Metric geometry

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

Summary

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.

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

Exercises

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.

There are also lecture notes from a mini course I gave in Jyväskylä.