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What isWhat isThe Poincaré Conjecture?
Alex KarassevAlex Karassev
ContentContent
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
Henri PoincarHenri Poincaréé((April 29, 1854 – July 17, 1912)April 29, 1854 – July 17, 1912)
Mathematician, physicist, philosopher
Created the foundations of Topology Chaos Theory Relativity Theory
Millennium ProblemsMillennium 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
The PoincarThe Poincaréé conjecture (1904) conjecture (1904)
Conjecture:Every closedsimply connected3-dimensional manifoldis homeomorphic to the3-dimensional sphere
What do these words mean?
Why is The Conjectue Important?Why is The Conjectue Important?
Geometry of The Universe
New directions in mathematics
The Study of SpaceThe 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!)
The Study of SpaceThe 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
Magellan'sMagellan's Journey Journey
August 10, 1519 — September 6, 1522
Start: about 250 men
Return: about 20 men
The Study of SpaceThe 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)
The Study of SpaceThe Study of Space
Universe has finite volume?
Bounded Universe?
However, no "edge"
A possible model:three-dimensional sphere!
What is 3-dim sphere?What is 3-dim sphere?
What is 2-dim sphere?
R
What is 3-dim sphere?What is 3-dim sphere?
The set of points in 4-dim spaceon the same distance from a given pointTake two solid balls and
glue their boundariestogether
WavesWaves
Amplitude
Wavelength
FrequencyFrequency
Short wavelength – High frequency
Long wavelength – Low frequency
high-pitched sound
low-pitched sound
Doppler EffectDoppler Effect
Stationarysource
Movingsource
Higher pitch
Wavelength and colorsWavelength and colors
Wavelength
RedshiftRedshift
Star at rest Moving Star
RedshiftRedshift
Distance
Expanding Universe?Expanding Universe?
The Big Bangtheory
Time
Alexander Friedman,1922
Georges-HenriLemaître, 1927
Edwin Hubble, 1929
Bounded and expanding?Bounded and expanding?
Spherical Universe?
Three-Dimensional sphere(balloon) is inflating
Infinite and Expanding?Infinite and Expanding?
Not quite correct!
(it appears that the Universe has an "edge")
Infinite and Expanding?Infinite and Expanding?
Big Bang
Distancesincrease – The Universestretches
Is a cylinder flat?Is a cylinder flat?
R
2πr
Triangle on a cylinderTriangle on a cylinder
α + β + γ = 180o
γ
β
αγ
β
α
Sphere is not flatSphere is not flat
γ
β
α
α + β + γ > 180o
90o
90o
90o
Sphere is not flatSphere is not flat
???
How to tell a sphere from planeHow 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
Flat and bounded?Flat and bounded?
Torus…
Flat and bounded?Flat and bounded?
Torus…and Flat Torus
A B
A B
3-dim Torus3-dim Torus
Section – flat torus
Torus UniverseTorus Universe
Assumptions about the UniverseAssumptions 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 everywhereSo it must have constant curvature
Shape of the Universe is the same everywhereSo it must have constant curvature
Constant curvature KConstant curvature K
Plane K =0 Sphere K>0
(K = 1/R2)
γ
β
α
α + β + γ >180o α + β + γ =180o α + β + γ < 180o
γβ
αγ
β
α
Pseudosphere (part of Hyperbolic plane)
K<0
Three geometries …Three geometries …and Three models of the Universeand Three models of the Universe
Plane K =0
K > 0
α + β + γ >180o α + β + γ =180o α + β + γ < 180o
Elliptic Euclidean Hyperbolic(flat)
K = 0 K < 0
What happens if we try to "flatten"What happens if we try to "flatten"a piece of pseudosphere?a piece of pseudosphere?
How to tell a torus from a How to tell a torus from a sphere?sphere?
First, compare a plane and a plane with a hole
?
Simply connected surfacesSimply connected surfaces
Simply connected Not simply connected
Homeomorphic Homeomorphic objectsobjects
continuous deformation of one object to another
≈ ≈ ≈
≈
≈ ≈
HomeomorphismHomeomorphism
≈
≈
HomeomorphismHomeomorphism
≈
HomeomorphismHomeomorphism
Can we cut?Can we cut?
Yes, if we glue after
So, a knotted circle is the same as So, a knotted circle is the same as usual circle!usual circle!
≈
The Conjecture…The Conjecture…
Conjecture:Every closedsimply connected3-dimensional manifoldis homeomorphic to the3-dimensional sphere
2-dimensional case2-dimensional case
Theorem (Poincare) Every closed
simply connected2-dimensional manifoldis homeomorphic to the2-dimensional sphere
Higher-dimensional versions of Higher-dimensional versions of the Poincare Conjecturethe 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
Perelman's proofPerelman'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
Fields MedalFields Medal
On August 22, 2006, Perelman was awarded the medal at the International Congress of Mathematicians in Madrid
Perelman declined to accept the award
Detailed ProofDetailed 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
Further readingFurther reading
"The Shape of Space"by Jeffrey Weeks
"The mathematics ofthree-dimensional manifolds"by William Thurston and Jeffrey Weeks(Scientific American, July 1984, pp.108-120)
Thank you!
http://www.nipissingu.ca/numeric
http://www.nipissingu.ca/topology