STRATIFIED SPACES TWIGS 1. Introduction These
... Another example is provided by the variety P ⊂ C3 defined by the vanishing of the polynomial f = x2 + y 3 + z 5 . The origin O is a singular point; observe that the gradient of f vanishes there. Otherwise P is a manifold. In fact the hypersurface P is a very interesting example. Take a small sphere ...
... Another example is provided by the variety P ⊂ C3 defined by the vanishing of the polynomial f = x2 + y 3 + z 5 . The origin O is a singular point; observe that the gradient of f vanishes there. Otherwise P is a manifold. In fact the hypersurface P is a very interesting example. Take a small sphere ...
Badih Ghusayni, Half a dozen famous unsolved problems in
... on hard problems that will fill up a million volumes, if not more, and have opened up new fields of mathematics. Taking up this strategy the author, for many years, has looked at some of these six hard problems and indeed, obtained some results. But, of course, more needs to be done and, hopefully, ...
... on hard problems that will fill up a million volumes, if not more, and have opened up new fields of mathematics. Taking up this strategy the author, for many years, has looked at some of these six hard problems and indeed, obtained some results. But, of course, more needs to be done and, hopefully, ...
Name
... e. Assigning true (T) or false (F) to a statement, only when the hypothesis is true and the conclusion is false f. The statement formed by negating the hypothesis and the conclusion g. a statement that is believed to be true h. the process of reasoning that a rule or statement is true because specif ...
... e. Assigning true (T) or false (F) to a statement, only when the hypothesis is true and the conclusion is false f. The statement formed by negating the hypothesis and the conclusion g. a statement that is believed to be true h. the process of reasoning that a rule or statement is true because specif ...
Poincaré conjecture
In mathematics, the Poincaré conjecture (/pwɛn.kɑːˈreɪ/ pwen-kar-AY; French: [pwɛ̃kaʁe]) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. The conjecture states: Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. An equivalent form of the conjecture involves a coarser form of equivalence than homeomorphism called homotopy equivalence: if a 3-manifold is homotopy equivalent to the 3-sphere, then it is necessarily homeomorphic to it.Originally conjectured by Henri Poincaré, the theorem concerns a space that locally looks like ordinary three-dimensional space but is connected, finite in size, and lacks any boundary (a closed 3-manifold). The Poincaré conjecture claims that if such a space has the additional property that each loop in the space can be continuously tightened to a point, then it is necessarily a three-dimensional sphere. The analogous conjectures for all higher dimensions had already been proven.After nearly a century of effort by mathematicians, Grigori Perelman presented a proof of the conjecture in three papers made available in 2002 and 2003 on arXiv. The proof built upon the program of Richard Hamilton to use the Ricci flow to attempt to solve the problem. Hamilton later introduced a modification of the standard Ricci flow, called Ricci flow with surgery to systematically excise singular regions as they develop, in a controlled way, but was unable to prove this method ""converged"" in three dimensions. Perelman completed this portion of the proof. Several teams of mathematicians verified that Perelman's proof was correct.The Poincaré conjecture, before being proven, was one of the most important open questions in topology. In 2000, it was named one of the seven Millennium Prize Problems, for which the Clay Mathematics Institute offered a $1,000,000 prize for the first correct solution. Perelman's work survived review and was confirmed in 2006, leading to his being offered a Fields Medal, which he declined. Perelman was awarded the Millennium Prize on March 18, 2010. On July 1, 2010, he turned down the prize saying that he believed his contribution in proving the Poincaré conjecture was no greater than Hamilton's (who first suggested using the Ricci flow for the solution). As of 2015, the Poincaré conjecture is the only solved Millennium problem.On December 22, 2006, the journal Science honored Perelman's proof of the Poincaré conjecture as the scientific ""Breakthrough of the Year"", the first time this honor was bestowed in the area of mathematics.