Topics in Geometric Analysis: Rectifiable sets and rectifiable measures

This is a monographic course on the topic of rectifiability, a concept introduced first by Besicovitch in the particular case of 1-dimensional sets in the plane. After introducing the Hausdorff dimension, the Hausdorff measures, the course covers Besicovitch's theory and touches upon the famous Besicovitch 1/2 conjecture, which is almost a century old and still unsolved. It then defines rectifiable sets and rectifiable measures in general dimension and codimension and covers the most basic and widely used rectifiability criteria.