Primes, Knots and the Scaling Site

The talk will develop on the inter-relation between a very specific Grothendieck topos, the scaling site, and the well-known analogy going back to 1963 between knots and prime numbers. In 2014, in a joint work with A. Connes, we proved that the points of the scaling site can be naturally identified with the points of a space that Connes introduced in 1996 in non-commutative geometry, in relation to the Riemann zeta function.  In algebraic geometry, A. Grothendieck with his theory of the etale fundamental group,  extended Galois theory from the context of fields to that of schemes. The main new result I will present in the talk (a joint work with A. Connes) is that the scaling site and the adele class space of the rationals provide an extension of the class field theory isomorphisms which traditionally relate groups of adelic nature to Galois groups. Exactly as Grothendieck extended Galois theory to schemes, the adele class space of the rationals gives, as a covering of the scaling site, the corresponding extension of the class field isomorphism for Q, to schemes intimately related to Spec Z. The corresponding spaces are understood respectively at the adelic and at the topos level. This construction exhibits, in particular, the linking number of a rational prime p with all other rational primes.