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“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!”