Until recently, topology was seen as being among the most abstract fields of mathematics, one that bore out Henry John Stephen Smith’s 19th century toast: “Pure mathematics — may it never be of use to anyone!” But now the field, which deals with the shape of many-dimensional objects, has unexpectedly proved its usefulness in, of all places, medicine. Researchers have used topology to discover a new subgroup of breast cancer patients with a 100 percent survival rate. More generally, the method may prove powerful for making sense of the massive, high-dimensional, noisy datasets modern science is producing.
Genetics experiments can produce vast quantities of data — determining the activity of each of the approximately 20,000 genes in a sample of breast cancer tissue, for example. Each sample can be seen as a point in 20,000-dimensional space. But these readings aren’t absolutely accurate, so each point may not be in exactly the right place. That makes plucking information out of that sea of data particularly challenging.
One key is to recognize that “data has shape, and that shape matters,” says mathematician Gunnar Carlsson of Stanford University.
Topology turns out to be especially useful for identifying the shape of noisy data, because it characterizes shapes in a flexible, qualitative way. Squish, twist or enlarge an object and topology will consider it unchanged, as long as you don’t punch holes or glue bits together. So from the perspective of topology, a coffee cup and a doughnut have the same shape: By squishing the cup down, the handle turns into a doughnutlike ring. This qualitative understanding turns out to deal perfectly with the noisiness of data sets, since the precise location of data points doesn’t matter.