Category Archives: general

in medieval architecture, signs of noncommutative geometry?

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MOSAIC SOPHISTICATION A quasi-crystalline Penrose pattern at the Darb-i Imam shrine in Isfahan, Iran

A few days ago I noticed this article in NYT science section that reports on a recent paper by Lu and Steinhardt in Science (see here and here for the full article; thanks to `thomas1111′). Their abstract says: “The conventional view holds that girih (geometric star-and-polygon, or strapwork) patterns in medieval Islamic architecture were conceived by their designers as a network of zigzagging lines, where the lines were drafted directly with a straightedge and a compass. We show that by 1200 C.E. a conceptual breakthrough occurred in which girih patterns were reconceived as tessellations of a special set of equilateral polygons (“girih tiles”) decorated with lines. These tiles enabled the creation of increasingly complex periodic girih patterns, and by the 15th century, the tessellation approach was combined with self-similar transformations to construct nearly perfect quasi-crystalline Penrose patterns, five centuries before their discovery in the West”.

Interestingly enough the occurrence of quasi periodic tilings in old Persian art was also extensively commented on, last year, in Alain and Matilde’s article “A walk in the noncommutative garden” (see Section 9 on tilings). The first four pics are from their article. (see also lieven le bruyn’s weblog where the NYT article is commented at). We look forward to comments by people in NCG, operator algebras, and those working on quasi periodic crystals.

good mathematics?

There are two interesting but quite different general discussions about “quality of maths” available on the web… There is the recent paper by Tao on “what is good mathematics?” and Serre’s talk on “how to write mathematics badly”…
I strongly recommend to listen to Serre’s talk which will result for sure in a definite improvement of the writing style of the listener. The talk is clear, funny, and makes a number of well taken points. As an example Serre explains the distinction between a proof and a “Bourbaki proof” (a term often used with a pejorative connotation): a proof is understandable by experts, a Bourbaki proof is understandable by non-experts (and of course that’s much better).
It is hard to comment on Tao’s paper, the second part on the specific case of Szemeredi’s theorem is nice and entertaining, but the first part has this painful flavor of an artist trying to define beauty by giving a list of criteria. This type of judgement is so subjective that I really had the impression of learning nothing except the pretty obvious fact about arrogance and hubris…
I was asked last year by Tim Gowers to write some advise for beginner mathematicians and reluctantly made an attempt. My main point is that mathematicians are so “singular”, (and behave like fermions as opposed to the physicists who behave like bosons) that making general statements about them often produces something obviously wrong or devoid of any content.

Be wise, quantize!

Perhaps a better title for this series of posts would be “Quantization and Noncommutative Geometry”. This is a huge topic and certainly takes a lot of time and contributions by many people to do justice to the subject. In a nutshell I would say the revolution brought in physics by the advent of quantum mechanics in the hands of Heisenberg, Dirac, Schrodinger and others in the years 1925-1926 is in many ways echoed in mathematics through noncommutative geometry. It took almost 55 years (1925 to 1980, roughly, since Connes already in 1978 was talking about the foliation algebra of a foliation and proved an index theorem for them) to reach to the current phase of development of NCG (= noncommutative geometry). It is a long time and it is certainly interesting to know why it took so long, but that is another issue.

I would like to invite all those who are interested to contribute to the following issues or to a related topic of their choice.
1. Dirac quantization rules and NCG
2. No go theorems
3. Various quantization schemes: geometric quantization, deformation quantization, Berezin and Toeplitz quantization, etc….
3. Semiclassical limits
4. Applications to mathematics, e.g. to index theorems,
5. Quantum groups
6. Noncommutative geometry techniques, e.g. the role of groupoids, strict deformations
7. The role of operator algebras, and original ideas of von Neumann
8. Second quantization
9. Quantum field theory and NCG
As I said this list is incomplete, so feel free to add topics, and also discuss!