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Car crashes in scheme theory

What do you get when two cars crash head on at full speed?

A heap of twisted metal.

What do you get when two tiny cars crash head on at full speed?

A smaller heap of twisted metal.

In the limit, what do you get when two point cars crash head on at full speed?

A point of twisted metal?

No, you get a point car with a better GPS system!

Huh?

All a point can see of a function is its own value. Call this a basic, grade one GPS.

When two points collide you’ll get a grade two GPS: not only does it give you its value, but also the tendency of the function in a tiny neighbourhood of the point.

It doesn’t matter whether the two point-cars collide along a line, or in the plane or in 24-dimensional space, you’ll always get the same grade two GPS.

carcrash

Things become more interesting when three or more point-cars crash together.

For three point-cars colliding in the plane a crash-scene investigator can tell from the resulting grade three GPS whether they crashed along a curve, or more randomly.

Again, if this crash happened in a higher dimensional space, you’d still have only hese two types of grade three GPS’s.

And if 4 point-cars pile up?

Then, there are 4 possible types of grade four GP-systems. For crashes of 5 point cars there are 9 possible types, and so on.

Always a finite number?

Hah, no! Seven or more point-cars can pile up in infinitely many ways.

How on earth would you prove such a thing?

A bit of classical geometry, and an extra bit of GIT.

If you’d try to describe all possible 7-car configurations up to isomorphisms, one of the subproblems you run into is to classify all possible intersections of two quadrics in $\mathbb{P}^3$. These usually give you an elliptic curve, and there is a 1-parameter family of those.

Wow!

It only gets better.

Let’s say a grade n GPS knows $n$ behavioural facts about functions, among which its value, in some locally closed neighborhood of the point.

Does such a GPS necessarily comes from the collision of n point cars in some high dimensional space?

Or put differently, can you unravel any GPS into n distinct point cars, just before they crash?

The way you ask this I suspect you’re going to tell me it’s not always possible. But, I haven’t the faintest clue on how to approach such a problem.

You’re right, it’s not always possible.

There exist grade 8 GPS’s which you can’t get from smashing 8 point cars in a 4 or higher dimensional space.

And you’ll need some scheme theory to prove this.

It’s pretty easy to show that the component containing n distinct point-cars in the scheme of all possible n-point configurations has dimension $n^2$. Follows because the automorphism group of n distinct point-cars is just the group of all permutations of the n points.

All grade n GPS’s made from massive car collisions are points in that component.
But then, the tangent space in such a point must have dimension at least $n^2$.

And then, all one has to do is to engineer a grade n GPS with a smaller dimensional tangent space.

A Berkeley group of 9 geometrical engineers did just that: Jonah Blasiak, Dustin Cartwright, David Eisenbud, Daniel Erman, Mark Haiman, Bjorn Poonen, Bernd Sturmfels, Mauricio Velasco, and Bianca Viray.

Here’s their simplest example:

$\mathbb{C}[a, b, c, d]/(a^2, ab, b^2, c^2, cd, d^2, ad − bc)$

Details can be found here:

B. Poonen : The moduli space of commutative algebras of finite rank.

B. Poonen : Isomorphism types of commutative algebras of finite rank over an algebraically closed field.

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NaNoWriMo (1)

Some weeks ago I did register to be a participant of NaNoWriMo 2016. It’s a belated new-year’s resolution.

When PS (pseudonymous sister), always eager to fill a 10 second silence at family dinners, asked

(PS) And Lieven, what are your resolutions for 2016?

she didn’t really expect an answer (for decades my generic reply has been: “I’m not into that kinda nonsense”)

(Me) I want to write a bit …

stunned silence

(PS) … Oh … good … you mean for work, more papers perhaps?

(Me) Not really, I hope to write a book for a larger audience.

(PS) Really? … Ok … fine … (appropriate silence) … Now, POB (pseudonymous other brother), what are your plans for 2016?

nanowrimo_2016_webbanner_participant

If you don’t know what NaNoWriMo is all about: the idea is to write a “book” (more like a ‘shitty first draft’ of half a book) consisting of 50.000 words in November.

We’re 5 days into the challenge, and I haven’t written a single word…

Part of the problem is that I’m in the French mountains, and believe me, there always more urgent or fun things to do here than to find a place of my own and start writing.

A more fundamental problem is that I cannot choose between possible book-projects.

Here’s one I will definitely not pursue:

“The Grothendieck heist”

“A group of hackers uses a weapon of Math destruction to convince Parisian police that a terrorist attack is imminent in the 6th arrondissement. By a cunning strategy they are then able to enter the police station and get to the white building behind it to obtain some of Grothendieck’s writings.

A few weeks later three lengthy articles hit the arXiv, claiming to contain a proof of the Riemann hypothesis, by partially dismantling topos theory.

Bi-annual conferences are organised around the globe aimed at understanding this weird new theory, etc. etc. (you get the general idea).

The papers are believed to have resulted from the Grothendieck heist. But then, similar raids are carried out in Princeton and in Cambridge UK and a sinister plan emerges… “

Funny as it may be to (ab)use a story to comment on the current state of affairs in mathematics, I’m not known to be the world’s most entertaining story teller, so I’d better leave the subject of math-thrillers to others.

Here’s another book-idea:

“The Bourbaki travel guide”

The idea is to hunt down places in Paris and in the rest of France which were important to Bourbaki, from his birth in 1934 until his death in 1968.

This includes institutions (IHP, ENS, …), residences, cafes they frequented, venues of Bourbaki meetings, references in La Tribu notices, etc.

This should lead to some nice Parisian walks (in and around the fifth arrondissement) and a couple of car-journeys through la France profonde.

Of course, also some of the posts I wrote on possible solutions of the riddles contained in Bourbaki’s wedding announcement and the avis de deces will be included.

Here the advantage is that I have already a good part of the raw material. Of course it still has to be followed up by in-situ research, unless I want to turn it into a ‘virtual math traveler ’s guide’ so that anyone can check out the places on G-maps rather than having to go to France.

I’m still undecided about this project. Is there a potential readership for this? Is it worth the effort? Can’t it wait until I retire and will spend even more time in France?

