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

17$\begingroup$ "The site works best for welldefined questions: math questions that actually have a specific answer. You'll notice that there is the occasional question making a list of something, asking about the workings of the mathematical community, or something else which isn't really a math question. Such questions can be helpful to the community, but it is extremely tricky to ask them in a way that produces a useful response. So if you're new to the site, we suggest you stick to asking precise math questions until you learn about the quirks of the community and the strengths of the medium." $\endgroup$– JBLJul 31, 2010 at 4:06

10$\begingroup$ @Andrew: if you think this question should be reopened, you should start a thread about it on meta. $\endgroup$– Qiaochu YuanJul 31, 2010 at 4:18

8$\begingroup$ 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 ShimuraTaniyama, 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 SatoTate 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 ... $\endgroup$– EmertonJul 31, 2010 at 12:53

9$\begingroup$ 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). $\endgroup$– EmertonJul 31, 2010 at 13:00

9$\begingroup$ Dear Victor, I don't think that this is a question of gossip. Especially from an amateur interested in FLT, but incapable of understanding the proof (I don't know whether or not the OP is such), or even from a grad student or mathematician in another field, it is a reasonable question. Since accepting that FLT is proved in that case is a matter of deferring to authority, it is reasonable to ask how widely disseminated the understanding of the proof is. $\endgroup$– EmertonJul 31, 2010 at 21:16
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 ShimuraTaniyama, are at the heart of much modern work in algebraic number theory and automorphic forms, such as (in addition to the proofs of ShimuraTaniyama and FLT) the proofs of Serre's conjectures and the SatoTate 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 JacquetLanglands, LanglandsTunnell, and basechange, and also carefully studied the details of the padic 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/TaylorWiles but have improved upon them and pushed them further.)

5$\begingroup$ 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. $\endgroup$ Aug 1, 2010 at 16:25

$\begingroup$ re: padic 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 padic 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 padic Hodge theory (e.g., comparison theorems using rings of padic periods). $\endgroup$– T..Aug 1, 2010 at 18:55

$\begingroup$ (or are you referring to the proof of the full modularity conjecture which used heavier padic technology?) $\endgroup$– T..Aug 1, 2010 at 18:59

1$\begingroup$ 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 padic Hodge theory (more precisely, Tate's pdivisible 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 padic Hodge theory box. So perhaps one shouldn't take my invocation of padic Hodge theory too seriously, but just take it as a metaphor for a certain flavour of technique. $\endgroup$– EmertonAug 1, 2010 at 23:47