MAT 678. Analysis on metric spaces.

MAT 678: Analysis on metric spaces.

Instructor: Raanan Schul
- Office: Math 4-102.
- Office hours: by appointment
Class: Mon, Wed 2:20pm- 3:40pm in Physics P128
Topics will be a mixture of
- Lectures on Analysis on metric spaces. By Juha Heinonen. (Book)
- Nonsmooth Calculus. By Juha Heinonen. Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 2, 163-232. (Survey paper) link
- Differentiable structures on metric measure spaces: a primer. By Bruice Kleiner and John Mackay. link
- Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9 (1999), no. 3, 428-517. link
- Quasiconformal maps in metric spaces with controlled geometry. Acta Math., 181 (1998), 1-61. link
The level of the course will be adjusted to fit the level of the participants.

We will review some analysis in Euclidean spaces and then continue to generalize it to the setting of metric spaces. One goal will be to understand Cheeger's result, saying that metric spaces which are not too bad, admit differentiable structures. Another goal will be to understand some basic facts about quasisymmetric maps. On route will will need to go over things such as covering theorems, Lipschitz functions, doubling measures, maximal functions, modulus, upper gradient, Sobolev spaces, Poincare inequality, quasisymmetric maps, etc.