Tuesday, June 14, 2016

What is a functional equation?

Like all number theorists I am fascinated (to say the least) with the functional equation of 
classical  L-series. Years ago, I came up with a simple characterization of functional equations basically using only complex conjugation. This point being that, via a canonical change of variables (going back to Riemann), such L-series are, up to a nonzero scalar, given by real power series with the expectation that the zeroes are also real. In characteristic p the best one can hope is also that the zeroes will be as rational as the coefficients (though this statement needs to be modified to take care of standard factorizations as well as the great generality of Drinfeld's base rings A).

For those interested, a two page pdf can be found at the following link: https://drive.google.com/open?id=0BwCbLZazAtweTmNIa1ZSc0h2UEE

Saturday, March 26, 2016

An indirect consequence of the famous Lucas congruence...

So, in the course of function field arithmetic, one runs into the binomial coefficients (like one does most everywhere in mathematics); or rather the coefficients modulo a prime p. The primary result about binomial coefficients modulo p is of course the congruence of Lucas. In function field arithmetic
one seems to be unable to avoid the group obtained by permuting p-adic (or q-adic) coefficients of a number. I recently discovered a congruence using these permutations and the product of two binomial coefficients that I decided to blog about. The proof is an indirect consequence of Lucas and is perhaps more interesting than the result itself. One is then led to look for something related with the Carlitz polynomials, which are the function field analog of the binomial coefficients.

I put all of this into a three page pdf which, if you are interested, you can find here:

Sunday, November 22, 2015

Review of "Arithmetic of characteristic p special L-values" by B. Anglès and L. Taelman

Bruno Anglès and Lenny Taelman have published a profound study of finite characteristic special values as listed in the title. This appeared in Proc. London Math. Soc. (3) 110 (2015) 1000-1032. The analogy with classical theory is clear throughout the paper, as well as some potentially interesting differences; these results bode very well for the future of the subject. I have written a long review of this paper as well as pointing to a very new one on the web. This is been submitted to Math Reviews/MathSciNet and appears here with their explicit permission.

(Added 12-11-2015: A revised version of the paper by Anglès and Tavares Rebeiro cited in the review (along with a third author Tuan Ngo Dac) has been put on the arXiv as arXiv:1511.06209v2. In this new version the result is established in complete generality for all q.)



A url for a pdf of this review is


Friday, October 30, 2015

C. Armana's Formula for the coefficients of $h$

(full pdf is at https://drive.google.com/file/d/0BwCbLZazAtweOC01dFgyQjBmUlU/view?usp=sharing)

Tuesday, August 11, 2015

Grand Unification in the Spectral Pati-Salam Model

Last week we (Chamseddine-Connes-van Suijlekom) posted a preprint on grand unification in the spectral Pati–Salam model which I summarize here (and here).
The paper builds on two recent discoveries in the noncommutative geometry approach to particle physics: we showed how  to obtain inner fluctuations of the metric without having to assume the order one condition on the Dirac  operator. Moreover the original argument by classification of finite geometries F that can provide the fine structure of Euclidean space-time as a product M×F (where M is a usual 4-dimensional Riemannian space) has now been replaced by a much stronger uniqueness statement. This new result shows that the algebra
where are the quaternions, appears uniquely when writing the higher analogue of the Heisenberg commutation relations. This analogue is written in terms of the basic ingredients of noncommutative geometry where one takes a spectral point of view, encoding geometry in terms of operators on a Hilbert space . In this way, the inverse line element is an unbounded self-adjoint operator D. The operator D is the product of the usual Dirac operator on M and a `finite Dirac operator’ on F, which is simply a hermitian matrix DF. The usual Dirac operator involves gamma matrices which allow one to combine the momenta into a single operator. The higher analogue of the Heisenberg relations puts the spatial variables on similar footing by combining them into a single operator Y using another set of gamma matrices and it is in this process that the above algebra appears canonically and uniquely in dimension 4.
This leads without arbitrariness to the Pati–Salam gauge group SU(2)R×SU(2)L×SU(4), together with the corresponding gauge fields and a scalar sector, all derived as inner perturbations of D. Note that the scalar sector can not be chosen freely, in contrast to early work on Pati–Salam unification. In fact, there are only a few possibilities for the precise scalar content, depending on the assumptions made on the finite Dirac operator.
From the spectral action principle, the dynamics and interactions are described by the spectral action,
where Λ is a cutoff scale and f an even and positive function. In the present case, it can be expanded using heat kernel methods,
where F4,F2,F0 are coefficients related to the function f and ak are Seeley deWitt coefficients, expressed in terms of the curvature of M and (derivatives of) the gauge and scalar fields. This action is interpreted as an effective field theory for energies lower than Λ.
One important feature of the spectral action is that it gives the usual Pati–Salam action with unification of the gauge couplings. Indeed, the scale-invariant term F0a4 in the spectral action for the spectral Pati–Salam model contains the terms
Normalizing this to give the Yang–Mills Lagrangian demands
which requires gauge coupling unification. This is very similar to the case of the spectral Standard Model where there is unification of gauge couplings. Since it is well known that the SM gauge couplings do not meet exactly, it is crucial to investigate the running of the Pati–Salam gauge couplings beyond the Standard Model and to find a scale Λ where there is grand
This would then be the scale at which the spectral action is valid as an effective theory. There is a hierarchy of three energy scales: SM, an intermediate mass scale mR where symmetry breaking occurs and which is related to the neutrino Majorana masses (10111013GeV), and the GUT scale Λ.
In the paper, we analyze the running of the gauge couplings according to the usual (one-loop) RG equation. As mentioned before, depending on the assumptions on DF, one may vary to a limited extent the scalar particle content, consisting of either composite or fundamental scalar fields. We will not limit ourselves to a specific model but consider all cases separately. This leads to the following three figures:
Running of coupling constants for the spectral Pati--Salam model with composite Higgs fields
Running of coupling constants for the spectral Pati–Salam model with composite Higgs fields
Running of coupling constants for the spectral Pati–Salam model with fundamental Higgs fields
Running of  coupling constants for the left-right symmetric spectral Pati--Salam model.

