talk 20040160 poincare
TRANSCRIPT
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7/30/2019 Talk 20040160 Poincare
1/76
PoincaresConjecture
Kim Taeik
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
Poincares ConjectureA picture for shape of Universe
Kim Taeik
KAIST Undergratuated
December 11, 2009
http://find/ -
7/30/2019 Talk 20040160 Poincare
2/76
PoincaresConjecture
Kim Taeik
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares ConjectureProof
Geometry
Euclidean
Riemannean
Geometrization
Question
What is the shape of the Universe?How can we know the shape of the Universe?
http://find/ -
7/30/2019 Talk 20040160 Poincare
3/76
PoincaresConjecture
Kim Taeik
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares ConjectureProof
Geometry
Euclidean
Riemannean
Geometrization
Shape of Earth
We have the world map
If we sail to east we will return
http://find/ -
7/30/2019 Talk 20040160 Poincare
4/76
PoincaresConjecture
Kim Taeik
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares ConjectureProof
Geometry
Euclidean
Riemannean
Geometrization
Shape of Earth
We have the world map
If we sail to east we will return
http://find/http://goback/ -
7/30/2019 Talk 20040160 Poincare
5/76
PoincaresConjecture
Kim Taeik
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares ConjectureProof
Geometry
Euclidean
Riemannean
Geometrization
Shape of Earth
We have the world map
If we sail to east we will return
http://find/http://goback/ -
7/30/2019 Talk 20040160 Poincare
6/76
PoincaresConjecture
Kim Taeik
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares ConjectureProof
Geometry
Euclidean
Riemannean
Geometrization
Shape of Earth
If we sail to north,we will return
http://find/ -
7/30/2019 Talk 20040160 Poincare
7/76
PoincaresConjecture
Kim Taeik
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares ConjectureProof
Geometry
Euclidean
Riemannean
Geometrization
Shape of Earth
If we sail to north,we will return
http://find/ -
7/30/2019 Talk 20040160 Poincare
8/76
PoincaresConjecture
Kim Taeik
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares ConjectureProof
Geometry
Euclidean
Riemannean
Geometrization
Shape of Earth
If we sail to north,we will return
http://find/ -
7/30/2019 Talk 20040160 Poincare
9/76
PoincaresConjecture
Kim Taeik
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares ConjectureProof
Geometry
Euclidean
Riemannean
Geometrization
Problem
Which is the shape of Earth?
http://find/ -
7/30/2019 Talk 20040160 Poincare
10/76
PoincaresConjecture
Kim Taeik
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares ConjectureProof
Geometry
Euclidean
Riemannean
Geometrization
Problem
Which is the shape of Earth?
Solution
Escape to the Universe! Earth is sphere!
http://find/ -
7/30/2019 Talk 20040160 Poincare
11/76
PoincaresConjecture
Kim Taeik
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares ConjectureProof
Geometry
Euclidean
Riemannean
Geometrization
Question
What is the shape of the Universe?How can we know the shape of the Universe?
http://find/ -
7/30/2019 Talk 20040160 Poincare
12/76
PoincaresConjecture
Kim Taeik
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares ConjectureProof
Geometry
Euclidean
Riemannean
Geometrization
Question
What is the shape of the Universe?How can we know the shape of the Universe?Solution
Escape from the universe!
http://find/ -
7/30/2019 Talk 20040160 Poincare
13/76
PoincaresConjecture
Kim Taeik
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares ConjectureProof
Geometry
Euclidean
Riemannean
Geometrization
Question
What is the shape of the Universe?How can we know the shape of the Universe?Solution
Escape from the universe!
