博文
费马大定理证明的背后
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When I came to the institute, my collaborator Kay invited me for a dinner in the new
town together with his friend Thilo and other guys. The city is wisely divided
into two functioning areas, old town and new town. All the ancient buildings like
King’s summer palace, women’s church and opera house are in the old town;
while
the restaurants, bars and other modern buildings are in the new town.
Each is in harmony with its environment and cultural atmosphere. We went to an
Italian restaurant. The sentence on its menu board reads:”think globally, eat
locally”, which roughly means that even it is a local restaurant, it offers global
food options. It is true. After ordering the food, it turns out that six of
us ordered totally different food: Kay ordered American style fish salad with
chips, Thilo ordered Greece style salad with bread, Martin ordered beef steak cooked in North European way, I ordered Chinese style fried beef and vegetable with
rice. It was still summer then. We sit around a long table under the shade of
the vine in the backyard of the house. It was green everywhere, with green plants
covering the wall, lovely flowers blooming in the small Garden. There was a well
for fetching water, just like any ordinary local residence here. Several big
candles were burning since it is getting dark. A man in the casual dress was
playing his guitar and singing lovely Italian songs.
After the dinner, with beers in hands, Thilo began to tell his stories when he once
was a researcher in the math department of Princeton University. He said,
academic jobs in the university are quite demanding. You have to play a multiple
roles in one person, as a researcher at first hand, also as a lecturer to teach
courses for undergraduates and postgraduates; a fund raiser to get money for your
research group; a manager to handle with quite a few Postdocs and Phds in your
group; a faculty member for the service in the department; a paper reviewer or
an editor for your academic community. You don’t need to be smart to know how
busy an academia is. “publish or perish”, the cliche still works now days.
Without enough papers published, without enough founding, you will fail before this
strict rule, “up or out” , if you don’t get your tenure within six years.
“ Then how those big names do their research?”, somebody asked.
“ Today I will reveal a secret”, he paused.
“ What secret? Don’t be mean, please tell us”, we said.
“ OK. Do you guys know Fermat's Last Theorem?”
“ Yes, we do”
“ Do you know who proved this theorem?”
“ Aha, you call it secret? Everyone here knows that is it is Prof. Andrew Wiles
who spent about 10 years to prove it. The final proof from him came in publication
in 1994”
“ It is correct. However, do you guys know how Prof. Andrew Wiles found these 10 years
to dedicate himself to the Great Fermat theorem?”, he signed, “Prof. Andrew Wiles
told me by himself, in order to focus on the proof of the Fermat's Last Theorem,
there was one year in which he worked extremely hard to write 20 papers and locked
them up in his desk drawer. Then he would pick up two to publish each year.
In this way, he gained precious ten years to allow himself to do nothing else but Fermat's Last Theorem”
Nobody spoke anything for a while. Everybody was pondering on something. Now days in
the era of science research being measured by SCI, Impact Factors, funding
committees etc., most researchers would not risk themselves to focus on some
true problems which demands some deep insights and long commitments. This is a
simple fact in our current funding system. No wonder, in a report to the US president by President’s Information Technology Advisory Committee, named as
“Computational Science: Ensuring America’s Competitiveness”, a group of leading
scientists expressed their concerns:
“Based on its analysis of Federal R&D agency activities, PITAC concluded that Federal
support for computational science research has been overly focused on
short-term, low-risk activities. In the long term, this is actually a high-risk
strategy that is less likely to yield the high-payoff, strategic innovations
needed for the future.”
Now we are in the age of competition: everything is required to be done faster;
everybody is required to produce more with less time. Nobody knows what the end of
this road is; Nobody knows whether it is the right way. It is more or less to
make people to feel nostalgia about the golden days of science in the past time,
before NSF or any other funding committees are established. For example,
in Cambridge University, after becoming a member of the faculty, you have the freedom
to do whatever you like to do within the university’s resources. You don’t need apply
any special funding for it. Nobody will evaluate your research every 2~3 years.
However, those golden time is gone. Now we can not undo what we had already done.
More importantly, we can not back to the age of doing science without complicated
devices and giant machines, which are essential for the progress of bio-science
and nano-technologies etc.
What could we do? It is impossible to ask researchers to work harder, like
Prof. Wiles did his pre-work before the proof of the Fermat's Last Theorem. Some young
researchers have already died prematurely for their overworking, such as
Dr. He Yong in Zhejiang University. We also could not change the system of competitive
application for funding and the peer-review, which have been proved to be the
pillars for the current progress of science. Maybe there is NO solution for
everybody in general. Each one has to create your own means to meet your own ends. Just like what Prof. Wiles did, we should know what our real interest is and hold it
dear to our heart always, and then try to do the best from the least.
