# 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