Coarse geometry and dimension theory
Coarse (or “fractal”) sets are sets which exhibit complex geometric properties at arbitrarily small scales. One common way to understand the fine scaling properties of a set is from the perspective of dimension theory, which attempts to assign and study various dimension-like invariants for sets, and to relate the properties of these dimensions to the geometry of the underlying set.
One family of dimensions which I have focused on are given by the Assouad-type dimensions, which are a family of dimensions which capture the “coarsest” scaling properties of a set. This area has been very active recently, with a large amount of interest in this field from various points of view, such as conformal geometry, embedding theory, and fractal geometry.
Another concept my research has focused on is the notion of dimension interpolation. In this scheme, one takes two commonly studied notions of dimension (for example, Hausdorff and box dimensions) and attempts to define a continuously parametrized family of dimensions which interpolate between the endpoints. Some examples include the (generalized) intermediate dimensions, the Assouad spectrum, the generalized Assouad dimensions, and the Fourier dimension spectrum
Here is a summary my work in this area:
- I have proven classification results for intermediate dimensions (with Amlan Banaji) and Assouad spectra.
- I have studied the Assouad dimension of overlapping self-affine carpets in my joint paper with Jonathan Fraser.
- I have studied the generalized Assouad dimensions, both in a general setting and for sets associated with branching processes and iterated function systems, in a joint paper with Amlan Banaji and Sascha Troscheit).
- Tangents and pointwise Assouad dimension of invariant sets
With: Antti Käenmäki Journal: Preprint (submitted) Links: pdfarxiv
- Interpolating with generalized Assouad dimensions
With: Amlan Banaji, Sascha Troscheit Journal: Preprint (submitted) Links: pdfarxiv
- Assouad-type dimensions of overlapping self-affine sets
With: Jonathan M. Fraser Journal: Preprint (submitted) Links: pdfarxiv
- Attainable forms of Assouad spectra
Journal: Indiana Univ. Math. J. (to appear) Links: pdfarxiv
- Attainable forms of intermediate dimensions
With: Amlan Banaji Journal: Ann. Fenn. Math. 47 (2022), 939–960 Links: pdfdoizblarxiv