Whenever I read the Steve Jobs’ commencement address in Stanford, I could not help
being moved by his “three stories”, which are linked together to make his life
shine like a diamond. He is a real man, a true hero who sticks to his ideals
always. At the end his speech, he said:
“When I was young, there was an amazing publication called The Whole Earth Catalog,
which was one of the bibles of my generation. It was created by a fellow named
Stewart Brand not far from here in Menlo Park, and he brought it to life with his
poetic touch. This was in the late 1960's, before personal computers and desktop
publishing, so it was all made with typewriters, scissors, and polaroid cameras.
It was sort of like Google in paperback form, 35 years before
Google came along: it was idealistic, and overflowing with neat tools and great notions.
Stewart and his team put out several issues of The Whole Earth Catalog, and then
when it had run its course, they put out a final issue. It was the mid-1970s,
and I was your age. On the back cover of their final issue was a photograph of
an early morning country road, the kind you might find yourself hitchhiking on
if you were so adventurous. Beneath it were the words: "Stay Hungry. Stay
Foolish." It was their farewell message as they signed off. Stay Hungry. Stay
Foolish. And I have always wished that for myself. And now, as you graduate to
begin anew, I wish that for you.
Stay Hungry. Stay Foolish. “
”
Reference:
1. Report to the US president,
“ Computational Science: Ensuring America’s Competitiveness”,
President’s Information Technology Advisory Committee,
http://www.nitrd.gov/pitac/reports/20050609_computational/computational.pdf
2. Report about He Yong, http://www.39.net/HotSpecial/zz/89198.html
3. 'You've got to find what you love', Commencement address by Steve Jobs,
CEO of Apple Computer and of Pixar Animation Studios, delivered on June 12, 2005, http://news-service.stanford.edu/news/2005/june15/jobs-061505.html
4. Jobs Speech, Chinese Translation: http://v35.blog.sina.com.cn/panshiyi#serial_4679dbbf050000f1
Appendix:
Andrew Wiles
From Wikipedia, the free encyclopedia
Source: http://en.wikipedia.org/wiki/Andrew_Wiles
Fermat's Last Theorem states that no nontrivial integer solutions exist for the equation: xn + yn = zn if n is an integer greater than two.
____________________________________
Andrew Wiles' most famous mathematical result is that all rational semistable
elliptic curves are modular which, in particular, implies Fermat's Last Theorem.
Wiles was introduced to Fermat's Last Theorem at the age of ten. He tried to prove
the theorem using textbook methods and later studied the work of mathematicians
who had tried to prove it. When he began his graduate studies he stopped trying to
prove it and began studying elliptic curves under the supervision of John Coates.
In the 1950s and 1960s a connection between elliptic curves and modular forms was
conjectured by the Japanese mathematician Goro Shimura based on some ideas that
Yutaka Taniyama posed. In the West it became well known through a paper by André Weil.
With Weil giving conceptual evidence for it, it is sometimes called the
Shimura-Taniyama-Weil conjecture. It states that every rational elliptic curve
is modular. The full conjecture was proven by Christophe Breuil, Brian Conrad,
Fred Diamond, and Richard Taylor in 1998 using many of the methods that Andrew Wiles
used in his 1995 published papers.
The bridge between Fermat and Taniyama If p is an odd prime and a, b, and c are
positive integers such that ap+bp=cp, then a corresponding equation y2 = x(x - ap)(x + bp)
defines a hypothetical elliptic curve, called the Frey curve, which must exist
if there is a counterexample to Fermat's Last Theorem. Following on work by
Yves Hellegouarch who first considered this curve, Frey pointed out that if such a
curve existed it had peculiar properties, and suggested in particular that it might not be modular.
A connection between Taniyama-Shimura and Fermat was made by Ken Ribet, following on
work by Barry Mazur and Jean-Pierre Serre, with his proof of the epsilon
conjecture showing that Frey's idea that the Frey curve could not be modular
was correct. In particular, this showed that a proof of the semistable case of
the Taniyama-Shimura conjecture would imply Fermat's Last Theorem. Wiles made
the decision that he would work exclusively on the Taniyama-Shimura conjecture shortly after he had learned that Ribet had proven the epsilon conjecture in 1986.
While many mathematicians thought the Taniyama-Shimura conjecture was inaccessible,
Wiles resolved to follow that approach.
When Wiles first began studying Taniyama-Shimura, he would casually mention Fermat to
people, but he found that doing so created too much interest. He wanted to be
able to work on his problem in a concentrated fashion, and if people were expressing
too much interest then he would not have been able to focus on his problem. Consequently he let only Nicholas Katz know what he was working on. Wiles did not do
any research that was not related to Taniyama-Shimura, though of course he did
continue in his teaching duties at Princeton University; continuing to attend
seminars, lecture undergraduates, and give tutorials.
https://m.sciencenet.cn/blog-3468-9859.html
上一篇：英国的合唱艺术
下一篇：遥远的水乡不再遥远
town together with his friend Thilo and other guys. The city is wisely divided
into two functioning areas, old town and new town. All the ancient buildings like
King’s summer palace, women’s church and opera house are in the old town;
while
the restaurants, bars and other modern buildings are in the new town.
