There’s one part of the “Scholze’s perfectoid spaces are the right idea” story that isn’t really mentioned here that I don’t quite understand, but I think is important for understanding it’s story. As it happens, Scholze was not the only one to discover the concept of perfectoid spaces and the tilting correspondence—Kedlaya and Liu also develop essentially the same theory in “Relative p-adic hodge theory: Foundations”, but with a different proof of the main result (the tilting correspondence).
The aftermath looks a lot like the result of a historian’s “social power struggle”. By all accounts the two works were independent. According to the introduction of Kedlaya-Liu, these authors had begun to disseminate their work before Scholze’s work went public in 2010. At that point, Scholze’s work “went viral”, causing Scholze to become a widely known mathematician. The other two authors reformulated their paper in terms of Scholze’s language and it was put on the arXiv in 2013, well after Scholze’s formulation was widely popular. Nowadays, most people give all of the credit for the discovery to Scholze, not even mentioning Kedlaya and Liu, much like the above article does. And off the record one can sometimes hear grumbles from researchers who are aware of Kedlaya-Liu’s work. Scholze is cooler, younger, and whiter that either of the other two authors.
As far as I can tell, the potential advantages that Scholze’s formalism has are
1) Slightly simpler foundations
2) The name “perfectoid” to illustrate a certain concept, which did not have a name in Kedlaya-Liu
3) Some simpler notation
4) The application to the Weight-Monodromy conjecture
So my question is the following: what exactly makes Scholze’s formalism the “right one” as compared to Kedlaya-Liu? Is it something from (1)-(4)? Is it something I’ve missed? Is it the fact that Scholze disseminated his research much quicker than Kedlaya-Liu? Or is it a more “social” concern, the tyranny of the “young white hotshot mathematician” or something like that? A combination of these factors?
But your basic point is right, and it's very difficult to separate a theory as influential as this one from the person who did the most to make it influential.
Perhaps this is he key: upon reflection, that one can have both things at once. The theory that we now call “perfectoid spaces” could have gone by a different name or had a slightly different formalism, but either way, the idea (which, if I understand the comments of the experts, is something like “these infinite highly ramified covers have nice properties, try to use them instead of ramification theory”) turns out to be *right* idea that was this culmination of years of work by many experts.
Perhaps this is your point: it’s possible to formalize the definition of a perfectoid space in Lean, but it’s unclear how something like Lean could help one to discover any formalism that fits this “right idea”, much less replace the human process choosing between various ideas and formalisms.
On the other hand, upon finding some idea, there is going to be some social process, “power struggle” or maybe “influence struggle” if you will, to decide who gets credit, how much the community values the work, and so on. And there need not even be any animosity between the various people involved, as you point out here all the authors involved are very careful to give credit to the others.
Those are all legitimate questions, and I am too distant from the specialty to be able to offer a reliable answer. I can share some information that may be relevant. Kedlaya spoke at the 2010 conference in Paris, which I attended, in honor of Fontaine. I believe he announced the construction, possibly with Liu, of a very general version of p-adic symmetric spaces. After the talk I overheard several experts expressing doubt that such spaces could exist, but in the Arizona Winter School book on Perfectoid Spaces Kedlaya wrote up an account of Scholze's construction of shtuka moduli spaces, so it's likely the experts were mistaken.
My understanding is that much of the theory covered in the AWS book was the culmination of several decades of work on p-adic Hodge theory by numerous authors, and unified this work in a way that soon led to the resolution of a number of open questions. It was with this in mind that I wrote that perfectoid spaces were the "right" theory. Scholze's introduction to the AWS book attributes the foundations of the theory to Kedlaya-Liu as well as to himself. You'll have to ask the experts how far the subsequent development of the theory could have been built on the basis of the Kedlaya-Liu work.
Scholze thanks a long list of people, including Kedlaya and Liu, in his IHES paper on perfectoid spaces.
