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Can one explain schemes to hipsters?

Published January 5, 2015 by lievenlb

Nature (the journal) asked David Mumford and John Tate (of Fields and Abel fame) to write an obituary for Alexander Grothendieck.

Probably, it was their first experience ever to get a paper… rejected!

What was their plan?

How did they carry it out?

What went wrong?

And, can we learn from this?

the plan

Mumford and Tate set themselves an ambitious goal. Although Nature would have been happiest with a purely biographical note, they seized the opportunity to explain three ‘simple’ things to a wider audience: (1) schemes, (2) category theory, and, (3) cohomology…

“Since the readership of Nature were more or less entirely made up of non-mathematicians, it seemed as though our challenge was to try to make some key parts of Grothendieck’s work accessible to such an audience. Obviously the very definition of a scheme is central to nearly all his work, and we also wanted to say something genuine about categories and cohomology.”

1. Schemes

Here, the basic stumbling block, as Mumford acknowledged afterwards, is of course that most people don’t know what a commutative ring is. If you’ve never encountered a scheme before in broad daylight, I’m not certain this paragraph tells you how to recognise one:

“… In simplest terms, he proposed attaching to any commutative ring (any set of things for which addition, subtraction and a commutative multiplication are defined, like the set of integers, or the set of polynomials in variables x,y,z with complex number coefficients) a geometric object, called the Spec of the ring (short for spectrum) or an affine scheme, and patching or gluing together these objects to form the scheme. …”

2. Categories

Here they do a pretty good job, I think. They want to explain Grothendieck’s ‘functor of points’ and the analogy they used with several measuring experiments is neat:

“… Grothendieck used the web of associated maps — called morphisms — from a variable scheme to a fixed one to describe schemes as functors and noted that many functors that were not obviously schemes at all arose in algebraic geometry.

This is similar in science to having many experiments measuring some object from which the unknown real thing is pieced together or even finding something unexpected from its influence on known things….”

3. Cohomology

Here, Mumford “hoped that the inclusion of the unit 3-sphere in $\mathbb{C}^2- \{ (0,0) \}$ would be fairly clear to most scientists and so could be used to explain the Mike Artin’s breakthrough that $H^3_{et}(\mathbb{A}^2 – \{ (0,0) \}) \not= 0$.”

I’d love to know the fractional odds an experienced bookmaker would set in case someone (not me!) wants to bet on them successfully getting this message across.

“… Using complex coordinates (z,w), a plane has four real dimensions and taking out a point, what’s left is topologically a three dimensional sphere. Following the inspired suggestions of Grothendieck, Artin was able to show how with algebra alone that a suitably defined third cohomology group of this space has one generator, that is the sphere lives algebraically too. Together they developed what is called étale cohomology at a famous IHES seminar. …”

the aftermath

The good news is that Nature will still publish the Tate-Mumford obit, is some form or another, next week, on januari 15th. According to Mumford they managed to sneak in three examples of commutative rings in passing: polynomials, dual numbers and finite fields.

what went wrong?

The usual?

We mathematicians are obsessed with getting definitions right. We truly believe that no-one can begin to understand the implications of an idea if we don’t teach them the nitty gritty details of our treasured definitions first.

It appears that we are alone on this.

Did physicists smack us in the face with the full standard-model Lagrangian, demanding us to digest the minute details of it first, before they could tell us they had discovered the Higgs boson?

No, most scientists know how to get a message across. You need 3 things:

– a catchy name (the ‘God Particle’)

– good graphics (machines at CERN, collision pictures)

– a killer analogy (the most popular, in relation to the Higgs particle, seems to be “like Maggie Tatcher walking into a room”…)

can we learn from this?

Of course we can.

And frankly, I’m somewhat surprised Mumford missed this chance.

After all, he dreamed up the graphics and the killer analogy…

Further reading

– Mumford’s original rant : Can one explain schemes to biologists?

– John Baez’ follow-up post : Can one explain schemes to biologists?

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Grothendieck’s gribouillis

Published January 3, 2015 by lievenlb

A math-story well worth following in 2015.

What will happen to Grothendieck’s unpublished notes, or as he preferred, his “gribouillis” (scribbles)?

Here’s the little I know about this:

1. The Mormoiron scribbles

During the 80ties Grothendieck lived in ‘Les Aumettes’ in Mormoiron

In 1991, just before he moved to the Pyrenees he burned almost all of his personal notes in the garden. He phoned Jean Malgoire:

“Si tu ne viens pas chercher mon bordel mathématique, il va brûler avec le reste.”

Malgoire sped to Mormoiron and rescued 5 boxes containing about 20.000 pages. The next 20 years he kept them in his office, not exactly knowing what to do with them.

On january 3rd 2010, Grothendieck wrote his (in)famous letter forbidding others to share or publish any of his writings. (Picture via the SecretBloggingSeminar)

Malgoire feared that Grothendieck would soon ask him to destroy the Mormoiron-gribouillis and decided to donate them to the University of Montpellier.

They are kept somewhere in their archives, the exact location known only to Jean Malgoire, Luc Gomel (who is in charge of the patrimonium of the University of Montpellier) and the person who put the boxes away.

After Grothendieck’s death on november 13th, FranceTV3 did broadcast a short news-item.

If Grothendieck’s children agree, the University of Montpellier intends to make an inventory of the 5 boxes and will make them available, at least to researchers.

2. The Lasserre scribbles

The final 23 years of his life, Grothendieck lived in the small village of Lasserre in the French Pyrenees.

Here he could be glimpsed blurrily through the window as he wrote for hours during the night.(Picture via the GrohendieckCircle)

Leila Schneps and her husband Pierre Lochak did visit the house and met with Grothendieck’s family the week after his death.

