Monday, August 13, 2007

Harmonic mean

This "post" is mainly an attempt to see if one can manage to use formulas in the blog and discuss some real stuff somehow. The formulas should be really visible, a bit like with transparencies. So as a pretext, I'll start by discussing an issue related to the basic Ansatz:

which gives the operator theoretic line element in terms of the Dirac operator in the general framework of "metric" noncommutative geometry. The kernel of the operator D is finite dimensional and one takes ds to vanish on that kernel. As was already discussed here, the knowledge of D gives back the metric. Moreover the noncommutative integral, in the form of the Dixmier trace, gives back the volume form. Thus the integral of a function f in dimension n is simply given by

where the "cut" integral is the Dixmier trace ie the functional that assigns to an infinitesimal of order one the coefficient of the logarithmic divergency in the series that gives the sum of its eigenvalues.

I will not try to justify the heuristic definition of the line element any further. It is more interesting to put it to the test, to question it, and I will discuss an example of an issue which left me perplex for quite sometime but has a pretty resolution.

The point is to understand what happens when one takes the product of two noncommutative geometries. One gets the following relation for the squares of the corresponding Dirac operators:

where we abuse notations by removing the tensor product by the identity operator that normally goes with each of the operators D_j. Now this relation is quite different from the simple Pythagorean relation of the classical line elements whose square simply add up and it thus raises the question of reconciling the above Ansatz with the simple formula of addition of the squares of the Dirac operators. More generally, one can consider a bunch of NC spaces with Dirac operators D_mu and combine them as follows: One starts with a positive matrix of operators in Hilbert space:

and one extends the above formula giving D^2 for a product of two spaces and forms the following sum:

We make no commutativity hypothesis and even drop the self-adjointness of D_mu which is not needed. We want a formula for the inverse of the square of D ie for:

in terms of the inverse matrix:
which plays a role similar to the g\mu\nu of Riemannian geometry, and of the operators
where the notation with z stresses the fact that we do not even assume self-adjointness of the various D_\mu.

It sounds totally hopeless since one needs a formula for the inverse of a sum of noncommuting operators. Fortunately it turns out that there is a beautiful simple formula that does the job in full generality. It is reminiscent of the definition of distances as an infimum. It is given by:

The infimum is taken over all decompositions of the given vector as a sum:

Note that this formula suffices to determine the operator ds^2 completely, since it gives the value of the corresponding positive quadratic form on any vector in Hilbert space. The proof of the formula is not difficult and can be done by applying the technique of Lagrange multipliers to take care of the above constraint on the free vectors \xi^\mu.

Tuesday, August 7, 2007

The Editorial Board of 'K-Theory' has resigned.

This just in. The editorial board of `K-theory' has resigned and a new journal titled `Journal
of K-theory' has been established. In recent years `K-theory' was published by Springer. `Journal of K-theory' will be published by the Cambridge University Press. The board of editors of the new `Journal of K-theory' has issued an open letter to the mathematics community that we quote in part:

from the Board of Editors
of the Journal of K-theory

Dear fellow mathematicians,

The Editorial Board of 'K-Theory' has resigned. A new journal titled
'Journal of K-theory' has been formed, with essentially the same Board
of Editors............The new journal is to be distributed by Cambridge University Press.
The price is 380 British pounds, which is significantly less than
half that of the old journal. Publication will begin in January 2008.
We ask for your continued support, in particular at the current time.
Your submissions are welcome and may be sent to any of the editors.

Board of Editors
Journal of K-theory''

Friday, August 3, 2007

Paul's Seventies

I am just back from a very nice event around Paul Baum's seventies, which took place in Warsaw last Monday, thanks in particular to Piotr Hajac. I have known Paul since the summer of 1980 when we first met in Kingston. I really had, when I first met him, the impression of meeting l'"Homologie en personne". The more I got to know him through our very long collaboration, the better I enjoyed his clarity of mind and his relentless quest for simplicity and beauty. In many ways he succeeds in doing something very difficult, which Grothendieck advised in "Récoltes et Semailles", namely to keep "une innocence enfantine" in front of mathematics.

The dinner on monday night was comparable in intensity to the memorable one in Martin Walter's place, in Boulder, for Paul's sixties when the team Paul Baum---Raoul Bott forced Martin to search (again and again) his cellar for more bottles of wine to keep up with their drinking ability!.

Raoul Bott died in December 2005. Not long before, Paul went all the way to California to visit him and they talked together for an entire day. This type of faithfulness in friendship and understanding of what really matters, is an attitude towards life which Paul has and which I truly admire.