Gregory Chambers, University of Chicago
“Problems in Quantitative Geometry”
Quantitative geometry is a broad field concerned with geometric problems in which quantitative properties play a central role. I will discuss two types of problems from this field; those in quantitative topology, and those in geometric analysis. Questions in quantitative topology seek to understand the geometric properties of objects whose existence is guaranteed by topological methods. In this talk, I will describe some of these problems, sketch some of the ideas involved in their solutions, and discuss future directions.
The questions that I will answer include the following. Suppose that two simple closed curves in a Riemannian surface are homotopic through curves of length less than L. Is there an isotopy between these curves also through curves of length of length less than L? Suppose that f : X → Y is a null- homotopic map with Lipschitz constant L. Under what conditions does there exist an L-Lipschitz null-homotopy? If M is a null-cobordant manifold, then how large must a manifold which fills M be? If the Coulomb energy of a set is close to maximal, then does the set have to be close to a ball? Do non-compact complete manifolds of finite volume contain minimal hypersurfaces?
MC 4041
Refreshments will be served in MC 5403 following the lecture. Everyone is welcome to attend.