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Begging your pardon for indulging in some personal history (perhaps personal propaganda), I will explain how I ended up applying R\'ecollte et Semaille. I do apologize in advance for interpreting the question in such a self-centered fashion!

I didn't come anywhere near to reading the whole thing, but I did spend many hours dipping into various portions while I was a graduate student. Serge Lang had put his copy into the mathematics library at Yale, a very cozy place then for hiding among the shelves and getting lost in thoughts or words. Even the bits I read of course were hard going. However, one thing was clear even to my superficial understanding: Grothendieck, at that point, was dissatisfied with motives. Even though I wasn't knowledgeable enough to have an opinion about the social commentary in the book, I did wonder quite a bit if some of the discontent could have a purely mathematical source.

A clue came shortly afterwards, when I heard from Faltings Grothendieck's ideas on anabelian geometry. I still recall my initial reaction to the section conjecture: Surely there are more splittings than points!' to which Faltings replied with a characteristically brief question:' Why?' Now I don't remember if it's in R&S as well, but I did read somewhere or hear from someone that Grothendieck had been somewhat pleased that the proof of the Mordell conjecture came from outside of the French school. Again, I have no opinion about the social aspect of such a sentiment (assuming the story true), but it is interesting to speculate on the mathematical contentcontext.

There were in Orsay and Paris some tremendously powerful people in arithmetic geometry. Szpiro, meanwhile, had a very lively interest in the Mordell conjecture, as you can see from his writings and seminars in the late 70's and early 80's. But somehow, the whole thing didn't come together. One suspects that the habits of the Grothendieck school, whereby the six operations had to be established first in every situation where a problem seemed worth solving, could be enormously helpful in some situations, and limiting in some others. In fact, my impression is that Grothendieck's discussion of the operations in R&S has an ironical tinge. [This could well be a misunderstanding due to faulty French or faulty memory.] Years later, I had an informative conversation with Jim McClure at Purdue on the demise of sheaf theory in topology. [The situation has changed since then.] But already in the 80's, I did come to realize that the motivic machinery didn't fit in very well with homotopy theory.

To summarize, I'm suggesting that the mathematical content of Grothendieck's strong objection to motives was inextricably linked with his ideas on homotopy theory as appeared in 'Pursuing Stacks' and the anabelian letter to Faltings, and catalyzed by his realization that the motivic philosophy had been of limited use (maybe even a bit of an obstruction) in the proof of the Mordell conjecture. More precisely, motives were inadequate for the study of points (the most basic maps between schemes!) in any non-abelian setting, but Faltings' pragmatic approach using all kinds of Archimedean techniques may not have been quite Grothendieck's style either. Hence, arithmetic homotopy theory.

Correct or not, this overall impression was what I came away with from the reading of R&S and my conversations with Faltings, and it became quite natural to start thinking about a workable approach to Diophantine geometry that used homotopy groups. Since I'm rather afraid of extremes, it was pleasant to find out eventually that one had to go back and find some middle ground between the anabelian and motivic philosophies to get definite results.

This is perhaps mostly a story about inspiration and inference, but I can't help feeling like I did apply R&S in some small way. (For a bit of an update, see my paper with Coates here.)

Added, 14 December: I've thought about this question on and off since posting, and now I'm quite curious about the bit of R&S I was referring to, but I no longer have access to the book. So I wonder if someone knowledgeable could be troubled to give a brief summary of what it is Grothendieck really says there about the six operations. I do remember there was a lot, and this is a question of mathematical interest.

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Begging your pardon for indulging in some personal history (perhaps personal propaganda), I will explain how I ended up applying R\'ecollte et Semaille. I do apologize in advance for interpreting the question in such a self-centered fashion!

I didn't come anywhere near to reading the whole thing, but I did spend many hours dipping into various portions while I was a graduate student. Serge Lang had put his copy into the mathematics library at Yale, a very cozy place then for hiding among the shelves and getting lost in thoughts or words. Even the bits I read of course were quite hard going. However, one thing was quite clear even to my superficial understanding: Grothendieck, at that point, was dissatisfied with motives. Even though I wasn't knowledgeable enough to have an opinion about the social commentary in the book, I did wonder quite a bit if some of the discontent could have a purely mathematical source.

A clue came shortly afterwards, when I heard from Faltings Grothendieck's ideas on anabelian geometry. I still recall my initial reaction to the section conjecture: Surely there are more splittings than points!' to which Faltings replied with a characteristically brief question:' Why?' Now I don't remember if it's somewhere in R&S as well, but I did read somewhere or hear from someone that Grothendieck had been somewhat pleased that the proof of the Mordell conjecture came from outside of the French school. Again, I have no opinion about the social side aspect of such a sentiment (assuming the story true), but it is interesting to speculate on the mathematical contextcontent.

