As the deformation from nonnegative to positive curvature is not well understood, the soul conjecture was proposed. In 1994, Perelman gave a short and elegant proof of the conjecture by establishing that in the general case , Sharafutdinov's retraction is a submersion. Three notable papers of Perelman's from 1994 to 1997 deal with the construction of various interesting Riemannian manifolds with positive Ricci curvature. The Poincaré conjecture, proposed by French mathematician Henri Poincaré in 1904, was one of the key problems in topology. Any loop on a 3-sphere—as exemplified by the set of points at a distance of 1 from the origin in four-dimensional Euclidean space—can be contracted into a point. The Poincaré conjecture asserts that any closed three-dimensional manifold, such that any loop can be contracted into a point, is topologically a 3-sphere. The analogous result has been known to be true in dimensions greater than or equal to five since 1960 as in the work of Stephen Smale. The four-dimensional case resisted longer, finally being solved in 1982 by Michael Freedman. But the case of three-manifolds turned out to be the hardest of them all. Roughly speaking, this is because in topologically manipulating a three-manifold there are too few dimensions to move "problematic regions" out of the way without interfering with something else. The most fundamental contribution to the three-dimensional case had been produced by Richard S. Hamilton. The role of Perelman was to complete the Hamilton program. In November 2002, Perelman posted the first of three preprints to the arXiv, in which he claimed to have outlined a proof of the geometrization conjecture, of which the Poincaré conjecture is a particular case. This was followed by the two other preprints in 2003. Perelman modified Richard S. Hamilton's program for a proof of the conjecture. The central idea is the notion of the Ricci flow. Hamilton's fundamental idea is to formulate a "dynamical process" in which a given three-manifold is geometrically distorted, with the distortion process governed by a differential equation analogous to the heat equation. The heat equation (which much earlier motivated Riemann to state his Riemann hypothesis on the zeros of the zeta function) describes the behavior of scalar quantities such as temperature. It ensures that concentrations of elevated temperature will spread out until a uniform temperature is achieved throughout an object.