What is a rough order of magnitude estimate? $$ $$ There is a thread on Meta about this question, http://mathoverflow.tqft.net/discussion/567/rapid-closing-of-questions/#Item_0
[Math] How many people fully understand the proof of Fermat’s Last Theorem
ag.algebraic-geometrynt.number-theorysoft-question
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The world's output of scientific papers increased exponentially from 1700 to 1950.
One online source is this article (which is concerned with what has happened since then). The author displays a graph (whose source is a 1961 book entitled "Science since Babylon" by Derek da Solla Price) showing exponential increase in the cumulative number of scientific journals founded; an increase by a factor of 10 every 50 years or so, with around 10 journals recorded in 1750.
Perhaps someone can locate similar statistics specific to mathematics, but it's reasonable to expect the same pattern. If so, it is a long time since any individual could follow the primary mathematical literature in anything close to its entirety.
But then, gobbling papers is not how leading mathematicians (or scientists) actually operate.
By making judicious choices of what to pursue when, and with sufficient brilliance and vision, it is possible even today to make decisive contributions to many fields. Serre has done so in, and between, algebraic topology, complex analytic geometry, algebraic geometry, commutative algebra and group theory, and continues to do so in algebraic number theory/representation theory/modular forms.
Actually, I am harder-pressed to say anything the supposedly simpler question. For Question 1, I would just add that we need the existence of an infinite computably enumerable set of primes satisfying FLT, and that this ought to suffice. (Since the set P of primes NOT satisfying FLT is obviously $\exists$-definable and hence c.e., another way to say this is that we need to show that P is not a simple set.) I would not have guessed that any odd prime was known to satisfy FLT, up until Wiles's proof, but I'm ready to be corrected on that.
For Question 2, I do think that one could come up with a proof avoiding FLT entirely. I can't do so myself; I'm in computability, and when I was thinking about computable categoricity for fields of infinite transcendence, I realized that we needed multivariable polynomials with known finite numbers of rational solutions, and I thought of the Fermat polynomials because I didn't know any other candidates. (Of course, this condition was not all that was needed, but clearly it's necessary.) I'm still a computability theorist, and I still don't know any other candidates, but field theorists have told me that they could come up with other such polynomials fairly readily. Whether those would satisfy the more difficult requirements (basically Theorem 3.1 in the paper) is not so clear, but I suspect that it can be done with other polynomials. Bjorn Poonen suggested at one point that the Fermat polynomials were actually a bad choice, because their symmetry creates an extra solution whenever one adds a single transcendental solution to the field.
As a related question: is there a computably stable field of infinite transcendence degree? A computable structure $\mathcal{A}$ is computably stable if, for every computable structure $\mathcal{B}$, every isomorphism from $\mathcal{A}$ onto $\mathcal{B}$ is computable. (Example: $\mathbb{Z}$ under the successor function.) A common way to build computably stable structures is to make them computably categorical and rigid, i.e. with no nontrivial automorphisms, so that the isomorphism from $\mathcal{A}$ onto any computable copy must be unique (by rigidity), hence computable (by categoricity). I would conjecture that it is possible to mimic the construction in the Fermat paper, with different polynomials, and to get a computably stable field of infinite transcendence degree, but I certainly don't know offhand what polynomials one might use.
Best 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.)