Is it possible?
http://find/http://goback/ -
7/30/2019 Talk 20040160 Poincare
14/76
PoincaresConjecture
Kim Taeik
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares ConjectureProof
Geometry
Euclidean
Riemannean
Geometrization
Earth Problem
How can we determine the shape without escaping theEarth?
http://find/ -
7/30/2019 Talk 20040160 Poincare
15/76
PoincaresConjecture
Kim Taeik
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares ConjectureProof
Geometry
Euclidean
Riemannean
Geometrization
Property of Spherical shell
P i
http://find/http://goback/ -
7/30/2019 Talk 20040160 Poincare
16/76
PoincaresConjecture
Kim Taeik
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares ConjectureProof
Geometry
Euclidean
Riemannean
Geometrization
Property of Spherical shell
a rope
P i
http://find/ -
7/30/2019 Talk 20040160 Poincare
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PoincaresConjecture
Kim Taeik
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares ConjectureProof
Geometry
Euclidean
Riemannean
Geometrization
Property of Spherical shell
shrink
slide on surface
Poincares
http://find/ -
7/30/2019 Talk 20040160 Poincare
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Poincare sConjecture
Kim Taeik
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares ConjectureProof
Geometry
Euclidean
Riemannean
Geometrization
Property of Spherical shell
Poincares
http://find/ -
7/30/2019 Talk 20040160 Poincare
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Poincare sConjecture
Kim Taeik
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares ConjectureProof
Geometry
Euclidean
Riemannean
Geometrization
Property of Spherical shell
Poincares
http://find/ -
7/30/2019 Talk 20040160 Poincare
20/76
Poincare sConjecture
Kim Taeik
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares ConjectureProof
Geometry
Euclidean
Riemannean
Geometrization
Property of Spherical shell
Poincares
http://find/ -
7/30/2019 Talk 20040160 Poincare
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Poincare sConjecture
Kim Taeik
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares ConjectureProof
Geometry
Euclidean
Riemannean
Geometrization
Property of Spherical shell
Rope can shrink to point
Poincares
http://find/ -
7/30/2019 Talk 20040160 Poincare
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Conjecture
Kim Taeik
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares ConjectureProof
Geometry
Euclidean
Riemannean
Geometrization
Torus Surface
Poincares
http://find/ -
7/30/2019 Talk 20040160 Poincare
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Conjecture
Kim Taeik
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares ConjectureProof
Geometry
Euclidean
Riemannean
Geometrization
Torus Surface
Poincares
http://find/ -
7/30/2019 Talk 20040160 Poincare
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Conjecture
Kim Taeik
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares ConjectureProof
Geometry
Euclidean
Riemannean
Geometrization
Torus Surface
Poincares
http://find/ -
7/30/2019 Talk 20040160 Poincare
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Conjecture
Kim Taeik
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares ConjectureProof
Geometry
Euclidean
Riemannean
Geometrization
Torus Surface
PoincaresC j
http://find/ -
7/30/2019 Talk 20040160 Poincare
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Conjecture
Kim Taeik
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
Torus Surface
PoincaresC j t
http://find/http://goback/ -
7/30/2019 Talk 20040160 Poincare
27/76
Conjecture
Kim Taeik
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
Torus Surface
the rope cannot shrink
PoincaresConject re
http://find/http://goback/ -
7/30/2019 Talk 20040160 Poincare
28/76
Conjecture
Kim Taeik
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
Simply Connected
Spherical Shell = simply connected
PoincaresConjecture
http://find/ -
7/30/2019 Talk 20040160 Poincare
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Conjecture
Kim Taeik
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
Simply Connected
Spherical Shell = simply connected
Torus simply connected
PoincaresConjecture
http://find/ -
7/30/2019 Talk 20040160 Poincare
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Conjecture
Kim Taeik
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
2D Simply Connected
simply connected 2-D spherical shell.
PoincaresConjecture
http://find/ -
7/30/2019 Talk 20040160 Poincare
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Conjecture
Kim Taeik
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
Poincares Conjecture
closed(compact) 3-D : simply connected 3D sphere?
