I don't get the point of the title. 4 pages + a PhD in math is needed.. lol.
Here's even a shorter summary:
Wiles proved a special case of the Taniyama-Shimura conjecture, ie establishing that for every rational elliptic curve there is a modular form with the same Dirichlet l-series. Faltings previously proved that if there was a counterexample to Fermat's Last Theorem, there'd be a certain kind of elliptic curve that is NOT modular - thus Wiles, by proving the modularity theorem for this class, proved FLT. But his proof had an essential gap, so he had to invent a certain kind of Iwasawa theory as an alternative method, to complete the proof.
In otherwords, "summarized" really is an important word.
I was doing a Master degree in Algebraic Geometry, though I was working on projective geometry rather than number theory. Because of this, I can follow this quite easily right up until it goes into intricacies of modular forms, after that there are some familiar concepts like deformations or Gorenstein modules, but the whole picture is completely blurry.
What I find interesting though is that behind every single sentence I could I understand from the first half of this article, there were literally hours of time spent on learning the concepts. For instance:
>With this addition, the solution set
has the structure of an abelian group, with ∞
as the neutral element. The inverse of (x, y) is
(x, −y), and the sum of three points vanishes if
they lie on a line.
This just reminds the hours I spent learning about group of Weil linear divisors, how they correspond to Cartier divisors on nonsingular varieties, why O(a) != O(b) for a != b if a, b are codimension 1 subvarieties (i.e. points) of eliptic curves, and so on, and so on. Literally every single sentence brings back memories of hours of study.
Now, reading sentences from the half I don't understand makes it completely obvious to me that just like the hours I spent to understand the first half, I need more hours to understand the second half. I feel sad I don't do math anymore.
I was obsessed with trying to understand it ever since I read the book.
How many years of math studies one need to make sense of it? (I'm doing a CS master degree but I'm not even close to feeling that I know anything about math, imposter syndrome or more likely not enough math skills...)
Any chances? Or should I probably give it up?
It took Wiles seven years to prove the thing (and the proof needed to be patched!), so I wouldn't be discouraged if it takes a while to understand. A nice thing about mathematics is, it waits for you.
If you're interested in the mathematics behind this, I'm not sure a direct attack on the FLT proof is the best route to take. The paper that proves a famous conjecture is normally sitting on a mountain of prior work, which means the final paper (1) assumes familiarity with that mountain and (2) is highly technical because all of the understandable things have already been tried.
So instead, why not start learning about the mountain?
The truth of FLT follows from the two claims:
(1) Taniyama--Shimura--Weil conjecture: "Every elliptic curve is modular."
(2) Ribet's Theorem: "If FLT has a counterexample, then such and such an elliptic curve is not modular."
As it happens, TSW was originally believed to be too difficult to prove, but I suppose the connection with FLT motivated people. Taylor and Wiles proved the absolute minimum of TSW that they could get by with and still get a contradiction from Ribet's theorem. (My understanding is that TSW is now fully proved -- the "modularity theorem".)
If you're wanting to "get" FLT, I'd encourage you to look into elliptic curves, modular forms, and their relationship. I wonder if working on the mathematics of elliptic curve cryptography might be a good way to get a feel for elliptic curves.
However, if you do want to take the direct route, I believe that Faltings's highly compressed article provides a syllabus. Once you can read and completely understand every sentence of that article, it is highly likely that the Wiles and Taylor and Wiles papers will make sense. I... would really not recommend this route.
Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem by Simon Singh
The proof of Fermat's last theorem has a long and intriguing history, and Singh's writing is accessible and entertaining for anyone with an interest in math and science, regardless of education level.
Modern algebraic number theory is pretty hard even for non-specialist mathematical audiences so if even if you're steeped in mathematical lore and culture it still can be hard to grasp.
And algebraic topology, and algebraic geometry, (..and I'm sure analysis and basically every other subset of mathematics). http://matt.might.net/articles/phd-school-in-pictures/ You get far enough into any field (and it doesn't even have to be in the hard sciences,) be it law, medicine, linguistics, or something as seemingly trivial as making industrial bearings, and you'll need not only a graduate level education but an additional 5 or 10 years to get up to par with the rest of the field. The world is so catastrophically complicated at this point, it often overwhelms me.
Here's even a shorter summary:
Wiles proved a special case of the Taniyama-Shimura conjecture, ie establishing that for every rational elliptic curve there is a modular form with the same Dirichlet l-series. Faltings previously proved that if there was a counterexample to Fermat's Last Theorem, there'd be a certain kind of elliptic curve that is NOT modular - thus Wiles, by proving the modularity theorem for this class, proved FLT. But his proof had an essential gap, so he had to invent a certain kind of Iwasawa theory as an alternative method, to complete the proof.
In otherwords, "summarized" really is an important word.
[edit: fixed spelling]