Quantitative Topology II
We will focus in this 4-lecture series on the following two questions.
1. Given a null-homotopic map f from S^m to S^n, what is the best Lipschitz constant of a null homotopy of f in terms of the Lipschitz constant Lip(f) of f?
2. If a manifold M is null-cobordant, what is the best lower bound of the geometric complexity of a manifold W bounding M in terms of the geometric complexity of M?
(Give M a metric with bounded local geometry, the geometric complexity of M is measured by the volume of M.)
Gromov found that in general the classical methods give bounds that are towers of exponentials (in Lip(f) or Vol(M)) but suggested that the true bounds are probably linear. We will attempt to explain a recent break through of Chambers, Dotterrer, Manin and Weinberger (CDMW) in getting the linear bound for the first question and an almost linear bound for the second.