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Category: geometry

Topology and the symmetries of roots

We know embarrassingly little about the symmetries of the roots of all polynomials with rational coefficients, or if you prefer, the absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$.

In the title picture the roots of polynomials of degree $\leq 4$ with small coefficients are plotted and coloured by degree: blue=4, cyan=3, red=2, green=1. Sums and products of roots are again roots and by a symmetry we mean a map on all roots, sending sums to sums and products to products and leaving all the green dots (the rational numbers) fixed.

John Baez has an excellent post on the beauty of roots, including a picture of all polynomials of degree $\leq 5$ with integer coefficients between $-4$ and $4$ and, this time, colour-coded by: grey=2, cyan=3, red=4 and black=5.

beauty of roots

In both pictures there’s a hint of the unit circle, black in the title picture and spanning the ‘white gaps’ in the picture above.

If we’d only consider the sub-picture of all (sums and products of) roots including the rational numbers on the horizontal axis and the roots of unity on the unit circle we’d get the cyclotomic field $\mathbb{Q}_{cyc} = \mathbb{Q}(\mu_{\infty})$. Here we know all symmetries: they are generated by taking powers of the roots of unity. That is, we know all about the Galois group $Gal(\mathbb{Q}_{cyc}/\mathbb{Q})$.

The ‘missing’ symmetries, that is the Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q}_{cyc})$ remained a deep mystery, until last week…

[section_title text=”The oracle speaks”]

On september 15th, Robert Kucharczyk and Peter Scholze (aka the “oracle of arithmetic” according to Quanta-magazine) arXived their paper Topological realisations of absolute Galois groups.

Peter Scholze

They discovered a concrete compact connected Hausdorff space $M_{cyc}$ such that Galois extensions of $\mathbb{Q}_{cyc}$ correspond to connected etale covers of $M_{cyc}$.

Let’s look at a finite field $\mathbb{F}_p$. Here, Galois extensions of $\mathbb{F}_p$ (and there is just one such extension of degree $n$, upto isomorphism) correspond to connected etale covers of the circle $S^1$.

An etale map $X \rightarrow S^1$ is such that every circle point has exactly $n$ pre-images. Here again, up to homeomorphism, there is a unique such $n$-fold cover of $S^1$ (the picture on the left gives the cover for $n=2$).

.

If we replace $\mathbb{F}_p$ by the cyclotomic field $\mathbb{Q}_{cyc}$ then the compact space $M_{cyc}$ replaces the circle $S^1$. So, if we take a splitting polynomial of degree $n$ with coefficients in $\mathbb{Q}_{cyc}$, then there is a corresponding etale $n$-fold cover $X \rightarrow M_{cyc}$ such that for a specific point $p$ in $M_{cyc}$ its pre-images correspond to the roots of the polynomial. Nice!

Sadly, there’s a catch. Even though we have a concrete description of $M_{cyc}$ it turns out to be a horrible infinite dimensional space, it is connected but not path-connected, and so on.

Even Peter Scholze says it’s unclear whether new results can be proved from this result (see around 39.15 in his Next Generation Outreach Lecture).

Btw. if your German is ok, this talk is a rather good introduction to classical Galois theory and etale fundamental groups, including the primes=knots analogy.



[section_title text=”the imaginary field with one element”]

Of course there’s no mention of it in the Kucharczyk-Scholze paper, but this result is excellent news for those trying to develop a geometry over the imaginary field with one element $\mathbb{F}_1$ and hope to apply this theory to problems in number theory.

As a side remark, some of these people have just published a book with the EMS Publishing House: Absolute arithmetic and $\mathbb{F}_1$-geometry



The basic idea is that the collection of all prime numbers, $\mathbf{Spec}(\mathbb{Z})$ is far too large an object to be a terminal object (as it is in schemes). One should therefore extend the setting of schemes to so called $\mathbb{F}_1$-schemes, in which $\mathbf{Spec}(\mathbb{Z})$ is some higher dimensional object.

