The conformal dimension of a metric space $(X,d)$ measures its optimal shape from the perspective of quasiconformal geometry. It is defined by infimizing dimension over metrics in the quasisymmetric equivalence class of $d$. Introduced by Pierre Pansu in 1989, conformal Hausdorff dimension played an important early role in the development of analysis on metric spaces by linking this emerging subject with the coarse geometry of Gromov hyperbolic groups. In later years a variant notion, conformal Assouad dimension, gained prominence. Assouad dimension—which bounds Hausdorff dimension from above—is a scale-invariant, quantitative measurement of the size of optimal coverings of a space. Intermediate between these two is Minkowski (box-counting) dimension. Dimension interpolation is an emerging program of research in fractal geometry which identifies geometrically natural one-parameter dimension functions interpolating between existing concepts. Two exemplars are the Assouad spectrum (Fraser-Yu, 2015), which interpolates between box-counting and Assouad dimension, and the intermediate dimensions (Falconer-Fraser-Kempton, 2020), which interpolate between Hausdorff and box-counting dimension.

In this talk, I’ll highlight a few significant milestones in the theory of conformal dimension, concluding with recent applications of dimension interpolation to the quasiconformal classification of sets and the range of conformal Assouad spectrum. The latter results are based on joint work with Efstathios Chrontsios Garitsis (Univ Tennessee) and Jonathan Fraser (Univ of St. Andrews).