Mathematical Breakthrough of the Century
13 views
Skip to first unread message
JohnEB
Mar 25, 2012, 7:23:25 AM3/25/12
to classica...@googlegroups.com
While Bohr's QM has led us to the totally useless Multiverse, see http://www.math.columbia.edu/~woit/wordpress/?cat=10
Einstein's work has led to 4-space unification of physics and fantastic progress in mathematics:
On May 24, 2000, the Clay Mathematics Institute established The Millennium Prize Problems:
--------------------------------------------------------------------------------
In
order to celebrate mathematics in the new millennium, The Clay Mathematics
Institute of Cambridge, Massachusetts (CMI) established seven Prize Problems.
The Prizes were conceived to record some of the most difficult problems with
which mathematicians were grappling at the turn of the second millennium; to
elevate in
the consciousness of the general public the fact that in mathematics,
the frontier is still open and abounds in important unsolved problems; to
emphasize the importance
of working towards a solution of the deepest, most
difficult problems; and to recognize achievement in mathematics of historical
magnitude.
The prizes were announced at a meeting in Paris, held on May 24, 2000 at the Collage de France. Three lectures were presented: Timothy Gowers spoke on
The prizes were announced at a meeting in Paris, held on May 24, 2000 at the Collage de France. Three lectures were presented: Timothy Gowers spoke on
The Importance of Mathematics; Michael Atiyah and John Tate
spoke on the problems themselves. The seven Millennium Prize Problems
were chosen by the founding
Scientific Advisory Board of CMI, which conferred
with leading experts worldwide. The focus of the board was on important classic
questions that have resisted solution
for many years.
Follwing the decision of the Scientific Advisory Board, the Board of Directors of CMI designated a $7 million prize fund for the solution to these problems, with $1 million
Follwing the decision of the Scientific Advisory Board, the Board of Directors of CMI designated a $7 million prize fund for the solution to these problems, with $1 million
allocated to the solution of each problem.
It is of note that one of the seven Millennium Prize Problems, the Riemann hypothesis, formulated in 1859, also appears in the list of twenty-three problem discuss in the
address
given in Paris by David Hilbert on August 9, 1900.
http://www.claymath.org/millennium/
--------------------------------------------------------------------------------
Einstein's work on relativity led to the concept of Ricci flow which in turn led to the Mathematical Breakthrough of the Century:
Grigory Perelman, who did the Breakthrough of the Century, is explained in the book named
PERFECT RIGOR
A Genius and the Mathematical Breakthrough of the Century
by MASHA GESSEN
MASHA GESSEN opens for us two windows, one into the mind of a genius, and the other into the world of mathematics in Russia.
http://www.claymath.org/millennium/
--------------------------------------------------------------------------------
Einstein's work on relativity led to the concept of Ricci flow which in turn led to the Mathematical Breakthrough of the Century:
Grigory Perelman, who did the Breakthrough of the Century, is explained in the book named
PERFECT RIGOR
A Genius and the Mathematical Breakthrough of the Century
by MASHA GESSEN
MASHA GESSEN opens for us two windows, one into the mind of a genius, and the other into the world of mathematics in Russia.
Both offer
fascinating views -- Mario Livio, author of Is God a Mathematician? and The
Golden Ratio
The following is from PERFECT RIGOR:
PROLOGUE
A Problem for a Million Dollars
Numbers cast a magic spell over all of us, but mathematicians are especially skilled at imbuing figures with meaning. In the year 2000, a group of the world's leading
mathematicians gathered in Paris for a meeting that they believed would be momentous. They would use this occasion to take stock of their field. They would discuss
the sheer beauty of mathematics—a value that would be understood and appreciated by everyone present. They would take the time to reward one another with praise
and, most critical, to dream. They would together try to envision the elegance, the substance, the importance of future mathematical accomplishments.
The Millennium Meeting had been convened by the Clay Mathematics Institute, a nonprofit organization founded by Boston-area businessman Landon Clay and his wife,
Lavinia, for the purposes of popularizing mathematical ideas and encouraging their professional exploration. In the two years of its existence, the institute had set up
a beautiful office in a building just outside Harvard Square in Cambridge, Massachusetts, and had handed out a few research awards. Now it had an ambitious plan for
the future of mathematics, "to record the problems of the twentieth century that resisted challenge most successfully and that we would most like to see resolved,"
as Andrew Wiles, the British number theorist who had famously conquered Fermat's Last Theorem, put it. "We don't know how they'll be solved or when: it may be five
years or it may be a hundred years. But we believe that somehow by solving these problems we will open up whole new vistas of mathematical discoveries and landscapes:'
As though setting up a mathematical fairy tale, the Clay Institute named seven problems, a magic number in many folk traditions, and assigned the fantastical value of one
The following is from PERFECT RIGOR:
PROLOGUE
A Problem for a Million Dollars
Numbers cast a magic spell over all of us, but mathematicians are especially skilled at imbuing figures with meaning. In the year 2000, a group of the world's leading
mathematicians gathered in Paris for a meeting that they believed would be momentous. They would use this occasion to take stock of their field. They would discuss
the sheer beauty of mathematics—a value that would be understood and appreciated by everyone present. They would take the time to reward one another with praise
and, most critical, to dream. They would together try to envision the elegance, the substance, the importance of future mathematical accomplishments.
