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.
(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 :
Technology Trigger
Peak of Inflated Expectations
Trough of Disillusionment
Slope of Enlightenment
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.
What might these phases be in the hype cycle of a mathematical idea?
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.
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.
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.
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.
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.
January 13th, Gallimard published Grothendieck’s text Recoltes et Semailles in a fancy box containing two books.
Here’s a G-translation of Gallimard’s blurb:
“Considered the mathematical genius of the second half of the 20th century, Alexandre Grothendieck is the author of Récoltes et semailles, a kind of “monster” of more than a thousand pages, according to his own words. The mythical typescript, which opens with a sharp criticism of the ethics of mathematicians, will take the reader into the intimate territories of a spiritual experience after having initiated him into radical ecology.
In this literary braid, several stories intertwine, “a journey to discover a past; a meditation on existence; a picture of the mores of a milieu and an era (or the picture of the insidious and implacable shift from one era to another…); an investigation (almost police at times, and at others bordering on the swashbuckling novel in the depths of the mathematical megapolis…); a vast mathematical digression (which will sow more than one…); […] a diary ; a psychology of discovery and creation; an indictment (ruthless, as it should be…), even a settling of accounts in “the beautiful mathematical world” (and without giving gifts…)”.”
All literary events, great or small, are cause for the French to fill a radio show.
The embedded YouTube above starts at 12:06, when Bourguignon describes Grothendieck’s main achievements.
Clearly, he starts off with the notion of schemes which, he says, proved to be decisive in the further development of algebraic geometry. Five years ago, I guess he would have continued mentioning FLT and other striking results, impossible to prove without scheme theory.
Now, he goes on saying that Grothendieck laid the basis of topos theory (“to define it, I would need not one minute and a half but a year and a half”), which is only now showing its first applications.
Grothendieck, Bourguignon goes on, was the first to envision the true potential of this theory, which we should take very seriously according to people like Lafforgue and Connes, and which will have applications in fields far from algebraic geometry.
Topos20 is spreading rapidly among French mathematicians. We’ll have to await further results before Topos20 will become a pandemic.
Another interesting fragment starts at 16:19 and concerns Grothendieck’s gribouillis, the 50.000 pages of scribblings found in Lasserre after his death.
Bourguignon had the opportunity to see them some time ago, and when asked to describe them he tells they are in ‘caisses’ stacked in a ‘libraire’.
Here’s a picture of these crates taken by Leila Schneps in Lasserre around the time of Grothendieck’s funeral.
If you want to know what’s in these notes, and how they ended up at that place in Paris, you might want to read this and that post.
If Bourguignon had to consult these notes at the Librairie Alain Brieux, it seems that there is no progress in the negotiations with Grothendieck’s children to make them public, or at least accessible.
