lieven le bruyn's blog
lazy blogging
Grothendieck’s survival talks
Aug 6th
The Grothendieck circle is a great resource to find published as well as unpublished texts by Alexander Grothendieck.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It is a bit strange that this text is listed among Grothendieck’s unpublished texts as Gaeta writes on page 3 : “GROTHENDIECK himself does not assume any responsability for the publication of these notes”.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 : …. ??
Gina says (continued)
Aug 5th
Via the Arcadian functor I’ve grabbed the full text of Gil Kalai’s book Gina Says: Adventures in the Blogsphere String War (part 1 and part 2) and read it on a lazy sunny afternoon.
Arguably the best paragraph is the final one, and, it might be sensible to start the book by reading it first as it clarifies Kalai’s point in collecting these comment-threads from a number of popular blogs (including not even wrong, the n-category cafe and asymptotia) :
This book is not about string theory. It is more about delicate boundaries between greatness and megalomania, between humility and arrogance, between fantasy and reality, between wisdom and bullshit, between people of different stature and standing, between skepticism and harassment, between sanity and its loss, and between truths and fallacies. These are delicate boundaries that we witness in academics and in science and even in blog discussions. This story offers a little salute to people’s passion for understanding their logical and physical reality, as well as for understanding themselves.
Now here’s an idea
Jun 23rd
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…
noncommutative space quiz
May 21st
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.
- the image is taken from Cran’s fractal art [↩]
Pollock your own noncommutative space
May 19th
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…
- 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 [↩]
- 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 [↩]
- that is, the variety corresponding to a huge noncommutative algebra such as free algebras, group algebras of arithmetic groups or fundamental groups [↩]
- technically, think of them as the connected components of isomorphism classes of finite dimensional simple representations of your favorite noncommutative algebra [↩]
- technically, the size of the connection is the dimension of the ext-group between generic simples in the components [↩]
Hi, I'm a professor of mathematics at the University of Antwerp, in Belgium. My research areas are non-commutative algebra and non-commutative geometry. Sometimes, you can also find me here :