Each is in harmony with its environment and cultural atmosphere. We went to an
Italian restaurant. The sentence on its menu board reads:”think globally, eat
locally”, which roughly means that even it is a local restaurant, it offers global
food options. It is true. After ordering the food, it turns out that six of
us ordered totally different food: Kay ordered American style fish salad with
chips, Thilo ordered Greece style salad with bread, Martin ordered beef steak cooked in North European way, I ordered Chinese style fried beef and vegetable with
rice. It was still summer then. We sit around a long table under the shade of
the vine in the backyard of the house. It was green everywhere, with green plants
covering the wall, lovely flowers blooming in the small Garden. There was a well
for fetching water, just like any ordinary local residence here. Several big
candles were burning since it is getting dark. A man in the casual dress was
playing his guitar and singing lovely Italian songs.
After the dinner, with beers in hands, Thilo began to tell his stories when he once
was a researcher in the math department of Princeton University. He said,
academic jobs in the university are quite demanding. You have to play a multiple
roles in one person, as a researcher at first hand, also as a lecturer to teach
courses for undergraduates and postgraduates; a fund raiser to get money for your
research group; a manager to handle with quite a few Postdocs and Phds in your
group; a faculty member for the service in the department; a paper reviewer or
an editor for your academic community. You don’t need to be smart to know how
busy an academia is. “publish or perish”, the cliche still works now days.
Without enough papers published, without enough founding, you will fail before this
strict rule, “up or out” , if you don’t get your tenure within six years.
“ Then how those big names do their research?”, somebody asked.
“ Today I will reveal a secret”, he paused.
“ What secret? Don’t be mean, please tell us”, we said.
“ OK. Do you guys know Fermat's Last Theorem?”
“ Yes, we do”
“ Do you know who proved this theorem?”
“ Aha, you call it secret? Everyone here knows that is it is Prof. Andrew Wiles
who spent about 10 years to prove it. The final proof from him came in publication
in 1994”
“ It is correct. However, do you guys know how Prof. Andrew Wiles found these 10 years
to dedicate himself to the Great Fermat theorem?”, he signed, “Prof. Andrew Wiles
told me by himself, in order to focus on the proof of the Fermat's Last Theorem,
there was one year in which he worked extremely hard to write 20 papers and locked
them up in his desk drawer. Then he would pick up two to publish each year.
In this way, he gained precious ten years to allow himself to do nothing else but Fermat's Last Theorem”
Nobody spoke anything for a while. Everybody was pondering on something. Now days in
the era of science research being measured by SCI, Impact Factors, funding
committees etc., most researchers would not risk themselves to focus on some
true problems which demands some deep insights and long commitments. This is a
simple fact in our current funding system. No wonder, in a report to the US president by President’s Information Technology Advisory Committee, named as
“Computational Science: Ensuring America’s Competitiveness”, a group of leading
scientists expressed their concerns:
“Based on its analysis of Federal R&D agency activities, PITAC concluded that Federal
support for computational science research has been overly focused on
short-term, low-risk activities. In the long term, this is actually a high-risk
strategy that is less likely to yield the high-payoff, strategic innovations
needed for the future.”
Now we are in the age of competition: everything is required to be done faster;
everybody is required to produce more with less time. Nobody knows what the end of
this road is; Nobody knows whether it is the right way. It is more or less to
make people to feel nostalgia about the golden days of science in the past time,
before NSF or any other funding committees are established. For example,
in Cambridge University, after becoming a member of the faculty, you have the freedom
to do whatever you like to do within the university’s resources. You don’t need apply
any special funding for it. Nobody will evaluate your research every 2~3 years.
However, those golden time is gone. Now we can not undo what we had already done.
More importantly, we can not back to the age of doing science without complicated
devices and giant machines, which are essential for the progress of bio-science
and nano-technologies etc.
What could we do? It is impossible to ask researchers to work harder, like
Prof. Wiles did his pre-work before the proof of the Fermat's Last Theorem. Some young
researchers have already died prematurely for their overworking, such as
Dr. He Yong in Zhejiang University. We also could not change the system of competitive
application for funding and the peer-review, which have been proved to be the
pillars for the current progress of science. Maybe there is NO solution for
everybody in general. Each one has to create your own means to meet your own ends. Just like what Prof. Wiles did, we should know what our real interest is and hold it
dear to our heart always, and then try to do the best from the least.