Initially, one hoped that $\mathbf{Spec}(\mathbb{Z})/\mathbb{F}_1$ might look like a curve, so that one could try to mimick Weil’s proof of the Riemann hypothesis for curves to prove the genuine Riemann hypothesis.

But, over the last decade it became clear that $\mathbf{Spec}(\mathbb{Z})/\mathbb{F}_1$ looks like an infinite dimensional space, a bit like the space $M_{cyc}$ above.

I’ll spare this to a couple of follow-up posts, but for now I’ll leave you with the punchline:

The compact connected Hausdorff space $M_{cyc}$ of Kucharczyk and Scholze is nothing but the space of complex points of $\mathbf{Spec}(\mathbb{Q}_{cyc})/\mathbb{F}_1$!

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Grothendieck’s gribouillis (2)

We left the story of Grothendieck’s Lasserre notes early 2015, uncertain whether they would ever be made public.

Some things have happened since.

Georges Maltsiniotis gave a talk at the Gothendieck conference in Montpellier in june 2015 having as title “Grothendieck’s manuscripts in Lasserre”, raising perhaps even more questions.

Philippe Douroux, a journalist at the French newspaper “Liberation”, had a few months ago his book out “Alexandre Grothendieck, sur les traces du dernier genie des mathematiques”. In the first and final couple of chapters he gives details on Grothendieck’s years in Lasserre.


In chapter 46 “Que reste-t-il du tresor de Grothendieck?” (what is left of Grothendieck’s treasure?) he recounts what has happened to the ‘Lasserre gribouillis’ and this allows us to piece together some of the jigsaw-puzzle.

Maltsiniotis’ talk

These days you don’t have to be present at a conference to get the gist of a talk you’re interested in. That is, if at least one of the people present is as helpful as Damien Calaque was in this case. A couple of email exchanges later I was able to get this post out on Google+:

Below is the relevant part of the picture taken by Edouard Balzin, mentioned in the post.

Maltsiniotis blackboard Grothendieck conference

The first three texts are given with plenty of details and add up to say 5000 pages. The fifth text is only given the approximate timing 1993-1998, although they present the bulk of the material (30000 pages).

A few questions come to mind:

– Why didn’t Maltsiniotis give more detail on the largest part of the collection?
– There seem to be at least 15000 pages missing in this roundup (previously, the collection was estimates at about 50000 pages). Were they destroyed?
– What happened to the post-1998 writings? We know from a certain movie that Grothendieck kept on writing until the very end.

Douroux’ book

If you have read Scharlau’s biographical texts on Grothendieck’s life, the middle part of Douroux’ book “Alexandre Grothendieck, sur les traces du dernier genie des mathematiques” will not be too surprising.

However, the first 5 and final 3 chapters contain a lot of unknown information (at least to me) about his life in Lasserre. The story of ‘his last friend Michel’ is particularly relevant.

Michel is a “relieur” (book-binder) and Grothendieck used his services to have carton boxes made, giving precise specifications as to their dimensions in mms, to contain his writings.

In the summer of 2000 there’s a clash between the two, details in chapter 4 “la brouille du relieur”. As a result, all writings from 2000-2014 are not as neatly kept as those before.

Each box is given a number, from 1 to the last one: 41.

In chapter 46 we are told that Georges Maltsiniotis spend two days in Lasserre consulting the content of the first 16 boxes, written between 1992 and 1994. He gives also additional information on the content:

Carton box 1 : “Geometrie elementaire schematique” contains 1100 pages of algebra and algebraic geometry which Maltsiniotis classifies as “assez classique” but which Douroux calls ‘this is solid mathematics on which one has to work hard to understand’ and a bit later (apparently quoting Michel Demasure) ‘we will need 50 years to transform these notes into accessible mathematics’.