Here’s yet another idea:

“Mr. Yoneda takes the Tokyo-subway”

This is just a working title, others are “the shape of prime numbers”, or “schemes for hipsters”, or “toposes for fashionistas”, or …

This should be a work-out of the sga4hipsters meme. Is it really possible to explain schemes, stacks, Grothendieck topologies and toposes to a ‘general’ public?

At the moment I’m re-reading Eugenia Cheng’s “Cakes, custard and category theory”. As much as I admire her fluent writing style it is difficult for me to believe that someone who didn’t knew the basics of it before would get an adequate understanding of category theory after reading it.

It is often frustrating how few of mathematics there is in most popular maths books. Can’t one do better? Or is it just inherent in the format? Can one write a Cheng-style book replacing the recipes by more mathematics?

The main problem here is to find good ‘real-life’ analogies for standard mathematical concepts such as topological spaces, categories, sheaves etc.

The tentative working title is based on a trial text I wrote trying to explain Yoneda’s lemma by taking a subway-network as an example of a category. I’m thinking along similar lines to explain topological spaces via urban-wide wifi-networks, and so on.

But al this is just the beginning. I’ll consider this a success only if I can get as far as explaining the analogy between prime numbers and knots via etale fundamental groups…

If doable, I have no doubt it will be time well invested. My main problem here is finding an appropriate ‘voice’.

At first I wanted to go along with the hipster-gimmick and even learned some the appropriate lingo (you know, deck, fin, liquid etc.) but I don’t think it will work for me, and besides it would restrict the potential readers.

Then, I thought of writing it as a series of children’s stories. It might be fun to try to explain SGA4 to a (as yet virtual) grandchild. A bit like David Spivak’s short but funny text “Presheaf the cobbler”.

Once again, all suggestions or advice are welcome, either as a comment here or privately over email.

Perhaps, I’ll keep you informed while stumbling along NaNoWriMo.

At least I wrote 1000 words today…

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Where’s Bourbaki’s tomb?

In according to Groth IV.22 we tried to solve one of the riddles contained in Roubaud’s announcement of Bourbaki’s death.

Today, we’ll try our hands on the next one: where was Bourbaki buried?

The death announcement gives this fairly opaque clue:

“The burial will take place in the cemetery for Random Functions (metro stations Markov and Gödel) on Saturday, November 23, 1968 at 3 o’clock in the afternoon.”

What happened on November 23rd 1968?

Bourbaki died on November 11th, 1968 (exactly 50 years after the end of WW1). Perhaps an allusion to the mandatory retirement age for members of Bourbaki, as suggested by the Canulars Bourbaki.

Be that as it may, I believe this date was chosen because it is conveniently close to the intended time of the burial.

But then, what’s so special about November 23rd, 1968?

Well, is there a more suitable moment to burry Bourbaki than during a Seminaire Bourbaki? And, yes, in the fall of 1968 the seminar was organised from saturday 23rd till monday 25th of november:


So, where would all of Bourbaki’s close family be at 3 o’clock on that particular saturday? Right, at l’Institut Henri Poincare.

But, it’s hard to view the IHP as a cemetery. Besides, it’s nowhere close to two metro stations as a quick look on the map shows. The closest one is the RER-station at the Luxembourg gardens, but the RER-line didn’t exist in 1968.

(True Parisians may object that the Gare du Luxembourg was at the time the terminus of the Ligne de Sceaux which has a fascinating history, but let’s try to remain on track…)

If the first clue is the Institut Henri Poincare, then if we are looking for a cemetery, we might ask:

Where’s Poincare’s tomb?

Jules Henri Poincare is burried in the family tomb at the Montparnasse cemetery

He’s not the only mathematician buried there. Évariste Galois, Jean Victor Poncelet, Joseph Liouville, Charles Hermite, and Gaston Darboux also found their last resting place in Montparnasse.

In fact, there are at least 104 mathematicians buried at Montparnasse.

This is hardly surprising as the Montparnasse cemetery is close to the IHP, the Collège de France, the Sorbonne, the “rue d’Ulm” aka the ENS, l’Observatoire and until 1976 l’École polytechnique.

Here’s a map with pointers to some of these tombs:

So, the Montparnasse cemetery appears to be a plausible place to host Bourbaki’s tomb.

But, what about the other “clues”?

“Cemetery of random functions (metro stations Markov and Gödel)”

There are several references lo logic, set theory and applied mathematics in Bourbaki’s death announcement. Why?

Roubaud (and many with him) feel that the Bourbaki enterprise failed miserably in these areas.

He writes on page 49 of his book Mathematics, a novel:

“But Bourbaki, that ‘collective mathematician”, as Raymond Queneau put it, also had a good knowledge of the current state of mathematics at the time when his Treatise was being composed; with, of course, a few “gaps”:

for example, probability, which was considered to be just an “applied” brand of measure theory”; and logic, especially logic, which was made almost a pariah because of (so it was rumored) the premature death of Herbrand, who, in the generation of founders, Normaliens to a man, had studied under Hilbert, and thus had been associated with his meteoric rise; in sum, logic had died in a climbing accident along with Herbrand.”

This might explain the cemetery of “random functions” and the metro stations named after the logicians and set theorists Kurt Gödel and A.A. Markov or the father of stochastic processes Andrey Markov.

Is there more into these references?

Probably not, but just to continue with our silly game, the two metro stations closest to the Montparnasse cemetery are Raspail and Edgar Quinet.

Now, François-Vincent Raspail was a French chemist, naturalist, physician, physiologist, attorney, and socialist politician.

More relevant to our quest is that the Centre d’analyse et de mathématique sociales (CAMS) was based at 54, boulevard Raspail. The mission statement on their website tells that this institute is clearly devoted to all applications of mathematics. That is, “Raspail” may be another pointer to applied mathematics and random functions.

As for the other metro station, Edgar Quinet was a French historian and intellectual. Is there a connection to logic or set theory? Well, sort of. The Encyclopedia Britannica has this to say about Edgar Quinet:

“His rhetorical power was altogether superior to his logical power, and the natural consequence is that his work is full of contradictions.”

I rest my case.

