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Quantitative Rectifiability: from Vitushkin's conjecture to Manifold Learning

CORDIS · observation · Publication date unknown

ravelling Salesman Theorem (ATST). Recent work by the PI and collaborators suggest that fundamental questions at the interface between Geometric Measure Theory (GMT), Harmonic Analysis (HA), PDEs and Machine Learning (ML) have at their core establishing higher dimensional analogues of Jones' ATST. This proposal takes up this challenge by focussing onto three concrete investigations: 1) We aim at solving a long-standing and notoriously difficult conjecture of Vitushkin on the connection between analytic capacity and Favard length. As a result of our strategy, we will prove a quantification of the classical Besicovitch-Federer projections theorem. 2) We study the interplay between the geometry and the differentiability structure a set can support, resulting in a) a geometric characterisation of domains admitting a Sobolev trace theorem, and b) a geometric converse of Rademacher's theorem, which answers a notable open question in the David-Semmes theory of uniform rectifiability. 3) We study the geometry of point clouds by developing a corona-type construction which tests whether the d

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recordType

award

status

SIGNED

region

EU

value

165312.96

unit

EUR