This question is partly motivated by Never appeared forthcoming papers.
Grothendieck's "Récoltes et Semailles" has been cited on various occasions on this forum. See for instance the answers to Good papers/books/essays about the thought process behind mathematical research or Which mathematicians have influenced you the most?. However, these citations reflect only one aspect of "Récoltes et Semailles", namely the nontechnical reflexion about Mathematics and mathematical activity. Putting aside the wonderful "Clef du Yin et du Yang", which is a great reading almost unrelated to Mathematics, I remember reading in "Récoltes et Semailles" a bunch of technical mathematical reflexions, almost all of which were above my head due to my having but a smattering of algebraic geometry. However, I recall for instance reading Grothendieck's opinion that standard conjectures were false, and claiming he had in mind a few related conjectures (which he doesn't state precisely) which might turn out to be the right ones. I still don't even know what the standard conjectures state and thus didn't understand anything, but I know many people are working hard to prove these conjectures. I've thus often wondered what was the value of Grothendieck's mathematical statements (which are not limited to standard conjectures) in "Récoltes et Semailles".
The questions I'd like to ask here are the following:
Have the mathematical parts of "Récoltes et Semailles" proved influential? If so, is there any written evidence of it, or any account of the development of the mathematical ideas that Grothendieck has expressed in this text? If the answer to the first question is negative, what are the difficulties involved in implementing Grothendieck's ideas?
In the latter case, I could come up with some possible explanations:
- Those who could have developed and spread these ideas didn't read "Récoltes et Semailles" seriously and thus nobody was aware of their existence.
- Those people took the mathematical content seriously but it was beyond anyone's reach to understand what Grothendieck was trying to get at because of the idiosyncratic writing style.
Should one of these two suppositions be backed by evidence, I'd appreciate a factual answer.
- The ideas were already outdated or have been proven wrong.
If this is the case, I'd appreciate a reference.
Given that "Pursuing Stacks" and "Les Dérivateurs" were written approximately in 1983 and 1990 respectively and have proved influential (see Maltsiniotis's page for the latter text, somewhat less known), I would be surprised should Grothendieck's mathematical ideas expressed around 1985 be worthless.