Dimension theory in the toolbox of a computational topologist: the case study of the Gromov-Hausdorff distance.

In topological data analysis (TDA), datasets are often summarised through topological signatures—compact descriptors capturing features such as connected components or holes. These signatures are typically embedded into Hilbert spaces for statistical analysis or low-dimensional visualisation. However, such embeddings may inherently distort the geometry of the space of signatures. Over the past decade, tools from dimension theory have been developed to study the complexity of these signature spaces and to quantify the intrinsic limitations of such representations.

In this talk, we explore the role of dimension theory—specifically, topological, Assouad, and asymptotic dimensions—in studying the space of compact metric spaces equipped with the Gromov–Hausdorff distance. This distance, which quantifies how far two metric spaces are from being isometric, serves as a foundational notion in shape comparison and the theoretical underpinnings of computational topology. We present recent results on the dimension of this space and discuss their implications in computational topology.