7=0 $

In the 8-dimensional affine space $\mathfrak{sl}_3 $ these matrices form a singular quadric with top the zero-matrix. So, it is better to go projective. That is, any non-zero matrix $X \in \mathfrak{sl}_3 $ determines a point in 7-dimensional projective space $\mathbb{P}^7 $ with homogeneous coordinates

$\overline{X} = [x_0:x_1:x_2:x_3:x_4:x_5:x_6:x_7] \in \mathbb{P}^7 $

and the points $\overline{X} $ corresponding to solutions of $Tr(X^2)=0 $ form a smooth 6-dimensional quadric $Q \subset \mathbb{P}^7 $ with homogeneous equation

$Q = \mathbb{V}(x_0^2+x_4^2+x_1x_3+x_2x_6+x_5x_7) $

6-dimensional quadrics may be quite hard to visualize, so it may help to recall the classic situation of lines on a 2-dimensional quadric (animated gif taken from surfex).

A 2-dimensional quadric contains two families of lines, often called the ‘blue lines’ and the ‘red lines’, each of these lines isomorphic to $\mathbb{P}^1 $. The rules-of-intersection of these are :

  • different red lines are disjoint as are different blue lines
  • any red and any blue line intersect in exactly one point
  • every point of the quadric lies on exactly one red and one blue line

The lines in either family are in one-to-one correspondence with the points on the projective line. We therefore say that there is a $\mathbb{P}^1 $-family of red lines and a $\mathbb{P}^1 $-family of blue lines on a 2-dimensional quadric.

A 6-dimensional quadric $Q \subset \mathbb{P}^7 $ contains two families of ‘3-planes’. That is, there is a family of red $\mathbb{P}^3 $’s contained in Q and a family of blue $\mathbb{P}^3 $’s. Can we determine these red and blue 3-planes explicitly?

Yes we can, using the Okubo algebra-product on $\mathfrak{sl}_3 $. Take $X \in \mathfrak{sl}_3 $ defining the point $\overline{X} \in Q $ (that is, $Tr(X^2)=0 $). then all 3×3 matrices one obtains by taking the Okubo-product with left X-factor form a 4-dimensional linear subspace in $\mathfrak{sl}_3 $

$L_X = { X \ast Y~|~Y \in \mathfrak{sl}_3 } \simeq \mathbb{C}^4 \subset \mathfrak{sl}_3 $

so its non-zero matrices determine a 3-plane in $\mathbb{P}^7 $ (consisting of all points with homogeneous coordinates $[p_{11}:p_{12}:p_{13}:p_{21}:p_{22}:p_{23}:p_{31}:p_{32}] $, using the above formulas) which actually lies entirely in the quadric Q. These are precisely the bLue 3-planes in Q. That is, the family of all bLue 3-planes consists precisely of the 3-planes

$\mathbb{P}(L_X) $ with $X \in \mathfrak{sl}_3 $ satisfying $Tr(X^2)=0 $

Phrased differently, any point $\overline{X} \in Q $ determines a blue 3-plane $\mathbb{P}(L_X) $.

Similarly, any point $\overline{X} \in Q $ determines a Red 3-plane by taking Okubo-products with Right X-factor, that is, $\mathbb{P}(R_X) $ is a 3-plane for Q where

$R_X = { Y \ast X~|~Y \in \mathfrak{sl}_3 } \simeq \mathbb{C}^4 \subset \mathfrak{sl}_3 $

and all Red 3-planes for Q are of this form. But, this is not all… these correspondences are unique! That is, any point on the quadric defines a unique red and a unique blue 3-plane, or, phrased differently, there is a Q-family of red 3-planes and a Q-family of blue 3-planes in Q. This is a consequence of triality.

To see this, note that the automorphism group of a 6-dimensional smooth quadric is the rotation group $SO_8(\mathbb{C}) $ and this group has Dynkin diagram $D_4 $, the most symmetrical of them all!

