My Research Diary

I have been reading recently about Dr. Grigory Perelman and also about poincare conjecture but I am not able to understand the problem itself. The problem statement is for 3 dimensional manifold poincare conjecture. It seems problem of conjecture with 4 or more manifolds has already been solved by Dr. Stephen Smale in 1960. This is what I have understood so far. manifold in mathematical sense here refers to mathematical space similar to Euclidean space with complex dimensional geometry. I would presume as the dimensions increase the space will be more complicated. So like line would be one dimensional and plane would be two-dimensional. If that is the case what makes 3 dimensional manifolds special ? Poincare conjecture states that any closed simply-connected 3 dimensional manifold is homomorphic (continuous stretching and bending of the object into a new shape. Thus, a square and a circle are homeomorphic) to the standard 3 dimensional sphere. I think the problem is to prove if any simply-connected 3 dimensional object can be continuously stretched to form 3 dimensional sphere without ripping it. So you can not transform a doughnut into a sphere without ripping it but other 3 dimensional shapes can be transformed into 3D sphere. Dr. Pereleman has published 3 papers on this outlining how to prove this. They are available here , here and here . Clay institute of mathematics has laid out millennium grand challenges, one of which is poincare conjecture. Read more here .