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