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from chocolate bars to constructivism

A fun way to teach first year students the different methods of proof is to play a game with chocolate bars, Chomp.

The players take turns to choose one chocolate block and “eat it”, together with all other blocks that are below it and to its right. There is a catch: the top left block contains poison, so the first player forced to eat it dies, that is, looses the game.

chocolate

Let’s prove some results about Chomp illustrating the strengths and weaknesses of different methods of proof.

[section_title text=”Direct proof”]

By far the most satisfying method is the ‘direct proof’. Once you understand it, the truth of the statement is unescapable.

Theorem 1: The first player to move has a winning strategy for a $2 \times n$ chocolate bar.

Here’s the winning move: you take the lower right block!

What can the other player do? She can bring the bar again in rectangular shape of strictly smaller size (in which case you eat again the lower right block), or she can eat more blocks from the lower row (but then, after her move, you eat as many blocks from the top row bringing the bar again in a shape in which the top row has exactly one more block than the lower one. You’re bound to win!

Sadly, it is not always possible to prove what you want in such a direct way. That’s why we invented another tactic:

[section_title text=”Proof by induction”]

A proof by induction is less satisfactory because it involves a lot more work, but still everyone considers it as a ‘fair’ method: it shows how to arrive at the solution, if only you had enough time and energy…

Theorem 2: Given any Chomp position, either the first player to move has a winning strategy or the second player has one.

A proof by induction relies on simplifying the situation at hand. Here, each move reduces the number of blocks in the chocolate bar, so we can apply induction on the number of blocks.

If there is just one block, it must contain poison and so the first player has to eat it and looses the game. That is, here the second player has a winning strategy and we indicate this by labelling the position with $0$. This one block position is the ‘basis’ for our induction.

Now, take a position $P$ for which you want to prove the claim. Look at all possible positions $X$ you get after one move. All these positions have strictly less blocks, so ‘by induction’ we may assume that the result is true for each one of them. So we can label each of these positions with $0$ if it is a second player win, or with $\ast$ if it is a first player win.

How to label $P$? Well, if all positions in $X$ are labeled $\ast$ we label $P$ with $0$ because it is a second player win. The first player has to move to a starred position (which is a first player win) and so the second player has a winning strategy from it (by moving to a $0$-position). If there is at least one position in $X$ labeled $0$ then we label $P$ with $\ast$. Indeed, the first player can move to the $0$-position and then has a winning strategy, playing second.

Here’s the chart of the first batch of positions:

But then, it may take your lifetime to work through the strategy for a complicated starting position…

What to do if neither a direct proof is found, nor a reduction argument allowing a proof by induction? Virtually all professional mathematicians have no objection whatsoever to resort to a very drastic tactic: proof by contradiction.

[section_title text=”Proof by contradiction”]

The law of the excluded third (or ‘tertium non datur’) says that for any proposition $P$ either $P$ or not $P$ is true. In 1927 David Hilbert stated that not much of mathematics would remain without the use of this rule:

[quote name=”David Hilbert”]To prohibit existence statements and the principle of excluded middle is tantamount to relinquishing the science of mathematics altogether.[/quote]

Theorem 3: The first player has a winning strategy for a rectangular starting position.

We will prove this by contradiction. That is, we will assume that it is false, that is, the first player has not a winning strategy, and derive a contradiction from it. Therefore $\neg \neg P$ is true which is equivalent to $P$ by the law of the excluded
middle.

If the first player does not have a winning strategy for that rectangular chocolate bar, the second player must have one by Theorem2. That is, for every possible first move the second player has a winning response.

Assume the first player eats the lower right block.

The second player must have some winning response to this. Whatever her move, the resulting position could have been obtained by the first player already after the first move.

So, the second player cannot have a winning strategy.

This strategy stealing argument is a cheap cheat. We have not the slightest idea of what the winning strategy for the first player is or how to find it.

Still, one shouldn’t stress this fact too much to first year students. They’ll have to work through plenty of proofs by contradiction in the years ahead…

[section_title text=”Accepting constructive mathematics?”]

Sooner or later in their career they will hear arguments in favour of ‘constructive mathematics’, which does not accept the law of the excluded third.

Andrej Bauer has described was happens next. They will go through the five stages people need to come to terms with life’s traumatising events: denial, anger, bargaining, depression, and finally acceptance.

His paper is just out in the Bulletin of the AMS Five stages of accepting constructive mathematics.

This paper is an extended version of a talk he gave at the IAS.

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Gitte exhibits in Ghent

Gitte Le Bruyn, the artist previously known as PD1 on this blog, exhibits some of her work in Ghent.

The exhibition VGC Visual Art Gitte Le Bruyn is hosted by the Van Crombrugghe’s Genootschap, Huidevetterskaai 39, 9000 Gent, Belgium.

Gitte shows hundreds of stills from her animation projects. She has a laborious working method: for every motion she needs about 30 aquarel paintings which she then scans and turns into an animation movie.

Here’s the result of all that work: Space Communication

Previously she made oil paintings on a glass plate and turned photographs of them into a movie. Here’s her first animation project: the clip for Silver Junkie’s Maria.

She was interviewed about the production process on national (flemish) television (in dutch): Canvas Cobra TV

If you’re in Ghent this or the coming week-end you can visit the exhibition (free access) from 14.00 till 20.30 on saturday and sunday.

If you’re interested in her work, please visit her website gittte.be.

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Hasse = “le P. Adique, de l’Ordre des Diophantiens”

The Bourbaki wedding invitation is probably the most effective branding- and marketing-campaign in the history of mathematics.

It contains this, seemingly opaque, paragraph:

The trivial isomorphism will be given to them by P. Adic, of the Diophantine Order, at the Principal Cohomology of the Universal Variety, the 3 Cartember, year VI, at the usual hour.

It was pretty easy to decode the date of the wedding “3 Cartember, year VI” to be June 3rd, 1939, and (a bit more difficult) the wedding place “the Principal Cohomology of the Universal Variety” as the l’église royale Notre-Dame du Val-de-Grâce in Paris.

The identity of the celebrating priest “P. Adic, of the Diophantine Order” remained unclear. The most likely suspect was Helmut Hasse, but I couldn’t place him in Paris on June 3rd, 1939.



