Andrew Wiles page on Wikipedia
- From: "Timothy Clemans" <clemacetc@xxxxxxx>
- Date: 14 Apr 2006 01:14:19 -0700
I am assuming that there are people who know about Andrew Wiles and
might like to look at http://en.wikipedia.org/wiki/Andrew_Wiles and
help expand it.
Here is the content:
:''Andrew Wiles should not be confused with [[André Weil]], another
famous mathematician who, like Wiles, had done important work in
[[elliptic curve]]s.''
'''Sir Andrew John Wiles''' (born [[April 11]], [[1953]]) is a [[United
Kingdom|British]] [[mathematician]] living in the [[United States]]. He
was educated at [[The Leys School]] Cambridge and in 1974 he graduated
from the [[University of Oxford]]. He then completed his [[Doctor of
Philosophy|Ph.D.]] at [[ Clare College, Cambridge|Clare College]] of
the [[University of Cambridge]] in 1979 and is currently a Professor
and the chair of the department of [[mathematics]] at [[Princeton
University]].
Wiles is well known for his proof of [[Fermat's Last Theorem]] and
before he proved it, he developed his repution as a brilliant number
theorist when he worked under the supervision of [[John Coates]] on
[[elliptic curve|elliptic curves]]. Working on elliptic curves he with
John Coates made some of the first breakthroughs on the famous [[Birch
and Swinnerton-Dyer conjecture]]([[Clay Mathematics Institute|one of
the Millennium Prize Problems]]) and did important work on the main
conjecture of [[Iwasawa theory]].
Andrew Wiles is an international expert on the [[Birch and
Swinnerton-Dyer
conjecture]][http://www.claymath.org/millennium/Birch_and_Swinnerton-Dyer_Conjecture/BSD.pdf].
He still writes papers on elliptic curves.
This page focuses on Wiles' work on Fermat's Last Theorem. We will
discuss what [[Fermat's Last Theorem]] states and why Wiles was
interested in it, the [[Taniyama-Shimura theorem|connection]] between
rational [[elliptic curve|elliptic curves]] and [[modular form|modular
forms]], the connection between Fermat's equation and elliptic curves,
the techiquences Wiles used, lectures Wiles gave on his work, and his
published paper.
== Interest in mathematics at a young age ==
Wiles' loves mathematics and his love of mathematics dates from his
eary childhood days where he grew in Cambridge in England. He loved
working on the mathematical problems from school and he would make up
his own problems that were new to him.The best problem he has ever
comes across is [[Fermat's Last Theorem]], because it is very simple to
understand but the great mathematics before him had not been able to
solve it.
== The statement of Fermat's Last Theorem ==
[[Fermat's Last Theorem]] (FLT) statements that it is impossible for
any number as a power such as <math>x^n</math> to be the sum of two
numbers of the same exponent n greater than 2 where x, y, z, and n are
integers.
It can also be stated as <math>x^n + y^n = z^n and x, y, z, and n
element of the set of the integers and n > 2</math> and is called the
Fermat equation.
=== Wiles' introduction and interest in the theorem ===
[[Pierre de Fermat]] did not discover his equation, but he did say that
he had a proof of the fact that there is no solution to his equation.
Fermat's proof has never been recovered. [[Eric Temple Bell|E.T.
Bell's]] wrote a book on the problem titled, ''The Last Problem''. When
Wiles was 10 years old he found the book in his local library and he
tried to slove himself, thinking that all the people in the world
expect Fermat who had tried to prove it missed something and maybe he
would not miss it. He studied all the techiqenes that had been used and
decided that none of them were really going to work. In graduate school
Wiles stopped working on the problem, and began working with John
Coates on elliptic curves.
== The connection between elliptic curves and modular forms ==
In the 1950s and 1960s a connection between elliptic curves and modular
forms was conjectured by the two japanese mathematicians [[Yutaka
Taniyama|Taniyama]] and [[Goro Shimura|Shimura]]. In the west it became
well known through a paper [[Andrew Weil|Weil]] wrote. Weil gave
conceptual edidence for it. The conjecture is now a theorem and was
proved in 1999. This theorem is called the [[Taniyama-Shimura
theorem]]. It states that every rational elliptic curve's given
j-invariant is modular. While it was not proved until 1999, papers were
published saying what would be the results of the conjecture being
correct. The problem with that is if it were wrong than anything that
relied on the conjecture would be too.
=== The connection between the Fermat equation and elliptic curves ===
The following equation, <math>y^2 = x(x + u^p)(x - v^p)</math>, is a
hyptothetical [[elliptic curve]]. While it had been studied long before
the connection between elliptic curves, the Fermat equation, and
modular forms was made, [[Gerhard Frey|Frey]] was the first to suggest
that if it had any solutions it was not modular and it represented the
Fermat equation as an elliptic curve. Serre created the priecise
machanism that related modular forms with Fermat's equation. It was
called the epsilon conjection and was proved by [[Ken Ribet|Ribet]].
The work Ribet did told mathematicians not only that the epsilon
conjecture was correct but that only a proof of the semistable elliptic
curve case of Taniyama-Shimura would impie Fermat's Last Theorem.
== What Wiles studied ==
Andrew Wiles may have been the only person who thought that they might
be able to prove Fermat's Last Theorem by proving the conjecture laid
out in the text above in the 1980s. No one had been able to find a way
to count all rational elliptic curves and their cosponding modular
form. He started working on the problem very soon after being told by a
friend that Ribet had proved the epsilon conjecture. He studied the
problem by studying not elliptic curves but Galois representations
while still studying j-invariants and modular forms.
