1 / 17

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.

carrolld
Download Presentation

Bagels, beach balls, and the Poincar é Conjecture

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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”

More Related