The Millennium Meeting had been convened by the Clay Mathematics Institute, a nonprofit organization founded by Boston-area businessman Landon Clay and his wife,
Lavinia, for the purposes of popularizing mathematical ideas and encouraging their professional exploration. In the two years of its existence, the institute had set up
a beautiful office in a building just outside Harvard Square in Cambridge, Massachusetts, and had handed out a few research awards. Now it had an ambitious plan for
the future of mathematics, "to record the problems of the twentieth century that resisted challenge most successfully and that we would most like to see resolved,"
as Andrew Wiles, the British number theorist who had famously conquered Fermat's Last Theorem, put it. "We don't know how they'll be solved or when: it may be five
years or it may be a hundred years. But we believe that somehow by solving these problems we will open up whole new vistas of mathematical discoveries and landscapes:'
As though setting up a mathematical fairy tale, the Clay Institute named seven problems, a magic number in many folk traditions, and assigned the fantastical value of one
million dollars for
each one's solution. The reigning kings of mathematics gave lectures summarizing
the problems. Michael Francis Atiyah, one of the previous century's most
influential mathematicians, began by outlining the Poincare Conjecture,
formulated by Henri Poincare in 1904. The problem was a classic of mathematical
topology. "It's
been worked on by many famous mathematicians, and it's still
unsolved," stated Atiyah. "There have been many false proofs. Many people have
tried and have made mistakes.
Sometimes they discovered the mistakes
themselves, sometimes their friends discovered the mistakes." The audience,
which no doubt contained at least a couple of people
who had made mistakes while
tackling the Poincare, laughed.
Atiyah suggested that the solution to the problem might come from physics. "This is a kind of clue or hint, by the teacher who cannot solve the problem to the student who is
trying to solve it," he joked. Several members of the audience were indeed working on problems that they hoped might move mathematics closer to a victory over the Poincare.
But no one thought a solution was near.
True, some mathematicians conceal their preoccupations when they're working on famous problems, as Wiles had done while he was working on Fermat's Last, but generally
Atiyah suggested that the solution to the problem might come from physics. "This is a kind of clue or hint, by the teacher who cannot solve the problem to the student who is
trying to solve it," he joked. Several members of the audience were indeed working on problems that they hoped might move mathematics closer to a victory over the Poincare.
But no one thought a solution was near.
True, some mathematicians conceal their preoccupations when they're working on famous problems, as Wiles had done while he was working on Fermat's Last, but generally
they stay abreast of one
another's research. And though putative proofs of the Poincare Conjecture had
appeared more or less annually, the last major breakthrough dated
back almost
twenty years, to 1982, when the American Richard Hamilton laid out a blueprint
for solving the problem. He had found, however, that his own plan for the
solution,
what mathematicians call a program, was too difficult to follow, and
no one else had offered a credible alternative. The Poincare Conjecture, like
Clay's other Millennium Problems,
might never be solved.
Solving any one of these problems would be nothing short of a heroic feat. Each had claimed decades of research time, and many a mathematician had gone to the grave
Solving any one of these problems would be nothing short of a heroic feat. Each had claimed decades of research time, and many a mathematician had gone to the grave
having
failed to solve the problem with which he or she had struggled for years. "The
Clay Mathematics Institute really wants to send a clear message, which is that
mathematics is mainly valuable because of these immensely difficult problems,
which are like the Mount Everest or the Mount Himalaya of mathematics," said the
French
mathematician Alain Connes, another twentieth-century giant. "And if we
reach the peak, first of all, it will be extremely difficult; we might even pay
the price of our lives or
something like that. But what is true is that when we
reach the peak, the view from there will be fantastic."
As unlikely as it was that anyone would solve a Millennium Problem in the foreseeable future, the Clay Institute nonetheless laid out a clear plan for giving each award.
As unlikely as it was that anyone would solve a Millennium Problem in the foreseeable future, the Clay Institute nonetheless laid out a clear plan for giving each award.
The
rules stipulated that the solution to the problem would have to be presented in
a refereed journal, which was, of course, standard practice. After publication,
a two-year
waiting period would begin, allowing the world mathematics community
to examine the solution and arrive at a consensus on its veracity and
authorship. Then a committee
would be appointed to make a final recommendation
on the award. Only after it had done so would the institute hand over the
million dollars. Wiles estimated that it would
take at least five years to
arrive at the first solution, assuming that any of the problems was actually
solved, so the procedure did not seem at all cumbersome.
