A non-linear Besicovitch–Federer projection theorem for metric spaces

Abstract: This talk will present a characterisation of purely $n$-unrectifiable subsets $S$ of a complete metric space with finite $n$-dimensional Hausdorff measure by studying non-linear projections (i.e. $1$-Lipschitz functions) into some fixed Euclidean space. We will show that a typical (in the sense of Baire category) non-linear projection maps $S$ to a set of zero $n$-dimensional Hausdorff measure. Conversely, a typical non-linear projection maps an $n$-rectifiable subset to a set of positive $n$-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.

analysis of PDEsdifferential geometry

Audience: researchers in the topic

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NCTS international Geometric Measure Theory seminar

Series comments: We envisage an event built around virtual presentations on progress in geometric measure theory by external speakers. Every researcher is free to register as a participant and thus gain access to a virtual facility which is complete with lobby, lecture hall, and areas with boards for discussion. Thus, it shall recreate the exchange possibilities found at international conferences.

Focus: regularity and singularity theories for submanifolds of Riemannian manifolds and some of its applications.

Frequency: one presentation every other month.

Registration: required for new participants, go to the seminar website (allow at least one working day for processing).

Virtual venue: HyHyve space NCTS iGMT seminar (only for registered participants, opened one hour before the events).

You might want to consult the description of the premises and instructions.

Former organiser: Guido De Philippis (till March 2022).

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