Hasse is the central figure in the picture above, taken in Oberwolfach in 1952, before one of his cars. Here’s another picture of car-freak Hasse (trains were to Andre Weil what cars were to Helmut Hasse). Both pictures are from the MFO photo collection.

Thanks to Peter Roquette’s publishing of Helmut Hasse’s letters we can now prove that Hasse was not in Paris on that particular day (however, he was there a couple of days earlier) but Weil had every reason to believe he might be there at the time he wrote the wedding invitation.

When was the wedding invitation written?

Frank Smithies recalls the spring 1939 period in Cambridge as follows :

“The climax of the academic year, as far as we were concerned, came in the Easter term. André Weil, Claude Chabauty, and Louis Bouckaert (from Louvain) were all in Cambridge, and the proposal was mooted that a marriage should be arranged between Bourbaki’s daughter Betti and Hector Pétard; the marriage announcement was duly printed in the canonical French style – on it Pétard was described as the ward of Ersatz Stanislas Pondiczery – and it was circulated to the friends of both parties. A couple of weeks later the Weils, Louis Bouckaert, Max Krook (a South African astrophysicist), Ralph and myself made a river excursion to Grantchester by punt and canoe to have tea at the Red Lion; there is a photograph of Ralph and myself, with our triumphantly captured lion between us and André Weil looking benevolently on.”

We know that this picture is taken on May 13th 1939 so the wedding-invitation was drawn up around mid april 1939.

“What did Weil know about Hasse’s visit to Paris?”

Hasse had been invited by Julia to give a series of lectures at the Institut Henri Poincare in 1938, but Hasse postponed his trip to Paris until May 1939.

In his letter to Hasse of January 20th 1939, Andre Weil writes:

“It is quite unfortunate that you couldn’t accept your invitation to Paris before this year, because last year all our number-theorists would have been present. By a sad coincidence all of us will be on travel this coming May (except for Chevalley perhaps who might have returned from the US by then). Pisot will be in Gottingen, Chabauty in Manchester visiting Mordell and I will be in Cambridge as I obtained a travel grant for England and Scandinavia.”

Clearly, Weil was aware of the upcoming visit of Hasse to Paris at the end of May, and there was no reason for him to assume that he wouldn’t be able to stay a weekend longer.

What do we know of Hasse’s visit to Paris?

Because Julia was exhausted and was on a three months sick leave, Elie Cartan took over the job of organising Hasse’s lecture series. In a letter of April 25th 1939 he proposes some possible dates, to which Hasse replies on April 30th 1939:

In it he fixes for the first time the dates of his talks which will be on “New results in the arithmetic of algebraic function fields” and consist of three lectures:

– On Friday 19th 1939: “Generalities: the group of divisor classes and the multiplier ring”

– On Saturday 20th 1939: “Rational and integral points on algebraic curves over the integers”

– On Tuesday 23rd 1939: “Rational points on algebraic curves with coefficient mod p”

He also mentions that he would stay for 15 days in Paris, arriving on May 17th, in time for the Jubilee Conference for Elie Cartan, scheduled on May 18th.

Weil must have known that Hasse would be present at the Cartan-fest and give a series of lectures in the following weeks. He had every reason to believe that Hasse would still be in Paris on Saturday June 3rd.

Where was Hasse on June 3rd 1939?

Back at home, as on that very day he wrote a letter to Henri Cartan, thanking him for an enjoyable day’s stay in Strasbourg, on the way back from Paris, on June 1st 1939:

If you want to catch up with previous posts on the Bourbaki wedding, you might want to download the booklet The Bourbaki Code.

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Toposes alive and kicking at IHES

After 50 years, vivid interest in topos theory seems to have returned to one of the most prestigious research institutes, the IHES. Last november, there was the meeting Topos a l’IHES.

At the meeting, Celine Loozen filmed a documentary which is supposed to have as its title “Unifying Worlds”. Its very classy trailer is now on YouTube (via +David Roberts).

How did topos theory, a topic considered by most to be far too abstract to be useful to main stream mathematics, suddenly return in such force?

It always helps when a couple of world-class mathematicians become interest in the topic, for their own particular reasons. Clearly, the topic gathers considerable momentum if these people are all permanent members of the IHES.

A lot of geometric information is contained in the category of all sheaves on the geometric object. Topos theory offers a way to construct ‘geometries’ out of nothing, that is, out of arbitrary categories.

Take your favourite category $\mathbf{C}$, then “presheaves” on $\mathbf{C}$ are defined to be contravariant functors $\mathbf{C} \rightarrow \mathbf{Sets}$. For any Grothendieck topology on $\mathbf{C}$ one can then restrict to the sub-category of “sheaves” for this topology, and that’s your typical topos.

Alain Connes got interested in topos theory because he observed that even for the most trivial of categories, such as the monoid category with just one object and endomorphisms the multiplicative semigroup $\mathbb{N}_{\geq 1}^{\times}$, and taking the coarsest of all Grothendieck topologies, one gets interesting objects of baffling complexity.

One of the ‘invariants’ one can associate to a topos is its collection of “points”. Together with Katia Consani, Connes computed in Geometry of the Arithmetic Site that the collection of points of this simple presheaf topos is exactly the set of adele classes $\mathbb{Q}^{\ast}_+ \backslash \mathbb{A}^f_{\mathbb{Q}} / \hat{\mathbb{Z}}^{\ast}$.

Here’s what Connes himself said about this revelation (followed by an attempted translation):

——————————————————

(50.36)

And,in this example, we saw the wonderful notion of a topos, developed by Grothendieck.

It was sufficient for me to open SGA4, a book written at the beginning of the 60ties or the late fifties.

It was sufficient for me to open SGA4 to see that all the things that I needed were there, say, how to construct a cohomology on this site, how to develop things, how to see that the category of sheaves of Abelian groups is an Abelian category, having sufficient injective objects, and so on … all those things were there.

This is really remarkable, because what does it mean?

It means that the average mathematician says: “topos = a generalised topological space and I will never need to use such things. Well, there is the etale cohomology and I can use it to make sense of simply connected spaces and, bon, there’s the chrystaline cohomology, which is already a bit more complicated, but I will never need it, so I can safely ignore it.”

