Bagels, beach balls, and the Poincar Conjecture

1
Bagels, beach balls, and the Poincaré Conjecture

• Emily Dryden
• Bucknell University

2
(No Transcript)
3
Poincaré
4
Confused topologists
5
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

6
Two dimensions surfaces

• Smooth no jagged peaks or ridges
• Compact can put it in a box
• Orientable distinguishable top and bottom
• No boundary

7
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?

8
Spheres of many dimensions
?
1-sphere
2-sphere
3-sphere
9
Distinguishing objects homeomorphic to 3-sphere

• Count holes?
• 2-sphere simple closed curves
• Torus loop that cannot be deformed to a point?

10
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

11
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...

12
A million bucks
13
An elusive character
arXivmath/0211159 (39 pages)
arXivmath/0303109 (22 pages)
arXivmath/0307245 (7 pages)
Perelman
14
The full story
http//www.arxiv.org/abs/math/0605667 (200 pages)
15
The intrigue
16
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

17
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