This talk will present a characterisation of purely -unrectifiable subsets of a complete metric space with finite -dimensional Hausdorff measure by studying non-linear projections (i.e. -Lipschitz functions) into some fixed Euclidean space. We will show that a typical (in the sense of Baire category) non-linear projection maps to a set of zero n dimensional Hausdorff measure. Conversely, a typical non-linear projection maps an -rectifiable subset to a set of positive -dimensional Hausdorff measure. These results provide a replacement for the classical Besicovitch–Federer projection theorem, which is known to be false outside of Euclidean spaces.

Time permitting, we will discuss some recent consequences of this characterisation.

Registration through the seminar website https://sites.google.com/ncts.ntu.edu.tw/international-gmt-seminar required.