# 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