QuRe-ViMaL
Financiado
Quantitative Rectifiability: from Vitushkin's conjecture to Manifold Learning
For compact planar sets, an analogue to the classic travelling salesman problem is: when can all points in a compact set E be traversed by a rectifiable curve? and how long should such a curve be? P. Jones came up with an answer i...
For compact planar sets, an analogue to the classic travelling salesman problem is: when can all points in a compact set E be traversed by a rectifiable curve? and how long should such a curve be? P. Jones came up with an answer in his influential Analyst's Travelling 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 data points lie near a parametrisable surface; this is a way of testing the manifold hypothesis, relied upon by most nonlinear dimensionality reduction algortihms in data analysis.
Our framework provide a common language within which we tackle these diverse issues. Hence, achieving our objectives will not only result in major subject-specific breakthroughs, but, just as importantly, will develop and expand this `language', thus providing fertile ground for multidisciplinary interactions to take place.
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Duración del proyecto: 32 meses
Fecha Inicio: 2023-04-17
Fecha Fin: 2025-12-31
Fecha Fin: 2025-12-31
Línea de financiación: concedida
El organismo HORIZON EUROPE notifico la concesión del proyecto el día 2023-04-17
Línea de financiación objetivo
El proyecto se financió a través de la siguiente ayuda:
HORIZON-MSCA-2022-PF-01-01:
MSCA Postdoctoral Fellowships 2022
Cerrada
hace 2 años
Presupuesto
El presupuesto total del proyecto asciende a
165K€
Líder del proyecto
UNIVERSIDAD DEL PAIS VASCO/EUSKAL HERRIKO UNI...
No se ha especificado una descripción o un objeto social para esta compañía.
Total investigadores
45
Total investigadores
45