Carton boxes 2-4 : “Structure de la psyche” (3700 pages) also being (according to Douroux) ‘a mathematical text in good form’.

Carton boxes 5-16 : Philosophical and mystical reflexions, among which “Psyche et structure” and “Probleme du mal” (7500 pages).

That is, we have an answer to most of the questions raised by Maltsiniotis talk. He only consulted the first 16 boxes, had a quick look at the other boxes and estimated they were ‘more of the same’ and packaged them all together in approximately 30000 pages of ‘Probleme du mal’. Probably he underestimated the number of pages in the 41 boxes containing all writings upto the summer of 2000.

Remains the problem to guess the amount of post 2000 writings. Here’s a picture taken by Leila Schneps days after Grothendieck’s death in Lasserre:

Grothendieck boxes in Lasserre

You will notice the expertly Michel-made carton boxes and a quick count of the middle green and rightmost red metallic box reveals that one could easily pack these 41 carton boxes in 3 metallic cases.

So, a moderate guess on the number of post 2000 pages is : 35000.

Why? Read on.

What does this have to do with the Paris attacks?

Grothendieck boxes in Lasserre

November 13th 2015 is to the French what 9/11 is to Americans (and 22 March 2016 is to Belgians, I’m sad to add).

It is also precisely one year after Grothendieck passed away in Saint-Girons.

On that particular day, the family decided to hand the Grothendieck-collection over to the Bibliotheque Nationale. (G’s last wishes were that everything he ever wrote was to be transferred to the BNF, thereby revoking his infamous letter of 2010, within 7 months after his death, or else had to be destroyed. So, to the letter of his will everything he left should have been destroyed by now. But fortunately none of it is, because 7 months is underestimating the seriousness with which the French ‘notaires’ carry out their trade, I can testify from personal experience).

While the attacks on the Bataclan and elsewhere were going on, a Mercedes break with on board Alexandre Jr. and Jean-Bernard, a librarian specialised in ancient writings, was approaching Paris from Lasserre. On board: 5 metallic cases, 2 red ones, 1 green and 2 blues (so Leila’s picture missed 1 red).

At about 2 into the night they arrived at the ‘commissariat du Police’ of the 6th arrondissement, and delivered the cases. It is said that the cases weighted around 400 kg (that is 80kg/case). As in all things Grothendieck concerned, this seems a bit over-estimated.

Anyway, that’s the last place we know to hold Grothendieck’s Lasserre gribouillis.

There’s this worrying line in Douroux’ book : ‘Who will get hold of them? The BNF? An american university? A math-obsessed billionaire?’

Let’s just hope for the best. That the initial plan to open up the gribouillis to the mathematical community at large will become a reality.

If I counted correctly, there are at least two of these metallic cases full of un-read post 2000 writings. To be continued…

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The Log Lady and the Frobenioid of $\mathbb{Z}$

“Sometimes ideas, like men, jump up and say ‘hello’. They introduce themselves, these ideas, with words. Are they words? These ideas speak so strangely.”

“All that we see in this world is based on someone’s ideas. Some ideas are destructive, some are constructive. Some ideas can arrive in the form of a dream. I can say it again: some ideas arrive in the form of a dream.”

Here’s such an idea.

It all started when Norma wanted to compactify her twisted-prime-fruit pies. Norma’s pies are legendary in Twin Peaks, but if you never ate them at Double R Diner, here’s the concept.

Start with a long rectangular strip of pastry and decorate it vertically with ribbons of fruit, one fruit per prime, say cherry for 2, huckleberry for 3, and so on.

For elegance, I argued, the $p$-th ribbon should have width $log(p)$.

“That may very well look natural to you,” she said, “but our Geometer disagrees”. It seems that geometers don’t like logs.

Whatever. I won.

That’s Norma’s basic pie, or the $1$-pie as we call it. Next, she performs $n$ strange twists in one direction and $m$ magical operations in another, to get one of her twisted-pies. In this case we would order it as her $\frac{m}{n}$-pie.

