Posts tagged Marcolli

Pollock your own noncommutative space

May 19th

Posted by lievenlb in geometry

4 comments

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 information1. 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’2.

For this reason, my own mental picture of a genuinely noncommutative space 3 looks more like the picture below

The colored blops you see are really sets of points which you might view as, say, a FacebookGroup4. Some chatter may occur between two distinct FacebookGroups, the more chatter the thicker the connection depicted5. 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…

  1. 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 [↩]
  2. 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 [↩]
  3. that is, the variety corresponding to a huge noncommutative algebra such as free algebras, group algebras of arithmetic groups or fundamental groups [↩]
  4. technically, think of them as the connected components of isomorphism classes of finite dimensional simple representations of your favorite noncommutative algebra [↩]
  5. technically, the size of the connection is the dimension of the ext-group between generic simples in the components [↩]
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Views of noncommutative spaces

May 18th

Posted by lievenlb in geometry

3 comments

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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best of 2008 (1) : wiskundemeisjes

Jan 2nd

Posted by lievenlb in general

1 comment

Of course, excellent math-blogs exist in every language imaginable, but my linguistic limitations restrict me to the ones written in English, French, German and … Dutch. Here a few links to Dutch (or rather, Flemish) math-blogs, in order of proximity : Stijn Symens blog, Rudy Penne’s wiskunde is sexy (math is sexy), Koen Vervloesem’s QED.

My favorite one is wiskundemeisjes (‘math-chicks’ or ‘math-girls’), written by Ionica Smeets and Jeanine Daems, two reasearchers at Leiden University. Every month they have a post called “the favorite (living) mathematician of …” in which they ask someone to nominate and introduce his/her favorite colleague mathematician. Here some examples : Roger Penrose chooses Michael Atiyah, Robbert Dijkgraaf chooses Maxim Kontsevich, Frans Oort chooses David Mumford, Gunther Cornelissen chooses Yuri I. Manin, Hendrik Lenstra chooses Bjorn Poonen, etc. the full list is here or here. This series deserves a wider audience. Perhaps Ionica and Jeanine might consider translating some of these posts?

I’m certain their English is far better than mine, so here’s a feeble attempt to translate the one post in their series they consider a complete failure (it isn’t even listed in the category). Two reasons for me to do so : it features Matilde Marcolli (one of my own favorite living mathematicians) and Matilde expresses here very clearly my own take on popular-math books/blogs.

The original post was written by Ionica and was called Weg met de ‘favoriete wiskundige van…’ :

“This week I did spend much of my time at the Fifth European Mathematical Congress in Amsterdam. Several mathematicians suggested I should have a chat with Matilde Marcolli, one of the plenary speakers. It seemed like a nice idea to ask her about her favorite (still living) mathematician, for our series.

Marcolli explained why she couldn’t answer this question : she has favorite mathematical ideas, but it doesn’t interest her one bit who discovered or proved them. And, there are mathematicians she likes, but that’s because she finds them interesting as human beings, independent of their mathematical achievements.

In addition, she thinks it’s a mistake to focus science too much on the persons. Scientific ideas should play the main role, not the scientists themselves. To her it is important to remember that many results are the combined effort of several people, that science doesn’t evolve around personalities and that scientific ideas are accessible to anyone.

Marcolli also dislikes the current trend in popular science writing: “I am completely unable to read popular-scientific books. As soon as they start telling anecdotes and stories, I throw away the book. I don’t care about their lives, I care about the real stuff.”

She’d love to read a popular science-book containing only ideas. She regrets that most of these books restrict to story-telling, but fail to disseminate the scientific ideas.”

Ionica then goes on to defend her own approach to science-popularization :

“… Probably, people will not know much about Galois-theory by reading about his turbulent life. Still, I can imagine people to become interested in ‘the real stuff’ after reading his biography, and, in this manner they will read some mathematics they wouldn’t have known to exist otherwise. But, Marcolli got me thinking, for it is true that almost all popular science-books focus on anecdotes rather than science itself. Is this wrong? For instance, do you want to see more mathematics here? I’m curious to hear your opinion on this.”

Even though my own approach is somewhat different, Ionica and Jeanine you’re doing an excellent job: “houden zo!”

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F_un hype resulting in new blog

Oct 3rd

Posted by lievenlb in groups

1 comment

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 F1″.

