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
- Minicourse notes: Quantitative
differentiability and
rectifiability (updated 2025-06-10)
- Course notes: Notes on quantitative rectifiability and
differentiability (updated 2020-03-10)
- Jeff Cheeger, Differentiability
of Lipschitz Functions on Metric Measure Spaces, Geometric
and Functional Analysis, 9 (1999), 428-517.
- 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