Wiles converted elliptic curves by their j-invariant to their
corresponding Galois representation to study elliptic curves. He did
studies on resentations according to partations and said in his paper
on this, "suppose for the moment that p_3 is irreducible". He tells us
why 3 is important.
He found a supprising link between '''[[Galois representation|Galois
representations]] and [[modular forms]]''' and '''the interpretation of
special values of [[L-functions]]'''. The proof is based on that link
and is used to prove a hypothesis that p_3 is semistable at 3 by
linking some of [[commutative algebra]] with a well-know type for a
[[class number]] problem.
=== The class number formula problem ===
{{expandsect}}
== Why Wiles stop attending many lectures and worked mostly at home on
his problem ==
{{expandsect}}
== Peparing to announce his proof and his Isaac Newton Instutute
lecture series ==
{{expandsect}}
== Submission and refering of the manuscript written before the lecture
series ==
{{expandsect}}
=== Attempeted repair and later replacement of the Euler system
argument ===
{{expandsect}}
== Release of the final papers ==
{{expandsect}}
<hr>
He began working on the problem again after [[Ken Ribet]] proved the
[[epsilon conjecture]] in 1985. This proof established that FLT would
follow from the [[Taniyama-Shimura theorem|conjecture]] of [[Yutaka
Taniyama|Taniyama]], [[Goro Shimura|Shimura]] and [[André Weil|Weil]]
that every [[elliptic curve]] can be parametrized by [[modular form]]s.
This conjecture was in turn inspired by the ideas of [[Jean-Pierre
Serre]] and [[Gerhard Frey]], and although it is less well-known than
Fermat's Last Theorem, it reaches down to even deeper currents in
number theory. Ribet's work showed that one merely has to prove that
every semi-stable elliptic curve is modular in order to prove FLT.
Wiles was uncharacteristically dramatic in revealing the proof. He
arranged to give three [[lecture]]s at the [[Isaac Newton Institute]],
[[Cambridge]], [[England]], in June of [[1993]]. He did not announce
the topic of the lectures in advance, and as the audience and the world
became aware of where the lectures were headed, the audience swelled so
that the third lecture was given to a very crowded room. At the end of
the third lecture, he announced "... this proves Fermat's Last Theorem.
I'll stop here", and received a standing ovation.
In the following months, the manuscript of the proof was circulated
only to a small number of mathematicians while the world waited for its
general publication. The first version of the proof depended on the
construction of an object called an [[Euler system]], and this aspect
proved problematical, as a flaw emerged during [[peer review]] of the
deep and subtle mathematics involved. For almost a year Wiles thought
that the flaw might be fatal, and that although he had made many
important discoveries, the ultimate goal had eluded him. He was on the
point of giving up, when he decided to make one last attempt to solve
the last remaining problem in his proof in collaboration with [[Richard
Taylor (mathematician)|Richard Taylor]], one of his former PhD students
in 1994. He commented:
:"... suddenly, totally unexpectedly, I had this incredible revelation.
It was the most important moment of my working life. Nothing I ever do
again will mean as much ... it was so indescribably beautiful, it was
so simple and so elegant, and I just stared in disbelief for twenty
minutes, then during the day I walked round the department. I'd keep
coming back to my desk to see it was still there – it was still
there."
The final version of Wiles' proof, which therefore differs from his
original one, was published in the ''[[Annals of Mathematics]]'' 141
(1995), pp. 443–551, together with another, supporting article by
Wiles and Taylor titled "[[Ring (mathematics)|Ring]]-theoretic
properties of certain [[Hecke algebra]]s" (''Annals of Mathematics''
141 (1995), pp. 553–572) relating to the final step of discovery.
==Awards==
Wiles has been awarded several major prizes in mathematics: [[Schock
Prize]] ([[1995]]), [[Royal Medal]] (1996), [[Cole Prize]] (1996),
[[Wolf Prize in Mathematics|Wolf Prize]] ([[1996]]), a silver plate
from the [[International Mathematical Union]] ([[1998]]), [[Faisal of
Saudi Arabia|King Faisal]] Prize ([[1998]]), [[Clay Research Award]]
([[1999]]) and [[Shaw Prize]] ([[2005]]). He became a [[Knight of the
British Empire]] in 2000. Wiles cannot receive the [[Fields Medal]] as
the award can only be given to those below 40 years of age (Wiles was
born in 1953 and proved the theorem in 1994), a rule strictly adhered
to.
== Further reading ==
* ''[http://math.stanford.edu/~lekheng/flt/wiles.pdf Modular elliptic
curves and Fermat's Last Theorem]'' – [[Annals of Mathematics]],
1995 (the published paper of his results).
*[[Simon Singh]], ''Fermat's Last Theorem'', ISBN 1841157910. A
best-selling book about Wiles and the story of his discovering of the
proof.
*Charles J Mozzochi, ''The Fermat Diary'', ISBN 0821826700
== External links ==
* [[Nova (TV series)|Nova]]
[http://www.pbs.org/wgbh/nova/transcripts/2414proof.html "The Proof"
Transcript] [[PBS]] Airdate: October 28, 1997
* {{MacTutor Biography|id=Wiles}}
* [http://fermatslasttheorem.blogspot.com Fermat's Last Theorem Blog]
– blog that traces the history of Fermat's Theorem from Fermat to
Andrew Wiles.
[[Category:1953 births|Wiles, Andrew]]
[[Category:Living people|Wiles, Andrew]]
[[Category:20th century mathematicians|Wiles, Andrew]]
[[Category:21st century mathematicians|Wiles, Andrew]]
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[[Category:Fellows of the Royal Society|Wiles, Andrew]]
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