A little pause, but some words on Q-lattices before that

Ok I’ll stop blogging until mid-april otherwise I can really see myself failing my exams… (Update: well, except maybe a few short posts, like the one above.)

Let me just mention nevertheless that Alain Connes himself (!) recently adressed , among other things, a naive question I had posed at the Noncommutative Geometry Blog: can the respective aperiodicity of primes (or Gaussian primes) and of quasi-crystals be related ?

Let me see if I can make sense of his answer. For the moment I’ll just rewrite it more or less.

Definition 1: a \mathbb{Q}-lattice of dimension n is a pair of a lattice \Lambda \subset\mathbb{R}^n and an homomorphism of additive (abelian) groups \phi : \mathbb{R}^n/\mathbb{Z}^n \rightarrow \mathbb{Q}\Lambda/\Lambda.

Definition 2: two \mathbb{Q}-lattices (\Lambda_1,\phi_1) and (\Lambda_2,\phi_2) are commensurable iff \Lambda_1 and \Lambda_2 are commensurable (i.e. \Lambda_1 + \Lambda_2 is also a lattice) and \forall x\in \mathbb{Q}^n/\mathbb{Z}^n we have (\phi_2 - \phi_1 )(x) \in \Lambda_1+\Lambda_2. This induces an equivalence relation.

From now on n=1. One has two natural actions on the space X of \mathbb{Q}-lattices modulo commensurability : a scaling action, and an action of Aut(\mathbb{Q}/\mathbb{Z}) by composition. The space X is a noncommutative one and is thus studied by looking at a related noncommutative algebra.

So far ok, but all that happens next is way over my head for the moment. Namely, the cyclic cohomology of this algebra is then computed in terms of distributions on X, and the nontrivial zeros of \zeta appear naturally when looking at the representation of the scaling group in this cohomology. Plus the explicit formula of number theory which relates the distrubution of primes to such zeros of \zeta is obtained straight away by applying the Lefchetz formula to that scaling operator. This clearly sounds incredibly interesting!

Still I don’t quite see the relation with the aperiodicity of quasicrystals (other than saying that X is a certain noncommutative space — similarly to the space of quasicrystals — from which zeros of \zeta emerge naturally). I had hoped for something more direct, like relating a particular quasicrystal to the Gaussian primes. I’ll return to this later…

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