# Grothendieck seminar at the ENS

Next week, the brand new séminaire « Lectures grothendieckiennes » will kick off on Tuesday October 24th at 18hr (h/t Isar Stubbe).

There will be one talk a month, on a tuesday evening from 18hr-20hr. Among the lecturers are the ‘usual suspects’:

Pierre Cartier (October 24th) will discuss the state of functional analysis before Grothendieck entered the scene in 1948 and effectively ‘killed the subject’ (said Dieudonné).

Alain Connes (November 7th) will talk on the origins of Grothendieck’s introduction of toposes.

In fact, toposes will likely be a recurrent topic of the seminar.

Laurant Laforgue‘s title will be ‘La notion de vérité selon Grothendieck'(January 9th) and on March 6th there will be a lecture by Olivia Caramello.

Also, Colin McLarty will speak about them on May 3rd: “Nonetheless one should learn the language of topos: Grothendieck on building houses”.

The closing lecture will be delivered by Georges Maltsiniotis on June 5th 2018.

Further Grothendieck news, there’s the exhibition of a sculpture by Nina Douglas, the wife of Michael Douglas, at the Simons Center for Geometry and Physics (h/t Jason Starr).

It depicts Grothendieck as shepherd. The lambs in front of him have Riemann surfaces inserted into them and on the staff is Grothendieck’s ‘Hexenkuche’ (his proof of the Riemann-Roch theorem).

# How to dismantle scheme theory?

In several of his talks on #IUTeich, Mochizuki argues that usual scheme theory over $\mathbb{Z}$ 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 $\mathbb{F}_1$.

The usual $\mathbb{F}_1$ 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…

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

$F_{+,\times}~:~\mathbf{rings} \rightarrow \mathbf{sets}$

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

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

$F_+~:~\mathbf{rings} \rightarrow \mathbf{monoids} \quad \text{and} \quad F_{\times}~:~\mathbf{rings} \rightarrow \mathbf{abelian~groups}$

Are these functors representable too?

Clearly, ring maps from $\mathbb{Z}[x]$ to our ring $R$ give us again the elements of $R$. 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 $\mathbb{Z}[x]$, namely a comultiplication $\Delta$ and a counit $\epsilon$

$\Delta~:~\mathbb{Z}[x] \rightarrow \mathbb{Z}[x] \otimes \mathbb{Z}[x] \quad \text{and} \quad \epsilon~:~\mathbb{Z}[x] \rightarrow \mathbb{Z}$

The idea of the comultiplication being that if we have two elements $r,s \in R$ with corresponding ring maps $f_r~:~\mathbb{Z}[x] \rightarrow R \quad x \mapsto r$ and $f_s~:~\mathbb{Z}[x] \rightarrow R \quad x \mapsto s$, composing their tensorproduct with the comultiplication

$f_v~:~\mathbb{Z}[x] \rightarrow^{\Delta} \mathbb{Z}[x] \otimes \mathbb{Z}[x] \rightarrow^{f_r \otimes f_s} R$

determines another element $v \in R$ which we can take either the product $v=r.s$ or sum $v=r+s$, depending on the comultiplication map $\Delta$.

The role of the counit is merely sending $x$ 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 $\mathbb{Z}[x]$ the structure of a biring

$\Delta(x) = x \otimes x \quad \text{and} \quad \epsilon(x) = 1$

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

The functor $F_{\times}$ forgetting the multiplication but retaining the addition is represented by the Hopf-ring $\mathbb{Z}[x]$, this time with

$\Delta(x) = x \otimes 1 + 1 \otimes x \quad \text{and} \quad \epsilon(x) = 0$

(that is, this time $x$ 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 $\mathbb{F}_1$ to be birings over $\mathbb{Z}$, with the co-ring structure playing the role of descent data from $\mathbb{Z}$ to $\mathbb{F}_1$.

As, for example, in my note The coordinate biring of $\mathbf{Spec}(\mathbb{Z})/\mathbb{F}_1$.

# The geometry of football

Soon, we will be teaching computational geometry courses to football commentators.

If a player is going to be substituted we’ll hear sentences like: “no surprise he’s being replaced, his Voronoi cell has been shrinking since the beginning of the second half!”

David Sumpter, the author of Soccermatics: Mathematical Adventures in the Beautiful Game, wrote a nice article over at Medium The geometry of attacking football.

As an example, he took an attack of Barcelona against Panathinaikos.

and explained the passing possibilities in terms of the Delaunay triangulation between the Barca-players (the corresponding Voronoi cell decomposition is in the header picture).

He concludes: “It is not only their skill on the ball, but also their geometrically accurate positioning that allows them to make the pass.”

Jaime Sampaoi produced a short video of changing Voronoi cells from kick-off by the blue team, with the red team putting pressure until a faulty pass is given, leading to a red-attack and a goal. All in 29 seconds.

I’d love to turn this feature on when watching an actual game.

Oh, and please different cell-colours for the two teams.

And, a remote control to highlight the Voronoi cell of a particular player.

# The subway singularity

The Boston subway is a complex system, spreading out from a focus at Park Street.

On March 3rd, the Boylston shuttle went into service, tying together the seven principal lines, on four different levels.

A day later, train 86 went missing on the Cambridge-Dorchester line.

The Harvard algebraist R. Tupelo suggested the train might have hit a node, a singularity. By adding the Boylston shuttle, the connectivity of the subway system had become infinite…

Never heard of this tragic incident?

Time to read up on A.J. Deutsch’s classic ‘A subway named Moebius’ from 1950. A 12 page pdf of this short story is available via the Rio Rancho Math Camp.

The ‘explanation’ given in the story is that the Moebius strip has a singularity. Before you yell that this is impossible, have a look at this or that.

A ‘non spatial network’ where ‘an exclusion principle operates’, Deutsch’s story says.

Here’s another take.

The train took the exceptional fiber branch, instead of remaining on the desingularisation?

Whatever really happened, it’s a fun read, mathematics clashing with bureaucracy.

In 1996 Gustavo Mosquera directed the film ‘Moebius’, set in Buenos Aires, loosely based on Deutsch’s story.

Here’s the full version (90 min.), with subtitles. Have fun!