In general, every node in a Dynkin diagram has an interesting projective variety associated to it, a so called homogeneous space. I’ll just mention what these spaces are corresponding to the 4 nodes of $D_4 $. Full details can be found in chapter 23 of Fulton and Harris’ Representation theory, a first course.

The left-most node corresponds to the orthogonal Grassmannian of isotropic 1-planes in $\mathbb{C}^8 $ which is just a fancy way of viewing our quadric Q. The two right-most nodes correspond to the two connected components of the Grassmannians of isotropic 4-planes in $\mathbb{C}^8 $, which are our red resp. blue families of 3-planes on the quadric. Now, as the corresponding dotted Dynkin diagrams are isomorphic



there corresponding homogeneous spaces are also isomorphic. Thus indeed, there is a one-to-one correspondence between points of the quadric Q and red 3-planes on Q (and similarly with blue 3-planes on Q).

Okay, so the Okubo-product allows us to associate to a point on the 6-dimensional quadric Q a unique red 3-plane and a unique blue 3-plane (much as any point on a 2-dimensional quadric determines a unique red and blue line). Do these families of red and blue 3-planes also satisfy ‘rules-of-intersection’?

Yes they do and, once again, the Okubo-product clarifies them. Here they are :

  • two different red 3-planes intersect in a unique line (as do different blue 3-planes)
  • the bLue 3-plane $\mathbb{P}(L_X) $ intersects the Red 3-plane $\mathbb{P}(R_Y) $ in a unique point if and only if the Okubo-product $X \ast Y \not= 0 $
  • the bLue 3-plane $\mathbb{P}(L_X) $ intersects the Red 3-plane $\mathbb{P}(R_Y) $ in a unique 2-plane if and only if the Okubo-product $X \ast Y = 0 $

That is, Right and Left Okubo-products determine the Red and bLue families of 3-planes on the 6-dimensional quadric as well as their intersections!

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Connes & Consani go categorical

Today, Alain Connes and Caterina Consani arXived their new paper Schemes over $ \mathbb{F}_1$ and zeta functions. It is a follow-up 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 paper, the main point of the first 25 pages of the new paper is to repace abelian groups by abelian monoids in the definition, making it more in tune with other approaches, most notably that of Anton Deitmar. The novelty, if you want, is that they package the two functors $\mathbf{rings} \rightarrow \mathbf{sets} $ and $\mathbf{ab-monoid} \rightarrow \mathbf{sets} $ into one functor $\mathbf{ring-monoid} \rightarrow \mathbf{sets} $ by using the ‘glued category’ $\mathbf{ring-monoid} $ (an idea they attribute to Pierre Cartier).

In general, if you have two categories $\mathbf{cat} $ and $\mathbf{cat’} $ and a pair of adjoint functors between them, then one can form the glued-category $\mathbf{cat-cat’} $ by taking as its collection of objects the disjoint union of the objects of the two categories and by defining the hom-sets between two objects the hom-sets in either category (if both objects belong to the same category) or use the adjoint functors to define the new hom-set when they do not (the very definition of adjoint functors makes that this doesn’t depend on the choice).

Here, one uses the functor $\mathbf{ab-monoid} \rightarrow \mathbf{rings} $ assigning to a monoid $M $ its integral monoid-algebra $\mathbb{Z}[M] $, having as its adjoint the functor $\mathbf{rings} \rightarrow \mathbf{ab-monoid} $ forgetting the additive structure of the commutative ring.

In the second part of the paper, they first prove some nice results on zeta-functions of Noetherian $\mathbb{F}_1 $-schemes and extend them, somewhat surprisingly, to settings which do not (yet) fit into the $\mathbb{F}_1 $-framework, namely elliptic curves and the hypothetical $\mathbb{F}_1 $-curve $\overline{\mathbf{spec}(\mathbb{Z})} $.