Whenever I read the Steve Jobs’ commencement address in Stanford, I could not help
being moved by his “three stories”, which are linked together to make his life
shine like a diamond. He is a real man, a true hero who sticks to his ideals
always. At the end his speech, he said:
“When I was young, there was an amazing publication called The Whole Earth Catalog,
which was one of the bibles of my generation. It was created by a fellow named
Stewart Brand not far from here in Menlo Park, and he brought it to life with his
poetic touch. This was in the late 1960's, before personal computers and desktop
publishing, so it was all made with typewriters, scissors, and polaroid cameras.
It was sort of like Google in paperback form, 35 years before
Google came along: it was idealistic, and overflowing with neat tools and great notions.
Stewart and his team put out several issues of The Whole Earth Catalog, and then
when it had run its course, they put out a final issue. It was the mid-1970s,
and I was your age. On the back cover of their final issue was a photograph of
an early morning country road, the kind you might find yourself hitchhiking on
if you were so adventurous. Beneath it were the words: "Stay Hungry. Stay
Foolish." It was their farewell message as they signed off. Stay Hungry. Stay
Foolish. And I have always wished that for myself. And now, as you graduate to
begin anew, I wish that for you.
Stay Hungry. Stay Foolish. “
”
Reference:
1. Report to the US president,
“ Computational Science: Ensuring America’s Competitiveness”,
President’s Information Technology Advisory Committee,
http://www.nitrd.gov/pitac/reports/20050609_computational/computational.pdf
2. Report about He Yong, http://www.39.net/HotSpecial/zz/89198.html
3. 'You've got to find what you love', Commencement address by Steve Jobs,
CEO of Apple Computer and of Pixar Animation Studios, delivered on June 12, 2005, http://news-service.stanford.edu/news/2005/june15/jobs-061505.html
4. Jobs Speech, Chinese Translation: http://v35.blog.sina.com.cn/panshiyi#serial_4679dbbf050000f1
Appendix:
Andrew Wiles
From Wikipedia, the free encyclopedia
Source: http://en.wikipedia.org/wiki/Andrew_Wiles
Fermat's Last Theorem states that no nontrivial integer solutions exist for the equation: xn + yn = zn if n is an integer greater than two.
____________________________________
Andrew Wiles' most famous mathematical result is that all rational semistable
elliptic curves are modular which, in particular, implies Fermat's Last Theorem.
Wiles was introduced to Fermat's Last Theorem at the age of ten. He tried to prove
the theorem using textbook methods and later studied the work of mathematicians
who had tried to prove it. When he began his graduate studies he stopped trying to
prove it and began studying elliptic curves under the supervision of John Coates.
In the 1950s and 1960s a connection between elliptic curves and modular forms was
conjectured by the Japanese mathematician Goro Shimura based on some ideas that
Yutaka Taniyama posed. In the West it became well known through a paper by André Weil.
With Weil giving conceptual evidence for it, it is sometimes called the
Shimura-Taniyama-Weil conjecture. It states that every rational elliptic curve
is modular. The full conjecture was proven by Christophe Breuil, Brian Conrad,
Fred Diamond, and Richard Taylor in 1998 using many of the methods that Andrew Wiles
used in his 1995 published papers.
The bridge between Fermat and Taniyama If p is an odd prime and a, b, and c are
positive integers such that ap+bp=cp, then a corresponding equation y2 = x(x - ap)(x + bp)
defines a hypothetical elliptic curve, called the Frey curve, which must exist
if there is a counterexample to Fermat's Last Theorem. Following on work by
Yves Hellegouarch who first considered this curve, Frey pointed out that if such a
curve existed it had peculiar properties, and suggested in particular that it might not be modular.
A connection between Taniyama-Shimura and Fermat was made by Ken Ribet, following on
work by Barry Mazur and Jean-Pierre Serre, with his proof of the epsilon
conjecture showing that Frey's idea that the Frey curve could not be modular
was correct. In particular, this showed that a proof of the semistable case of
the Taniyama-Shimura conjecture would imply Fermat's Last Theorem. Wiles made
the decision that he would work exclusively on the Taniyama-Shimura conjecture shortly after he had learned that Ribet had proven the epsilon conjecture in 1986.
While many mathematicians thought the Taniyama-Shimura conjecture was inaccessible,
Wiles resolved to follow that approach.
When Wiles first began studying Taniyama-Shimura, he would casually mention Fermat to
people, but he found that doing so created too much interest. He wanted to be
able to work on his problem in a concentrated fashion, and if people were expressing
too much interest then he would not have been able to focus on his problem. Consequently he let only Nicholas Katz know what he was working on. Wiles did not do
any research that was not related to Taniyama-Shimura, though of course he did
continue in his teaching duties at Princeton University; continuing to attend
seminars, lecture undergraduates, and give tutorials.
https://m.sciencenet.cn/blog-3468-9859.html
上一篇：英国的合唱艺术
下一篇：遥远的水乡不再遥远