And (s)he puts the notion of a topos in a certain category of things which are generalisations of things, developed only to be generalisations…

But in fact, reality is completely different!

In our work with Katia Consani we saw not only that there is this epicyclic topos, but in fact, this epicyclic topos lies over a site, which we call the arithmetic site, which itself is of a delirious simplicity.

It relies only on the natural numbers, viewed multiplicatively.

That is, one takes a small category consisting of just one object, having this monoid as its endomorphisms, and one considers the corresponding topos.

This appears well … infantile, but nevertheless, this object conceils many wonderful things.

And we would have never discovered those things, if we hadn’t had the general notion of what a topos is, of what a point of a topos is, in terms of flat functors, etc. etc.

(52.27)

——————————————————-

Pierre Cartier has a very wide interest in mathematical theories, the wilder the better: Witt rings, motifs, cosmic Galois groups, toposes…

He must have been one of the first people to speak about toposes at the Bourbaki seminar. In february 1978 he gave the talk Logique, categories et faisceaux, d’apres F. Lawvere et M. Tierney (and dedicated to Grothendieck’s 50th birthday).

He also gave the opening lecture of the Topos a l’IHES conference.

In this fragment of an interview with Stephane Dugowson and Anatole Khelif in 2014 he plays down his own role in the development of topos theory, compared to his contributions in other fields, such as motifs.

——————————————————-

(46:24)

Well, I didn’t invest much time in topos theory.

Except, I once gave a talk at the Bourbaki seminar on the use of topos theory in logic, such as the independence of the axiom of choice, that is, on the idea of forcing.

But, it was just this talk, I didn’t do anything original in it.

Then there is nonstandard analysis, where one can formulate certain things in terms of topos theory. When I got interested in nonstandard analysis, I had this possible application of topos theory in mind.

At the moment when you have a nonstandard model of the integers or more generally of set theory, then one has two models of set theory, that is two different toposes, and then one obviously tries to compare them.

In that sense, I was completely aware of the fact that everything I was doing could be expressed in the language of toposes,or at least in the philosophy of toposes.

I haven’t made any important contributions in that theory, for me it merely remained a tool.

(47:49)

——————————————————-

Laurent Lafforgue says he spend hundredths and hundredths of hours talking to Olivia Caramello about topos theory.

She must have been quite convincing. The last couple of years Lafforgue is a fierce advocate of Caramello’s work.

Her basic idea is that the same topos can arise from two very different mathematical settings (that is, two different categories with Grothendieck topologies can have equivalent categories of sheaves).

The hope then is to translate results from one theory to the other, or as she expresses it, toposes can be used as “bridges” between different mathematical topics.

At the moment though, is seems a bit far fetched for this idea to be relevant to the Langlands programme.

Caramello and Lafforgue have just a paper out: Sur la dualit´e des topos et de leurs pr´esentations et ses applications : une introduction.

The paper is based on a lecture Lafforgue gave in April in Nantes. Here’s the video:

In the introduction they write:

“It is our conviction that the theory of toposes and their representations, with its essential and structural ambiguity, is destined to have an impact on mathematics comparable to the impact group theory has had from the moment, some decades after its discovery by Galois, the mathematical community began to understand it.”

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The group algebra of all algebraic numbers

Some weeks ago, Robert Kucharczyk and Peter Scholze found a topological realisation of the ‘hopeless’ part of the absolute Galois group $\mathbf{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. That is, they constructed a compact connected space $M_{cyc}$ such that etale covers of it correspond to Galois extensions of the cyclotomic field $\mathbb{Q}_{cyc}$. This gives, at least in theory, a handle on the hopeless part of the Galois group $\mathbf{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}_{cyc})$, see the previous post in this series.

Here, we will get halfway into constructing $M_{cyc}$. We will try to understand the topology of the prime ideal spectrum $\mathbf{Spec}(\mathbb{C}[\overline{\mathbb{Q}}^{\times}])$ of the complex group algebra of the multiplicative group $\overline{\mathbb{Q}}^{\times}$ of all non-zero algebraic numbers.

[section_title text=”Pontryagin duals”]

Take an Abelian locally compact group $A$ (for example, an Abelian group equipped with the discrete topology), then its Pontryagin dual $A^{\vee}$ is the space of all continuous group morphisms $A \rightarrow \mathbb{S}^1$ to the unit circle $\mathbb{S}^1$ endowed with the compact open topology.

There are these topological properties of the locally compact group $A^{\vee}$:

– $A^{\vee}$ is compact if and only if $A$ has the discrete topology,

– $A^{\vee}$ is connected if and only if $A$ is a torsion free group,

– $A^{\vee}$ is totally disconnected if and only if $A$ is a torsion group.

If we take the additive group of rational numbers with the discrete topology, the dual space $\mathbb{Q}^{\vee}$ is the one-dimensional solenoid

It is a compact and connected group, but is not path connected. In fact, it path connected components can be identified with the finite adele classes $\mathbb{A}_f/\mathbb{Q} = \widehat{\mathbb{Z}}/\mathbb{Z}$ where $\widehat{\mathbb{Z}}$ is the ring of profinite integers.

Keith Conrad has an excellent readable paper on this fascinating object: The character group of $\mathbb{Q}$. Or you might have a look at this post.

[section_title text=”The multiplicative group of algebraic numbers”]

A torsion element $x$ in the multiplicative group $\overline{\mathbb{Q}}^{\times}$ of all algebraic numbers must satisfy $x^N=1$ for some $N$ so is a root of unity, so we have the exact sequence of Abelian groups

$0 \rightarrow \pmb{\mu}_{\infty} \rightarrow \overline{\mathbb{Q}}^{\times} \rightarrow \overline{\mathbb{Q}}^{\times}_{tf} \rightarrow 0$

where the last term is the maximal torsion-free quotient of $\overline{\mathbb{Q}}^{\times}$. By Pontryagin duality this gives us an exact sequence of compact topological groups

$0 \rightarrow (\overline{\mathbb{Q}}^{\times}_{tf})^{\vee} \rightarrow (\overline{\mathbb{Q}}^{\times})^{\vee} \rightarrow \pmb{\mu}^{\vee}_{\infty} \rightarrow 0$

Here, the left-most space is connected and $\pmb{\mu}^{\vee}_{\infty}$ is totally disconnected. That is, the connected components of $(\overline{\mathbb{Q}}^{\times})^{\vee}$ are precisely the translates of the connected subgroup $(\overline{\mathbb{Q}}^{\times}_{tf})^{\vee}$.