PoincaresConjecture
http://find/ -
7/30/2019 Talk 20040160 Poincare
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C j u
Kim Taeik
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
Question
What is the shape of the Universe?How can we know the shape of the Universe?Solution
Escape from the universe!
Is it possible?
PoincaresConjecture
http://find/ -
7/30/2019 Talk 20040160 Poincare
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j
Kim Taeik
Shape of theUniverse
Shape of the
UniverseShape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
Question
What is the shape of the Universe?How can we know the shape of the Universe?
Solution
Check simply connected.
PoincaresConjecture
G C
http://find/ -
7/30/2019 Talk 20040160 Poincare
34/76
Kim Taeik
Shape of theUniverse
Shape of the
UniverseShape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
Geometization Conjecture
Too many, too complicated...
PoincaresConjecture
G i i C j
http://find/ -
7/30/2019 Talk 20040160 Poincare
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Kim Taeik
Shape of theUniverse
Shape of the
UniverseShape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
Geometization Conjecture
Too many, too complicated...
Are there some structures?
PoincaresConjecture
G
http://find/http://goback/ -
7/30/2019 Talk 20040160 Poincare
36/76
Kim Taeik
Shape of theUniverse
Shape of the
UniverseShape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
Geometry
S3
E3
H
3
S2 R
H2 R
SL2
(R)Nil
Sol
PoincaresConjecture
G
http://find/ -
7/30/2019 Talk 20040160 Poincare
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Kim Taeik
Shape of theUniverse
Shape of the
UniverseShape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
Geometry
For exampleS3
E3
H
3
S2 R
H2 R
SL2
(R)Nil
Sol
http://find/ -
7/30/2019 Talk 20040160 Poincare
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PoincaresConjecture
L th f Li
-
7/30/2019 Talk 20040160 Poincare
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Kim Taeik
Shape of theUniverse
Shape of the
UniverseShape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
Length of Line
x
y
i
f
f(x)
PoincaresConjecture
L th f Li
http://find/ -
7/30/2019 Talk 20040160 Poincare
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Kim Taeik
Shape of theUniverse
Shape of the
UniverseShape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
Length of Line
x
y
i
f
f(x)
To calculate the length
fi
ds
PoincaresConjecture
Length of Line
http://find/http://goback/ -
7/30/2019 Talk 20040160 Poincare
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Kim Taeik
Shape of theUniverse
Shape of the
UniverseShape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
Length of Line
x
y
i
f
f(x)
To calculate the length
fi
ds
What is ds?
PoincaresConjecture
Euclidean Geometry
http://find/ -
7/30/2019 Talk 20040160 Poincare
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Kim Taeik
Shape of theUniverse
Shape of the
UniverseShape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
Euclidean Geometry
Axioms
PoincaresConjecture
Euclidean Geometry
http://find/ -
7/30/2019 Talk 20040160 Poincare
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Kim Taeik
Shape of theUniverse
Shape of the
UniverseShape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
Euclidean Geometry
Axioms1. to draw a straight line from any point to any point.2. to produce a finite straight line3. To describe a circle with any center and distance4. That all right angles are equal to one another5. The parallel postulate
PoincaresConjecture
Euclidean Geometry
http://find/ -
7/30/2019 Talk 20040160 Poincare
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Kim Taeik
Shape of theUniverse
Shape of the
UniverseShape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
Euclidean Geometry
5. The parallel postulate
PoincaresConjecture
Euclidean Geometry
http://find/ -
7/30/2019 Talk 20040160 Poincare
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Kim Taeik
Shape of theUniverse
Shape of the
UniverseShape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
Euclidean Geometry
5. The parallel postulate
Pythagoras theorem (ds)2 = (dx)2 + (dy)2
PoincaresConjecture
Ki T ik Euclidean Geometry
http://goforward/http://find/http://goback/ -
7/30/2019 Talk 20040160 Poincare
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Kim Taeik
Shape of theUniverse
Shape of the
UniverseShape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
Euclidean Geometry
5. The parallel postulate
Pythagoras theorem (ds)2 = (dx)2 + (dy)2
The sum of angles in triangle is 180
PoincaresConjecture
Ki T ik Euclidean Geometry
http://find/http://goback/ -
7/30/2019 Talk 20040160 Poincare
47/76
Kim Taeik
Shape of theUniverse
Shape of the
UniverseShape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
Euclidean Geometry
5. The parallel postulate
Pythagoras theorem (ds)2 = (dx)2 + (dy)2
The sum of angles in triangle is 180
Does the postulate need?
