Bagels, beach balls, and the Poincaré Conjecture Emily Dryden Bucknell University.

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Presentation transcript:

Bagels, beach balls, and the Poincaré Conjecture Emily Dryden Bucknell University

Poincaré

Confused topologists

Homeomorphisms A homeomorphism is a continuous stretching and bending of an object into a new shape. Poincaré Conjecture is about objects being homeomorphic to a sphere in three dimensions

Two dimensions: surfaces Smooth: no jagged peaks or ridges Compact: can put it in a box Orientable: distinguishable “top” and “bottom” No boundary:

Classifying such surfaces Genus: “number of holes” Example of surface with 0 holes? Example of surface with 1 hole? Example of surface with 2 holes? And so on..... What about classifying higher- dimensional objects?

Spheres of many dimensions ? 1- sphere 2- sphere 3- sphere

Distinguishing objects homeomorphic to 3-sphere Count holes? 2-sphere: simple closed curves Torus: loop that cannot be deformed to a point?

Poincaré asks... If a compact 3-dimensional object* M has the property that every simple closed curve within the object* can be deformed continuously to a point, does it follow that M is homeomorphic to the 3-sphere? Poincaré Conjecture: answer is yes

More, more, more! Dimensions 5 and higher: proved in 1960s by Smale, Stallings, Wallace Dimension 4: proved in 1980s by Freedman Dimension 3: lots of people tried...

A million bucks

An elusive character Perelman arXiv:math/ (39 pages) arXiv:math/ (22 pages) arXiv:math/ (7 pages)

The full story /abs/math/ (200 pages)

The intrigue

How did they do it? Metric: way to measure distance Curvature: how much does object bend? (line, circle, plane, sphere) Ricci flow: solutions to a certain differential equation, says metric changes with time so that distances decrease in directions of positive curvature

Ricci what? Think heat equation: heat one end of cold rod, heat flows through rod until have even temperature distribution Ricci flow: positive curvature spreads out until, in the limit, manifold has constant curvature Perelman: dealt with singularities that could arise during flow, showed they were “nice”