[section_title text=”Prime ideal spectra”]

The short exact sequence of Abelian groups gives a short exact sequence of the corresponding group schemes

$0 \rightarrow \mathbf{Spec}(\mathbb{C}[\overline{\mathbb{Q}}^{\times}_{tf}]) \rightarrow \mathbf{Spec}(\mathbb{C}[\overline{\mathbb{Q}}^{\times}] \rightarrow \mathbf{Spec}(\mathbb{C}[\pmb{\mu}_{\infty}]) \rightarrow 0$

The torsion free abelian group $\overline{\mathbb{Q}}^{\times}_{tf}$ is the direct limit $\underset{\rightarrow}{lim}~M_i$ of finitely generated abelian groups $M_i$ and as the corresponding group algebra $\mathbb{C}[M_i] = \mathbb{C}[x_1,x_1^{-1},\cdots, x_k,x_k^{-1}]$, we have that $\mathbf{Spec}(\mathbb{C}[M_i])$ is connected. But then this also holds for

$\mathbf{Spec}(\mathbb{C}[\overline{\mathbb{Q}}^{\times}_{tf}]) = \underset{\leftarrow}{lim}~\mathbf{Spec}(\mathbb{C}[M_i])$

The underlying group of $\mathbb{C}$-points of $\mathbf{Spec}(\mathbb{C}[\pmb{\mu}_{\infty}])$ is $\pmb{\mu}_{\infty}^{\vee}$ and is therefore totally disconnected. But then we have

$\pi_0(\mathbf{Spec}(\mathbb{C}[\overline{\mathbb{Q}}^{\times}]) \simeq \pi_0(\mathbf{Spec}(\mathbb{C}[\pmb{\mu}_{\infty}]) \simeq \pmb{\mu}_{\infty}^{\vee}$

and, more importantly, for the etale fundamental group

$\pi_1^{et}(\mathbf{Spec}(\mathbb{C}[\overline{\mathbb{Q}}^{\times}],x) \simeq \pi_1^{et}(\mathbf{Spec}(\mathbb{C}[\overline{\mathbb{Q}}^{\times}_{tf}],y)$

So, we have to compute the latter one. Again, write the torsion-free quotient as a direct limit of finitely generated torsion-free Abelian groups and recall that connected etale covers of $\mathbf{Spec}(\mathbb{C}[M_i])=\mathbf{Spec}(\mathbb{C}[x_1,x_1^{-1},\cdots,x_k,x_k^{-1}])$ are all of the form $\mathbf{Spec}(\mathbb{C}[N])$, where $N$ is a subgroup of $M_i \otimes \mathbb{Q}$ that contains $M_i$ with finite index (that is, adjoining roots of the $x_i$).

Again, this goes through the limit and so a connected etale cover of $\mathbf{Spec}(\mathbb{C}[\overline{\mathbb{Q}}^{\times}_{tf}])$ would be determined by a subgroup of the $\mathbb{Q}$-vectorspace $\overline{\mathbb{Q}}^{\times}_{tf} \otimes \mathbb{Q}$ containing $\overline{\mathbb{Q}}^{\times}_{tf}$ with finite index.

But, $\overline{\mathbb{Q}}^{\times}_{tf}$ is already a $\mathbb{Q}$-vectorspace as we can take arbitrary roots in it (remember we’re using the multiplicative structure). That is, $\mathbf{Spec}(\mathbb{C}[\overline{\mathbb{Q}}^{\times}])$ is simply connected!

[section_title text=”Bringing in the Galois group”]

Now, we’re closing in on the mysterious space $M_{cyc}$. Clearly, it cannot be the complex points of $\mathbf{Spec}(\mathbb{C}[\overline{\mathbb{Q}}^{\times}])$ as this has no proper etale covers, but we still have to bring the Galois group $\mathbf{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}_{cyc})$ into the game.

The group algebra $\mathbb{C}[\overline{\mathbb{Q}}^{\times}]$ is a commutative and cocommutative Hopf algebra, and all the elements of the Galois group act on it as Hopf-automorphisms, so it is natural to consider the fixed Hopf algebra

$H_{cyc}=\mathbb{C}[\overline{\mathbb{Q}}^{\times}]^{\mathbf{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}_{cyc})}$

This Hopf algebra has an interesting alternative description as a subalgebra of the Witt ring $W(\mathbb{Q}_{cyc})$, bringing it into the realm of $\mathbb{F}_1$-geometry.

This ring of Witt vectors has as its underlying set of elements $1 + \mathbb{Q}_{cyc}[[t]]$ of formal power series in $\mathbb{Q}_{cyc}[[t]]$. Addition on this set is defined by multiplication of power series. The surprising fact is that we can then put a ring structure on it by demanding that the product $\odot$ should obey the rule that for all $a,b \in \mathbb{Q}_{cyc}$ we have

$(1-at) \odot (1-bt) = 1 – ab t$

In this mind-boggling ring the Hopf algebra $H_{cyc}$ is the subring consisting of all power series having a rational expression of the form

$\frac{1+a_1t+a_2t^2+ \cdots + a_n t^n}{1+b_1 t + b_2 t^2 + \cdots + b_m t^m}$

with all $a_i,b_j \in \mathbb{Q}_{cyc}$.

We can embed $\pmb{\mu}_{\infty}$ by sending a root of unity $\zeta$ to $1 – \zeta t$, and then the desired space $M_{cyc}$ will be close to

$\mathbf{Spec}(H_{cyc} \otimes_{\mathbb{Z}[\pmb{\mu}_{\infty}]} \mathbb{C})$

but I’ll spare the details for another time.

