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MIGSAA Mini-course: Rademacher's theorem in metric measure spaces and Alberti representations

Kindly organised by Jonas Azzam, this short mini course will be an insight into Cheeger's Theorem in Geometric Measure Theory.

Venue: 5.20 ICMS Lecture Theatre, 5th Floor, 47 Potterow, Edinburgh EH8 9BT.

Time: 9 am - 11 am

Day:  Monday 22nd and Wednesday 24th October 2018

Speaker: David Bate (University of Helsinki)

Abstract: Rademacher’s theorem states that any Lipschitz $f \colon \mathbb R^n \to \mathbb R$ is differentiable Lebesgue almost everywhere. It is a fundamental result in geometric measure theory.  In 1999 Cheeger gave a very deep generalisation of Rademacher's theorem which replaces the domain with a doubling metric measure space that satisfies a Poincaré inequality.

This minicourse will give an overview of a new proof of Cheeger's theorem which uses modern techniques in the field of analysis on metric spaces.  These techniques consider a rich structure of Lipschitz curves in the metric space, known as an “Alberti representation”, which allow us to form a partial derivative of any Lipschitz function.  By considering many such families of curves, we are able to form a derivative and hence deduce Cheeger’s theorem.