There’s one part of the “Scholze’s perfectoid spaces are the right idea” story that isn’t really mentioned here that I don’t quite understand, but I think is important for understanding it’s story. As it happens, Scholze was not the only one to discover the concept of perfectoid spaces and the tilting correspondence—Kedlaya and Liu also develop essentially the same theory in “Relative p-adic hodge theory: Foundations”, but with a different proof of the main result (the tilting correspondence).
The aftermath looks a lot like the result of a historian’s “social power struggle”. By all accounts the two works were independent. According to the introduction of Kedlaya-Liu, these authors had begun to disseminate their work before Scholze’s work went public in 2010. At that point, Scholze’s work “went viral”, causing Scholze to become a widely known mathematician. The other two authors reformulated their paper in terms of Scholze’s language and it was put on the arXiv in 2013, well after Scholze’s formulation was widely popular. Nowadays, most people give all of the credit for the discovery to Scholze, not even mentioning Kedlaya and Liu, much like the above article does. And off the record one can sometimes hear grumbles from researchers who are aware of Kedlaya-Liu’s work. Scholze is cooler, younger, and whiter that either of the other two authors.
As far as I can tell, the potential advantages that Scholze’s formalism has are
1) Slightly simpler foundations
2) The name “perfectoid” to illustrate a certain concept, which did not have a name in Kedlaya-Liu
3) Some simpler notation
4) The application to the Weight-Monodromy conjecture
So my question is the following: what exactly makes Scholze’s formalism the “right one” as compared to Kedlaya-Liu? Is it something from (1)-(4)? Is it something I’ve missed? Is it the fact that Scholze disseminated his research much quicker than Kedlaya-Liu? Or is it a more “social” concern, the tyranny of the “young white hotshot mathematician” or something like that? A combination of these factors?
But your basic point is right, and it's very difficult to separate a theory as influential as this one from the person who did the most to make it influential.
Perhaps this is he key: upon reflection, that one can have both things at once. The theory that we now call “perfectoid spaces” could have gone by a different name or had a slightly different formalism, but either way, the idea (which, if I understand the comments of the experts, is something like “these infinite highly ramified covers have nice properties, try to use them instead of ramification theory”) turns out to be *right* idea that was this culmination of years of work by many experts.
Perhaps this is your point: it’s possible to formalize the definition of a perfectoid space in Lean, but it’s unclear how something like Lean could help one to discover any formalism that fits this “right idea”, much less replace the human process choosing between various ideas and formalisms.
On the other hand, upon finding some idea, there is going to be some social process, “power struggle” or maybe “influence struggle” if you will, to decide who gets credit, how much the community values the work, and so on. And there need not even be any animosity between the various people involved, as you point out here all the authors involved are very careful to give credit to the others.
Those are all legitimate questions, and I am too distant from the specialty to be able to offer a reliable answer. I can share some information that may be relevant. Kedlaya spoke at the 2010 conference in Paris, which I attended, in honor of Fontaine. I believe he announced the construction, possibly with Liu, of a very general version of p-adic symmetric spaces. After the talk I overheard several experts expressing doubt that such spaces could exist, but in the Arizona Winter School book on Perfectoid Spaces Kedlaya wrote up an account of Scholze's construction of shtuka moduli spaces, so it's likely the experts were mistaken.
My understanding is that much of the theory covered in the AWS book was the culmination of several decades of work on p-adic Hodge theory by numerous authors, and unified this work in a way that soon led to the resolution of a number of open questions. It was with this in mind that I wrote that perfectoid spaces were the "right" theory. Scholze's introduction to the AWS book attributes the foundations of the theory to Kedlaya-Liu as well as to himself. You'll have to ask the experts how far the subsequent development of the theory could have been built on the basis of the Kedlaya-Liu work.
Scholze thanks a long list of people, including Kedlaya and Liu, in his IHES paper on perfectoid spaces.