In case you want to know more about the title-picture, quoting from John Baez’ post The Beauty of Roots:

“Sam Derbyshire decided to to make a high resolution plot of some roots of polynomials. After some experimentation, he decided that his favorite were polynomials whose coefficients were all 1 or -1 (not 0). He made a high-resolution plot by computing all the roots of all polynomials of this sort having degree ≤ 24. That’s $2^{24}$ polynomials, and about $24 \times 2^{24}$ roots — or about 400 million roots! It took Mathematica 4 days to generate the coordinates of the roots, producing about 5 gigabytes of data.”

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according to Groth. IV.22

At the Bourbaki Seminar in November 1968 the participants were handed the following (premature) announcement of Bourbaki’s death.



The French text can be found at the Canulars Bourbaki, and the English translation below is from Maurice Mashaal’s book Bourbaki, a secret society of mathematicians, page 115.

I’ve underlined a couple of riddles in the text.

———-

The Cantor, Hilbert, and Noether families;
The Cartan, Chevalley, Dieudonne, and Weil families;
The Bruhat, Dixmier, Godement, Samuel, and Schwartz families;
The Cartier, Grothendieck, Malgrange, and Serre families;
The Demazure, Douady, Giraud, and Verdier families;
The Right-Filtering and Strict-Epimorphism families;
Mesdemoiselles Adele and Idele;

regret to announce the death of Monsieur

NICOLAS BOURBAKI

Respectively their father, brother, son, grandson, great-grandson, and grand-cousin.

He died piously in his home on November 11, 1968 (on the anniversary of great victory) in his home in Nancago.

The burial will take place in the cemetery for Random Functions (metro stations Markov and Goedel) on Saturday, November 23, 1968 at 3 o’clock in the afternoon.

A reception will be held at the bar The Direct Products, at the crossroads of the Projective Resolutions (formerly Koszul square).

Following the wish of the departed, His Eminence the Cardinal Aleph I will hold a mass in Our Lady of Universal Problems in the presence of representatives from all equivalence classes and from all (algebraically closed) fields. The students from l’Ecole Normale Superieure and the Chern classes will observe a minute of silence.

No flowers or wreath products.

For God is the Alexandrov compactification of the universe.” Groth. IV.22

———-

This announcement is clearly inspired by the faire-part of Betti Bourbaki’s wedding (with Hector Petard), written by Andre Weil and Claude Chabauty in the spring of 1939.

Some years ago I wrote a couple of posts on possible solutions of the riddles contained in that faire-part, a pdf-version can be downloaded as the Bourbaki code. (Note to self: repost some of those and add new material!)

Whereas the wedding announcement was concocted by members of Bourbaki, this is not the case for this death announcement. It was written by the mathematician and writer Jacques Roubaud, a member of the literary group OuLiPo.

In 1997 he wrote the novel ‘Mathematique’ (now available in English translation). In it, he recalls his mathematical years, from his first lecture at the IHP in 1952 till the 70ties. It contains an insiders view on Parisian mathematics in the 50ties and 60ties, dominated largely by Bourbaki, and offers clues to decrypt some of the riddles in the death announcement.

Today, we’ll consider the final one

For God is the Alexandrov compactification of the universe.Groth. IV.22

Can we make sense of the ‘reference’ Groth. IV.22?

Does it refer to EGA IV?

Roubaud’s motif (pardon the expression) for writing the announcement of Bourbaki’s death in 1968 can be read between the lines in his book Mathematics, a novel from which all quotes below are taken.

page 146: “I was invited by Raymond Queneau to join the Oulipo and I met FLL in the fall of 1966. By then, I had reached the end of my passion for Bourbaki, after being one of their most faithful and credulous readers for many years.”

page 73: “The “biography” of that many-headed beast, Bourbaki, is still to be written. It would be a fascinating but arduous task. Here, I shall say only what is strictly necessary to my own entreprise. Having reached his dotage after 1968, “he” is for all intents and purposes now dead.”

By 1968, Bourbaki had become an institution dominating French mathematics and so had to die after the May 1968 revolt.

But, Roubaud had found a new prophet to follow…

page 284: “It was a book of mathematics. It had just been published. It was in a large format, with a blue cover. Its title was Elements of Algebraic Geometry (affectionately and familiarly abbreviated, in French, to EGA). Its author: Grothendieck.

page 285: “For I had so immersed mself in Bourbakism that such a text, the fruit of its final flowering, the monumental work of he who could be considered as Dr. Frankenstein-Bourbaki’s Monster, and which had been drafted according to the group’s inimitable stylistic norms, here applied, in its prose, in a heightened, frenetic way, ran through my mind like honey, no, like nectar, an intellectual ambrosia. Just thinking about it now fills me with stupefaction. I was someone who managed to read EGA with pleasure – worse, with delight. For any normal mathematician today, such an affirmation would seem as perverse as adoring an American soft drink.”

Roubaud was reading EGAs like others would read Nicki French thrillers, one per year:

(1960) : “Éléments de géométrie algébrique: I. Le langage des schémas”

(1961) : “Éléments de géométrie algébrique: II. Étude globale élémentaire de quelques classes de morphismes”

(1961) : “Eléments de géométrie algébrique: III. Étude cohomologique des faisceaux cohérents, Première partie”

(1963) : “Éléments de géométrie algébrique: III. Étude cohomologique des faisceaux cohérents, Seconde partie”

(1964) : “Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Première partie”

(1965) : “Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Seconde partie”

(1966) : “Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Troisième partie”

(1967) : “Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie”

It was now november 1968, and Roubaud was hoping that another sequel would be published soon. As the last one ended with section IV.21, this new volume would start with IV.22, and, no doubt, contain more divine mathematics…

However, Pieter Belmans objected that it was planned from the outset for EGA4 to consist of 21 chapters, and no more. Surely, Roubaud knew about this…

ADDED october 4th: Pieter has done some further digging on this issue in his post According to Groth IV.22.

Can it refer to SGA IV?

Luckily, there is another option. Grothendieck ran the Séminaire de Géométrie Algébrique du Bois Marie at the IHES from 1962 to 1969.

SGA4 was about “Théorie des topos et cohomologie étale des schémas” (Topos theory and étale cohomology) and ran in 1963–1964. A decade later the notes were published in Springer’s Lecture Notes in Mathematics 269, 270 and 305, 1972/3.

