Metric geometry

Metric geometry: Rademacher’s theorem in metric measure spaces and Alberti representations.


This topics course concerns the study of Rademacher’s differentiation theorem in the setting of general metric spaces. Specifically, we will present a new proof of Cheeger’s celebrated generalisation of Rademacher’s theorem when the domain is replaced with a metric measure space that satisfies a Poincaré inequality.

This proof relies on recently developed techniques in analysis on metric spaces. We will also cover more standard aspects of this theory, in particular Gromov-Hausdorff convergence and weak tangent spaces.

We will also look at Kirchheim’s differentiation theory when the image is replaced by an arbitrary metric space.

Lecture notes are provided and will be updaded throughout the course.


Other notes

There are lecture notes from a mini course I gave in Edinburgh. These contain only the parts of the course that are relevant for the proof of Cheeger’s theorem. However, there is a significant difference: these notes deal with curve fragments in a metric space, rather than (connected) curves. The benefit of this approach is that it does not require embeddings into a Banach space for the extra structure. Conceptually, the proof is exactly the same.