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 element pointing us to a YouTube-clip featuring Alain Connes explaining his paper with Katia Consani and Matilde Marcolli entitled “Fun with F_un”. Here’s the clip

Finally, as I’ll be running a seminar here too on F_un, we’ve set up a group blog with the people from MPI (clearly, if you are interested to join us, just tell!). At the moment there are just a few of my old F_un posts and a library of F_un papers, but hopefully a lot will be added soon. So, have a look at F_un mathematics

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Looking for F_un

Jun 3rd

Posted by lievenlb in geometry

6 comments

Fun with F_un

  1. Looking for F_un
  2. The F_un folklore
  3. Absolute linear algebra
  4. F_un and braid groups
  5. F_un with Manin

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 field with one element \mathbb{F}_1, or in French : “F-un”. The topic must have reached a level of maturity as there was a conference dedicated entirely to it : NONCOMMUTATIVE GEOMETRY AND GEOMETRY OVER THE FIELD WITH ONE ELEMENT.

In this series I’d like to find out what the fuss is all about, why people would like it to exist and what it has to do with noncommutative geometry. However, before we start two remarks :

The field \mathbb{F}_1 does not exist, so don’t try to make sense of sentences such as “The ‘field with one element’ is the free algebraic monad generated by one constant (p.26), or the universal generalized ring with zero (p.33)” in the wikipedia-entry. The simplest proof is that in any (unitary) ring we have 0 \not= 1 so any ring must contain at least two elements. A more highbrow version : the ring of integers \mathbb{Z} is the initial object in the category of unitary rings, so it cannot be an algebra over anything else.

The second remark is that several people have already written blog-posts about \mathbb{F}_1. Here are a few I know of : David Corfield at the n-category cafe and at his old blog, Noah Snyder at the secret blogging seminar, Kea at the Arcadian functor, AC and K. Consani at Noncommutative geometry and John Baez wrote about it in his weekly finds.

The dream we like to keep alive is that we will prove the Riemann hypothesis one fine day by lifting Weil’s proof of it in the case of curves over finite fields to rings of integers.

Even if you don’t know a word about Weil’s method, if you think about it for a couple of minutes, there are two immediate formidable problems with this strategy.

For most people this would be evidence enough to discard the approach, but, we mathematicians have found extremely clever ways for going into denial.

The first problem is that if we want to think of \wis{spec}(\mathbb{Z}) (or rather its completion adding the infinite place) as a curve over some field, then \mathbb{Z} must be an algebra over this field. However, no such field can exist…

No problem! If there is no such field, let us invent one, and call it \mathbb{F}_1. But, it is a bit hard to do geometry over an illusory field. Christophe Soule succeeded in defining varieties over \mathbb{F}_1 in a talk at the 1999 Arbeitstagung and in a more recent write-up of it : Les varietes sur le corps a un element.

We will come back to this in more detail later, but for now, here’s the main idea. Consider an existent field k and an algebra k \rightarrow R over it. Now study the properties of the functor (extension of scalars) from k-schemes to R-schemes. Even if there is no morphism \mathbb{F}_1 \rightarrow \mathbb{Z}, let us assume it exists and define \mathbb{F}_1-varieties by requiring that these guys should satisfy the properties found before for extension of scalars on schemes defined over a field by going to schemes over an algebra (in this case, \mathbb{Z}-schemes). Roughly speaking this defines \mathbb{F}_1-schemes as subsets of points of suitable \mathbb{Z}-schemes.

But, this is just one half of the story. He adds to such an \mathbb{F}_1-variety extra topological data ‘at infinity’, an idea he attributes to J.-B. Bost. This added feature is a \mathbb{C}-algebra \mathcal{A}_X, which does not necessarily have to be commutative. He only writes : “Par ignorance, nous resterons tres evasifs sur les proprietes requises sur cette \mathbb{C}-algebre.”

The algebra \mathcal{A}_X originates from trying to bypass the second major obstacle with the Weil-Riemann-strategy. On a smooth projective curve all points look similar as is clear for example by noting that the completions of all local rings are isomorphic to the formal power series k[[x]] over the basefield, in particular there is no distinction between ‘finite’ points and those lying at ‘infinity’.

The completions of the local rings of points in \wis{spec}(\mathbb{Z}) on the other hand are completely different, for example, they have residue fields of different characteristics… Still, local class field theory asserts that their quotient fields have several common features. For example, their Brauer groups are all isomorphic to \mathbb{Q}/\mathbb{Z}. However, as Br(\mathbb{R}) = \mathbb{Z}/2\mathbb{Z} and Br(\mathbb{C}) = 0, even then there would be a clear distinction between the finite primes and the place at infinity…

Alain Connes came up with an extremely elegant solution to bypass this problem in Noncommutative geometry and the Riemann zeta function. He proposes to replace finite dimensional central simple algebras in the definition of the Brauer group by AF (for Approximately Finite dimensional)-central simple algebras over \mathbb{C}. This is the origin and the importance of the Bost-Connes algebra.

We will come back to most of this in more detail later, but for the impatient, Connes has written a paper together with Caterina Consani and Matilde Marcolli Fun with \mathbb{F}_1 relating the Bost-Connes algebra to the field with one element.

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