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

2. 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

3. 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

4. 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
18/07/2019
Dr Ray Adams

5. 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

6. ≠ 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

7. 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

8. 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

9. 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.
18/07/2019
Dr Ray Adams

10. 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

11. 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

12. 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

13. 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

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

15. 18/07/2019
Dr Ray Adams

16. 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
Dr Ray Adams
18/07/2019

17. G. Perelman, The entropy formula for the Ricci flow and its geometric applications, 2002{
G. Perelman, Ricci flow with surgery on three-manifolds, 2003 {
G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, 2003 {
Bruce Kleiner and John Lott, Notes on Perelman's Papers (May 2006) (fills in the details of Perelman's proof of the geometrization conjecture).
{ }
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: /AJM.2006.v10.n2.a2. Retrieved   Cite uses deprecated parameters
Revised version (December 2006): Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture
18/07/2019
Dr Ray Adams

18. 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

19. AN ADVERT (page 1 of 3) 18/07/2019 Dr Ray Adams
18/07/2019
Dr Ray Adams

20. AN ADVERT (page 2 of 3) 18/07/2019 Dr Ray Adams
18/07/2019
Dr Ray Adams

21. AN ADVERT (page 3 of 3) 18/07/2019 Dr Ray Adams
18/07/2019
Dr Ray Adams