Poincare’s Conjecture Awesome Topology 18/07/2019 Dr Ray Adams

The Poincaré Conjecture Explained “Every simply connected, closed 3** – manifold is homeomorphic to the 3** – sphere.” Simply connected space. This is a space that has no holes.  (x) = 0 Closed space. This is a space that is finite and has no boundaries. A 2 - sphere (s2) is a closed space. Manifold. A manifold is a small neighbourhood that approximates Euclidean space. Homeomorphic. Two spaces are homeomorphic to each other if you can continuously deform one to the other. ** The number indicates the number of dimensions 18/07/2019 Dr Ray Adams

Simply connected A topological space is called simply connected (or 1-connected) if it is path-connected and every path between two points can be continuously transformed, staying within the space, into any other such path while preserving the two endpoints.   A sphere is simply connected because every loop can be contracted (on the surface) to a point. Four-Dimensional Sphere 18/07/2019 Dr Ray Adams

Genus -- from Wolfram MathWorld mathworld.wolfram.com/Genus.html Genus. A topologically invariant property of a surface defined as the largest number of nonintersecting simple closed curves that can be drawn on the surface without separating it. It can be seen as the number of holes in a surface or as a parameter for a closed surface in topology equal to the number of handles added to a sphere to form the surface. A sphere has genus 0, a torus, genus 1, etc Genus -- from Wolfram MathWorld mathworld.wolfram.com/Genus.html http://www.rwgrayprojects.com/Lynn/DoubleTorus/dt01.html 18/07/2019 Dr Ray Adams

Henri Poincaré Conjecture The Poincaré Conjecture has been a holy grail for mathematicians since 1904, only solved by Perelman in 2002. “Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.” OR “Any three dimensional body floating in four dimension that contains no holes and is not twisted can be morphed into a three dimensional sphere also floating in four dimensions.” Szpiro, G. (2007). Poincaré’s Prize. New York. Dutton.   → 18/07/2019 Dr Ray Adams

≠ A sphere versus a ball Spherical surface ball 18/07/2019 Spherical surface is a geometrical locus in a space, that is a totality of all points, equally removed from one point O, which is called a centre ( Fig.90 ). It has a radius and a circumference. Ball (sphere) is a solid body, bounded by a spherical surface. It also has a radius and a circumference. ≠ Spherical surface ball 18/07/2019 Dr Ray Adams

A three dimensional ball has a two dimensional boundary 2 – sphere (s2) A three dimensional ball has a two dimensional boundary   An ordinary sphere (or 2-sphere) is a two-dimensional surface that forms the boundary of a ball that is in three dimensions. 18/07/2019 Dr Ray Adams

3 – sphere (s3) A four dimensional ball has a three dimensional boundary i.e. a 3 – sphere A 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space. Just as an ordinary sphere (or 2-sphere) is a two-dimensional surface that forms the boundary of a ball in three dimensions, a 3-sphere is an object with three dimensions that forms the boundary of a ball in four dimensions. 18/07/2019 Dr Ray Adams

Henri Poincaré Conjecture If a closed 3-dimensional manifold has a trivial fundamental group, must it be homeomorphic to the 3-sphere? A fundamental group is the set of parameters of a topological space. A trivial fundamental group relates to a simply connected group.   Any finite 3-dimensional space, which doesn’t have any “holes” in it, can be continuously deformed into a 3-sphere. http://laplacian.wordpress.com/2010/03/13/the-poincare-conjecture/ 18/07/2019 Dr Ray Adams

Perelman’s solution The fundamental group of the n-sphere is equivalent to the group with one element (the identity element), because any path can always “shrink” down into a single point due to the fact the n-sphere has no hole. If a space’s fundamental group is trivial, then that means it is simply connected, as described above. Perelman used Ricci flow plus two other techniques designed to overcome Hamilton’s problems with singularities and cigars.   18/07/2019 Dr Ray Adams

The proof The conjecture asks whether the same is true for the 3-sphere, which is an object living naturally in four dimensions. This question motivated much of modern mathematics, especially in the field of topology. 18/07/2019 Dr Ray Adams

Perelman’s solutions Ricci flow tends to make curvature in 2D more uniform, approaching a sphere, a torus or a multiple genus torus. A bumpy surface smoothes out. In 3D, the process is complicated by the creation of singularities such as narrow necks or horns that can form and pinch off.   18/07/2019 Dr Ray Adams

  In 3D, the Ricci flow process can be complicated by the creation of singularities such as narrow necks or horns that can form and pinch off. 18/07/2019 Dr Ray Adams

singularities 18/07/2019 Dr Ray Adams Perelman proved that any singularity can be avoided 18/07/2019 Dr Ray Adams

18/07/2019 Dr Ray Adams

Where is topology useful????? Image: “Kaluza - Klein Space” by Professor Brad Dowden, California State University, Sacramento This figure illustrates the concept of (7 + 4 =11) dimensions, of which seven are compactified. Shown here are a set of spheres placed at several points on a plane. Each spheres is associated with one point on the plane. Return to start http://dissertationreviews.org/archives/1773 Dr Ray Adams 18/07/2019

G. Perelman, The entropy formula for the Ricci flow and its geometric applications, 2002 {http://arxiv.org/abs/math/0211159} G. Perelman, Ricci flow with surgery on three-manifolds, 2003 {http://arxiv.org/abs/math/0303109} G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, 2003 {http://arxiv.org/abs/math/0307245} Bruce Kleiner and John Lott, Notes on Perelman's Papers (May 2006) (fills in the details of Perelman's proof of the geometrization conjecture). {http://arxiv.org/abs/math/0605667 } Cao, Huai-Dong; Zhu, Xi-Ping (June 2006). "A Complete Proof of the Poincaré and Geometrization Conjectures: Application of the Hamilton-Perelman theory of the Ricci flow". Asian Journal of Mathematics 10 (2): 165–498. doi:10.4310/AJM.2006.v10.n2.a2. Retrieved 2006-07-31.  Cite uses deprecated parameters Revised version (December 2006): Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture http://arxiv.org/pdf/math/0612069v1.pdf 18/07/2019 Dr Ray Adams

A polychoron (plural: polychora) is a figure in four dimensions A polychoron (plural: polychora) is a figure in four dimensions. The word comes from Greek poly, which means many and choros which means room, or space. RETURN TO START The tesseract* is the best known polychoron, containing eight cubic cells, three around each edge. It is viewed here as a Schlegel diagram projection** into 3-space, distorting the regularity, but keeping its topological continuity. The eighth cell projects into the volume of space exterior to the boundary. * A Tesseract is a four-dimensional object, much like a cube is a three-dimensional object. ** projection of a polytope from a point beyond one of its facets or faces (Schlegel diagram). 18/07/2019 Dr Ray Adams

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