Quantitative rectifiability and harmonic measure beyond the Ahlfors-David-regular setting
Table of contents
Project description
The class of uniformly rectifiable (UR) measures plays a central role in geometric measure theory due to their connection to harmonic analysis and partial differential equations. The underlying assumption on these measures is that they are "n-AD-regular", which can be understood as a "quantitative n-dimensionality" property. In a recent work of X. Tolsa and D. DÄ…browski, a class of measures that generalizes UR measures to non-AD-regular setting was identified. The aim of this project is to obtain new characterizations of these measures, and then to use them to generalize some important recent results on harmonic measure to non-AD-regular setting. In parallel, some questions related to orthogonal projections will be pursued, most notably ones involving the Favard length and Furstenberg sets.
