10 oct 2024 -- 17:00   [open in google calendar]

Aula Riunioni - Dipartimento di Matematica

Abstract.

An Alberti representation of a (finite, Borel) measure is a decomposition into 1-rectifiable measures. By Fubini's theorem, Lebesgue measure on 0,1ⁿ has n "independent" Alberti representations, each one consisting of curves parallel to a coordinate axis. This naturally extends to representations of n-rectifiable sets in Euclidean space, where the tangents to curves span the approximate tangent space almost everywhere, and can be extended further to rectifiable subsets of a metric space. This talk will consider the converse statement. We will show that, if n-dimensional Hausdorff measure of a metric space X has n independent Alberti representations, then X is n-rectifiable. This result has numerous applications to rectifiability in metric spaces that will be discussed. We will also discuss a key element of the proof on the integrability of measuressatisfying a PDE constraint. This talk is based on joint work with Tuomas Orponen and Julian Weigt.

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