PoincaresConjecture
Kim Taeik Euclidean Geometry
http://goforward/http://find/http://goback/ -
7/30/2019 Talk 20040160 Poincare
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Kim Taeik
Shape of theUniverse
Shape of the
UniverseShape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
Euclidean Geometry
5. The parallel postulate
Pythagoras theorem (ds)2 = (dx)2 + (dy)2
The sum of angles in triangle is 180
Does the postulate need?
If the postulate false, how change ds?
PoincaresConjecture
Kim Taeik Without Postulate
http://find/http://goback/ -
7/30/2019 Talk 20040160 Poincare
49/76
Kim Taeik
Shape of theUniverse
Shape of the
UniverseShape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
Without Postulate
PoincaresConjecture
Kim Taeik Without Postulate
http://goforward/http://find/http://goback/ -
7/30/2019 Talk 20040160 Poincare
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Kim Taeik
Shape of theUniverse
Shape of the
UniverseShape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
Without Postulate
PoincaresConjecture
Kim Taeik Without Postulate
http://goforward/http://find/http://goback/ -
7/30/2019 Talk 20040160 Poincare
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Kim Taeik
Shape of theUniverse
Shape of the
UniverseShape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
Without Postulate
PoincaresConjecture
Kim Taeik Without Postulate
http://find/http://goback/ -
7/30/2019 Talk 20040160 Poincare
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Shape of theUniverse
Shape of the
UniverseShape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
the sum of angles larger than 180
PoincaresConjecture
Kim Taeik Without Postulate
http://find/ -
7/30/2019 Talk 20040160 Poincare
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Shape of theUniverse
Shape of the
UniverseShape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
The postulate not always true!
PoincaresConjecture
Kim Taeik Without Postulate
http://find/ -
7/30/2019 Talk 20040160 Poincare
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Shape of theUniverse
Shape of the
UniverseShape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
The postulate not always true!Space is characterized by curvature
PoincaresConjecture
Kim Taeik Without Postulate
http://find/ -
7/30/2019 Talk 20040160 Poincare
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Shape of theUniverse
Shape of the
UniverseShape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
ds must have terms
The postulate not always true!Space is characterized by curvature
PoincaresConjecture
Kim Taeik Without Postulate
http://find/ -
7/30/2019 Talk 20040160 Poincare
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Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
ds must have terms
How can we determine ?
The postulate not always true!Space is characterized by curvature
PoincaresConjecture
Kim Taeik Metric
http://find/ -
7/30/2019 Talk 20040160 Poincare
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Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
ds ds()
PoincaresConjecture
Kim Taeik Metric
http://find/ -
7/30/2019 Talk 20040160 Poincare
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Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
ds ds()(ds)
PoincaresConjecture
Kim Taeik Metric
http://find/ -
7/30/2019 Talk 20040160 Poincare
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Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
ds ds()(ds)
metric : a distance between elements of a set.
d(x,y)
PoincaresConjecture
Kim Taeik Metric
http://find/ -
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Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
ds ds()(ds)
metric : a distance between elements of a set.