The topic of SGA4 (topos theory) is clearly closer in spirit to the fake biblical quote on the topological nature of God than that of EGA4 which was about the local structure of schemes and their morphisms.

The original notes were published in fascicles by the IHÉS, most of which went through two or three revisions, and were published as the seminar proceeded. So, Roubaud had access to them in the later 60ties.

The original versions, as well as their re-published LaTeX versions can be found here.

Again, we face the problem that there are not enough chapters, only 19 in this case.

Fortunately, we can search the LaTeX-ed version for references to the Alexandroff compactification, and there is just a single one:

This is in the first lecture on Presheaves by Grothendieck and Verdier. More precisely, it is in section 2 (Univers et espèces de structures) of the Appendix, which is labeled

II. Appendice : Univers (by N. Bourbaki (*))

So, the paragraph on the Alexandroff compactification is in SGA IV,II.2, or, if we read 22 as II.2 this might explain Groth. IV.22.

We have found a reference in SGA IV including “Bourbaki”, “the univers” and “Alexandroff compactification”.

But then, who dreamed up this topological definition of God?

Jean-Paul Benzecri

Dieu est le compactifié d’Alexandrof de l’univers.Jean-Paul Benzecri

Jean-Paul Benzécri is a French statistician who has been professor at Université Pierre-et-Marie-Curie in Paris. In the 60ties he was a professor at the university of Rennes where he was a colleague of Roubaud.

Jacques Roubaud has another book on his reminiscences as a mathematician, Impératif catégorique. Unfortunately, this book is not (yet) translated into English.

In section 80, La déesse Fortune ne se montra pas envers moi avare de ses bienfaits, he tells about his years at the University of Rennes where also his friend and topos-theorist Jean Bénabou was at the time. Bénabou and Benzécri knew each other from their student days at the Ecole Normale.

Benzécri had a very strict catholic family background, and in the 50ties he attended the Centre Richelieu des étudiants catholiques.

.

He liked to explain his axiom as follows:

“Of course, God created the univers. But, he created it locally compact and not compact. That it, left on its own, the universe would suffer a serious structural defect which could only be repaired by introducing a point at infinity, which marks the presence of the divine.”

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Did Nöbeling discover toposes?

Chasing one story, one sometimes tumbles into a different one. For some time I’m trying to debunk the story that Wolfgang Krull was close to inventing the notion of schemes in the early 1930’s.

I guess my first encounter with it was through The Rising Sea: Grothendieck
on simplicity and generality I
by Colin McLarty which contains:

“From Emmy Noether’s viewpoint, then, it was natural to look at prime ideals instead of classical and generic points as we would more likely say today, to identify points with prime ideals. Her associate Wolfgang Krull did this. He gave a lecture in Paris before the Second World War on algebraic geometry taking all prime ideals as points, and using a Zariski topology. He did this over any ring, not only polynomial rings like $\mathbb{C}[x, y]$. The generality was obvious from the Noether viewpoint, since all the properties needed for the definition are common to all rings. The expert audience laughed at him and he abandoned the idea.”

The story seems to be due to Jurgen Neukirch’s ‘Erinnerungen an Wolfgang Krull’ published in ‘Wolfgang Krull : Gesammelte Abhandlungen’ (P. Ribenboim, editor).

This rumour is quickly ruled out as Parisian pre-war mathematical life only involves the Hadamard- and Julia-seminars and they are very well documented.

A more thorough investigation was carried out by Theo Raedschelders who contacted Karl-Otto Stöhr (a former student of Krull) and this is what he had to say about it:

“I remember that Prof. Krull once told to me, that in the early thirties he proposed in a talk that in algebraic geometry a larger number of points should be taken in consideration, namely points corresponding to the prime ideals of commutative rings. I always thought that this talk did happen at some place in Germany. He further mentioned that the mathematician Nöbeling in the audience argued that this idea would not be of any help to understand italian algebraic geometry.

I had never heard of Nöbeling, so here’s where this story takes a turn…

[section_title text=”The Vienna Mathematical Seminar”]

Wien 1938 und der Exodus der Mathematik is a fascinating account of Vienna mathematical life in the years leading up to WW2.

Karl menger was a central figure in the Vienna Mathematical Institute and founded its Mathematical Seminar. He gathered around him a brilliant group of young mathematicians including Kurt Gödel, Abraham Wald, Franz Alt and Olga Taussky.



Merger made important contributions to topology, including the “Menger sponge” and mathematical logic.

He seems to have been the first person to raise the idea of a point-free definition of the concept of topological space (aka ‘pointless topology’). In his 1928 book Dimensionstheorie, he defined the concept of space without referring to the points of an underlying set, but rather using pieces or, as he liked to say, “lumps”.

Georg Nöbeling was one of the first students and closest collaborators of Menger, finishing his Ph.D. in 1931 on a generalisation of Menger’s embedding problem.



In 1933 he moved to Erlangen, where Krull was a professor at the time. No doubt they discussed Krull’s invention of what we now know as the Zariski topology and Nöbeling may have said he didn’t believe it to be of any use in studying Italian geometry.

In Peter Johnstone’s historical account of the pre-history of topos theory The point of pointless topology there is no mention of Menger’s work. To him, the idea that points are secondary in a topological space required the prior development of lattice theory, which was developed in the mid 30-ties by Stone.

Stone’s lattice-theoretical approach to general topology found its final presentation in Georg Nöbeling’s 1954 book “Grundlagen der analytischen Topologie”. In fact, Nöbeling’s book could be seen as marking the end of the lattice-theoretical phase of pointless topology. A couple of years later locales and toposes where introduced.

So, did Nöbeling invent topos theory as some say Krull invented scheme theory? No, of course not, they both lacked the crucial ingredient of sheaf theory.

Still, it is fair to say that the Zariski topology was probably discovered by Krull in the early 30-ties and that Menger introduced ‘pointless topology’ in the late 20-ties, years ahead of the lattice-theoretic approach.

If you want to read more on this, please consult the paper by Mathieu Bélanger and Jean-Pierre MarquisMenger and Nöbeling on pointless topology.

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