October 01, 2019
5:00 PM - 6:00 PM
Math Tower 5-127
|Dennis Sullivan, Stony Brook University|
Why are three manifolds the way that they are?
The Poincare-Thurston picture implies the end compactification of the complete unwrapping of any closed three manifold is remarkably always the three sphere. This means one adds zero, one, two or a cantor set of points to the universal cover of any closed three manifold and the result is the three sphere. Perelman's geometric proof of Poincare'-Thurston culminating a century of efforts uses the Ricci flow which leads to the special geometries predicted by Thurston which we then used to construct the embedding of the universal cover into the three sphere. This construction is based on joint work last year with PhD student and knot theorist, Alice Kwon. (see arxiv note)
These geometrical pictures, in turn, when crossed with R and combined, can be regionally described in terms of special Lorentz geometries: AdS3xR [Anti-deSitter Three x R], AdS4 [ Anti-deSitter Four], Minkowski(3,1), dS4 [deSitter Four] and Schwarzschild.
In particular the Poincare fundamental group acts with a spatially compact fundamental domain on the complement of the special end-point-lines in the three Sphere x R and when viewed from afar appears approximately homogeneous. This suggests one answer to the question of the title might be. "So the century earned picture of three manifolds might be used as a resource in constructing possible models of space-time." This is work in progress.