Background notes
... Let M be a smooth manifold and assume dim M = n. (If different components of M have different dimensions, then make this construction one component at a time.) One of the most important consequences of the smooth structure is the tangent bundle, the collection of tangent spaces ...
... Let M be a smooth manifold and assume dim M = n. (If different components of M have different dimensions, then make this construction one component at a time.) One of the most important consequences of the smooth structure is the tangent bundle, the collection of tangent spaces ...
Boundary manifolds of projective hypersurfaces Daniel C. Cohen Alexander I. Suciu
... Suppose A is a finite-dimensional, graded, connected algebra over an algebraically closed field k of characteristic 0. For each a ∈ A1 , multiplication by a defines a cochain complex (A, a). The resonance varieties of A are the jumping loci for the cohomology of these complexes: Rkd (A) = {a ∈ A1 | ...
... Suppose A is a finite-dimensional, graded, connected algebra over an algebraically closed field k of characteristic 0. For each a ∈ A1 , multiplication by a defines a cochain complex (A, a). The resonance varieties of A are the jumping loci for the cohomology of these complexes: Rkd (A) = {a ∈ A1 | ...
Document
... Double-Edged Straightedge Conjecture If two parallel lines are intersected by a second pair of parallel lines that are the same distance apart as the first pair, then the parallelogram formed is a rhombus. ...
... Double-Edged Straightedge Conjecture If two parallel lines are intersected by a second pair of parallel lines that are the same distance apart as the first pair, then the parallelogram formed is a rhombus. ...
On Colimits in Various Categories of Manifolds
... computation, construction, and induction. Proofs often go by constructing some horrendously complicated object (usually via a tower of increasingly complicated objects) and then proving inductively that we can understand what’s going on at each step and that in the limit these steps do what is requi ...
... computation, construction, and induction. Proofs often go by constructing some horrendously complicated object (usually via a tower of increasingly complicated objects) and then proving inductively that we can understand what’s going on at each step and that in the limit these steps do what is requi ...
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.