Marketing-wise, these pies are problematic. They are infinite in length, so Norma can serve only a finite portion, limiting the number of fruits you can taste.

That’s why Norma wants to compactify her pies, so that you can hold the entire pastry in your hands, and taste the infinite richness of our local fruits.

“Wait!”, our Geometer warned, “You can never close them up with ordinary scheme-dough, the laws of scheme-pastry prohibit this!” He suggested to use a ribbon of marzipan, instead.

“Fine, then… Margaret, before you start complaining again, how much marzipan should I use?”, Norma asked.

“Well,” I replied, “ideally you’d want it to have zero width, but that wouldn’t close anything. So, I’d go for the next best thing, the log being zero. Take a marzipan-ribbon of width $1$.”

The Geometer took a $1$-pie, closed it with marzipan of width $1$, looked at the pastry from every possible angle, and nodded slowly.

“Yes, that’s a perfectly reasonable trivial bundle, or structure sheaf if you want. I’d sell it as $\mathcal{O}_{\overline{\mathbf{Spec}(\mathbb{Z})}}$ if I were you.”

“In your dreams!  I’ll simply call this  a $1$-pastry, and an $\frac{m}{n}$-pie closed with a $1$-ribbon of marzipan can be ordered from now on as an $\frac{m}{n}$-pastry.”

“I’m afraid this will not suffice,” our Geometer objected, ” you will have to allow pastries having an arbitrary marzipan-width.”

“Huh? You want me to compactify an $\frac{m}{n}$-pie  with marzipan of every imaginable width $r$ and produce a whole collection of … what … $(\frac{m}{n},r)$-pastries? What on earth for??”

“Well, take an $\frac{m}{n}$-pastry and try to unravel it.”

Oh, here we go again, I feared.

Whereas Norma’s pies all looked and tasted quite different to most of us, the Geometer claimed they were all the same, or ‘isomorphic’ as he pompously declared.

“Just reverse the operations Norma performed and you’ll end up with a $1$-pie”, he argued.

So Norma took an arbitrary $\frac{m}{n}$-pastry and did perform the reverse operations, which was a lot more difficult that with pies as now the marzipan-bit produced friction. The end-result was a $1$-pie held together with a marzipan-ribbon of width strictly larger or strictly smaller than $1$, but never gave back the $1$-pastry. Strange!

“Besides”, the Geometer added, “if you take two of your pastries, which I prefer to call $\mathcal{L}$ and $\mathcal{M}$, rather than use your numerical system, then their product $\mathcal{L} \otimes \mathcal{M}$ is again a pastry, though with variable marzipan-width.

In the promotional stage it might be nice to give the product for free to anyone ordering two pastries.”

“And how should I produce such a product-pastry?”

“Well, I’m too lazy to compute such things, it must follow trivially from elementary results in Picard-pastry. Surely, our log lady will work out the details in your notation. No doubt it will involve lots of logs…”

And so I did the calculations in my dreams, and I wrote down all formulas in the Double R Diner log-book, for Norma to consult whenever a customer ordered a product, or power of pastries.

A few years ago we had a Japanese tourist visiting Twin Peaks. He set up office in the Double R Diner, consulted my formulas, observed Norma’s pastry production and had endless conversations with our Geometer.

I’m told he categorified Norma’s pastry-bizness, probably to clone the concept to the Japanese market, replacing pastries by sushi-rolls.

When he left, he thanked me for working out the most trivial of examples, that of the Frobenioid of $\mathbb{Z}$…

Added december 2015:

I wrote this little story some time ago.

The last couple of days this blog gets some renewed interest in the aftermath of the IUTT-Mochizuki-Fest in Oxford last week.

I thought it might be fun to include it, if only in order to decrease the bounce rate.

If you are at all interested in the maths, you may want to start with this google+ post, and work your way back using the links curated by David Roberts here.

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