MATJ5114 MA2: On the Geometry of Rectifiable and Purely Unrectifiable Subsets of a Metric Space (JSS32) (2 op)

• Rectifiability in metric spaces: basic definitions and Kirchheim's description of rectifiable metric spaces. Comparison to classical rectifiability.
• Sufficient conditions for rectifiability: bi-Lipschitz decompositions of functions; rectifiability from topology; Alberti representations.
• Gromov--Hausdorff convergence of metric measure spaces and the various definitions of tangent metric measure spaces.
• Characterising rectifiable metric spaces in terms of flat tangent spaces.

The main objective is to give a concise introduction to the theory of rectifiability in an arbitrary metric space and to draw comparisons to classical rectifiability in Euclidean space. The students will also become familiar with concepts in analysis on metric spaces and Gromov--Hausdorff convergence.

Standard theory of undergraduate analysis and measure theory