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Bagels, beach balls, and the Poincar é Conjecture

Bagels, beach balls, and the Poincar é Conjecture. Emily Dryden Bucknell University. Poincar é. Confused topologists. Homeomorphisms. A homeomorphism is a continuous stretching and bending of an object into a new shape.

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Bagels, beach balls, and the Poincar é Conjecture

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

  2. Poincaré

  3. Confused topologists

  4. 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

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

  6. 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?

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

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

  9. 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

  10. 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...

  11. A million bucks

  12. An elusive character arXiv:math/0211159 (39 pages) arXiv:math/0303109 (22 pages) arXiv:math/0307245 (7 pages) Perelman

  13. The full story http://www.arxiv.org/abs/math/0605667 (200 pages)

  14. The intrigue

  15. 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

  16. 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”

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