
Scottish solids, final(?) comments
In the spring of 2009 I did spend a fortnight dogsitting in a huge house in the countryside, belonging to my parentsinlaw, who both passed away the year before. That particular day it was raining and thundering heavily. To distract myself from the sombre and spooky atmosphere in the house I began to surf the…

What is the knot associated to a prime?
Sometimes a MathOverflow question gets deleted before I can post a reply… Yesterday (NewYear) PD1&2 were visiting, so I merely bookmarked the What is the knot associated to a prime?topic, promising myself to reply to it this morning, only to find out that the page no longer exists. From what I recall, the OP interpreted…

The odd knights of the round table
Here’s a tiny problem illustrating our limited knowledge of finite fields : “Imagine an infinite queue of Knights ${ K_1,K_2,K_3,\ldots } $, waiting to be seated at the unitcircular table. The master of ceremony (that is, you) must give Knights $K_a $ and $K_b $ a place at an odd root of unity, say $\omega_a…

Pollock your own noncommutative space
I really like Matilde Marcolli’s idea to use some of Jackson Pollock’s paintings as metaphors for noncommutative spaces. In her talk she used this painting and refered to it (as did I in my post) as : Jackson Pollock “Untitled N.3”. Before someone writes a post ‘The Pollock noncommutative space hoax’ (similar to my own…

Views of noncommutative spaces
The general public expects pictures from geometers, even from noncommutative geometers. Hence, it is important for researchers in this topic to make an attempt to convey the mental picture they have of their favourite noncommutative space, … somehow. Two examples : This picture was created by Shahn Majid. It appears on his visions of noncommutative…

On2 : Conway’s nimarithmetics
Conway’s nimarithmetic on ordinal numbers leads to many surprising identities, for example who would have thought that the third power of the first infinite ordinal equals 2…

Mumford’s treasure map
In the series “Brave new geometries” we give an introduction to ‘strange’ but exciting new ideas. We start with Grothendieck’s schemerevolution, go on with Soule’s geometry over the field with one element, Mazur’s arithmetic topology, Grothendieck’s anabelian geometry, Connes’ noncommutative geometry etc.

ConnesConsani for undergraduates (3)
A quick recap of last time. We are trying to make sense of affine varieties over the elusive field with one element $\mathbb{F}_1 $, which by Grothendieck’s schemephilosophy should determine a functor $\mathbf{nano}(N)~:~\mathbf{abelian} \rightarrow \mathbf{sets} \qquad A \mapsto N(A) $ from finite Abelian groups to sets, typically giving pretty small sets $N(A) $. Using the…

ConnesConsani for undergraduates (2)
Last time we have seen how an affine $\mathbb{C} $algebra R gives us a maxifunctor (because the associated sets are typically huge) $\mathbf{maxi}(R)~:~\mathbf{abelian} \rightarrow \mathbf{sets} \qquad A \mapsto Hom_{\mathbb{C}alg}(R, \mathbb{C} A) $ Substantially smaller sets are produced from finitely generated $\mathbb{Z} $algebras S (therefore called minifunctors) $\mathbf{mini}(S)~:~\mathbf{abelian} \rightarrow \mathbf{sets} \qquad A \mapsto Hom_{\mathbb{Z}alg}(S, \mathbb{Z} A)…