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On2 : transfinite number hacking

In ONAG, John Conway proves that the symmetric version of his recursive definition of addition and multiplcation on the surreal numbers make the class On of all Cantor’s ordinal numbers into an algebraically closed Field of characteristic two : On2 (pronounced ‘Onto’), and, in particular, he identifies a subfield
with the algebraic closure of the field of two elements. What makes all of this somewhat confusing is that Cantor had already defined a (badly behaving) addition, multiplication and exponentiation on ordinal numbers.

Over the last week I’ve been playing a bit with sage to prove a few exotic identities involving ordinal numbers. Here’s one of them ($\omega $ is the first infinite ordinal number, that is, $\omega={ 0,1,2,\ldots } $),

$~(\omega^{\omega^{13}})^{47} = \omega^{\omega^7} + 1 $

answering a question in Hendrik Lenstra’s paper Nim multiplication.

However, it will take us a couple of posts before we get there. Let’s begin by trying to explain what brought this on. On september 24th 2008 there was a meeting, intended for a general public, called a la rencontre des dechiffeurs, celebrating the 50th birthday of the IHES.

One of the speakers was Alain Connes and the official title of his talk was “L’ange de la géométrie, le diable de l’algèbre et le corps à un élément” (the angel of geometry, the devil of algebra and the field with one element). Instead, he talked about a seemingly trivial problem : what is the algebraic closure of $\mathbb{F}_2 $, the field with two elements? My only information about the actual content of the talk comes from the following YouTube-blurb

Alain argues that we do not have a satisfactory description of $\overline{\mathbb{F}}_2 $, the algebraic closure of $\mathbb{F}_2 $. Naturally, it is the union (or rather, limit) of all finite fields $\mathbb{F}_{2^n} $, but, there are too many non-canonical choices to make here.

Recall that $\mathbb{F}_{2^k} $ is a subfield of $\mathbb{F}_{2^l} $ if and only if $k $ is a divisor of $l $ and so we would have to take the direct limit over the integers with respect to the divisibility relation… Of course, we can replace this by an increasing sequence of a selection of cofinal fields such as

$\mathbb{F}_{2^{1!}} \subset \mathbb{F}_{2^{2!}} \subset \mathbb{F}_{2^{3!}} \subset \ldots $

But then, there are several such suitable sequences! Another ambiguity comes from the description of $\mathbb{F}_{2^n} $. Clearly it is of the form $\mathbb{F}_2[x]/(f(x)) $ where $f(x) $ is a monic irreducible polynomial of degree $n $, but again, there are several such polynomials. An attempt to make a canonical choice of polynomial is to take the ‘first’ suitable one with respect to some natural ordering on the polynomials. This leads to the so called Conway polynomials.

Conway polynomials for the prime $2 $ have only been determined up to degree 400-something, so in the increasing sequence above we would already be stuck at the sixth term $\mathbb{F}_{2^{6!}} $…

So, what Alain Connes sets as a problem is to find another, more canonical, description of $\overline{\mathbb{F}}_2 $. The problem is not without real-life interest as most finite fields appearing in cryptography or coding theory are subfields of $\overline{\mathbb{F}}_2 $.

(My guess is that Alain originally wanted to talk about the action of the Galois group on the roots of unity, which would be the corresponding problem over the field with one element and would explain the title of the talk, but decided against it. If anyone knows what ‘coupling-problem’ he is referring to, please drop a comment.)

Surely, Connes is aware of the fact that there exists a nice canonical recursive construction of $\overline{\mathbb{F}}_2 $ due to John Conway, using Georg Cantor’s ordinal numbers.

