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Poincare Conjecture Solved

The PoincarÃ© Conjecture, a 100-year-old mathematical conundrum, has been solved. The proof by the Russian mathematician Grigori Perelman, which he first presented in 2002, has been confirmed by an international group of mathematicians whose findings were presented to participants at a conference held at the Abdus Salam International Centre for Theoretical Physics (ICTP) in Trieste, Italy, on 17 June. The 60 participants, more than half from the developing world, reaffirmed the approving judgement of the mathematicians.

Solving this endlessly complex problem, which is one of the seven Millennium Problems in mathematics announced by the Clay Mathematics Institute in the United States in 2000, would make Perelman eligible for a US$1 million cash prize.

The PoincarÃ© Conjecture, first stated by the great French mathematician and physicist Henri PoincarÃ© in 1904, focuses on the relationship of shapes, spaces and surfaces.

Consider a beach ball. Near each point on its surface it looks like a two-dimensional plane. But at a distance, of course, it is a round two-dimensional sphere. Mathematicians refer to the ball as a “two-dimensional manifold that is compact and connected".

Now slide a rubber band around the beach ball holding a finger firm 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 mathematicians view a two-dimensional sphere as a “compact, connected and simply connected two-dimensional manifold". PoincarÃ© Conjecture characterized abstract three-dimensional spheres in the same way.

From another perspective, think of Earth’s inhabitants 500 years ago. For them, the Earth was flat. Today, astrophysicists imagine our universe may be infinite. In fact, we can envision another possibility: that we are living on a three-dimensional sphere. PoincarÃ© characterized this sphere 100 years ago and now Perelman has provided the proof.


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