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…

Bloomsday has a tradition of bringing drastic changes to this blog.

Two years ago, it signaled a bloomsday-ending to the original neverendingbooks, giving birth (at least for a couple of months) to MoonshineMath.

Last year, the bloomsday 2 post was the first of several ‘conceptual’ blog proposals, voicing my conviction that a math-blog can only survive as a group-blog.

A few months later, I launched yet another proposal and promised that neverendingbooks would end on new-years eve, exactly five years after it started.

And, here we are again, half a year later, still struggling on … barely.

Well, don’t expect drastic statements from me today. I’ll continue to post when I do feel I’ve something to say (and won’t if I don’t) ¹. Also, there won’t be another pathetic cry-for-cooperation. I must have given up on that hope.

In fact, there isn’t much I can add to the post just mentioned (in particular my comment to it) to explain my present state of mind when it comes to blogging (and maths).

Let’s hope google wave will be released soon and that some of you will use it to make relevant waves. I promise to add blips when possible.

1. that is, apart from this silly post [↩]

Creating (or taking) an image and explaining how it depicts your mental picture of a noncommutative space is one thing. Ideally, the image should be strong enough so that other people familiar with it might have a reasonable guess what you attempt to depict.

But, is there already enough concordance in our views of noncommutative spaces? I doubt it, whence this experiment. Below my attempt¹ to depict one of the most popular noncommutative spaces around :

Can you guess what space this is? How does it agree with (resp. differ from) your own mental image of it?

Further, if you know of links to other depictions of noncommutative spaces, please leave a comment, or, send me an email.

1. the image is taken from Cran’s fractal art [↩]

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 [↩]

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.