
The Proof of Fermat’s Last Theorem by R.Taylor and A.Wiles (1995) [pdf] - luisb
http://www.ams.org/notices/199507/faltings.pdf
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dkural
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.

[edit: fixed spelling]

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xyzzyz
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.

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eranation
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?

~~~
cokernel
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.

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valuegram
If you haven't read it, I highly recommend:

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.

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chrispeel
I understood the abstract...

I am not only not one of the 'specialists', but am not one of the 'wider
mathematical audience', it appears.

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dkarapetyan
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.

~~~
iheartmemcache
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/](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.

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vortico
Does anyone have a mirror of this? I can't find this except for AMS's notices
board, which is what this links to.

~~~
ikeboy
[http://vtf-math.narod.ru/pdf/faltings.pdf](http://vtf-
math.narod.ru/pdf/faltings.pdf), several more at
[https://scholar.google.com/scholar?cluster=17113326573299277...](https://scholar.google.com/scholar?cluster=17113326573299277476&hl=en&as_sdt=0,33&sciodt=0,33)