In fact, in chapter 6 of his book On Numbers And Games, John Conway proves that the symmetric version of his recursive definition of addition and multiplcation on the surreal numbers make the class $\mathbf{On} $ of all Cantor’s ordinal numbers into an algebraically closed Field of characteristic two : $\mathbf{On}_2 $ (pronounced ‘Onto’), and, in particular, he identifies a subfield

$\overline{\mathbb{F}}_2 \simeq [ \omega^{\omega^{\omega}} ] $

with the algebraic closure of $\mathbb{F}_2 $. What makes all of this somewhat confusing is that Cantor had already defined a (badly behaving) addition, multiplication and exponentiation on ordinal numbers. To distinguish between the Cantor/Conway arithmetics, Conway (and later Lenstra) adopt the convention that any expression between square brackets refers to Cantor-arithmetic and un-squared ones to Conway’s. So, in the description of the algebraic closure just given $[ \omega^{\omega^{\omega}} ] $ is the ordinal defined by Cantor-exponentiation, whereas the exotic identity we started out with refers to Conway’s arithmetic on ordinal numbers.

Let’s recall briefly Cantor’s ordinal arithmetic. An ordinal number $\alpha $ is the order-type of a totally ordered set, that is, if there is an order preserving bijection between two totally ordered sets then they have the same ordinal number (or you might view $\alpha $ itself as a totally ordered set, namely the set of all strictly smaller ordinal numbers, so e.g. $0= \emptyset,1= { 0 },2={ 0,1 },\ldots $).

For two ordinals $\alpha $ and $\beta $, the addition $[\alpha + \beta ] $ is the order-type of the totally ordered set $\alpha \sqcup \beta $ (the disjoint union) ordered compatible with the total orders in $\alpha $ and $\beta $ and such that every element of $\beta $ is strictly greater than any element from $\alpha $. Observe that this definition depends on the order of the two factors. For example,$ [1 + \omega] = \omega $ as there is an order preserving bijection ${ \tilde{0},0,1,2,\ldots } \rightarrow { 0,1,2,3,\ldots } $ by $\tilde{0} \mapsto 0,n \mapsto n+1 $. However, $\omega \not= [\omega + 1] $ as there can be no order preserving bijection ${ 0,1,2,\ldots } \rightarrow { 0,1,2,\ldots,0_{max} } $ as the first set has no maximal element whereas the second one does. So, Cantor’s addition has the bad property that it may be that $[\alpha + \beta] \not= [\beta + \alpha] $.

The Cantor-multiplication $ \alpha . \beta $ is the order-type of the product-set $\alpha \times \beta $ ordered via the last differing coordinate. Again, this product has the bad property that it may happen that $[\alpha . \beta] \not= [\beta . \alpha] $ (for example $[2 . \omega ] \not=[ \omega . 2 ] $). Finally, the exponential $\beta^{\alpha} $ is the order type of the set of all maps $f~:~\alpha \rightarrow \beta $ such that $f(a) \not=0 $ for only finitely many $a \in \alpha $, and ordered via the last differing function-value.

Cantor’s arithmetic allows normal-forms for ordinal numbers. More precisely, with respect to any ordinal number $\gamma \geq 2 $, every ordinal number $\alpha \geq 1 $ has a unique expression as

$\alpha = [ \gamma^{\alpha_0}.\eta_0 + \gamma^{\alpha_1}.\eta_1 + \ldots + \gamma^{\alpha_m}.\eta_m] $

for some natural number $m $ and such that $\alpha \geq \alpha_0 > \alpha_1 > \ldots > \alpha_m \geq 0 $ and all $1 \leq \eta_i < \gamma $. In particular, taking the special cases $\gamma = 2 $ and $\gamma = \omega $, we have the following two canonical forms for any ordinal number $\alpha $

$[ 2^{\alpha_0} + 2^{\alpha_1} + \ldots + 2^{\alpha_m}] = \alpha = [ \omega^{\beta_0}.n_0 + \omega^{\beta_1}.n_1 + \ldots + \omega^{\beta_k}.n_k] $

with $m,k,n_i $ natural numbers and $\alpha \geq \alpha_0 > \alpha_1 > \ldots > \alpha_m \geq 0 $ and $\alpha \geq \beta_0 > \beta_1 > \ldots > \beta_k \geq 0 $. Both canonical forms will be important when we consider the (better behaved) Conway-arithmetic on $\mathbf{On}_2 $, next time.

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