Grigoriy Perelman 博士拒绝接收一百万美元奖金

克雷数学研究所(Clay
Mathematics Institute)网站

I would like to repeat
Jim's welcome to Landon and Lavinia Clay, to the entire Clay family, to members
of the Poincaré family and to all the mathematicians assembled here at this
historic occasion. At its meeting ten years ago here in

We made a conscious
choice, in contrast to Hilbert, to pick problems that were already formulated
and had stood the test of time. Of course, one big potential drawback might
have been that the problems would rest for a long time unsolved. Then
mathematics might appear static and the outside world might lose interest. The
time frame for the solution of the great problems of antiquity, such as the
squaring of the circle and the trisection of an angle was measured not in tens
or even hundreds of years, but rather in thousands of years. It has therefore
been a wonderful and perhaps unexpected surprise that we are able to be here
today to celebrate the first of the Clay problems to be solved. This is
especially true for this particular problem, which together with the Riemann
hypothesis, has been on everyone's top list of mathematical problems for all of
our lifetimes. For this we can thank Perelman, as well as his many
predecessors, including particularly

Famous Theorems define
the mathematical landscape. They beckon from afar, rising dimly in the mists,
an elusive challenge to the mathematical community. Are they accessible or is
there some vast ravine or raging torrent that has to be traversed ?

When, after many years of
exploration, pioneers reach the foothills, the ascent looks formidable or
impossible. Initial attempts lead to dead-ends yor a return to the starting
point. Explorations pack up and go home and prepare to tackle lesser heights.

But some mountaineers do
not give up. They examine every aspect of the climb, spending many years
identifying the optimal route. Then the final assault begins. Step by step the
paths are cleared, the rivers crossed and the cliffs scaled. Finally on a
glorious day the summit is reached, the mountain is tamed and a magnificent vista
is opened up.

Today we celebrate such
an event. A century after the death of Henri Poincaré, and in the city where he
lived and worked, the Conjecture which he bequeathed to us has been settled.
Grigory Perelman is the mountaineer who reached this pinnacle of the
3-dimensional world.

Exploration can be an end
in itself but it also offers the land for other developments. Crops can be
grown, cities built and art can flourish. Perelman has provided geometers with
a fruitful land to cultivate.

It is a very special
pleasure for me to have this occasion to publicly express my deep admiration
and appreciation for Grigori Perelman.

Over several years in the
1970's, I developed to a vision of 3-dimensional manifolds as fitting into a
beautiful geometric pattern, the geometrization conjecture, that became the
central focus of my life's work. At a symposium on Poincaré in 1980, I felt
emboldened to say that the geometrization conjecture put the Poincaré
conjecture into a fuller and more constructive context. I expressed confidence
that the geometrization conjecture is true, and I predicted it would be proven,
but whether in one year or 100 years I could not say – I hoped it would be
within my lifetime. I tried hard to prove it. I am truly gratified to see my
hope finally become reality.

Perelman, with tremendous
focus and virtuosity, constructed a beautiful proof where I and others failed.
It is a proof that I could not have done: some of Perelman's strengths are my
weaknesses. That the geometrization conjecture is true is not a surprise. That
a proof like Perelman's could be valid is not a surprise: it has a certain
rightness and inevitability, long dreamed of by many people (including me).
What is surprising, wonderful and amazing is that someone – Perelman –
succeeded in rigorously analyzing and controlling this process, despite the
many hurdles, challenges and potential pitfalls. His method begin with a
3-dimensional shape that is irregular, complicated and hard to analyze or take
in. The shape changes and evolves much like a bubble to even itself out,
quickly smoothing small-scale irregularities, following the Ricci flow as
developed by Richard Hamilton. Bubbles can pop: sometimes a bubble breaks up
and splits apart, but Perelman found ways to analyze and control this process,
to show that eventually all bubbles glide into a perfect form. Perelman's
accomplishment gives us a solid foundation to build higher levels of
understanding.

Perelman's aversion to
public spectacle and to riches is mystifying to many. I have not talked to him
about it and I can certainly not speak for him, but I want to say I have
complete empathy and admiration for his inner strength and clarity, to be able
to know and hold true to himself. Our true needs are deeper – -yet in our
modern society most of us reflexively and relentlessly pursue wealth, consumer
goods and admiration. We have learned from Perelman's mathematics. Perhaps we
should also pause to reflect on ourselves and learn from Perelman's attitude
toward life.

It is no mere convention
to say that it a great honour to be asked to speak here about the work of
Grigory Perelman. From the time when his preprints concerning the Poincaré and
Geometrisation Conjectures appeared, mathematicians around the world have been
united in expressing their appreciation, awe and wonder at his extraordinary
achievement, and I believe that I speak here as a representative of our whole
intellectual community.

There are many signal
qualities of Perelman's work. First, of course, it solves an outstanding,
century-old, problem: a problem that has done much to drive the development of
topology from its inception. Second, the work is, to the highest degree,
original and profound. He introduced an entirely new idea which cut the Gordian
knot that had held up the Ricci-flow approach – -bearing on the central
question of "collapsing" in Riemannian geometry. But that was just
the beginning – -Perelman developed a host of extremely subtle and novel
arguments: blending partial differential equations, differential geometry and
the theory of convergence of spaces. The whole edifice he created, in his
proof, is something unmatched, in its scope and depth, in this general area of
mathematics. The ideas and techniques will have ramifications in many other
problems for years to come. Third, there is a unique and romantic quality of
his work. In modern mathematics, as in other endeavours, much progress is
collaborative; either in the literal sense or, more generally, in developments
driven by resonance between the ideas of different workers. Perelman's
achievement is a testament to the continued power of the individual human mind
in bringing about the most fundamental advances in mathematics.

The great 19th century
mathematician Niels Henrik Abel wrote in his memoirs that in order to solve a
difficult problem one has to to correctly formulate it.

A correct formulation is,
usually, not as transparant and elementary as the original one – its
significance is seen only a posteriori when the problem is eventually solved.

For example, the great
achivement of 20th century mathematics – the proof of the
"elementary" Fermat Last Theorem was obtained via a reduction to the
Taniyama-Shimura-Weil conjecture, the very formulation of which relies on a
profound "non-elementary" structure underlying the naive concept of
an algebraic equation between integers. The solution of this conjecture by
Andrew Wiles provided an understanding of this structure uncomparably deeper
than the original message carried by the Fermat Theorem.

Similarly, Perelman has
done much more than proving the Poincaré conjecture or even the more
comprehensive Thurston's geometrization conjecture. What Perelman has achieved
is the fulfillment of Richard Hamilton's program, thus revealing a profound
structure in the space of 3-manifolds.

To get, by way of
analogy, an idea of what Perelman accomplished, imagine that you have no
overall picture of the geography of the Earth. You send one after another sea
expedition in which discover new lands. Eventually, six continents are
discovered. You keep sending hundreds of new expeditions, but they find nothing
besides these six. You conjecture that there are no other great land masses on
Earth. This is what Poincaré and Thurston say about the world of 3-manifolds:
there are no manifolds besides those which has been already discovered.

Perelman's theorem goes
far beyond proving this "non-existence" claim, just as Wiles' theorem
tells you much more than non-existence of integer solutions of certain
equations.

Perelman's work revealed
the laws of

It will probably take a
decade or so for the mathematical community to build up new edifices on the
land discovered by Perelman.