Bagels beach balls and the Poincar Conjecture Emily

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

A million bucks

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

The full story http: //www. arxiv. org /abs/math/0605667 (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”