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Author: lievenlb

Life on Gaussian primes

At the moment I’m re-reading Siobhan Roberts’ biography of John Horton Conway, Genius at play – the curious mind of John Horton Conway.

In fact, I’m also re-reading Alexander Masters’ biography of Simon Norton, The genius in my basement – the biography of a happy man.

If you’re in for a suggestion, try to read these two books at about the same time. I believe it is beneficial to both stories.

Whatever. Sooner rather than later the topic of Conway’s game of life pops up.

Conway’s present pose is to yell whenever possible ‘I hate life!’. Problem seems to be that in book-indices in which his name is mentioned (and he makes a habit of checking them all) it is for his invention of the game of Life, and not for his greatest achievement (ihoo), the discovery of the surreal numbers.

If you have an hour to spare (btw. enjoyable), here are Siobhan Roberts and John Conway, giving a talk at Google: “On His LOVE/HATE Relationship with LIFE”

By synchronicity I encounter the game of life now wherever I look.

Today it materialised in following up on an old post by Richard Green on G+ on Gaussian primes.

As you know the Gaussian integers Z[i]Z[i] have unique factorization and its irreducible elements are called Gaussian primes.

The units of Z[i]Z[i] are {±1,±i}{±1,±i}, so Gaussian primes appear in 44- or 88-tuples having the same distance from the origin, depending on whether a prime number pp remains prime in Z[i]Z[i] or splits.

Here’s a nice picture of Gaussian primes, taken from Oliver Knill’s paper Some experiments in number theory

Note that the natural order of prime numbers is changed in the process (look at the orbits of 33 and 55 (or 1313 and 1717).

Because the lattice of Gaussian integers is rectangular we can look at the locations of all Gaussian primes as the living cell in the starting position on which to apply the rules of Life.

Here’s what happens after one move (left) and after three moves (right):

Knill has a page where you can watch life on Gaussian primes in action.

Even though the first generations drastically reduce the number of life spots, you will see that there remains enough action, at least close enough to the origin.

Knill has this conjecture:

When applying the game of life cellular automaton to the Gaussian primes, there is motion arbitrary far away from the origin.

What’s the point?

Well, this conjecture is equivalent to the twin prime conjecture for the Gaussian integers Z[i]Z[i], which is formulated as

“there are infinitely pairs of Gaussian primes whose Euclidian distance is 22.”

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Grothendieck’s gallery No. 154

Since mid May the Montpellier part of Grothendieck’s gribouillis are online and for everyone available at the Archives Grothendieck.

The story is well-known.

End of June 1990, Grothendieck phoned Jean Malgoire warning him to come asap if he wanted to safeguard the better part of G’s mathematical archive, for he was making a bonfire…

A second handover apparently took place on July 28th 1995.

Malgoire kept these notes (in huge Pampers boxes!) until 2010 when he got cold feet as a result of Grothendieck’s letter. He then donated the boxes to the University of Montpellier in 2012.

After Grothendieck’s death in 2014, Montpellier started a project to scan each and every page and put them online, with the backing of Grothendieck’s children (and generous financial support from the local authorities).

So here we are now, and… nobody seems to care.

I’m aware only of this post on MathOverflow by someone who wants to LaTex some of the material on motives.

Perhaps this is due to the less than optimal presentation of the material, or more likely, Grothendieck’s terrible handwriting. Perhaps the University of Montpellier should partner up with the (older generation of) French pharmacists.

But then, there’s this artistic gem in the archive: cote 154 systemes the pseudo-droites written in 1983-84.

It is written on ancient matrix-plotter page. Here’s a typical example

Which mathematical department wouldn’t want to acquire a framed version of one of these original pages?

That’s the point I wanted to make early may in this G+-post, hoping to raise money to safeguard the Lasserre part of Grothendieck’s gribouillis.

When in need for a header picture, I’ll use a page of Grothendieck’s gallery No 154 from now on.

Here’s evidence that Grothendieck was working on GUTS! (literally).

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How to dismantle scheme theory?

In several of his talks on #IUTeich, Mochizuki argues that usual scheme theory over ZZ is not suited to tackle problems such as the ABC-conjecture.

The idea appears to be that ABC involves both the additive and multiplicative nature of integers, making rings into ‘2-dimensional objects’ (and clearly we use both ‘dimensions’ in the theory of schemes).

So, perhaps we should try to ‘dismantle’ scheme theory, and replace it with something like geometry over the field with one element F1F1.

The usual F1F1 mantra being: ‘forget all about the additive structure and only retain the multiplicative monoid’.

So perhaps there is yet another geometry out there, forgetting about the multiplicative structure, and retaining just the addition…

This made me wonder.

In the forgetting can’t be that hard, can it?-post we have seen that the forgetful functor

F+,× : ringssetsF+,× : ringssets

(that is, forgetting both multiplicative and additive information of the ring) is representable by the polynomial ring Z[x]Z[x].

So, what about our ‘dismantling functors’ in which we selectively forget just one of these structures:

F+ : ringsmonoidsandF× : ringsabelian groupsF+ : ringsmonoidsandF× : ringsabelian groups

Are these functors representable too?

Clearly, ring maps from Z[x]Z[x] to our ring RR give us again the elements of RR. But now, we want to encode the way two of these elements add (or multiply).

This can be done by adding extra structure to the ring Z[x]Z[x], namely a comultiplication ΔΔ and a counit ϵϵ

Δ : Z[x]Z[x]Z[x]andϵ : Z[x]ZΔ : Z[x]Z[x]Z[x]andϵ : Z[x]Z

The idea of the comultiplication being that if we have two elements r,sRr,sR with corresponding ring maps fr : Z[x]Rxrfr : Z[x]Rxr and fs : Z[x]Rxsfs : Z[x]Rxs, composing their tensorproduct with the comultiplication

fv : Z[x]ΔZ[x]Z[x]frfsRfv : Z[x]ΔZ[x]Z[x]frfsR

determines another element vRvR which we can take either the product v=r.sv=r.s or sum v=r+sv=r+s, depending on the comultiplication map ΔΔ.

The role of the counit is merely sending xx to the identity element of the operation.

Thus, if we want to represent the functor forgetting the addition, and retaining the multiplication we have to put on Z[x]Z[x] the structure of a biring

Δ(x)=xxandϵ(x)=1Δ(x)=xxandϵ(x)=1

(making xx into a ‘group-like’ element for Hopf-ists).

The functor F×F× forgetting the multiplication but retaining the addition is represented by the Hopf-ring Z[x]Z[x], this time with

Δ(x)=x1+1xandϵ(x)=0Δ(x)=x1+1xandϵ(x)=0

(that is, this time xx becomes a ‘primitive’ element).

Perhaps this adds another feather of weight to the proposal in which one defines algebras over the field with one element F1F1 to be birings over ZZ, with the co-ring structure playing the role of descent data from ZZ to F1F1.

As, for example, in my note The coordinate biring of Spec(Z)/F1Spec(Z)/F1.

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