We shall try to highlight papers on, or around, noncommutative geometry as they are posted. (except occasionally, I won’t comment on papers.) Let us know if you think a paper is left out….. Here are a few papers posted this month.

Cyclic homology with coefficients

Categorical aspects of bivariant K-theory

Homological algebra in bivariant K-theory and other triangulated categories

Lectures on noncommutative geometry

Notes on motives in finite characteristic

Quantum statistical mechanics and class field Theory

The quantum dilogarithm and unitary representations of the cluster mapping class groups

Quantum Groups and Twisted Spectral Triples

The bar derived category of a curved dg algebra

A Riemann-Roch-Hirzebruch formula for traces of differential operators

On Serre duality for compact homologically smooth DG algebras

K-theoretic construction of noncommutative instantons of all charges

# Monthly Archives: February 2007

# Physics in finite characteristic

I was delighted when Alain asked me to post on this blog and I came upon

the catchy title above. As a newcomer to noncommutative geometry, I am

impressed by the applications of concepts arising originally in physics

to number theory. An excellent instance of this is the expression

of the Riemann zeta function as a partition function in the work of

Bost and Connes.

For function fields over finite fields, the applications of ideas from

physics has long been a theme and I don’t really have a good idea why

such things work so well except to steal a bit from Feynman: I remember

reading in one of Feynman’s works his musings about how it is that

physics is able to handle so many different types of phenomena.

Feynman remarked, I believe, that this is due to the fact

that while the phenomena may be very different the differential equations

tend to be alike, thus cutting the work load greatly.

Well, to go a bit further, the

gods of mathematics were also quite frugal when they “created” mathematics.

Indeed, we see the same ideas occurring in many very different circumstances

and areas; this is in fact one of the real glories of mathematics.

For fields of finite characteristic, we see this phenomenon

very early on: Let k be a field of characteristic p and let

Fr be the p-th power morphism. It has long been known that Fr has many

similarities with differentiation D and this motivated early researchers

such as Ore. If we embed k into its perfect, one also has the

p-th root operator Fr* which is then analogous to integration. The field

of constants for D gets replaced by the fixed field of Fr, one has

adjoint operators etc.

If k is a function field over a finite field, we are free to pick

a fixed closed point infty and view it as the “infinite prime”. The

ring A of functions regular away from infty is a Dedekind domain

with finite class and unit groups. The ring A is then, by fiat, the

“bottom” for the theory of Drinfeld A-modules. A Drinfeld A-module phi is

essentially a representation of the ring A by polynomials in Fr; thus

given a in A one obtains a polynomial phi_a. The zeroes of phi_a

then become a finite A-module which must be isomorphic

the d-th Cartesian product of A/(a) with itself; this number d is the

“rank” of the Drinfeld module.

The noncommutative algebra involved already with the simplest Drinfeld module

of them all, the “Carlitz module” (discovered by Carlitz in the 1930’s),

already allowed A. Kochubei to define analogs of “creation” and

“annihilation” operators and the canonical commutation

relations of quantum mechanics.

In classical theory, function fields of course have no

bottom whereas the rational numbers are

obviously the bottom for number fields. Thus imposing a bottom allows us

to begin to model aspects of classical arithmetic in finite characteristic

that had been missed in earlier theories. In particular, due to L. Carlitz

and D. Hayes, one can create “cyclotomic” extensions of (k,infty) based

on the torsion points of certain Drinfeld modules of rank 1.

Based on the connection Fr has with D, Drinfeld was able to produce

an analogy to the work of Krichever on KdV; thus to give an

interpretation of his modules in terms of special coherent sheaves

called “shtuka”.

In the fall of 1987 Greg Anderson and Dinesh Thakur discovered a

fundamental relationship between the characteristic polynomial of

the Frobenius morphism on the Jacobian of a curve

and the type of products that arise in

the definition of characteristic p gamma functions. This arose

by analysis of a seminal example due to Robert Coleman. This and the

analogy with KdV led Greg to formulate “solitons” in characteristic

p. In turn this technology allowed Anderson, D. Brownawell and M.

