$pointcarre conjecture$

#Pointcarre conjecture series

It asserts that if any loop in a closed three-dimensional space without boundary can be shrunk to a point without tearing either the loop or the space, then the space is equivalent to a three-dimensional sphere. The PoincarГ© conjecture concerns the three-dimensional equivalent of this situation. We know then that the doughnut is not equivalent to the sphere. Through the hole, then there is no way of pulling the string so that it shrinks to a point without breaking either the string or the doughnut. On the other hand, if you lay the same string on the surface of a doughnut,

Imagine placing a piece of string around a football in a closed loop you can pull it back slowly and shrink the loop down to a point without tearing the ball or the string and without it leaving the surface. To understand the conjecture, think of two-dimensional spaces, like the surface of a football or of a doughnut. For example, a coffee mug (with a handle) and a doughnut are deemed to be the same because we can deform one into the other, without any tearing. It is sometimes described as 'rubber sheet geometry', since there is no notion of distance. So what does the conjecture say? It belongs to the field of topology, a branch of mathematics which deals with shapes.

This book provides a long-overdue popular account of the problem. The rest of the world, meanwhile, was left wondering what the conjecture actually means and why it is so important. Mathematicians around the world were left trying to expand To the frustration of the mathematical world, but much to the delight of the media, Perelman soon disappeared from the mathematical radar and even declined the most prestigious prize of mathematics: the Fields Medal. He embarked on a tour of the USA, explaining the main ideas contained in his papers - they seemed sound but lacked some detail.

#Pointcarre conjecture series

Mathematician Grigory Perelman posted a series of online papers claiming to have solved the problem. Acknowledged as one of the most important open questions in mathematics, and endowed by the Clay Mathematics Institute with a reward of $1,000,000 for the first correct solution, it had bugged mathematicians for over a century. The PoincarГ© conjecture is one of the few mathematical results that has managed to catch the interest of the mainstream media. The PoincarГ© Conjecture: In search of the shape of the universe Donal O'Shea