Subject: WSJ on the Poincare conjecture
From: "Ron Umble"
Date: Mon, 24 Jul 2006 10:33:06 -0400
[A shorter version appeared as an AP story.]
The Wall Street Journal
21 July 2006
SCIENCE JOURNAL
Major Math Problem Is Believed Solved By Reclusive Russian
By Sharon Begley
920 words
A9
FOR SIX YEARS, $7 million in prize money has lay unclaimed at the Clay
Mathematics Institute in Cambridge, Mass., waiting for someone to solve
any of
the seven "millennium prize problems," the oldest of which has been
kicking
around since 1859. Despite periodic claims, it looked like the institute
would
hold on to the cash until after the sun burned out.
But the math world is abuzz over the very real possibility that one
millennium
problem, the Poincare conjecture, has been proved by a mathematician in
Russia.
After nearly four years of scrutiny by other mathematicians, the work
holds up,
even though Grigori Perelman's work is decidedly unusual.
In 2002 and 2003, he posted two papers to an online archive. Usually, a
posting
serves a flag-planting function -- "I solved this first!" -- until the
paper is
published in a journal, which can take years. But as the math community
waited
for him to follow up his postings, a realization set in. Dr. Perelman,
long
affiliated with the Steklov Institute of Mathematics in St. Petersburg,
apparently has no intention of saying more. He probably feels he proved
the
Poincare conjecture, mathematicians surmise, and has no interest in the $1
million bounty. (He did not respond to emailed requests for comment.)
Dr. Perelman's style is reminiscent of the Sid Harris cartoon of a board
filled
with equations and, at a key step, the words, "then a miracle occurs." One
mathematician tells the other, "I think you should be more explicit here
in step
two."
THE CONJECTURE Henri Poincare posited in 1904 is the most famous problem
in
topology, the branch of math that analyzes the shape of objects and space.
He
claimed, "if a closed 3-dimensional manifold has trivial fundamental
group, [it
must be] homeomorphic to the 3-sphere," as John Milnor of Stony Brook
University
puts it.
Translated, that means that if you wrap one rubber band around the surface
of an
orange and another around a doughnut, and shrink down both, the rubber
bands act
differently. The one around the orange keeps shrinking without tearing or
leaving the surface. The one around the doughnut can't, without breaking
itself
or the doughnut. This difference says something profound about the
structure of
space itself.
Many mathematicians have claimed to prove Poincare, but the claims flamed
out
immediately, their fatal flaws obvious. Dr. Perelman's proof has survived.
The
dilemma for the Clay Institute is that, according to its rules, a proof
must be
published in a refereed math publication. The archives aren't refereed.
Putting his proof online rather than in a journal is only one example of
Dr.
Perelman's iconoclasm [strange use of the word -pjf]. He admits that he
gives
only "a sketch of an eclectic proof of" a more general conjecture from
which
Poincare's follows; he never mentions Poincare. The papers are difficult
to
understand, and sketchy in the extreme. He asserts that one can prove
something
by a variation on an earlier argument, but it isn't clear what the
variation
is. "Perelman's papers are written in a style rather different from what
would
appear in a journal," says mathematician Bruce Kleiner of Yale University.
The sketchiness may reflect how a genius interacts with mortals. Dr.
Perelman
may believe some things are so obvious he needn't bother to explain them
step by
step, say mathematicians. If readers are too dumb to fill in the blanks,
he
doesn't care. Or, he has better things to do than justify every tortuous
step,
as proofs must.
Others have taken it upon themselves to explicate his work -- and find no
major
flaws. Like Torah commentaries, they dwarf the original. Dr. Perelman's
2003
paper is 22 pdf pages; the 2002 paper is 39. But "Notes on Perelman's
Papers,"
in which Prof. Kleiner and John Lott of the University of Michigan explain
them
almost line-by-line, is 192 pages. A book on the papers is expected to top
300
pages. A "complete proof" of Poincare, based on Dr. Perelman's
breakthrough and
published last month in the Asian Journal of Mathematics (which Prof.
Milnor
describes as throwing "a monkey wrench" into the question of who gets
credit),
is 328 pages long.
ODDLY, EITHER THE book or the Kleiner-Lott paper might count as the
"refereed"
work the Clay Institute demands. If so, we would have the weird situation
in
which authors of the work that satisfies the prize requirement aren't the
people
who figured out the proof. But their efforts could win Dr. Perelman $1
million.
"It's definitely an unusual situation, but what's important is that the
person
who made the breakthrough put it out there so the community could
scrutinize and
analyze it," says institute president, James Carlson.
Dr. Perelman shuns the limelight, but is known through lectures in the
U.S. and
for getting a perfect score at the 1982 International Mathematical
Olympiad, at
age 16. He isn't expected at the quadrennial meeting of the International
Congress of Mathematicians, in Madrid. There, the Fields Medal, math's
Nobel
Prize, will be awarded to the "outstanding" mathematician 40 or under. Dr.
Perelman is the odds-on favorite.
And the millennium prizes? "I don't think the other six will be solved in
my
lifetime," says Dr. Carlson. "But then, I didn't think the Poincare
conjecture
would be solved either."