Second part in a series which I gave at a seminar in Ghent (April 28, 2011): “Scheme theory over the field of characteristic one”. (The first part being “Introduction to Absolute Arithmetic”.)

This is part 1 of a series of two lectures which I recently gave at a seminar in Ghent (April 28, 2011): “Introduction to Absolute Arithmetic”.

Here is the recently published paper “The hyperring of adèle classes” (by Connes and Consani). ConnesConsani2011

Matilde Marcolli: Cyclotomy and endomotives; arXived on January 20th, 2009: arXiv:0901.3167v1.

Slides are available of a talk given last week in Reims on Manin’s analytic F_un-geometry.

The prep-notes for the halloween-talk on “F_un and other ghost stories” at the Arts are available.

Katia Consani gave a talk “On the notion of geometry over F_un” at the Fields institute.

It is perhaps surprising that Alain Connes and Katia Consani, two icons of noncommutative geometry, restrict themselves to define commutative algebraic geometry over the field with one element. Remains the fact that their approach screams for a noncommutative extension.

Alain Connes and Katia Consani arXived the paper “On the notion of geometry over F_un” in which they refine and simplify Soule’s approach. This series tries to explain their construction to non-specialists in algebraic geometry.

Inspired by the equivalent definition of a scheme as a particular kind of functor, Soulé gives his definition of a gadget over F_un as a couple consisting on a functor, that takes the rôle of the functor of points, and an algebra, which encodes a ‘topology at infinity’ and gives us the extra information that is missing in the functor.