What is The Poincaré Conjecture? Alex Karassev

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

Henri Poincaré (April 29, 1854 – July 17, 1912) Mathematician, physicist, philosopher Created the foundations of Topology Chaos Theory Relativity Theory

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

The Poincaré conjecture (1904) Conjecture: Every closed simply connected 3-dimensional manifold is homeomorphic to the 3-dimensional sphere What do these words mean?

Why is The Conjectue Important? Geometry of The Universe New directions in mathematics

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!)

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

Magellan's Journey August 10, 1519 — September 6, 1522 Start: about 250 men Return: about 20 men

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)

The 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 2-dim sphere? R

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

Waves Amplitude Wavelength

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

Doppler Effect Stationary source Moving source Higher pitch

Wavelength and colors Wavelength

Redshift Star at rest Moving Star

Redshift Distance

Expanding Universe? Alexander Friedman,1922 The Big Bang theory Time Georges-Henri Lemaître, 1927 Edwin Hubble, 1929

Bounded and expanding? Spherical Universe? Three-Dimensional sphere (balloon) is inflating

Infinite and Expanding? Not quite correct! (it appears that the Universe has an "edge")

Infinite and Expanding? Distances increase – The Universe stretches Big Bang

Is a cylinder flat? R 2πr

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

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

Sphere is not flat ???

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

Flat and bounded? Torus…

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

3-dim Torus Section – flat torus

Torus Universe

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

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

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

What happens if we try to "flatten" a piece of pseudosphere?

How to tell a torus from a sphere? First, compare a plane and a plane with a hole ?

Simply connected surfaces Not simply connected

≈ ≈ ≈ ≈ ≈ ≈ Homeomorphic objects continuous deformation of one object to another ≈ ≈ ≈ ≈ ≈ ≈

Homeomorphism ≈ ≈

Homeomorphism ≈

Homeomorphism

Can we cut? Yes, if we glue after

So, a knotted circle is the same as usual circle! ≈

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

2-dimensional case Theorem (Poincare) Every closed simply connected 2-dimensional manifold is homeomorphic to the 2-dimensional sphere

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

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

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

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

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.108-120)

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