I guess one possible use of a blog, like this one, is as a space of freedom where one can tell things that would be out of place in a "serious" math paper. The finished technical stuff finds its place in these papers and it is a good thing that mathematicians maintain a high standard in the writing style since otherwise one would quickly loose control of what is proved and what is just wishful thinking. But somehow it leaves no room for the more profound source, of poetical nature, that sets things into motion at an early stage of the mental process leading to the discovery of new "hard" facts. Grothendieck expressed this in a vivid manner in Récoltes et semailles :
"L'interdit qui frappe le rêve mathématique, et à travers lui, tout ce qui ne se présente pas sous les aspects habituels du produit fini, prêt à la consommation. Le peu que j'ai appris sur les autres sciences naturelles suffit à me faire mesurer qu'un interdit d'une semblable rigueur les aurait condamnées à la stérilité, ou à une progression de tortue, un peu comme au Moyen Age où il n'était pas question d'écornifler la lettre des Saintes Ecritures. Mais je sais bien aussi que la source profonde de la découverte, tout comme la démarche de la découverte dans tous ses aspects essentiels, est la même en mathématique qu'en tout autre région ou chose de l'Univers que notre corps et notre esprit peuvent connaitre. Bannir le rêve, c'est bannir la source - la condamner à une existence occulte"
I shall try to involve on the post of Masoud about tilings and give a heuristic description of a basic qualitative feature of noncommutative spaces which is perfectly illustrated by the space T of Penrose tilings of the plane. Given the two basic tiles : the Penrose kites and darts (or those shown in the pictures), one can tile the plane with these two tiles (with a matching condition on the colors of the vertices) but no such tiling is periodic. Two tilings are the same if they are carried into each other by an isometry of the plane. There are plenty of examples of tilings which are not the same. The set T of all tilings of the plane by the above two tiles is a very strange set because of the following:
"Every finite pattern of tiles in a tiling by kites and darts does occur, and infinitely many times, in any other tiling by the same tiles''.
This means that it is impossible to decide locally with which tiling one is dealing. Any pair of tilings can be matched on arbitrarily large patches and there is no way to tell them apart by looking only at finite portions of each of them. This is in sharp contrast with real numbers for instance since if two real numbers are distinct their decimal expansions will certainly be different far enough. I remember attending quite long ago a talk by Roger Penrose in which he superposed two transparencies with a tiling on each and showed the strange visual impression one gets by matching large patches of one of them with the other... he expressed the intuitive feeling one gets from the richness of these "variations on the same point" as being similar to "quantum fluctuations". A space like the space T of Penrose tilings is indeed a prototype example of a noncommutative space. Since its points cannot be distinguished from each other locally one finds that there are no interesting real (or complex) valued functions on such a space which stands apart from a set like the real line R and cannot be analyzed by means of ordinary real valued functions. But if one uses the dictionary one finds out that the space T is perfectly encoded by a (non-commutative) algebra of q-numbers which accounts for its "quantum" aspect. See this book for more details.
In a comment to the post of Masoud on tilings the question was formulated of a relation between aperiodic tilings and primes. A geometric notion, analogous to that of aperiodic tiling, that indeed corresponds to prime numbers is that of a Q-lattice. This notion was introduced in our joint work with Matilde Marcolli and is simply given by a pair of a lattice L in R together with an additive map from Q/Z to QL/L. Two Q-lattices are commensurable when the lattices are commensurable (which means that their sum is still a lattice) and the maps agree (modulo the sum). The space X of Q-lattices up to commensurability comes naturally with a scaling action (which rescales the lattice and the map) and an action of the group of automorphisms of Q/Z by composition. Again, as in the case of tilings the space X is a typical noncommutative space with no interesting functions. It is however perfectly encoded by a noncommutative algebra and the natural cohomology (cyclic cohomology) of this algebra can be computed in terms of a suitable space of distributions on X, as shown in our joint work with Consani and Marcolli.
There are two main points then, the first is that the zeros of the Riemann zeta function appear as an absorption spectrum (ie as a cokernel) from the representation of the scaling group in the above cohomology, in the sector where the group of automorphisms of Q/Z is acting trivially (the other sectors are labeled by characters of this group and give the zeros the corresponding L-functions).
The second is that if one applies the Lefschetz formula as formulated in the distribution theoretic sense by Guillemin and Sternberg (after Atiyah and Bott) one obtains the Riemann-Weil explicit formulas of number theory that relate the distribution of prime numbers with the zeros of zeta.
A first striking feature is that one does not even need to define the zeta function (or L-functions), let alone its analytic continuation, before getting at the zeros which appear as a spectrum. The second is that the Riemann-Weil explicit formulas involve rather delicate principal values of divergent integrals whose formulation uses a combination of the Euler constant and the logarithm of 2 pi, and that exactly this combination appears naturally when one computes the operator theoretic trace, thus the equality of the trace with the explicit formula can hardly be an accident.
After the initial paper an important advance was done by Ralf Meyer who showed how to prove the explicit formulas using the above functional analytic framework (instead of the Cauchy integral).
