Suppose that f is a contractible map from the unit m-sphere to the unit n-sphere with Lipschitz constant L. Is it possible to choose a null-homotopy with Lipschitz constant bounded by a reasonable function of L? Gromov posed this question about twenty years ago. For special choices of m and n, he constructed homotopies with Lipschitz constant at most C(m,n) L. But for most dimensions, the bounds that were known until recently were astronomical, towers of exponentials in L. In the last year, Chambers, Dotterrer, Manin, Ferry, and Weinberger constructed null-homotopies with nearly sharp Lipschitz constants. I will give a little background about the problem and then discuss their work.
This video is part of Harvard University‘s conference JDG 2017.
