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Yet more topos news

Every topos has its own internal language, the so called Mitchell-Bénabou language, allowing us to speak about formulas and their truth values.

Sadly, Jean Bénabou died last week.

Here’s a nice interview with Bénabou (in French) on category theory, Grothendieck, logic, and a rant on plagiarism among topos theorists (starting at 1:00:16).

Yesterday, France Culture’s ‘La methode scientifique’ hosted Alain Connes, Laurent Lafforgue and Olivia Caramello in a special programme Grothendieck: la moisson (Grothendieck, the harvest), dedicated to the recent publication of ‘Récoltes et Semailles’.

An interesting item is ‘le reportage du jour’ by Céline Loozen in which she manages to have a look at the 60.000 pages of Grothendieck’s Lasserre notes, stocked in the cellars of the Librairie Alain Brieux, and talks to Jean-Bernard Gillot who is commissioned by Grothendieck’s son to appraise the work (starts at 36:40).

Perhaps the publication of ‘Récoltes et Semailles’ is part of a deal with the family to make these notes available, at last.

Towards the end of the programme Connes, Caramello and Lafforgue lament that topos theory is still not taken seriously by the mathematical community at large, whereas it is welcomed warmly by the engineers of Huawei.

In more topos news, I learn from the blog of Olivia Caramello, that Laurent Lafforgue is going to give an online course on toposes as ‘bridges’ at the University of Warwick, the first talk starts today at 14hrs London time.

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The hype cycle of an idea

These three ideas (re)surfaced over the last two decades, claiming to have potential applications to major open problems:

  • (2000) $\mathbb{F}_1$-geometry tries to view $\mathbf{Spec}(\mathbb{Z})$ as a curve over the field with one element, and mimic Weil’s proof of RH for curves over finite fields to prove the Riemann hypothesis.
  • (2012) IUTT, for Inter Universal Teichmuller Theory, the machinery behind Mochizuki’s claimed proof of the ABC-conjecture.
  • (2014) topos theory : Connes and Consani redirected their RH-attack using arithmetic sites, while Lafforgue advocated the use of Caramello’s bridges for unification, in particular the Langlands programme.

It is difficult to voice an opinion about the (presumed) current state of such projects without being accused of being either a believer or a skeptic, resorting to group-think or being overly critical.

We lack the vocabulary to talk about the different phases a mathematical idea might be in.

Such a vocabulary exists in (information) technology, the five phases of the Gartner hype cycle to represent the maturity, adoption, and social application of a certain technology :

  1. Technology Trigger
  2. Peak of Inflated Expectations
  3. Trough of Disillusionment
  4. Slope of Enlightenment
  5. Plateau of Productivity

This model can then be used to gauge in which phase several emerging technologies are, and to estimate the time it will take them to reach the stable plateau of productivity. Here’s Gartner’s recent Hype Cycle for emerging Artificial Intelligence technologies.



Picture from Gartner Hype Cycle for AI 2021

What might these phases be in the hype cycle of a mathematical idea?

  1. Technology Trigger: a new idea or analogy is dreamed up, marketed to be the new approach to that problem. A small group of enthusiasts embraces the idea, and tries to supply proper definitions and the very first results.
  2. Peak of Inflated Expectations: the idea spreads via talks, blogposts, mathoverflow and twitter, and now has enough visibility to justify the first conferences devoted to it. However, all this activity does not result in major breakthroughs and doubt creeps in.
  3. Trough of Disillusionment: the project ran out of steam. It becomes clear that existing theories will not lead to a solution of the motivating problem. Attempts by key people to keep the idea alive (by lengthy papers, regular meetings or seminars) no longer attract new people to the field.
  4. Slope of Enlightenment: the optimistic scenario. One abandons the original aim, ditches the myriad of theories leading nowhere, regroups and focusses on the better ideas the project delivered.

    A negative scenario is equally possible. Apart for a few die-hards the idea is abandoned, and on its way to the graveyard of forgotten ideas.

  5. Plateau of Productivity: the polished surviving theory has applications in other branches and becomes a solid tool in mathematics.

It would be fun so see more knowledgable people draw such a hype cycle graph for recent trends in mathematics.

Here’s my own (feeble) attempt to gauge where the three ideas mentioned at the start are in their cycles, and here’s why:

  • IUTT: recent work of Kirti Joshi, for example this, and this, and that, draws from IUTT while using conventional language and not making exaggerated claims.
  • $\mathbb{F}_1$: the preliminary programme of their seminar shows little evidence the $\mathbb{F}_1$-community learned from the past 20 years.
  • Topos: Developing more general theory is not the way ahead, but concrete examples may carry surprises, even though Gabriel’s topos will remain elusive.

