Download What is the Poincaré Conjecture?

What is
The Poincaré Conjecture?
Alex Karassev
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
Henri Poincaré
(April 29, 1854 – July 17, 1912)
Mathematician, physicist,
philosopher
Created the foundations of
 Topology
 Chaos Theory
 Relativity Theory
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
The Poincaré conjecture (1904)
Conjecture:
Every closed
simply connected
3-dimensional manifold
is homeomorphic to the
3-dimensional sphere
What do these words mean?
Why is The Conjectue Important?
Geometry of The Universe
New directions in mathematics
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!)
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
Magellan's Journey
August 10, 1519 — September 6, 1522
Start: about 250 men
Return: about 20 men
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), threedimensional, and "flat"
(mathematical model: Euclidean 3-space)
The 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 2-dim sphere?
R
What is 3-dim sphere?
The set of points in 4-dim space
on theTake
same
distance
fromand
a given point
two
solid balls
glue their boundaries
together
Waves
Amplitude
Wavelength
Frequency
high-pitched sound
Short wavelength – High frequency
low-pitched sound
Long wavelength – Low frequency
Doppler Effect
Stationary
source
Higher pitch
Moving
source
Wavelength and colors
Wavelength
Redshift
Star at rest
Moving Star
Redshift
Distance
Expanding Universe?
Alexander Friedman,1922
The Big Bang
theory
Georges-Henri
Lemaître, 1927
Time
Edwin Hubble, 1929
Bounded and expanding?
Spherical Universe?
Three-Dimensional sphere
(balloon) is inflating
Infinite and Expanding?
Not quite correct!
(it appears that the Universe has an "edge")
Infinite and Expanding?
Big Bang
Distances
increase –
The Universe
stretches
Is a cylinder flat?
R
2πr
Triangle on a cylinder
α + β + γ = 180
β
β
α
γ
α
γ
o
Sphere is not flat
α + β + γ > 180
o
90o
β
α γ
90o
90o
Sphere is not flat
???
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
Flat and bounded?
Torus…
Flat and bounded?
Torus… and Flat Torus
A
B
A
B
3-dim Torus
Section – flat torus
Torus Universe
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
Constant curvature K
Pseudosphere
Plane K =0
Sphere K>0
(part of Hyperbolic plane)
(K = 1/R2)
K<0
β
α
β
γ
α + β + γ >180
α
o
β
α
γ
α + β + γ =180
o
γ
α + β + γ < 180
o
Three geometries …
and Three models of the Universe
Plane K =0
Elliptic
Euclidean
Hyperbolic
(flat)
K=0
K>0
α + β + γ >180
o
α + β + γ =180
K<0
o
α + β + γ < 180
o
What happens if we try to "flatten"
a piece of pseudosphere?
How to tell a torus from a
sphere?
 First, compare a plane and a plane with a hole
?
Simply connected surfaces
Simply connected
Not simply connected
Homeomorphic objects
continuous deformation of one object to another
≈
≈
≈
≈
≈
Homeomorphism
≈
Homeomorphism
≈
Homeomorphism
Can we cut?
Yes, if we glue after
So, a knotted circle is the same as
usual circle!
≈
The Conjecture…
Conjecture:
Every closed
simply connected
3-dimensional manifold
is homeomorphic to the
3-dimensional sphere
2-dimensional case
Theorem (Poincare)
 Every closed
simply connected
2-dimensional manifold
is homeomorphic to the
2-dimensional sphere
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
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
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
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
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.108-120)
Thank you!
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