Periodic Reporting for period 2 - STMAGMT (Structure Theorems for Modern Aspects of Geometric Measure Theory)
Berichtszeitraum: 2022-04-01 bis 2023-09-30
The aim of this research project 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.
Do the characterisations obtained from Theme (A) have natural counter parts for uniformly rectifiable sets?
(C): The structure of currents.
In particular, the project will follow a path towards solving the flat chain conjecture of Ambrosio--Kirchheim on metric currents in Euclidean space.
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.
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.
Do the characterisations obtained from Theme (A) have natural counter parts for uniformly rectifiable sets?
(C): The structure of currents.
In particular, the project will follow a path towards solving the flat chain conjecture of Ambrosio--Kirchheim on metric currents in Euclidean space.
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.
In Theme (A) the project has characterised rectifiable metric spaces in terms of (Gromov-Hausdorff) tangent spaces that consist of n-dimensional Banach spaces. The project has also investigated how this tangent structure determines how a the natural Hausdorff measure on a rectifiable set can be deformed under Lipschitz images.
In Theme (B) the project has demonstrated how uniformly rectifiable metric spaces are characterised in terms of a Gromov-Hausdorf variant of the Bi-Lateral Weak Geometric Lemma of David and Semmes. This is a quantitative analogue of the first result from Theme (A) mentioned above. In fact, the project has introduced analogues of several constructions of the David-Semmes theory tailored for analysis on metric spaces, and characterised uniformly rectifiable metric spaces in terms of these constructions.
Theme (C) has begun initial investigations of the nature of metric currents in Euclidean space.
In Theme (B) the project has demonstrated how uniformly rectifiable metric spaces are characterised in terms of a Gromov-Hausdorf variant of the Bi-Lateral Weak Geometric Lemma of David and Semmes. This is a quantitative analogue of the first result from Theme (A) mentioned above. In fact, the project has introduced analogues of several constructions of the David-Semmes theory tailored for analysis on metric spaces, and characterised uniformly rectifiable metric spaces in terms of these constructions.
Theme (C) has begun initial investigations of the nature of metric currents in Euclidean space.
Further progress on Theme (A) will include new characterisations of rectifiable sets in terms of Alberti representations, an intricate structure of curves in a metric space; and investigations of the image of measures on purely unrectifiable sets under Lipschitz functions. The project will also adapt the theory of upper gradients and test plans, typically used to study Sobolev functions on a metric space, to describe the space of Lipschitz functions.
More attention will turn to Theme (C), driving a path towards the flat chain conjecture.
More attention will turn to Theme (C), driving a path towards the flat chain conjecture.