Topology asks questions like "are these two manifolds diffeomorphic?" or "can these two maps be deformed into one another?".  But what happens if one only has bounded resources for constructing diffeomorphisms (e.g. if the rubber sheets are really made out of rubber and will tear if pulled too much) or for building homotopies (e.g. if there is an energy level one can't go beyond in constructing a homotopy)?   The question is ultimately one of the comprehensibility of (understood) topology.  

Insights originating in logic, topological data analysis, approximation theory as well as homotopy theory, geometric measure theory, probability and harmonic analysis have been shed light on aspects of this theme.  I will try to describe some of these ideas (due to very many people), and also some problems that are close to the surface, yet remain out of reach.