Topics in Quantitative Rectifiability: Traveling Salesmen, Lipschitz Decompositions, Densities, and Big Pieces

This dissertation presents and proves a series of results in the field of quantitative rectifiability. It explores several key areas: the quantitative rectifiability of Jordan arcs in Hilbert spaces, with a focus on a traveling salesman beta number estimate; the existence of Lipschitz decompositions for domains with quantitatively flat boundaries; the regularity of Hausdorff measure on uniformly rectifiable metric spaces, proving the weak constant density condition; and the iteration of the big pieces operator in Ahlfors regular metric spaces, demonstrating stabilization after two iterations through a general extension theorem.

Curriculum

This dissertation is structured into five main chapters, each addressing a specific area within quantitative rectifiability. The chapters cover theoretical advancements and proofs related to traveling salesman theorems, Lipschitz decompositions, density conditions, and iterative operators.

1. Chapter 1: Introduction
   • The analyst’s traveling salesman theorem for Jordan arcs
   • Lipschitz decompositions of domains with bilaterally flat boundaries
   • Uniformly rectifiable metric spaces satisfy the weak constant density condition
   • Iterating the big pieces operator
2. Chapter 2: A traveling salesman theorem for Jordan curves in Hilbert space
   • Introduction and Overview
   • Related Results and Questions
   • Preliminaries
   • Large-scale balls: G_L
   • Non-flat arcs: G₁, G₃, ∆₂.₂
   • Almost flat arcs: ∆₁*, ∆₂.₁
   • Theorem B
3. Chapter 3: Lipschitz decompositions of domains with bilaterally flat boundaries
   • Introduction, Outlines of the paper and proofs of the theorems
   • Preliminaries: Conventions, Reifenberg parameterizations, Whitney cubes, Coronizations
   • The proof of Theorem C
   • The proofs of Theorems D and E
   • Lipschitz graph domains
   • Controlling the change in the derivative of Reifenberg parameterizations
   • Graph coronizations for Reifenberg flat sets
4. Chapter 4: Uniformly rectifiable metric spaces satisfy the weak constant density condition
   • Introduction
   • Preliminaries
   • Oscillation of means of L² functions
   • Bi-Lipschitz images satisfy the weak constant density condition
   • Stability of the weak constant density condition under big pieces
5. Chapter 5: Iterating the big pieces operator and larger sets
   • Introduction
   • A Theorem

Careers

The skills and knowledge gained through doctoral research in Mathematics, particularly in areas like quantitative rectifiability, prepare graduates for advanced roles in academia and research-intensive industries.

• Academic Researcher
• University Professor
• Applied Mathematician
• Data Scientist
• Quantitative Analyst