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

Shapes, Spaces and Spheres

Take 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.

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