This hopefully will shed some light on the comment of Masoud which hinged on the tricky topic of the use of noncommutative geometry in an approach to RH. It is a delicate topic because as soon as one begins to discuss anything related to RH it generates some irrational attitudes. For instance I was for some time blinded by the possibility to restrict to the critical zeros, by using a suitable function space, instead of trying to follow the successful track of André Weil and develop noncommutative geometry to the point where his argument for the case of positive characteristic could be successfully transplanted. We have now started walking on this track in our joint paper with Consani and Marcolli, and while the hope of reaching the goal is still quite far distant, it is a great incentive to develop the missing noncommutative geometric tools. As a first goal, one should aim at translating Weil's proof in the function field case in terms of the noncommutative geometric framework. In that respect both the paper of Benoit Jacob and the paper of Consani and Marcolli that David Goss mentionned in his recent post open the way.
I'll end up with a joke inspired by the European myth of Faust, about a mathematician trying to bargain with the devil for a proof of the Riemann hypothesis. This joke was told to me some time ago by Ilan Vardi and I happily use it in some talks, here I'll tell it in French which is a bit easier from this side of the atlantic, but it is easy to translate....
La petite histoire veut qu'un mathématicien ayant passé sa vie à essayer de résoudre ce problème se décide à vendre son âme au diable pour enfin connaître la réponse. Lors d'une première rencontre avec le diable, et après avoir signé les papiers de la vente, il pose la question "L'hypothèse de Riemann est-elle vraie ?" Ce à quoi le diable répond "Je ne sais pas ce qu'est l'hypothèse de Riemann" et après les explications prodiguées par le mathématicien "hmm, il me faudra du temps pour trouver la réponse, rendez vous ici à minuit, dans un mois". Un mois plus tard le mathématicien (qui a vendu son âme) attend à minuit au même endroit... minuit, minuit et demi... pas de diable... puis vers deux heures du matin alors que le mathématicien s'apprête à quitter les lieux, le diable apparaît, trempé de sueur, échevelé et dit "Désolé, je n'ai pas la réponse, mais j'ai réussi à trouver une formulation équivalente qui sera peut-être plus accessible!"
I like very much the faust story.
Probably all mathematicians that dreamed on such an exiting problem as the RH have had some day the desire to sell their own soul to the devil... but this did not give us
the solution, which proves by the experience the validity of your parabola.
I have noticed that number theory as a generalized quantum physics is one of visions making itself visible in postings.
I have been personally working with a dual of this vision: the vision about physics as a generalized number theory. I am of course just a physicist and the standards of mathematical rigor are many orders of magnitude lower. But because poetic self expression is encouraged, I decided to make a brief comment.
One ends up with the notion of number theoretic universality from the requirement that physics is extended so that it becomes a fusion of real physics with physics associated with various p-adic number fields and their algebraic extensions. The fusion relies on a generalization of number concept so that reals and p-adics have rationals and possibly some algebraic numbers as common: a generalization of notion of manifold is implied. S-matrix elements and various mathematical quantities needed to deduce physical predictions should make sense in both real and p-adic number fields or their appropriate algebraic extensions so that these quantities should be algebraic numbers.
This poses powerful constraint on a theory constrained already by other symmetries such as generalization of super-conformal symmetry. A hierarchy of physics based on extensions or rationals is predicted and Galois groups become good candidates for symmetries of physical system.
One might say that the model represented in the article "Quantum Statistical Mechanics and Class Field Theory" provides a realization of the number theoretic universality at the infinite temperature limit. The values of states in the quantum thermodynamical model for maximal Abelian extension of rationals are indeed algebraic numbers generating this extension and corresponding Galois group acts as symmetries of system.
More extensive comments in my blog.
Best Wishes for the new blog, Matti Pitkanen
hi, thanks so much for your posts. As much as I like its math content(which is way over my head!) I also liked a lot the Grothendieck quote
and the `Faustian joke'. It reminded me of a story invloving Hardy and the RH..... Speaking of the RH I like to put this question to visitors of this blog: do you think a proof (by any means) is in sight??
I think there is no proof in sight because even such simple things as adeles, ideles, are not enough well understood geometrically.
The previous lack of a serious geometry that put all absolute values on an equality footing is also part of the problem.
It is clear from the automorphic or spectral point of view that zeta functions are not talking about primes but about interaction of absolute values. A symmetry question can not be answered if it is not asked in a symmetric way.
Mathematicians should come back to the old gifted days of absolute values and work with them in a symmetric fashion for an evil better formulation of the problem,
I would like to thank Alain for this enlightening post.
Here is a short expository
piece, written about a year ago, on the approach to the Riemann
hypothesis that was mentioned by Alain in his post. There was a
numerical error (log 2) in my original version that is now corrected
(many thanks to Alain for pointing this out).
As a parallel to the ideas of Alain Connes on tilings, eg, by Penrose tiles, it has long seemed to me that the Universality Theorem about the critical strip of the Riemann zeta function should have similar properties. By the Universality Theorem, I mean the theorem that says that a copy of every disc-shaped region of every analytic function can be found, to within arbitrary tolerance, within the critical strip of the zeta function. Of course, every tiling by Penrose tiles also has such a "universal" property: Every finite size patch from a Penrose tiling will be found repeated (an infinite number of times) in any other Penrose tiling.
Please forgive me if this connection - between Connes' ideas and the Universality Theorem of the zeta function - is well known.
Post a Comment