What is The Poincar é Conjecture?

1. What isThe 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 closedsimply connected3-dimensional manifoldis homeomorphic to the3-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 (Alook 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 • University has finite volume? • Bounded Universe? • However, no "edge" • A possible model:three-dimensional sphere!

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

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

14. Amplitude Wavelength Waves

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

16. Stationarysource Movingsource Doppler Effect Higher pitch

17. Wavelength and colors Wavelength

18. Redshift Moving Star Star at rest

19. Redshift Distance

20. Expanding Universe? Alexander Friedman,1922 The Big Bangtheory Time Georges-HenriLemaître, 1927Edwin 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? Distancesincrease – The Universestretches Big Bang

24. R Is a cylinder flat? 2πr

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

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

27. Sphere is not flat ???

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

29. Torus… Flat and bounded?

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

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

35. Three geometries …and Three models of the Universe Plane K =0 Elliptic Euclidean Hyperbolic (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 Simply connected 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 closedsimply connected3-dimensional manifoldis homeomorphic to the3-dimensional sphere

46. 2-dimensional case • Theorem (Poincare) • Every closedsimply connected2-dimensional manifoldis homeomorphic to the2-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 Huaidongpublished 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