A quiet Sydney
autumn weekend was a perfect opportunity to re-read Azcel’s 1996 book -
Fermat's Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem.
This book has been reviewed countless times with most expressing delight in how
Azcel manages to describe the making of the proof with all of its red herrings,
clues and suspense not too dissimilar to a mystery novel.
I also wanted to
look at the interview Andrew Wiles gave Nova in
2000. I found re-reading Azcel’s book alongside the insights shared in this
interview made the journey as exciting as the first time I read the book as
well as help clarify Wiles’ motivation, and how a chance conversation about
Ribet's Taniyama-Shimura and Fermat's Last Theorem connection with a friend would change
the course of his life.
Wiles
thought that nobody had any idea how to approach Taniyama-Shimura but at least
it was mainstream mathematics. He could try and prove results, which, even if
they didn't get the whole thing, would be worthwhile mathematics. So the romance
of Fermat, which had held him all his life, was now combined with a problem
that was professionally acceptable.
For seven years Wiles worked in isolation pursuing
the proof. He described the experience as of doing
mathematics in terms of a journey through a dark unexplored mansion. He says “You
enter the first room of the mansion and it's completely dark. You stumble
around bumping into the furniture, but gradually you learn where each piece of
furniture is. Finally, after six months or so, you find the light switch, you
turn it on, and suddenly it's all illuminated. You can see exactly where you
were. Then you move into the next room and spend another six months in the
dark. So each of these breakthroughs, while sometimes they're momentary,
sometimes over a period of a day or two, they are the culmination of—and
couldn't exist without—the many months of stumbling around in the dark that
proceed them.”
In 1993, Wiles made the crucial breakthrough. The New York Times exclaimed "At
Last Shout of 'Eureka!' in Age-Old Math Mystery," but unknown
to them, and to Wiles, there was an error in the proof. Wiles described the error to be in
a crucial part of the argument, but it was something so subtle that he’d missed
it completely until that point. The error is so abstract that it can't really
be described in simple terms. Even explaining it to a mathematician would
require the mathematician to spend two or three months studying that part of
the manuscript in great detail.
After a year of work and after inviting the
Cambridge mathematician Richard Taylor to work with him they repaired the
proof. Below is a copy of the proof from the Annals of Mathematics, Modular
elliptic curves and Fermat’s Last Theorem142 (1995), 443-551 [a hundred page
proof]. Also, I have provided the Taylor and Wiles “repair”, Ring-theoretic
properties of certain Hecke algebras, Annals of Mathematics, 141 (1995),
553-572.
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