Before she went, she was optimistic about the outcome as she emailed:

“I have already started modifying the Grothendieck circle website and it will of course eventually return completely. Plus many things will be added, as we will now have access to Grothendieck’s correspondence and many other papers.”

Her latest comment, from december 16th, left on the Grothendieck-circle bulletin board, is more pessimistic:

“Il a ecrit a Lasserre sans cesse pendant plus de 20 ans. Je n’ai pu que jeter un rapide coup d’oeil sur tout ce qu’il a laisse. Il y a de tout: des maths, des reflexions sur lui-meme, et des reflexions sur la nature humaine et sur l’univers. Rien n’est disponible pour le moment. Ces manuscrits finiront dans une bibliotheque et seront peut-etre un jour consultables.”

The good news is that there appears to be some mathematics among the Lassere-gribouillis. The sad news being that none of this is available at the moment, and perhaps never will.

So, what happened? Here’s my best guess:

Grothendieck’s children were pretty upset a private letter from one of them turned up in the French press, a couple of years ago.

Perhaps, they first want to make sure family related material is recovered, before they’ll consider donating the rest (hopefully to the University of Montpellier to be reunited with Grothendieck’s Mormoiron-notes).

This may take some time.

Further reading (in French):

Grothendieck, mon tresor (nationale)

Un génie mystérieux, un secret bien gardé

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$\mathbf{Ext}(\mathbb{Q},\mathbb{Z})$ and the solenoid $\widehat{\mathbb{Q}}$

Published July 31, 2014 by lievenlb

Note to self: check Jack Morava’s arXiv notes on a more regular basis!

It started with the G+-post below by +David Roberts:

Suddenly I realised I hadn’t checked out Morava‘s “short preprints with ambitious ideas, but no proofs” lately.

A couple of years ago I had a brief email exchange with him on the Habiro topology on the roots of unity, and, in the process he send me a 3 page draft with ideas on how this could be relevant to higher dimensional topological QFT (If my memory doesn’t fail me, I can’t find anything remotely related in the arXiv-list).

Being in a number-theory phase lately (yes, I also have to give next year, for the first time, in the second semester, a master-course on Number Theory) the paper A topological group of extensions of $\mathbb{Q}$ by $\mathbb{Z}$ caught my eyes.

The extension group $Ext(\mathbb{Q},\mathbb{Z})$ classifies all short exact sequences of Abelian groups

$0 \rightarrow \mathbb{Z} \rightarrow A \rightarrow \mathbb{Q} \rightarrow 0$

upto equivalence, that is commuting sequences with end-maps being identities.

The note by Boardman Some Common Tor and Ext Groups hs a subsection on this group/rational vector space, starting out like this:

“This subsection is strictly optional. The group $Ext(\mathbb{Q}, \mathbb{Z})$ is much more difficult to determine. It is easy to see that it is a rational vector space, simply from the presence of $\mathbb{Q}$, but harder to see what its dimension is. This group is not as mysterious as is sometimes claimed, but is related to adèle groups familiar to number theorists.”

Boardman goes on to show that this extension group can be identified with $\mathbb{A}^f_{\mathbb{Q}}/\mathbb{Q}$ where $\mathbb{A}^f_{\mathbb{Q}}$ is the ring of finite adèles, that is, sequence $(x_2,x_3,x_5,…)$ of $p$-adic numbers $x_p \in \widehat{\mathbb{Q}}_p$ with all but finitely many $x_p \in \widehat{\mathbb{Z}}_p$, and $\mathbb{Q}$ is the additive subgroup of constant sequences $(x,x,x,…)$.

Usually though, one considers the full adèle ring $\mathbb{A}_{\mathbb{Q}} = \mathbb{R} \times \mathbb{A}^f_{\mathbb{Q}}$ and one might ask for a similar interpretation of the adèle class-group $\mathbb{A}_{\mathbb{Q}}/\mathbb{Q}$.

This group is known to be isomorphic to the character group (or Pontrtrjagin dual) of the rational numbers, that is, to $\widehat{\mathbb{Q}}$ which are all group-morphisms $\mathbb{Q} \rightarrow S^1$ from the rational numbers to the unit circle. This group is sometimes called the ‘solenoid’ $\Sigma$, it is connected but not path connected and the path-component of the identity $\Sigma_0 = \mathbb{R}$.

A very nice and accessible account of the solenoid is given in the paper The character group of $\mathbb{Q}$ by Keith Conrad.

The point of Morava’s note is that he identifies the solenoid $\mathbb{A}_{\mathbb{Q}}/\mathbb{Q}$ with a larger group of ‘rigidified’ extensions $Ext_{\mathbb{Z}_0}(\mathbb{Q},\mathbb{Z})$.That is, one starts with a usual extension in $Ext_{\mathbb{Z}}(\mathbb{Q},\mathbb{Z})$ as above, but in addition, one fixes a splitting of the induced sequence

$0 \rightarrow \mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{R} \rightarrow A \otimes_{\mathbb{Z}} \mathbb{R} \rightarrow \mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{R} \rightarrow 0$

Forgetting the splitting this gives the exact sequence

$0 \rightarrow \mathbb{R} \rightarrow Ext_{\mathbb{Z}_0}(\mathbb{Q},\mathbb{Z}) \rightarrow Ext_{\mathbb{Z}}(\mathbb{Q},\mathbb{Z}) \rightarrow 0$

which is isomorphic to the sequence involving the path-component of the solenoid!

$0 \rightarrow \Sigma_0 = \mathbb{R} \rightarrow \Sigma=\widehat{Q} \rightarrow \mathbb{A}^f_{\mathbb{Q}}/\mathbb{Q} \rightarrow 0$

Morava ends with: “I suppose the proposition above has a natural reformulation
in Arakelov geometry; but I don’t know anything about Arakelov geometry”…

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