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# News from ICTP 113 - Features - Poincare

*The Poincaré Conjecture, one of the toughest problems
in mathematics, has apparently been solved.*

**Shapes, Spaces and Spheres**

**T**ake a ball. Any ball. Hold it.
Bounce it. Catch it. For most of us, a ball is a ball is a ball.

Now ask a mathematician to take a ball. For him or her, a ball---or
more precisely, a sphere---remains a sphere even when it is deflated-even
when its shape becomes unrecognisable, even when it resembles
a pancake that becomes difficult to move, let alone bounce.

That is because for mathematicians a sphere (the ball's surface)
is defined not only by its shape but also by the 'essence of its
shape', and that ultimately means it doesn't even have to be round.

This seemingly odd definition, which mathematicians refer to as
the topology of the sphere, is due to the fact that mathematicians
aim to describe---or, perhaps more importantly, identify---an
object such as a ball not only by its shape (or its 'essence of
shape'), but also algebraically.

That is to say mathematicians move between the world of numbers
and the world of physical shapes in elegant and seamless 'ways
of equivalence' that enable the findings of one world to accurately
represent the findings of the other world.

This is comparable in its dynamics to the simultaneous translation
of different languages. Such thinking is especially important
in mathematics because sometimes mathematicians can describe and
identify objects algebraically or graphically without being able
to see them in the physical world in which we all live---and sometimes
we can all see objects in our physical world to which mathematicians
cannot assign a mathematical or graphical equivalent. In short,
objects often get lost in translation.

That brings us to the Poincaré Conjecture, a mathematical
conundrum first stated by the great French mathematician and physicist
Henri Poincaré in 1904, which focusses on the relationship
of shapes, spaces and surfaces in spheres such as a ball---and
a desire to express this relationship in equivalent ways whether
the form of expression is algebraic or geometric.

*Henri Poimcaré*

With the Poincaré Conjecture in mind, let's take that ball
in hand (or, more precisely, in mind) again, not in an effort
to bounce it but to understand how the surface of the ball (a
sphere) can be perceived (translated, if you will) in all its
possible configurations.

To successfully enter this world, we must suspend our conventional
notion of what a ball is and instead define it abstractly as a
set of points that are the same distance from a fixed point.

A ball (or, more precisely, a sphere) on the playing field of
mathematics assumes the characteristics not just of a round object
but also of mathematical equations and diverse geometric forms.
Mathematicians, by turning to topology, can indeed have a sphere
that is as flat as a pancake---and, even more unusual, a sphere
that may indeed look flat even when round.

In Poincaré's world (and now ours), each point on the ball's
surface, close up, looks like a two-dimensional plane. Mathematicians
refer to the sphere as a 'two-dimensional manifold that is compact
and connected.'

Now, returning from topology to reality, slide a rubber band around
the ball holding a finger firmly on a single point. By pulling
the rubber band around the ball, you can retract it to the point
being held by your finger. Mathematicians call this property 'simple
connectedness,' which applies to all two-dimensional spheres.
It's for this reason that mathematicians view a two-dimensional
sphere as a 'compact, connected *and* simply connected two-dimensional
manifold.'

The Poincaré Conjecture characterises abstract three-dimensional
spheres in the same way as described above for two-dimensional
spheres---that is, it is locally like three-dimensional space
(or, as mathematicians prefer to say, it is a 3-manifold); it
is connected (you can find your way from one point to another);
it is compact (the distance between points is confined); and it
is *simply connected*.

However, since this abstract sphere resides in a dimension that
is beyond our perceived reality, Poincaré forced himself
and his colleagues to turn to algebra to state the conjecture---and
that's where challenge has resided for more than a century.

Indeed Poincaré first described this uncompromising, yet
unproven, characteristic of a 3-manifold sphere in 1904. The conundrum---how
to express algebraically what seemed irrefutable but impossible
to prove geometrically---remained unsolved until the Russian mathematician,
Grigori Perelman, first presented his proof in 2002.

