Topics in Geometry: Quantitative Differentiability and Rectifiability

Overview

Differentiability and rectifiability describe how well functions and sets can be linearly approximated at infinitesimal scales. For many problems, knowing the behavior of a set at small scales isn't enough; one needs quantitative versions of differentiability and rectifiability that bound how well functions and sets can be approximated at many different scales. In this course, we will study applications of quantitative differentiability and rectifiability in geometry and analysis.

Tentative outline:

• Coarse differentiation of curves
• From curves to spaces: Rademacher's theorem
• Pansu's theorem and metric embeddings
• Rectifiability and the Jones Traveling Salesman Problem
• Uniform rectifiability
• Surfaces in \(\mathbb{R}^n\)

Basic information

• Instructor: Robert Young (ryoung@cims.nyu.edu)
• Office: CIWW 601
• Office hours: by appointment
• Lectures: CIWW 1302, Thursdays, 1:25--3:15

Resources

• Course notes: Notes on quantitative rectifiability and differentiability (updated 2020-03-10)
• Piotr Hajłasz, Geometric Analysis
• Alex Eskin, David Fisher, Kevin Whyte, Quasi-isometries and rigidity of solvable groups, Pure and Applied Mathematics Quarterly, 3 (2007), no. 4
• Raanan Schul, Analyst's Traveling Salesman Theorems: a survey.
• Peter Jones, Rectifiable sets and the Traveling Salesman Problem, Inventiones mathematicae, 102.1 (1990): 1-16.
• Kate Okikiolu, Characterization of Subsets of Rectifiable Curves in \(\mathbb{R}^n\), Journal of the London Mathematical Society, series 2, 46 (1992), no. 2, 336–348.
• Guy David, Stephen Semmes, Singular integrals and rectifiable sets in \(\mathbb{R}^n\): Au-delà des graphes lipschitziens, Asterisque, 193 (1991).
• Slides: Filling multiples of embedded curves and quantifying nonorientability