But when might a coffee cup not be equivalent to a doughnut?
“Say we have a six-dimensional coffee cup, and we know we can stretch it into a doughnut, but if you’re only allowed to stretch at a constant rate, how long does that take?" Manin said. “Can we get a computer to decide whether you can do it?”
In some cases it’s guaranteed that two topologically equivalent objects will easily transform into one another, he said, but in others it’s not possible for even computers to tell whether two are topologically the same. His work examines the transformation in terms of both the geometric complexity (the deformation) and the algorithmic complexity (the computer’s ability to discern equality or — lack thereof).