Speaker: Hannah Alpert
Title: Macroscopic scalar curvature and the minimizing hypersurface trick
Abstract: If the curvature of a closed manifold has a positive lower bound, how does that constrain the topology? If it has a negative lower bound and complicated topology, how does that constrain the volume? We consider the macroscopic cousins of these questions, meaning that instead of a lower bound on curvature, we require an upper bound on volumes of balls of a given radius in the universal cover of the manifold. In many cases the macroscopic questions are better resolved than the original questions, and surprisingly, many of the proofs rely on One Weird Trick of locally comparing an area-minimizing hypersurface to a sphere.