Wednesday, April 28, 2021 3:15 PM
Hannah Alpert (UBC)

Gromov conjectured in 1983 that if an n-manifold has large positive scalar curvature at every point, then it can be mapped to an (n-2)-complex with every fiber of the map having small diameter.  We consider a "macroscopic" version of the conjecture, where the scalar curvature hypothesis is replaced by supposing that every ball of radius 10 has small volume and that every loop in a ball of radius 1 is null-homotopic in the concentric ball of radius 2.  For a macroscopic claim such as this one, the available approaches avoid geometric analysis and instead use a few elementary tools.  We describe these tools and give partial results on the macroscopic conjecture.  Joint work in progress with Alexey Balitskiy and Larry Guth.