What is What is The Poincaré Conjecture? Alex Karassev.

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!

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