Quantitative topology I
Tam Nguyen Phan
16:00 - 17:30
We will focus in this 4-lecture series on the following two
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
In the first lecture, we will discuss what was known before CDMW's break through.
The talk is based on an expository paper by Larry Guth