L et’s start with what’s probably the most tired, overused joke in math: A topologist is someone who can’t tell a coffee cup from a doughnut. Both, you see, have a hole in them. Topology is usually described as a sort of “rubber sheet” geometry in which two shapes are considered the same if one can be stretched or compressed into the other without tearing it. But this summary leaves out something essential: How do topologists, and the many other mathematicians who rely on their methods, rigorously account for all this stretching? They don’t look at a doughnut and a coffee cup, squint, and say to themselves, “Sure, I can intuitively see how to squeeze one into the other, so they must be the same.” Rather, they describe a shape in a way that can “forget” about distance while respecting the underlying structure in a flexible way, allowing it to bend and stretch.…