# Geometric Analysis and Topology Seminar

#### New Methods for Sobolev Spaces on Metric Spaces

**Speaker:**
Sylvester Eriksson-Bique, University of Oulu

**Location:**
Online

**Videoconference link:** https://nyu.zoom.us/j/95759976381

**Date:**
Friday, October 7, 2022, 11 a.m.

**Synopsis:**

Sobolev spaces on metric measure spaces were introduced by several authors including Hajlasz, Cheeger, Shanmugalingam and Ambrosio. I will discuss some recent new techniques to describe the structure of Sobolev spaces, and to approximate Sobolev functions. In particular, this gives a new way to prove the equivalence of two different definitions: a relaxation definition and an upper gradient definition. This approximation seems a convenient, and explicit, way of combining a number of different methods which have appeared in the literature. I will also discuss a way to define a differential for a Sobolev function, which is pointwise defined. The techniques have an application to tensorization, and lead to further questions, which I will mention.