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.

Dirac and integrality

In the first paper on "second quantization", namely the paper of Dirac called "The quantum theory of the emission and absorption of radiation" the process of second quantization is introduced and is related again to "integrality". This time it is not the Fredholm index that is behind the integrality but the following simple fact : if an operator a satisfies [a,a*]=1, then the spectrum of a*a is contained in \N, the set of positive integers (as follows from the equality of the spectra of a a* and a* a except possibly for 0).... Second quantization is obtained simply by replacing the ordinary complex numbers a_j which label the Fourier expansion of the electromagnetic field by non-commutative variables fulfilling [a_j,a_j*]=1....(more precisely the 1 is replaced by \hbar \nu where \nu is the frequency of the Fourier mode). This example shows of course that integrality and non-commutativity are deeply related... While the Fredholm index is a good model of relative integers (positive or negative), the a a* for [a,a*]=1
is a good model for positive integers...

Alain's comment

This has been posted as a comment to Masoud's post. Since it is almost invisible there, I am posting a copy of it here:

This topic of "quantization and NCG" is very relevant. The word `quantum', from the beginning, is not so much related to `non-commutativity' but rather to `integrality'. In the word `quantum' there is really this discovery by Planck, of the formula for blackbody radiation, from which he understood that energy had to be quantized in quanta of $\hbar \nu$. There is a confusion, created by people doing deformation theory who let one believe that quantizing an algebra just means deforming it to a non-commutative one. They take a commutative space and since they deform the product into a non-commutative algebra, they believe they are quantizing. But this is wrong: you succeed in quantizing a space only if you give a deformation into a very specific algebra : the algebra of compact operators. And then, there is an integrality, the integrality of the Fredholm index. The use of the wrong vocabulary, creates confusion and does not help at all to understand. That's why I am so reluctant to use the word `quantum' - instead of "non-commutative" and am against talking about "quantum spaces" or "quantum geometry".... this looks more flashy, perhaps, but the truth is that you are doing something quantum only in very particular cases, otherwise you are doing something non-commutative, that's all. Then this may be less fashionable at the linguistic level, but never mind: it is much closer to reality.

Tips on using blogger

Normally, you would get an invitation email, requesting you to join this blog page. The email would look like this:



One has to click on the link and keep following the directions, which should be routine. However, there is one place I have seen where there might be a bit of confusion. Suppose you are visiting the page http://noncommutativegeometry.blogspot.com and you are not logged into google in any other browser tab or window. Then the screen will have a link "sign in" on the top right. You click on it and keep following the directions, you will soon reach the following screen:




Merely typing your google username here does not work. You have to type your username with "@gmail.com" appended at the end.

To avoid all this unnecessary hassle or lengthy procedure at various points, best thing is to create a gmail account (which you can do now by going to the page http://www.gmail.com), log in to gmail and then open another browser tab and visit this page.

New books on noncommutative geometry

Two new books on noncommutative geometry recently appeared. An Introduction to Noncommutative Geometry, by J. Varilly is a short (about 100 pages) introduction to metric and spectral aspects of noncommutative geometry. Surveys in Noncommutative Geometry is edited by N. Higson and J. Roe. It is based on lectures given at the Clay Mathematics Institute Instructional Symposium on Noncommutative Geometry held at Mount Holyoke College in Massachusetts in 2000.

Speaking of noncommutative geometry books, I should mention that the main single text on the subject ( `the book') , that is Alain Connes' book, is freely available to download from Alain's website.

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!

First issue of Journal of Noncommutative Geometry

The very first issue of Journal of Noncommutative Geometry is now out:
http://www.ems-ph.org/journals/jncg/jncg.php
The Journal is published by the European Mathematical Society publications. There will be four issues per year. Congratulations to all those who took part in this initiative, including authors of the first issue!