Just two years later, in November 2002, a Russian mathematician posted his proof of the Poincare Conjecture on the Internet. He was not the first person to claim he'd solved
Just two years later, in November 2002, a Russian mathematician posted his proof of the Poincare Conjecture on the Internet. He was not the first person to claim he'd solved
the Poincare--he was not even the only Russian to post a putative proof
of the conjecture on the Internet that year, but his proof turned out to be
right.
And then things did not go according to plan, not the Clay Institute's plan or any other plan that might have struck a mathematician as reasonable. Grigory Perelman, the
And then things did not go according to plan, not the Clay Institute's plan or any other plan that might have struck a mathematician as reasonable. Grigory Perelman, the
Russian, did not publish his work in a
refereed journal. He did not agree to vet or even to review the explications of
his proof written by others. He refused numerous job
offers from the world's
best universities. He refused to accept the Fields Medal, mathematics' highest
honor, which would have been awarded to him in 2006. And then
he essentially
withdrew from not only the world's mathematical conversation but also most of
his fellow humans' conversation.
Perelman's peculiar behavior attracted the sort of attention to the Poincare Conjecture and its proof that perhaps no other story of mathematics ever had. The unprecedented
Perelman's peculiar behavior attracted the sort of attention to the Poincare Conjecture and its proof that perhaps no other story of mathematics ever had. The unprecedented
magnitude of the award
that apparently awaited him helped heat up interest too, as did a sudden
plagiarism controversy in which a pair of Chinese mathematicians claimed
they
deserved the credit for proving the Poincare. The more people talked about
Perelman, the more he seemed to recede from view; eventually, even people who
had once
known him well said that he had "disappeared," although he continued to
live in the St. Petersburg apartment that had been his home for many years. He
did occasionally pick
up the phone there, but only to make it clear that he
wanted the world to consider him gone.
When I set out to write this book, I wanted to find answers to three questions: Why was Perelman able to solve the conjecture; that is, what was it about his mind that set him
When I set out to write this book, I wanted to find answers to three questions: Why was Perelman able to solve the conjecture; that is, what was it about his mind that set him
apart from all the
mathematicians who had come before? Why did he then abandon mathematics and, to
a large extent, the world? Would he refuse to accept the Clay prize
money, which
he deserved and most certainly could use, and if so, why?
This book was not written the way biographies usually are. I did not have extended interviews with Perelman. In fact, I had no conversations with him at all. By the time I
This book was not written the way biographies usually are. I did not have extended interviews with Perelman. In fact, I had no conversations with him at all. By the time I
started working on this project, he had cut off communication with all
journalists and most people. That made my job more difficult. I had to imagine
a person I had literally
never met, but also more interesting: it was an
investigation. Fortunately, most people who had been close to him and to the
Poincare Conjecture story agreed to talk to me.
In fact, at times I thought it
was easier than writing a book about a cooperating subject, because I had no
allegiance to Perelman's own narrative and his vision of himself
except to try
to figure out what it
was.
--------------------------------------------------------------------------------
Poincare Conjecture
http://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture
--------------------------------------------------------------------------------
Poincare Conjecture
http://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture
Solution of the Poincare conjecture
http://en.wikipedia.org/wiki/Solution_of_the_Poincar%C3%A9_conjecture
Notes and commentary on Perelman's Ricci flow
papers
http://math.berkeley.edu/~lott/ricciflow/perelman.html
http://math.berkeley.edu/~lott/ricciflow/perelman.html
Ricci Flow and John Morgan - Choose Mathematician
http://en.wikipedia.org/wiki/John_Morgan
http://en.wikipedia.org/wiki/John_Morgan
Christina Sormani http://comet.lehman.cuny.edu/sormani/
is an expert on Riemannian Geometry which is the basis of Einstein's General Relativity. She played a key role in
Perelman's visit to the U.S. in 2003. In fact, she was one of the few people that Perelman would talk to.
Sormani's Notes on Perelman's Lectures
http://comet.lehman.cuny.edu/sormani/others/perelman/perelman.html
is an expert on Riemannian Geometry which is the basis of Einstein's General Relativity. She played a key role in
Perelman's visit to the U.S. in 2003. In fact, she was one of the few people that Perelman would talk to.
Sormani's Notes on Perelman's Lectures
http://comet.lehman.cuny.edu/sormani/others/perelman/perelman.html
Riemannian Geometry
http://en.wikipedia.org/wiki/Riemannian_geometry
http://en.wikipedia.org/wiki/Riemannian_geometry
Reply all
Reply to author
Forward
0 new messages