What is The Poincar

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What is The Poincaré Conjecture?

• Alex Karassev

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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

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Henri Poincaré(April 29, 1854 July 17, 1912)

• Mathematician, physicist, philosopher
• Created the foundations of
• Topology
• Chaos Theory
• Relativity Theory

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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

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The Poincaré conjecture (1904)

• ConjectureEvery closedsimply
  connected3-dimensional manifoldis homeomorphic
  to the3-dimensional sphere
• What do these words mean?

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Why is The Conjectue Important?

• Geometry of The Universe
• New directions in mathematics

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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!)

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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

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Magellan's Journey

• August 10, 1519 September 6, 1522
• Start about 250 men
• Return about 20 men

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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)

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The Study of Space

• Universe has finite volume?
• Bounded Universe?
• However, no "edge"
• A possible modelthree-dimensional sphere!

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What is 3-dim sphere?
What is 2-dim sphere?
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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
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Waves
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Frequency
high-pitched sound
Short wavelength High frequency
low-pitched sound
Long wavelength Low frequency
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Doppler Effect
Higher pitch
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Wavelength and colors
Wavelength
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Redshift
Star at rest
Moving Star
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Redshift
Distance
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Expanding Universe?
Alexander Friedman,1922
The Big Bangtheory
Time
Georges-HenriLemaître, 1927Edwin Hubble, 1929
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Bounded and expanding?

• Spherical Universe?
• Three-Dimensional sphere(balloon) is inflating

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Infinite and Expanding?
Not quite correct!(it appears that the Universe
has an "edge")
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Infinite and Expanding?
Distancesincrease The Universestretches
Big Bang
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Is a cylinder flat?
2pr
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Triangle on a cylinder
a ß ? 180o
ß
ß
?
a
?
a
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Sphere is not flat
a ß ? gt 180o
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Sphere is not flat
???
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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

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Flat and bounded?
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Flat and bounded?
and Flat Torus
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3-dim Torus
Section flat torus
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Torus Universe
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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
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Constant curvature K
Pseudosphere (part of Hyperbolic plane)
Klt0
Plane K 0
Sphere Kgt0(K 1/R2)
ß
?
a
a ß ? gt180o a ß ? 180o a
ß ? lt 180o
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Three geometries and Three models of the
Universe
Plane K 0
Elliptic Euclidean Hyperbolic
(flat)
K 0
K lt 0
K gt 0
a ß ? gt180o a ß ? 180o a
ß ? lt 180o
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What happens if we try to "flatten"a piece of
pseudosphere?
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How to tell a torus from a sphere?

• First, compare a plane and a plane with a hole

?
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Simply connected surfaces
Simply connected
Not simply connected
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Homeomorphic objects
continuous deformation of one object to another






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Homeomorphism


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Homeomorphism

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Homeomorphism
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Can we cut?
Yes, if we glue after
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So, a knotted circle is the same as usual circle!

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The Conjecture

• ConjectureEvery closedsimply
  connected3-dimensional manifoldis homeomorphic
  to the3-dimensional sphere

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2-dimensional case

• Theorem (Poincare)
• Every closedsimply connected2-dimensional
  manifoldis homeomorphic to the2-dimensional
  sphere

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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

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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

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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

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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

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Further 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)

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Thank you!
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