d(x,y)
Essential characteristic of space : metric
PoincaresConjecture
Kim Taeik Ricci Flow
http://find/ -
7/30/2019 Talk 20040160 Poincare
61/76
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
tgij = 2Rij
PoincaresConjecture
Kim Taeik Ricci Flow
http://find/ -
7/30/2019 Talk 20040160 Poincare
62/76
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
tgij = 2Rij
gij
PoincaresConjecture
Kim Taeik Ricci Flow
http://find/ -
7/30/2019 Talk 20040160 Poincare
63/76
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
tgij = 2Rij
gijRicci flow canonical form
PoincaresConjecture
Kim Taeik Ricci Flow
http://find/ -
7/30/2019 Talk 20040160 Poincare
64/76
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
tgij = 2Rij
gijRicci flow canonical form
Space
PoincaresConjecture
Kim Taeik Ricci Flow
http://find/ -
7/30/2019 Talk 20040160 Poincare
65/76
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
tgij = 2Rij
gijRicci flow canonical form
Space manifold with uniform metricmodel
PoincaresConjecture
Kim Taeik Ricci Flow
http://find/ -
7/30/2019 Talk 20040160 Poincare
66/76
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
tgij = 2Rij
gijRicci flow canonical form
Space manifold with uniform metricmodel
PoincaresConjecture
Kim Taeik Ricci Flow
http://find/ -
7/30/2019 Talk 20040160 Poincare
67/76
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
tgij = 2Rij
gijRicci flow canonical form
Space manifold with uniform metricconnected
Ricci flow
grouping
The eight geometries
PoincaresConjecture
Kim Taeik Geometry
http://find/ -
7/30/2019 Talk 20040160 Poincare
68/76
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
S3
E3
H3
S2 R
H2 R
SL2(R)
Nil
Sol
manifold with gij
PoincaresConjecture
Kim Taeik Geometry
http://find/ -
7/30/2019 Talk 20040160 Poincare
69/76
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
S3
E3
H3
S2 R
H2 R
SL2(R)
Nil
Sol
PoincaresConjecture
Kim Taeik Geometry
http://find/ -
7/30/2019 Talk 20040160 Poincare
70/76
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
S3
E3
H3
S2 R
H2 R
SL2(R)
Nil
Sol
Ricci flow:flatiron
PoincaresConjecture
Kim Taeik
Sh f h
Geometry
http://find/ -
7/30/2019 Talk 20040160 Poincare
71/76
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
S3
E3
H3
S2 R
H2 R
SL2(R)
Nil
Sol
PoincaresConjecture
Kim Taeik
Sh f th
Geometry
http://find/ -
7/30/2019 Talk 20040160 Poincare
72/76
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
S3
E3
H3
S2 R
H2 R
SL2(R)
Nil
Sol
Conversely
PoincaresConjecture
Kim Taeik
Shape of the
Geometry
http://find/ -
7/30/2019 Talk 20040160 Poincare
73/76
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
S3
E3
H3
S2 R
H2 R
SL2(R)
Nil
Sol
characterized by metric
PoincaresConjecture
Kim Taeik
Shape of the
Geometry
http://find/ -
7/30/2019 Talk 20040160 Poincare
74/76
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
S3
E3
H3
S2 R
H2 R
SL2(R)
Nil
Sol
Ricci flow1:wrinkling
PoincaresConjecture
Kim Taeik
Shape of the
Geometry
http://find/ -
7/30/2019 Talk 20040160 Poincare
75/76
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
S3
E3
H3
S2 R
H2 R
SL2(R)
Nil
Sol
Ricci flow1:wrinkling
PoincaresConjecture
Kim Taeik
Shape of the
The Poincare Conjecture
1 Sh f h U i
http://find/ -
7/30/2019 Talk 20040160 Poincare
76/76
Shape of theUniverse
Shape of theUniverse
Shape of Earth
Simply Connected
Spherical Shell
Simply Connected
PoincaresConjecture
Poincares Conjecture
Proof
Geometry
Euclidean
Riemannean
Geometrization
1 Shape of the UniverseShape of the UniverseShape of Earth
2 Simply ConnectedSpherical Shell
Simply Connected3 Poincares Conjecture
Poincares ConjectureProof
4 GeometryEuclideanRiemanneanGeometrization
http://find/