Variants of Gromov-Wasserstein distances

Gromov-Wasserstein (GW) distances comprise a family of metrics on the space of (isomorphism classes of) metric measure spaces. Driven by specialized applications, there have been a large number of variants of GW distance introduced in the literature in recent years, each of which is designed to provide meaningful comparisons between certain data objects with complex structure. These complex data objects include (attributed) graphs, hypergraphs, point clouds endowed with preferred persistent homology cycles, and many others. In this talk, I will survey some of these variants, focusing on those with connections to applied and computational topology. I will also describe recent joint work with Bauer, Mémoli and Nishino, which introduces a general framework that captures several of these variants, allowing us to derive broadly applicable theoretical properties.