Facundo Mémoli — The Gromov-Hausdorff distance between spheres

A very natural question in geometry (and in many applications) is: How can we measure how different two shapes are?
One way to do this is by using distances between metric spaces, such as the Gromov–Hausdorff distance, which is often invoked in areas like data classification. However, even though distances of this type are widely used, their exact values are known only in a few very special cases.
In this talk, I will focus on the case of spheres (with their usual round, geodesic distance). I will describe how one can obtain lower bounds for the Gromov–Hausdorff distance between spheres, and in certain cases prove that these bounds are sharp by constructing optimal correspondences. But 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 analysis.
Interestingly, these results connect with a classical topological result called the Borsuk–Ulam theorem—but in a new setting where we have to deal with discontinuous functions.