And now Perelman's proof has been confirmed by an international
group of mathematicians whose findings were presented to participants
at a two-week conference, titled "Summer School and Conference
on Geometry and Topology of 3-Manifolds," held at ICTP this
past June. The 60 participants, more than half from the developing
world, reaffirmed the judgement of Perelman through a rigorous
assessment conducted by a team of reviewers worldwide who meticulously
divided their labour to examine each specific aspect of Perelman's
complex proof looking for flaws and shortcomings. They found none.

Putting to rest the Poincaré Conjecture, defined as one
of the seven Millennium Prize Problems in mathematics by the US-based
Clay Mathematics Institute in 2000, makes Perelman eligible for
a US$1 million cash prize, which the reclusive mathematician has
yet to say he will accept.

While riches may await Perelman, what does the Poincaré
Conjecture mean for those of us who are not mathematicians? After
all, the vast majority of people cannot even begin to understand
the nature of the problem, let alone the proof. The best way to
understand the importance of this mind-bending puzzle is not to
think in terms of its utility but rather in terms of the intellectual
challenge and the new perspectives that it provides on how we
may live in the universe.

Think, for example, of the Earth's inhabitants 500 years ago.
For them, with their close-up perspective, the Earth was flat.
Today, astrophysicists, with their wide-angle views, augmented
by telescopes that can see deep into space, have concluded that
our universe may indeed be infinite.

In fact, the Poincaré Conjecture allows us to envision
another possibility: that we live on a three-dimensional sphere
that enables our perspective(s) to be both close-up and far away---in
effect to be both myopic and infinite, thus laying equal claims
to our ancestors and today's visionaries.

WHO IS POINCARÉ?

French-born Henri Poincare (1854-1912) is widely recognised as
one of the world's greatest mathematicians and physicists, an
ingenious scientist who made numerous contributions to mathematics,
celestial and fluid mechanics, dynamical systems, the special
theory of relativity and the philosophy of science. Much of his
research involved interactions between different mathematical
topics. Indeed his broad understanding of the full spectrum of
knowledge allowed him to address problems from many different
angles. He is perhaps best known as the father of algebraic topology.
Yet Poincaré also contributed mightily to our understanding
of algebraic geometry and number theory. In applied mathematics
he studied optics, electricity and telegraphy, and in physics
he made major contributions to thermodynamics, potential theory,
quantum theory, theory of relativity and cosmology. The breadth
of Poincaré's knowledge led him to be the only member of
*Académie des Sciences *to be elected to each of the
five sections of the Academy: geometry, mechanics, physics, geography
and navigation. Much of this information is derived from www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Poincare.html.
For more details, consult this website.

WHO IS GRIGORI PERELMAN?

Russian-born Grigori Perelman, who has studiously avoided the
public, works at the Steklov Institute of Mathematics in St. Petersburg,
Russia. Perelman's proof of the Poincaré Conjecture allows
him to claim credit for one of the most sought-after mathematical
proofs of the past century, elevating him to the pinnacle of his
profession. By all accounts, Perelman couldn't care less. His
proof was quietly posted on the web as a 'preprint' three years
ago and he's been heard from only occasionally ever since. He
seldom attends conferences and he answers email only from fellow
mathematicians. Indeed it is not even clear how old he is, which
makes it impossible to determine whether he is eligible for the
Fields Medal (winners must be younger than 40). There is also
some indication, as yet unconfirmed, that he will not accept the
US$1 million prize from the Clay Mathematics Institute in Cambridge,
Massachusetts, if it is offered. Indeed he has yet to officially
publish his proof in a professional journal. Perelman is a mathematician
obsessively dedicated to his work, and such devotion pays off
in ways that only he and those who share his passion can understand.

AND WHAT IS MILLION DOLLAR MATH?

The Clay Mathematics Institute, a non-profit foundation established
in 1998 by Boston, Massachusetts, businessman Landon T. Clay,
is dedicated to increasing and disseminating mathematical knowledge.
In 2000, the institute launched the Millennium Prize Problems.
A cheque for US$1 million dollars awaits anyone who can solve
one of seven "classic questions that have resisted solution
over the years." For additional information about the Clay
Mathematics Institute, including a complete list of the Millennium
Prize Problems, see www.claymath.org.