levels

Oct 8, 2008

Connes-Consani for undergraduates (2)

Posted by in Connes-Consani2008, undergraduate No comments

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.


Oct 6, 2008

Connes-Consani for undergraduates (1)

Posted by in Connes-Consani2008, featured, undergraduate 8 comments

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.


Oct 4, 2008

The skeleton of Soulé’s F_un geometry

Posted by in featured, graduate, Soule2004 No comments

Any possible definition of a variey over F_un should have a nice extension of scalars to the integers. In particular, we might expect to be able to control the behavior of such extension for the easiest family of rings defined over F_un: its finite abelian extensions F_un^n.


Oct 1, 2008

ceci n’est pas un corps

Posted by in featured, outreach, Tits1957 1 comment

To Gavin Wraiht a mathematical phantom is a “nonexistent entity which ought to be there but apparently is not; but nevertheless obtrudes its effects so convincingly that one is forced to concede a broader notion of existence”. Mathematics’ history is filled with phantoms getting the kiss of life.


Sep 22, 2008

F_un and braid groups

Posted by in KapranovSmirnov, undergraduate 2 comments

We have seen before that the symmetric group S(n) has a F-un interpretation as the linear group GL(n) over the field with one element. Hence, we can ask whether there is also a F-un interpretation of the n-string braid group and of the above group-morphism.


Sep 22, 2008

F_un linear algebra

Posted by in graduate, KapranovSmirnov No comments

Over F_un vectorspaces are pointed sets. One can also do linear algebra over the F_un extension fields F_un^n. Vectorspaces are then pointed sets having a free action of n-th roots of unity.


Sep 22, 2008

The F_un folklore

Posted by in graduate, KapranovSmirnov, Manin1995 1 comment

The followers of F-un have their own collection of semi-secret texts, surrounded by whispers, of which they try to decode every single line in search of enlightenment. Fortunately, you do not have to search the shelves of the Bibliotheque National in Paris, but the depths of the internet to find them as huge, bandwidth-unfriendly, scanned documents.


Sep 22, 2008

Looking for F_un

Posted by in Connes-C-M-2008, featured, Soule1999, Soule2004, undergraduate 1 comment

In recent years several people spend a lot of energy looking for properties of an elusive object : the field with one element or in French : “F-un”. One of the motivations is the hope to prove the Riemann hypothesis by mimicking Weil’s proof in the case of curves.