Topology meets discrete geometry: Proximity and incompressibility

Topology gives certificates that spaces are not the same or that one space does not embed into another. I will discuss geometric, quantitative versions of these ideas: If two spaces are not the same, to what extent do they differ? If a space cannot sit inside another, how much discontinuity is forced on any injective function? The methods that bound these quantities have connections to coverings and packings, optimal transport, and to the Borsuk-Ulam theorem. I will explain some of these connections, illustrated with various examples.