Title: The Gromov-Hausdorff distance between spheres

Abstract: A very natural question in geometry and in many applications is: How can we quantify how different two shapes are? One way to do this is by invoking distances between metric spaces, such as the Gromov–Hausdorff distance, which is used in areas like Riemannian geometry and data analysis. Nevertheless, exact values are known only in a few special cases. For finite metric spaces, the exact computation of this distance (and closely related distances) is NP-hard. Despite this hardness, practitioners develop heuristic algorithms to estimate it, which in turn motivates the search for theoretically determined benchmarks against which to evaluate such methods.

In this talk, I will focus on the case of unit round spheres equipped with the geodesic metric. I will describe how to obtain lower bounds for the Gromov–Hausdorff distance between spheres and, in certain cases, establish sharpness by constructing optimal correspondences, thereby determining the exact value of $d_{\mathrm{GH}}$ for the corresponding pairs of spheres.

Many challenges remain: for most pairs of spheres, the exact value of the distance is still unknown, and even the right techniques for tackling these cases are not yet clear. This makes the problem a rich source of open questions at the intersection of geometry, topology, and data analysis. Interestingly, these results connect with a classical topological result called the Borsuk–Ulam theorem—but in the unusual setting where we must deal with discontinuous functions.