SPEAKER: Jacob Honeycutt (UTK)

TITLE: Bi-Lipschitz Arcs

ABSTRACT: We generalize a bi-Lipschitz extension result of David and Semmes from Euclidean spaces to complete metric measure spaces with controlled geometry (Ahlfors regularity and supporting a Poincaré inequality). In particular, we find sharp conditions on metric measure spaces X so that any bi-Lipschitz embedding of a subset of the real line into X extends to a bi-Lipschitz embedding of the whole line. Along the way, we prove that if the complement of an open subset Y of X has small Assouad dimension, then it is a uniform domain, and we prove a quantitative approximation of continua in $X$ by bi-Lipschitz curves.