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What is What is The Poincaré Conjecture? Alex Karassev.

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1 What is 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 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 Short wavelength – High frequency Long wavelength – Low frequency high-pitched sound low-pitched sound

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? The Big Bang theory Time Alexander Friedman,1922 Georges-Henri Lemaître, 1927 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? Big Bang Distances increase – The Universe stretches

24 Is a cylinder flat? R 2πr2πr

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

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

27 Sphere is not flat ???

28 How to tell a sphere from plane 1 st method: Plane is unbounded 2 nd 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 Shape of the Universe is the same everywhere So it must have constant curvature

34 Constant curvature K Plane K =0 Sphere K>0 (K = 1/R 2 ) γ β α α + β + γ >180 o α + β + γ =180 o α + β + γ < 180 o γ β α γ β α Pseudosphere (part of Hyperbolic plane) K<0

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

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 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 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

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!

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