There were in Orsay and Paris some tremendously powerful people in arithmetic geometry. Szpiro, meanwhile, had a very lively interest in the Mordell conjecture, as you can see from his writings and seminars in the late 70's and early 80's. But somehow, the whole thing didn't come together. One suspects that the habits of the Grothendieck school, whereby the six operations had to be established first in every situation where a problem seemed worth solving, could be tremendously enormously helpful in some situations, and limiting in some others. In fact, my impression is that Grothendieck's discussion of the operations in R&S has an ironical tinge. [This could well be a misunderstanding due to faulty French or faulty memory.] Much Years later, I had an informative conversation with Jim McClure at Purdue on the demise of sheaf theory in topology. [The situation has changed since then.] But already in the 80's, I did come to realize that the motivic machinery didn't fit in very well with homotopy theory.

To summarize, I'm suggesting that the mathematical content of Grothendieck's strong objection to motives was intertwined inextricably linked with his ideas on homotopy theory as appeared in 'Pursuing Stacks' and the anabelian letter to Faltings, and catalyzed by his realization that the motivic philosophy had been of limited use (maybe even a bit of an obstruction) in the proof of the Mordell conjecture. More precisely, motives were inadequate for the study of points (the most basic maps between schemes!) in any non-abelian setting, but Faltings' pragmatic approach using all kinds of Archimedean techniques may not have been quite Grothendieck's style either. Hence, arithmetic homotopy theory.

Correct or not, this overall impression was what I came away with from my the reading of R&S and my conversations with Faltings, and it became quite natural to start thinking about a workable approach to Diophantine geometry that used homotopy groups. Since I I'm rather dislike afraid of extremes, it was pleasant to find out eventually that one had to go back and find some middle ground between the anabelian and motivic machineries philosophies to get definite results.

This is perhaps mostly a story about inspiration and inference, but I can't help feeling like I did apply R&S in some small way. (For a bit of an update, see my paper with Coates here.)

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Begging your pardon for indulging in some personal history (perhaps personal propaganda), I will explain how I ended up applying R\'ecollte et Semaille. I do apologize in advance for interpreting the question in such a self-centered fashion!

I didn't come anywhere near to reading the whole thing, but I did spend many hours dipping into various portions while I was a graduate student. Serge Lang had put his copy into the mathematics library at Yale, a very cozy place then for hiding among the shelves and getting lost in thoughts or words. Even the bits I read of course were quite hard going. However, one thing was quite clear even to my superficial understanding: Grothendieck, at that point, was dissatisfied with motives. Even though I wasn't knowledgeable enough to have an opinion about the social commentary in the book, I did wonder quite a bit if some of the discontent could have a purely mathematical source.

A clue came shortly afterwards, when I heard from Faltings Grothendieck's ideas on anabelian geometry. I still recall my initial reaction to the section conjecture: `Surely there are more splittings than points!' to which Faltings replied with a characteristically brief question:' Why?' Now I don't remember if it's somewhere in R&S as well, but I did read somewhere or hear from someone that Grothendieck had been somewhat pleased that the proof of the Mordell conjecture came from outside of the French school. Again, I have no opinion about the social side of such a sentiment (assuming the story true), but it is interesting to speculate on the mathematical context.

There were in Orsay and Paris some tremendously powerful people in arithmetic geometry. Szpiro, meanwhile, had a very lively interest in the Mordell conjecture, as you can see from his writings and seminars in the late 70's and early 80's. But somehow, the whole thing didn't come together. One suspects that the habits of the Grothendieck school, whereby the six operations had to be established first in every situation where a problem seemed worth solving, could be tremendously helpful in some situations, and limiting in some others. In fact, my impression is that Grothendieck's discussion of the operations in R&S has an ironical tinge. [This could well be a misunderstanding due to faulty French or faulty memory.] Much later, I had an informative conversation with Jim McClure at Purdue on the demise of sheaf theory in topology. [The situation has changed since then.] But already in the 80's, I did come to realize that the motivic machinery didn't fit in very well with homotopy theory.

To summarize, I'm suggesting that the mathematical content of Grothendieck's strong objection to motives was intertwined with his ideas on homotopy theory as appeared in 'Pursuing Stacks' and the anabelian letter to Faltings, and catalyzed by his realization that the motivic philosophy had been of limited use (maybe even a bit of an obstruction) in the proof of the Mordell conjecture. More precisely, motives were inadequate for the study of points, but Faltings' pragmatic approach using all kinds of Archimedean techniques may not have been quite Grothendieck's style either. Hence, arithmetic homotopy theory.

Correct or not, this overall impression was what I came away with from my reading of R&S and my conversations with Faltings, and it became quite natural to start thinking about a workable approach to Diophantine geometry that used homotopy groups. Since I rather dislike extremes, it was pleasant to find out eventually that one had to go back and find some middle ground between the anabelian and motivic machineries to get definite results.

This is perhaps mostly a story about inspiration and inference, but I can't help feeling like I did apply R&S in some small way. (For a bit of an update, see my paper with Coates here.)