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.
