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!

MOEBIUS dirigido por Gustavo Mosquera from Universidad del Cine on Vimeo.

]]>You can use structure sheaves. That is, compute all prime ideals of your ring and turn them into a space. Then, put a sheaf of rings on this space by localisation. You’ll get your ring back taking global sections.

Or, you might try the ‘functor of points’. That is, you take any other ring. Compute all ring-morphisms from yours to that one. You’ll recover your ring from Yoneda’s lemma.

And here’s the funny part.

Scheme-theorists claim there’s no differences between these two approaches. They are ‘equivalent’, as they prefer to say.

Do you believe them?

Let’s look at an example.

Take the ring of all polynomials with integer coefficients, $\mathbb{Z}[x]$.

Do you know all its prime ideals?

Sure, you’ll say.

There’s zero because it’s a domain. Then there are the ‘curves’. These are all prime numbers and all irreducible polynomials because it’s a UFD.

And then there are the ‘points’. They depend on a prime number $p$ and an irreducible polynomial which does not factor over $p$.

Not exactly rocket science, is it?

Okay, now let’s take them all together into a space.

Can you picture the intersection points of different curves? Let’s keep it simple. Take the curve given by a prime number $p$ and the one given by an irreducible polynomial $F(x)$. How do they intersect?

Easy! They are the factors of $F(x)$ modulo $p$.

Right, but can you picture this pattern for all prime numbers at once?

That depends on $F(x)$. David Mumford sketched the situation for $x^2+1$.

If $-1$ is a square modulo $p$, then $F(x)$ splits in two factors giving two points, such as along $5$. If not, $F(x)$ remains irreducible over $p$ and gives a thicker point like over $3$ or $7$. Except for the ‘odd’ case over $2$ where $F(x)$ is a square. Gauss knew already the situation for every prime.

But, what about arbitrary polynomials?

That’s a lot more difficult. Chebotarev knew how to get their Galois group from the factors at all primes.

So, you’ll need to solve deep problems in number theory before you can picture this space. The structure of the absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ to name one.

I’m afraid nobody understands the space of all prime ideals of $\mathbb{Z}[x]$ completely, let alone its structure sheaf.

What about the other approach? Let’s try to understand the functor of points of $\mathbb{Z}[x]$.

Take any ring $R$. We need to figure out all ring-maps $\mathbb{Z}[x] \rightarrow R$. But, we know such a map once we know the image of $x$. That is, there are as many ring-maps as there are elements in the set $R$.

Forgetting all about addition and multiplication on $R$. It is just the forgetful functor from rings to sets.

And they claim this is equivalent to solving deep problems in number theory?

Forgetting can’t be that hard, can it?

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