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What is 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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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 Short wavelength – High frequency Long wavelength – Low frequency high-pitched sound low-pitched sound

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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? The Big Bang theory Time Alexander Friedman,1922 Georges-Henri Lemaître, 1927 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? Big Bang Distances increase – The Universe stretches

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Is a cylinder flat? R 2πr2πr

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

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

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Sphere is not flat ???

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How to tell a sphere from plane 1 st method: Plane is unbounded 2 nd 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 Shape of the Universe is the same everywhere So it must have constant curvature

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Constant curvature K Plane K =0 Sphere K>0 (K = 1/R 2 ) γ β α α + β + γ >180 o α + β + γ =180 o α + β + γ < 180 o γ β α γ β α Pseudosphere (part of Hyperbolic plane) K<0

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Three geometries … and Three models of the Universe Plane K =0 K > 0 α + β + γ >180 o α + β + γ =180 o α + β + γ < 180 o Elliptic Euclidean Hyperbolic (flat) K = 0 K < 0

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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… 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

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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!

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