Sunday, March 29, 2009


I was in Baltimore last week attending the Jami Conference at JHU which was followed by a two days long workshop on F_1. Both events were coordinated mostly by Katia Consani, with the help of S. Mahanta (and myself). I went to all lectures and feel ready to make some comments for those talks which I had the impression to understand. For now I'll just talk about the first day.
The first talk was by Paul Baum, with his abstract and title:

Morita Equivalence Revisited
The essence of this talk, in Paul's very pleasant lecturing style, is that in their joint work with A.M.Aubert and R.J.Plymen, on representation theory of reductive p-adic groups, the authors deal with algebras which are finite extensions of commutative algebras but to which the tools of NCG, such as cyclic homology, apply succesfully. In order to formulate their conjectured geometric description of the primitive ideal space in the representation theory of reductive p-adic groups, the need for a suitable weakening of Morita equivalence of algebras has emerged. The new notion is defined in the general algebraic set-up and time will tell if it provides a useful comparison for algebras. One basic difficulty seems to be that the relation is defined by iterating noncommuting basic steps, so that the corresponding maps of cyclic cohomology groups depend upon the chain of steps implementing the equivalence of two objects. Thus in a way, it seems that they are defining a new category of algebras, where the objects are the same but the morphisms are obtained as composition of the various allowed steps in their equivalence relation.
The second talk was by Masoud Khalkhali, with his abstract and title

Holomorphic Structures on the Quantum Projective Line
The three authors of this work, Masoud Khalkhali, Gianni Landi and Walter Van Suijlekom, have started very recently this joint work and are in an "exploratory" stage with this q-deformation of the two sphere. It is a very interesting concrete case to discover how much of the miraculous structure of the two sphere as a complex curve, or equivalently as a conformal manifold, actually survive the q-deformation. Many tools of NCG are available there, including the abstract perturbation of conformal structures by Beltrami differentials as explained in Chapter 4, section 4, pages 339-346 of the ncg book. A great challenge is to prove an analogue of the measurable Riemann mapping theorem in the q-deformed case. The formalism of q-groups allows one to set-up a simple algebraic framework but the real challenge resides in the analysis.

That was all for the talks of the morning session. The first talk in the afternoon was given by David Goss with abstract and title

The group S(q) and indications of functional equations in finite characteristic
In this brilliant talk congruences were flying all over and from my own "amateur" point of view I learnt something I should have known for years, namely the congruence due to Lucas on binomial coefficients. It says that modulo a prime p, the binomial coefficient (n choose k) is the product of the binomial coefficients (n_j choose k_j) of the respective digits of n and k in base p. In particular this expression is invariant under the operation of arbitrarily permuting the digits. David explained in his talk how to define a group S(q) of homeomorphisms of the compact space Zp of p-adic integers. His group has the cardinality of the continuum and involves arbitrary infinite permutations of the p-adic digits. The computational evidence shows that this group should be involved in a functional equation for characteristic p zeta functions. Since David has written a post in this blog on this precise topic I will just refer to his explanations.
The next talk was by Sacha Goncharov with abstract and title

The quantum dilogarithm and quantization of cluster varieties The subject of the talk is the joint work of Sacha Goncharov with V. V. Fock. The talk was excellent but it created a quite uneasy feeling in me which I had a hard time to identify. At first I thought it was due to the usual difficulty I have to hear a talk on something called the "quantum torus" and which looked like a reincarnation of the work I had done in 1980 on the representation of the noncommutative torus in L^2(R) and on the duality I had discovered there between the torus for theta and 1/theta (cf line 10 of page 8 of the english translation of the note). But in fact this "reincarnation" was appearing in a very strange way, with a factor of i=square root(-1) in the exponents of the operators acting on L^2(R) and that was the real reason why I felt disturbed. What I have found since then, and checked in an email exchange with Sacha, is that the commutation relations of these self-adjoint operators are only "formal" and hold on a dense domain but these operators actually do not commute. If you take the simplest case where q=1 then the presentation of the "quantum torus" is simply:
------------------------------A=A*, B=B* and AB=BA---------------------------------
Thus you should get two commuting self-adjoint operators in L^2(R) given by Af(s) = exp(s) f(s) and Bf(s)= f(s+2\pi i). Now it is true that A and B are self-adjoint, since A is a multiplication operator by the (positive) function exp(s) and B is similar in Fourier. But the trouble is that they do not commute, even though they commute on a dense domain. Thus when you exponentiate to the corresponding one-parameter groups exp(itA) and exp(isB) just do not commute. This can be seen because the function s-> exp(s) is injective from R to C and hence the operator A generates the algebra which is maximal abelian in L^2(R). This is the algebra of all multiplication operators. Similarly the operator B generates the algebra of all translation operators.. and of course these two algebras not only do not commute but form an irrep in L^2(R). Another way to understand why the formal commutation on the dense domain is not enough is that the vectors in the dense domain are not analytic vectors and for instance the L^2 norm of A^n f for f(x)=exp(-x^2/2) grows like exp(n^2).... One could argue that the case q=1 is special and that requiring exact commutation is too demanding but when one considers the "dual torus" (corresponding to 1/theta) it is quite reasonable to require that it exactly commutes with the initial torus (and in fact generates its commutant as in my 1980 note). However, for the same reason as in the case q=1, this will just fail with the above representation of the "quantum torus"...
I checked with Sacha, who told me that he knew that the operators do not commute, that this is not a problem for what they are doing, namely the unitary group representation. But for the "algebra" it clearly indicates a kind of dichotomy between "formal computations" of deformation quantization and real Hilbert space stuff.
The final talk of this first day of the meeting was by Patrick Brosnan, with abstract and title

Essential dimension
I knew the author from his work with PRAKASH BELKALE on the conjecture of Kontsevich. In fact it was pretty useful for me to hear also Marc Levine playing the role of the "motivic expert" throughout the conference. The talk of Patrick Brosnan started by defining in a very simple and natural manner the notion of dimension using degree of transcendence of field extensions, as a classical way to count the "number of parameters". His talk ended with a beautiful result which I had the impression to understand, namely the evaluation of the essential dimension of the spinor group Spin_n which is the totally split form of the spin group over a field k. The result says, roughly, that this essential dimension is between 2^((n-1)/2) -n(n-1)/2 and 2^((n-1)/2) and hence has an "exponential" demand of new parameters. This is quite striking given that the previously known lower bound was linear in n.....