Clearly, you don’t agree, and that’s fine. We now have a common terminology, and you can point me to results or events I must have missed, forcing me to redraw my graph.

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Chevalley’s circle of friends

Last week, Danielle Couty ArXiVed her paper Friendly views on Claude Chevalley (in French).

From the abstract: “We propose to follow the itinerary of Claude Chevalley during the last twenty years of his life, through the words of Jacques Roubaud, Denis Guedj and Alexander Grothendieck. Our perspective is that of their testimonies filled with friendship.”

Claude Chevalley was one of the founding fathers of Bourbaki. Two of the four pre-WW2 Bourbaki-congresses were held in “La Massoterie”, the Chevalley family domain in Chancay (see this post, update: later I learned from Liliane Beaulieu that the original house was destroyed by fire).

In 1938 he left for Princeton and stayed there during the war, making it impossible to return to a position in France for a very long time. Only in 1957 he could return to Paris where he led a seminar which proved to be essential for the development of algebraic groups and algebraic geometry.



Picture from N. Bourbaki, an interview with C. Chevalley

The Couty paper focusses on the post-1968 period in which Chevalley distanced himself from Bourbaki (some of its members, he thought, had become ‘mandarins’ and ‘reactionaires’), became involved with the ecological movement ‘Survivre et vivre’ and started up the maths department of a new university at Vincennes.

The paper is based on the recollections of three of his friends.

1. Jacques Roubaud is a French poet, writer and mathematician.

On this blog you may have run into Roubaud as the inventor of Bourbaki’s death announcement, and the writer of the book with title $\in$.

He’s also a member of Oulipo, a loose gathering of (mainly) French-speaking writers and mathematicians. Famous writers such as Georges Perec and Italo Calvino were also Oulipo-members (see also Ouilpo’s use of the Tohoku paper).

Chevalley introduced Roubaud (and others) to the game of Go. From Couty’s paper this quote from Roubaud (G-translated):

“. . . it turns out that he had learned to play go in Japan and then, in Paris, he could not find a player […] I played go with him […] and then at a certain moment , we thought, Pierre Lusson and myself, it would still be good to create circumstances such that Chevalley could have players. And so, we had a lot of ambition, we said to ourselves: “We’re going to write a treatise on go, and then lots of people will start playing go”. »

The resulting Go-book is A short treatise inviting the reader to discover the subtle art of Go. Here’s Georges Perec (left) and Jacques Roubaud playing a game.



Picture from Petit traite invitant a la decouverte de l’art subtil du Go

2. Denis Guedj was a French novelist, mathematician and historian of science professor, perhaps best known for his book The Parrot’s Theorem.

In May 1968, Guedj was a PhD-student of Jean-Paul Benzecri (the one defining God as the Alexandroff compactification of the univers), working in the building where ‘Le Comité de Grève’ installed itself. Here he met Chevalley. A Guedj-quote from Couty’s paper (G-translated):

“Claude Chevalley was one of the three professors of the Faculty of Science to commit himself totally to the adventure until the end, occupying the premises with the students on the Quai Saint-Bernard […] and sleeping there frequently . That’s where I met him.

We had taken possession of this universe which until then had only been a place of study and knowledge, and which, in the mildness of this month of May, had become a place of life, of a life wonderfully exhilarating. The college was ours. At night we walked down the aisles yet? lined with tall trees, entered the empty lecture halls, slept under the stars. Needless to say that at the beginning of the school year, in the fall of 1968, it was impossible for us to find our place in these undressed spaces from which the magic had withdrawn. »



Picture from Décès de l’écrivain et universitaire Denis Guedj

In June 2008, Guedj was one of the guests at the special edition of France Culture on the occasion of Grothendieck’s 80th birthday, Autour d’Alexandre Grothendieck.

3. Alexander Grothendieck, mathematician and misogynist, deified by some of today’s ‘mandarins’.

The paper by Danielle Couty may shed additional light on Grothendieck’s withdrawal from Bourbaki and mathematics as a whole. A G-translated Grothendieck quote from the paper:

“It was Chevalley who was one of the first, with Denis Guedj whom I also met through Survivre, to draw my attention to this ideology (they called it “meritocracy” or a name like that), and what there was in her of violence, of contempt. It was because of that, Chevalley told me […] that he could no longer bear the atmosphere in Bourbaki and had stopped setting foot there. »

Claude Chevalley stayed at Vincennes until his retirement in 1978, he died on June 28th 1984.

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