## View abstract

### Session S13 - Harmonic Analysis, Fractal Geometry, and Applications

Thursday, July 15, 17:45 ~ 18:15 UTC-3

Many notions of dimension have been introduced to help understand the geometry of (often small') subsets of metric spaces, such as subsets of $\mathbb{R}^{n}$ of Lebesgue measure zero. Hausdorff and box dimensions are well known examples of such notions, and possibly less known are the upper and lower Assouad dimensions of a set $E$, which quantify the thickest' or thinnest' part of the space.
In the talk I will review some results related to a general class of intermediate dimensions', referred to as the upper and lower $\Phi$-dimensions, which can roughly be thought of as local refinements of the box-counting dimensions where one takes the most extreme local behaviour, at scales `ruled' by a function $\Phi$. These $\Phi$-dimensions provide a range of bi-Lipschitz invariant dimensions between the box and Assouad dimensions. As the box and Assouad dimensions for a given set can all be different, the intermediate $\Phi$-dimensions provide more refined information about the local geometry of the set, such as detailed information about the scales at which one can observe extreme local behaviour.