Doctoral Dissertations

Date of Award

5-2025

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematics

Major Professor

Vyron S. Vellis

Committee Members

Joan Lind, Conrad Plaut, Scott Zimmerman

Abstract

We extend several results on the geometric rectifiability of subsets in Euclidean spaces to metric measure spaces with controlled geometry (Ahlfors regularity and supporting a Poincar´e inequality). Specifically, we provide a strengthened version of the classical Denjoy-Riesz theorem, generalize MacManus’ result on the quasisymmetric rectifiability of uniformly disconnected subsets, and extend the work of David and Semmes on bi-Lipschitz rectifiability to metric measure spaces.

For the Denjoy-Riesz theorem, we prove that compact, totally disconnected subsets of metric measure spaces with controlled geometry can be captured by an arc that is locally bi-Lipschitz away from the totally disconnected set. In the case of MacManus’ result, we show that compact, uniformly disconnected subsets in these spaces can be captured by the image of a quasisymmetric embedding, quantitatively. For David and Semmes’ work, we provide sharp conditions on a metric measure space such that a subset which is bi-Lipschitz homeomorphic to a subset of the real line is captured by a bi-Lipschitz arc, quantitatively.

Finally, we establish several results concerning the co-uniformity of subsets in these spaces under various conditions, including bounds on Assouad dimension and uniform disconnectedness.

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