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What is a rough order of magnitude estimate? $$ $$ There is a thread on Meta about this question, http://tea.mathoverflow.net/discussion/567/rapid-closing-of-questions/#Item_0

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@Andrew: if you think this question should be reopened, you should start a thread about it on meta. –  Qiaochu Yuan Jul 31 '10 at 4:18
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Dear Michael, The methods introduced by Wiles, and by Taylor and Wiles, in the two papers that proved FLT, as well as the methods introduced by Ribet in his earlier paper reducing FLT to Shimura--Taniyama, are at the heart of much modern work in algebraic number theory and automorphic forms, so, such as the proofs of Serre's conj. and the Sato--Tate conj. Conferences/workshops in these fields typically attract on the order of magnitude of 100 or so particants, which gives you some sense of the number of students/researchers thinking about these questions: its in the tens or hundreds, but ... –  Emerton Jul 31 '10 at 12:53
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probably not in the thousands. Of course, not all these people know all the details, but the people at the top of the field surely do. (Of course, there is a question of what "understand" means exactly. I don't know how many people have carefully studied all the details of the trace formula that underly Jacquet--Langlands, Langlands--Tunnell, and base-change, and have also carefully studied the details of $p$-adic Hodge theory that is used in the arguments. But certainly the top people do understand the significance of these techinques, and are fluent in their use and application, ... –  Emerton Jul 31 '10 at 12:59
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and understand both the overall structure and strategy, as well the technical details, of the proof of FLT itself (and various more recent related results). –  Emerton Jul 31 '10 at 13:00
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1 Answer

Dear Michael,

The methods introduced by Wiles, and by Taylor and Wiles, in the two papers that proved FLT, as well as the methods introduced by Ribet in his earlier paper reducing FLT to Shimura--Taniyama, are at the heart of much modern work in algebraic number theory and automorphic forms, such as (in addition to the proofs of Shimura--Taniyama and FLT) the proofs of Serre's conjectures and the Sato--Tate conjecture.

Conferences/workshops in these fields typically attract on the order of magnitude of 100 or so particants, which gives you some sense of the number of students/researchers thinking about these questions: its in the tens or hundreds, but probably not in the thousands. Of course, not all these people know all the details, but the people at the top of the field surely do. (Of course, there is a question of what "understand" means exactly. I don't know how many people have both carefully studied all the details of the trace formula arguments that underly Jacquet--Langlands, Langlands--Tunnell, and base-change, and also carefully studied the details of the p-adic Hodge theory and other arithmetic geometry that is used in the arguments. But certainly the top people do understand the significance of these techinques, and are fluent in their use and application, and understand both the overall structure and strategy, as well the technical details, of the proof of FLT itself (and of various more recent related results).

Finally, let me note that the best evidence for the final claim of the previous paragraph is that this is currently an extremely vibrant area of research, which has progressed at a rapid clip over the last ten years or so. (The reason for this being that people have not only assimilated the arguments of Wiles/Taylor--Wiles but have improved upon them and pushed them further.)

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Dear Matthew: I'm glad you took it upon yourself to answer this question. On first reading it, I was reminded of one of those popular misconceptions like 'only five people in the world understand general relativity.' It's certainly worthwhile to help interested amateurs gain a more sophisticated understanding of the scientific process. Your last paragraph makes the point most clearly. –  Minhyong Kim Aug 1 '10 at 16:25
    
re: p-adic Hodge theory, is it used in the published version of Wiles' proof? The earlier version, with the gap that was later repaired, cited Faltings' paper on p-adic Hodge theory --- a work which was considered essentially correct but not completely clear in all of its arguments. But I thought the Wiles and Taylor published work has no logical dependence on Faltings or on what one usually calls p-adic Hodge theory (e.g., comparison theorems using rings of p-adic periods). –  T.. Aug 1 '10 at 18:55
    
(or are you referring to the proof of the full modularity conjecture which used heavier p-adic technology?) –  T.. Aug 1 '10 at 18:59
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Dear T., I think that you're certainly correct in one sense, but on the other hand, I think that the proof relies on Faltings's Theorem (Tate's conjecture), which uses p-adic Hodge theory (more precisely, Tate's p-divisible groups paper)(as well as many other things!), and the computation of the structure of the flat deformation ring uses some ideas that one could reasonably put (or at least, that I put) in the p-adic Hodge theory box. So perhaps one shouldn't take my invocation of p-adic Hodge theory too seriously, but just take it as a metaphor for a certain flavour of technique. –  Emerton Aug 1 '10 at 23:47
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