Andre Weil on the Riemann hypothesis

Don’t be fooled by introductory remarks to the effect that ‘the field with one element was conceived by Jacques Tits…’ Let’s have it out into the open : F_un mathematics’ goal is no less than proving the Riemann Hypothesis.

Connes-Consani for undergraduates (3)

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.

Gadgets a la Soulé

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.

ConnesConsani2011

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

Anyone interested?

I’m about to write a series of posts on Borger’s notion of geometry over the field with one element using lambda-rings. Problem remains : where should I post them? Here, or elsewhere? In other words, are there any humans left here (or are the 50 to 60 daily hits robot-performed), and, more importantly, are any [...]

The Hyperring of Adele Classes

Here is a link to a rather recent YouTube clip, where Alain Connes describes his joint paper with Consani The Hyperring of Adele Classes: .

Kapranov-Smirnov on F_un

“One can postulate, of course, that spec(F_un) is the absolute point, but the real problem is to develop non-trivial consequences of this point of view.”

Quomodocumque on F_un (Update 10-21)

In a follow-up post, F_un and the braid group – a note of skepticism, arguments are collected against the claim that the d-string braid group should be invertible dxd matrices over the polynomials over F_un.