The Flowering of Differential Topology
Overview
A number of important advances in understanding the curvature of surfaces in three- and higher dimensional space have occurred in the decades following 1950. A method of cutting up surfaces, called surgery on manifolds, enabled the resolution of some long-standing conjectures about surfaces in higher dimensional spaces. In ordinary three-dimensional space, computer-assisted investigators discovered families of new minimal area surfaces. René Thom's catastrophe theory claimed to provide a means of explaining abrupt changes in the stable behaviors of complex systems, but met a varied reception among scientists and mathematicians.
Background
Topology is concerned with the behavior of geometrical forms as they are stretched or squeezed or twisted. To a topologist a billiard ball and a soup bowl are related because they can be gradually transformed into each other without separating any points that were originally very close to each other. In the same sense, a donut and a teacup are topologically related to each other but not related to a billiard ball since they both involve an opening. One of the simplest objects studied by topologists is the Möbius strip, which is obtained when a long strip of paper is given a half twist and its ends pasted together.
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