Geometry MMA Seminar: Bi-Lipschitz arcs in metric spaces

Speaker:Scott Zimmerman, Ohio State University

Title:  Bi-Lipschitz arcs in metric spaces

Abstract:  A bi-Lipschitz arc in a metric space $X$ is the image of an interval in the real line under a bi-Lipschitz map. A natural question is as follows: when is a subset of a metric space contained in a bi-Lipschitz arc? In other words, given a set $K \subset \mathbb{R}$ and a bi-Lipschitz map $f:K \to X$, when is there a bi-Lipschitz extension $F:I \to X$ where $I$ is an interval containing $K$? This question was answered in the case $X = \mathbb{R}^n$ by David and Semmes for $n \geq 3$ and later by MacManus when $n = 2$. David and Semmes originally proved this result as part of their celebrated work in quantitative rectifiability. In this talk, I will discuss a recent preprint in which we prove this bi-Lipschitz extension result in a general setting when $X$ is one of a large class of metric spaces possessing certain geometric properties (namely Ahlfors regularity and supporting a Poincar\’{e} Inequality). This is joint work with Jacob Honeycutt and Vyron Vellis.

Time: Friday, March 29, 2024 – 1:30pm-2:30pm

Place: Exploratory Hall, Room 4208