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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)**

Conjecture: Every closed simply connected 3-dimensional manifold is homeomorphic to the 3-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 model: three-dimensional sphere!

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What is 3-dim sphere? What is 2-dim sphere? R

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**Take two solid balls and glue their boundaries together**

What is 3-dim sphere? The set of points in 4-dim space on the same distance from a given point Take two solid balls and glue their boundaries together

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Waves Amplitude Wavelength

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**Frequency high-pitched sound low-pitched sound**

Short wavelength – High frequency low-pitched sound Long wavelength – Low frequency

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Doppler Effect Stationary source Moving source 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 Bang theory**

Time Georges-Henri Lemaître, Edwin 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?**

Distances increase – The Universe stretches Big Bang

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Is a cylinder flat? R 2πr

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Triangle on a cylinder α + β + γ = 180o β β γ α γ α

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Sphere is not flat α + β + γ > 180o 90o γ β α

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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? Torus…

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Flat and bounded? Torus… and Flat Torus A B A B

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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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**Pseudosphere (part of Hyperbolic plane) K<0**

Constant curvature K Pseudosphere (part of Hyperbolic plane) K<0 Sphere K>0 (K = 1/R2) Plane K =0 γ β α γ β α β γ α α + β + γ >180o α + β + γ =180o α + β + γ < 180o

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**Three geometries … and Three models of the Universe**

Elliptic Euclidean Hyperbolic Plane K =0 (flat) K = 0 K > 0 K < 0 α + β + γ >180o α + β + γ =180o α + β + γ < 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**

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… Conjecture: Every closed simply connected 3-dimensional manifold is homeomorphic to the 3-dimensional sphere

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**2-dimensional case Theorem (Poincare)**

Every closed simply connected 2-dimensional manifold is homeomorphic to the 2-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 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 Чжу Сипин и Цао Хуайдун, Shing-Tung Yau

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**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 )

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**Thank you! http://www.nipissingu.ca/numeric**

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