STMAGMT | Structure Theorems for Modern Aspects of Geometric Measure Theory

Summary
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.
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More information & hyperlinks
Web resources: https://cordis.europa.eu/project/id/948021
Start date: 01-10-2020
End date: 30-09-2025
Total budget - Public funding: 1 421 562,00 Euro - 1 421 562,00 Euro
Cordis data

Original description

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.

Status

SIGNED

Call topic

ERC-2020-STG

Update Date

27-04-2024
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Horizon 2020
H2020-EU.1. EXCELLENT SCIENCE
H2020-EU.1.1. EXCELLENT SCIENCE - European Research Council (ERC)
ERC-2020
ERC-2020-STG