Papanikolas to prove analogs of well-known transcendency conjectures

for the function field (“geometric”) gamma function. This proof was

in the basic case of A=Fq[T] whereas the gamma functions exist for

all A. The difficulty is constructing the correct “Coleman functions”

in general. To solve this, Anderson reformulated things in an adelic

setting so as to be able to use harmonic analysis and, in particular,

Tate’s thesis. The point being that from a Schwartz function on the

adeles one can go one way to get solitons or another to get L-functions.

Recent papers of Anderson have put the general theory (for all A)

within reach.

One therefore sees how intertwined arithmetic arising from Drinfeld

modules is with the classical (Artin-Weil) zeta function of the field

k. It is therefore natural to ask whether this function itself can be

brought directly into the set-up of Drinfeld modules. This takes us

back to Bost and Connes! Indeed, in a paper (soon to appear in the

Journal of Noncommutative Geometry), B. Jacob uses the general cyclotomic

theory of rank one Drinfeld modules mentioned above to describe

a Bost-Connes system for (k,infty). In this case the partition function

is the Artin-Weil zeta function of k with the Euler factor at infty

removed!

However, the noncommutative geometry does not stop with recapturing

the classical zeta-function of k. Indeed, encoding the characteristic

polynomials of the Frobenius morphism leads naturally into characteristic

p valued L-series; for instance one can (beginning with Carlitz) prove

analogs here of Euler’s results on the values at positive even

integers of Riemann’s zeta function. In the Journal of Number Theory 123

(2007), C. Consani and M. Marcolli translate the machinery of Bost-Connes

into characteristic p analysis and thereby express the characteristic p

zeta function as a partition function!

Finally, Papanikolas reappears at this stage. Indeed, Matt has developed

the correct Tannakian theory in this situation and thus also the

appropriate geometric Galois groups. Using Matt’s technology,

Chieh-Yu Chang and Jing Yu have recently established that the

above mentioned zeta values ONLY satisfy the algebraic relations

given by the analog of Euler’s result AND the obvious one coming

from the p-th power mapping!

Well that finishes my necessarily very incomplete first post. This clearly

represents my take on things. It would be fabulous to hear, in their

own voices, from the other people involved with these results. No one

viewpoint ever describes everything when it comes to number theory

(and physics too?)!

ps: In a previous post, Alain gave the url for Serre’s talk at Harvard

on “how to write mathematics badly.” Fortunately about 30 years ago Serre

took me aside and gave me a talk on how to write math well! In the

early 90’s I wrote these hints down and got Serre’s opinion on them.

Then in 1998 I incorporated some input from E.G. Dunne and P.Vojta.

Finally, when I told Serre that I was going to mention these hints

on this blog, he sent me another minor change… Anyway, those

interested may find these hints here.

# Real and Complex

I would like to discuss the “next entry” in the parallel texts that Masoud was presenting in his post. On the function theory side we are talking about “real and complex variables”. A perfect book to get introduced to that is “real and complex analysis” by W. Rudin (McGraw-Hill). It is a classic and remains one of the best entrance doors to the subject. What one learns is the constant interplay between the “real variable” techniques such as the Lebesgue integral, differentiability almost everywhere, etc.. and the “complex variable” techniques. There is a saying of André Weil like “The complex world is beautiful, the real world is dirty”. One might then be tempted to ignore the “real world” and only work in the complex variable set-up where “any” function is holomorphic and hence infinitely differentiable etc… That’s fine, and one can go some distance with that, except that most of the deep results in complex analysis do rely on real analysis.

