Smale’s Fields Medal honored a famous piece of work in topology, a branch of mathematics that flourished in the twentieth century and had a particular heyday in the fifties. Topology studies the properties that remain unchanged when shapes are deformed by twisting or stretching or squeezing. Whether a shape is square or round, large or small, is irrelevant in topology, because stretching can change those properties. Topologists ask whether a shape is connected, whether it has holes, whether it is knotted. They imagine surfaces not just in the one–, two–, and three-dimensional universes of
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