Project description
Geometric measure theory could answer long-standing open problems in mathematics
The aim of the EU-funded STMAGMT project is to lay theoretical foundations in different areas of geometric measure theory to answer fundamental questions that relate to modern mathematical analysis. The project will expand on newly developed geometric measure theory techniques that address arbitrary metric spaces to solve seemingly unrelated problems in different fields of mathematical analysis, such as the calculus of variations, harmonic analysis and geometric function theory, and differential geometry. The project will concentrate on three main objectives: generalising classical characterisations of rectifiability to non-Euclidean settings; proving a quantitative analogue to the Besicovitch-Federer projection theorem; and solving the flat chain conjecture of Ambrosio and Kirchheim.
Objective
The aim of this research proposal is to develop the necessary theory of three areas of Geometric Measure Theory in order to solve several fundamental open questions. The origins of these questions can be found in recent advancements in various areas of modern analysis, such as the calculus of variations, harmonic analysis and geometric function theory, and differential geometry.
The project will use and expand upon techniques recently pioneered by the PI. These techniques demonstrated the viability of geometric measure theory in arbitrary metric spaces. One focus of this project will be to continue with the natural progression of this research. The other focus will be to developing these techniques in order to solve seemingly unrelated problems in new areas of analysis. These methods have been successfully used to solve many well known questions, but it is clear that their full potential has yet to be realised.
The main areas of interest are:
(A): Fundamental questions regarding geometric measure theory in metric spaces.
A central point of interest will be generalising classical characterisations of rectifiability to non-Euclidean settings.
(B): Characterisations of quantitative rectifiability.
The main goal is to prove a quantitative analogue to the Besicovitch--Federer projection theorem conjectured by David and Semmes.
(C): The structure of currents.
In particular, the project will follow a path towards solving the flat chain conjecture of Ambrosio--Kirchheim.
Each of these areas concerns difficult yet important problems. As with other fundamental results of GMT, it is expected that these techniques will find applications far beyond their original purpose.
Fields of science (EuroSciVoc)
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: https://op.europa.eu/en/web/eu-vocabularies/euroscivoc.
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: https://op.europa.eu/en/web/eu-vocabularies/euroscivoc.
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Funding Scheme
ERC-STG - Starting GrantHost institution
CV4 8UW COVENTRY
United Kingdom