![]() ![]() Manifold – At every small neighborhood on the space, it approximates Euclidean space. A disk is also not because even though it is finite, it has a boundary. A sphere (more technically a 2-sphere or ) is closed, but the plane ( ) is not because it is infinite. Technically we can say but I’ll explain this notation further on.Ĭlosed space – The space is finite and has no boundaries. Simply connected space – This means the space has no “holes.” A football is simply connected, but a donut is not. Let’s define each of the terms in the conjecture: This is a statement about topological spaces. Here is the statement of the conjecture from wikipedia:Įvery simply connected, closed 3-manifold is homeomorphic to the 3-sphere. Most explanations of the problem are overly-simplistic or overly-technical, but on this blog I try to hit a nice middle-ground. It was proved by Grigori Perelman who subsequently turned down the $1 million prize money, left mathematics, and moved in with his mother in Russia. ![]() The Poincaré Conjecture is first and only of the Clay Millennium problems to be solved. ![]()
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May 2023
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