Now what about the next entry in the parallel text? It is

Complex variable…………….Operator on Hilbert space

Real variable……………………..Self-adjoint operator

where I have slightly rewritten the previous entry

functions f: X -> C ……………..operators on Hilbert space

of Masoud’s post to stress that the right column gives an ideal model for what the loose notion of a “variable” is… The set of values of the variable is the spectrum of the operator, and the number of times a value is reached is the spectral multiplicity. Continuous variables (operators with continuous spectrum) coexist happily with discrete variables precisely because of non-commutativity of operators.

The holomorphic functional calculus gives a meaning to f(T) for all holomorphic functions f on the spectrum of T, and a deep result controls the spectrum of f(T). The really amazing fact is that while for general operators T in Hilbert space the only functions f(z) that can be applied to T are the holomorphic ones (on the spectrum of T), the situation changes drastically when one deals with self-adjoint operators: for T=T* the operator f(T) makes sense for **any** function f! You can take a pencil and draw the graph of a function, it does not need to be continuous…nor even piecewise continuous, just anything you can name will do….(at the technical level the only requirement on f is that it is universally measurable but nobody can construct explicitly a function which does not fulfill this condition!)…Moreover a bounded operator is a function of T (ie is of the form f(T) ) if and only if it shares all the symmetries of T (ie if it commutes with all operators that commute with T ).

I remember that, at a very early stage of my encounter with mathematics, it is this very fact that convinced me of the power of the Hilbert space techniques in close relation with the adjoint operation T -> T*. This was enough to resist the temptation of starting directly in the “complex world” of algebraic geometry which was attracting most beginners at that time, following the aura of Grothendieck, who described so well his first encounter with that world:

“Je me rappelle encore de cette impression saisissante (toute subjective certes), comme si je quittais des steppes arides et revèches, pour me retrouver soudain dans une sorte de “pays promis” aux richesses luxuriantes, se multipliant à l’infini partout où il plait à la main de se poser, pour cueillir ou pour fouiller….”

# the Rosetta Stone of noncommutative geometry

In any case, Alain mentioned that the spectrum of ab and ba are the same except possibly for 0 and this played a role in Dirac’s paper. Let me add this, and this will be my first entry into

this dictionary. The Spectrum can be regarded as the noncommutative analogue of the Range of a function. So let us make a small table:

Commutative ……………………………………………………………….Noncommutative

functions f: X to C ……………..operators on Hilbert space; elements of an algebra

range of a function…………………………..spectrum of an operator

Let me be more specific and recall some basic facts. Let A be an algebra with a unit (for example A can be: the algebra of, say continuous, functions on a space; the algebra of all n by n matrices; or the algebra of operators on Hilbert space etc.) The spectrum of an element a in A, denoted Spect (a) is the set of all numbers (scalars) lambda such that a-lambda 1 is not invertible.

Example 1. Let A= C(X) be the algebra of continuous functions on a compact space X, and let f in A. Obviously Spect (f) ={ f(x); x in X }= Range (f).

This is one reason we should think of spectrum as a noncommutative analogue of the range of a function i.e. the set of values attained by a function (a classical observable).

Example 2. For A=M_n (C) the algebra of n by n matrices the spectrum of a matrix is of course nothing but the set of its good old eigenvalues.

Classically, we have Range (fg) = Range (gf)

(because in fact fg=gf!). Now there is an almost perfect analogue of this fact in the noncommutative world. I said almost because there is a nuance: Spect (ab) and Spect (ba) are the same except for 0!

Here is a nice exercise in noncommutative algebra: show that in any unital algebra A, (ab-1) is invertible iff (ba-1) is invertible. From this of course follows that Spect(ab) = Spect (ba).

The fact that the two spectra are not exactly the same is in fact a good thing and the discrepancy is responsible for the index of Fredholm operators. Indeed assuming there is a b such that both (1-ba) and (1-ab) are trace class operators, it follows that a is Fredholm, and its index is given by index (a)= Tr (1-ba)- Tr (1-ab).

There are several other similar formulas for Fredholm index, e.g. the McKean-Singer formula, derived from the same principle.

# 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!