Running of coupling constants for the left-right symmetric spectral Pati–Salam model

In other words, we establish grand unification for all of the scenarios with unification scale of the order of 1016 GeV, thus confirming validity of the spectral action at the corresponding scale, independent of the specific form of DF.

Saturday, July 18, 2015

Uffe Haagerup

Uffe Haagerup was a wonderful man, with a perfect kindness and openness of mind, and a mathematician of incredible power and insight. 
His whole career is a succession of amazing achievements and of decisive and extremely influential contributions to the field of operator algebras, C*-algebras and von Neumann algebras. 
His first work (1973-80) concerned the theory of weights and more generally the modular theory of Tomita-Takesaki. Uffe  Haagerup began by solving a key open question, showing that semi-finite weights are indeed supremum of families of normal states. This allowed him to develop the standard form of von Neumann algebras, a basic result used over and over since then.  I remember vividly his first appearance in the field of operator algebras and the striking elegance, clarity and strength of his contributions.
He was the first to introduce the Lp-spaces associated to a type III von Neumann algebra. These new spaces have since then received a lot of attention both from operator algebraists and specialists of Banach spaces and operator spaces.
In a remarkable paper published in Inventiones at the end of the seventies, Haagerup was able to analyse the operator norm in the C*-algebra of the free group, to control it by suitable Sobolev norms in spite of the exponential growth of the group and to prove in particular that in spite of its lack of nuclearity the reduced C*-algebra has the Banach space approximation property. This turned out in the long run to be a breakthrough of major importance. It was extended by Haagerup and his collaborators to any discrete subgroup of a simple real rank one Lie-group. One corollary is the property RD of rapid decay which allows one to define the analogue of the Harish-Chandra algebra of smooth elements in the general context of Gromov hyperbolic groups.  This highly non-trivial result of Haagerup turned out to be the technical key in the proof of the Novikov conjecture for such groups. Moreover the study of discrete groups with the Haagerup property continues to play a major role in geometric group theory.  The associated approximation property for the corresponding factors of type II_1 (called the Haagerup property) also played a major role in the solution by Popa of the long-standing problem of factors N non-isomorphic to M_2(N), and of exhibiting a factor with trivial fundamental group. 
The next fundamental contribution of Haagerup is to the classification of injective factors (1983-87). In my work on the subject I had left one case completely open (in 76) and the classification was thus incomplete. After several years of extremely hard work Haagerup was able to settle this question by proving that there is up to isomorphism only one hyperfinite factor of type III_1. This is a wonderful achievement.
Then Haagerup turned to the theory of subfactors, and, once again, was able to make fundamental contributions such as his construction of new irreducible subfactors of small index > 4, and outclass the best specialists of the subject created by Vaughan Jones at the beginning of the eighties. 
Another key contribution of Haagerup is to Voiculescu's Free Probability and to random matrices which he was able to apply very successfully to the theory of C*-algebras and of factors of type II1. For instance he proved that, in all II1 factors fulfilling the approximate embedding property (which is true in all known cases) every operator T with a non-trivial Brown measure has a non-trivial closed invariant subspace affiliated with the von Neumann algebra generated by T. 
Uffe continued at the same relentless pace to produce amazing contributions and the few mentioned above only give a glimpse of his most impressive collection of breakthrough achievements.
Uffe Haagerup was a marvelous mathematician, well-known to operator algebraists, and to the general community of analysts.  From my own perspective an analyst is characterized by the ability of having ”direct access to the infinite” and Uffe Haagerup possessed that quality to perfection. His disparition is a great loss for all of us.