1. What is The Poincaré Conjecture?
Alex Karassev

2. Content Henri Poincaré Millennium Problems
Poincaré Conjecture – exact statement
Why is the Conjecture important …and what do the words mean?
The Shape of The Universe
About the proof of The Conjecture

3. Henri Poincaré (April 29, 1854 – July 17, 1912)
Mathematician, physicist, philosopher
Created the foundations of
Topology
Chaos Theory
Relativity Theory

4. Millennium Problems
The Clay Mathematics Institute of Cambridge, Massachusetts has named seven Prize Problems
Each of these problems is VERY HARD
Every prize is $ 1,000,000
There are several rules, in particular
solution must be published in a refereed mathematics journal of worldwide repute
and it must also have general acceptance in the mathematics community two years after

5. The Poincaré conjecture (1904)
Conjecture: Every closed simply connected 3-dimensional manifold is homeomorphic to the 3-dimensional sphere
What do these words mean?

6. Why is The Conjectue Important?
Geometry of The Universe
New directions in mathematics

7. The Study of Space
Simpler problem: understanding the shape of the Earth!
First approximation: flat Earth
Does it have a boundary (an edge)?
The correct answer "The Earth is "round" (spherical)" can be confirmed after first space travels (A look from outside!)

8. The Study of Space Nevertheless, it was obtained a long time before!
First (?) conjecture about spherical shape of Earth: Pythagoras (6th century BC)
Further development of the idea: Middle Ages
Experimental proof: first circumnavigation of the earth by Ferdinand Magellan

9. Magellan's Journey August 10, 1519 — September 6, 1522
Start: about 250 men
Return: about 20 men

10. The Study of Space What is the geometry of the Universe?
We do not have a luxury to look from outside
"First approximation": The Universe is infinite (unbounded), three-dimensional, and "flat" (mathematical model: Euclidean 3-space)

11. The Study of Space Universe has finite volume? Bounded Universe?
However, no "edge"
A possible model: three-dimensional sphere!

12. What is 3-dim sphere?
What is 2-dim sphere? R

13. Take two solid balls and glue their boundaries together
What is 3-dim sphere?
The set of points in 4-dim space on the same distance from a given point
Take two solid balls and
glue their boundaries together

14. Waves
Amplitude
Wavelength

15. Frequency high-pitched sound low-pitched sound
Short wavelength – High frequency
low-pitched sound
Long wavelength – Low frequency

16. Doppler Effect
Stationary source
Moving source
Higher pitch

17. Wavelength and colors
Wavelength

18. Redshift
Star at rest
Moving Star

19. Redshift
Distance

20. Expanding Universe? Alexander Friedman,1922 The Big Bang theory
Time
Georges-Henri Lemaître, Edwin Hubble, 1929

21. Bounded and expanding? Spherical Universe?
Three-Dimensional sphere (balloon) is inflating

22. Infinite and Expanding?
Not quite correct! (it appears that the Universe has an "edge")

23. Infinite and Expanding?
Distances increase –
The Universe stretches
Big Bang

24. Is a cylinder flat? R
2πr

25. Triangle on a cylinder
α + β + γ = 180o β β γ α γ α

26. Sphere is not flat
α + β + γ > 180o
90o γ β α

27. Sphere is not flat
???

28. How to tell a sphere from plane
1st method: Plane is unbounded
2nd method: Sum of angles of a triangle
What is triangle on a sphere?
Geodesic – shortest path

29. Flat and bounded?
Torus…

30. Flat and bounded?
Torus…
and Flat Torus A B A B

31. 3-dim Torus
Section – flat torus

32. Torus Universe

33. Assumptions about the Universe
Homogeneous
matter is distributed uniformly (universe looks the same to all observers)
Isotropic
properties do not depend on direction (universe looks the same in all directions )
Shape of the Universe is the same everywhere
So it must have constant curvature

34. Pseudosphere (part of Hyperbolic plane) K<0
Constant curvature K
Pseudosphere (part of Hyperbolic plane) K<0
Sphere K>0 (K = 1/R2)
Plane K =0 γ β α γ β α β γ α
α + β + γ >180o α + β + γ =180o α + β + γ < 180o

35. Three geometries … and Three models of the Universe
Elliptic Euclidean Hyperbolic
Plane K =0
(flat)
K = 0
K > 0
K < 0
α + β + γ >180o α + β + γ =180o α + β + γ < 180o

36. What happens if we try to "flatten" a piece of pseudosphere?

37. How to tell a torus from a sphere?
First, compare a plane and a plane with a hole ?

38. Simply connected surfaces
Not simply connected

39. ≈ ≈ ≈ ≈ ≈ ≈ Homeomorphic objects
continuous deformation of one object to another ≈ ≈ ≈ ≈ ≈ ≈

40. Homeomorphism ≈ ≈

41. Homeomorphism ≈

42. Homeomorphism

43. Can we cut?
Yes, if we glue after

44. So, a knotted circle is the same as usual circle!≈

45. The Conjecture…
Conjecture: Every closed simply connected 3-dimensional manifold is homeomorphic to the 3-dimensional sphere

46. 2-dimensional case Theorem (Poincare)
Every closed simply connected 2-dimensional manifold is homeomorphic to the 2-dimensional sphere

47. Higher-dimensional versions of the Poincare Conjecture
… were proved by:
Stephen Smale (dimension n ≥ 7 in 1960, extended to n ≥ 5) (also Stallings, and Zeeman) Fields Medal in 1966
Michael Freedman (n = 4) in 1982, Fields Medal in 1986

48. Perelman's proof
In 2002 and 2003 Grigori Perelman posted to the preprint server arXiv.org three papers outlining a proof of Thurston's geometrization conjecture
This conjecture implies the Poincaré conjecture
However, Perelman did not publish the proof in any journal

49. Fields Medal
On August 22, 2006, Perelman was awarded the medal at the International Congress of Mathematicians in Madrid
Perelman declined to accept the award

50. Detailed Proof
In June 2006, Zhu Xiping and Cao Huaidong published a paper "A Complete Proof of the Poincaré and Geometrization Conjectures - Application of the Hamilton-Perelman Theory of the Ricci Flow" in the Asian Journal of Mathematics
The paper contains 328 pages
Чжу Сипин и Цао Хуайдун, Shing-Tung Yau

51. Further reading "The Shape of Space" by Jeffrey Weeks
"The mathematics of three-dimensional manifolds" by William Thurston and Jeffrey Weeks (Scientific American, July 1984, pp )

52. Thank you! http://www.nipissingu.ca/numeric