The Flowering of Differential Topology
The Flowering of Differential Topology
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.
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. This object has only one surface and one edge, as can be seen by coloring the middle of the strip or the edge with a crayon. But topologists are also interested in the properties of objects that cannot be visualized in any usual sense.
As in other areas of mathematics, topologists try to be as general as possible in drawing conclusions. Often they do not restrict themselves to the two- or three-dimensional space of experience, but ask about the characteristics of objects in four, or five, or 500 dimensions. These studies are not necessarily sheer flights of fancy. An equation in five variables defines a surface or "hypersurface" in five-dimensional space. Topologists will want to know whether the surface is closed or infinite in extent, and about how curved different parts of the surface might be.
Differential topology is the study of the curvature of generalized surfaces, or, as topologists call them, manifolds. Measuring the curvature of a surface in a space of more than three dimensions is difficult and topologists often deal with the problem by taking what might be called an "ant's eye view." Suppose that a mathematically inclined ant is walking over the surface of a very large mound. Since he is small, he is unable to tell if the mound is flat, or has a bit of curvature to it. If he finds a bit of string, however, he can tuck one end into the ground and trace out a circle, and then measure, perhaps by counting paces, both the apparent circumference of the circle and its diameter. If the ratio of circumference to diameter equals the number π, that is 3.14159 . . . , he can conclude that the surface is flat. If it is larger, then the surface is said to have a negative curvature. If it is smaller, then it has a positive curvature. In either case he can use the difference between the ratio and π as a measure of the curvature.
One of the means by which topologists classify manifolds is by their connectedness. If one draws a circle on the surface of a sphere, one can deform it gradually so that it gets smaller and smaller and shrinks to a point. Because this is true for any circle that can be drawn on a sphere, or those other three-dimensional objects that a sphere can be transformed into, we say that the sphere is simply connected. This property is not shared by the torus (donut shape), because a circle drawn around the hole can never be shrunk to a point.
One of the founders of topology, the mathematician Jules-Henri Poincaré (1854-1912), asked whether the same shrinking circle argument could be used for a manifold that was the three-dimensional surface of a four-dimensional sphere; that is, could the so-called "three-spheres" and the topologically related objects be distinguished from all other three-dimensional manifolds by this shrinking circles test. This question has remained unanswered, mainly because topologists have not found a completely effective way of classifying three-dimensional manifolds. In 1961 the American mathematician Stephen Smale (1931- ) developed a method for classifying higher dimensional manifolds by breaking them up into small pieces that could be moved around. Smale was able to prove that the circle test would hold true for manifolds of five or higher dimensions, but was unable to find a proof for the three- or four-dimensional case. The study of higher dimensional manifolds has provided a number of surprises. In 1959 American mathematician John Milnor (1931- ) showed that a seven-dimensional sphere could be rearranged to form a smooth manifold in 28 different ways.
Another area of interest in differential topology is the existence of so called minimal surfaces. If one forms a closed curve out of wire and then dips it into a soap solution, the film that is formed when the wire is removed will be one of minimum area for the curve selected. For such a minimum area surface, it can be shown that the average of the curvature around any point will be zero. The property of average zero curvature can then be used to define a minimal surface of infinite extent. For many years topologists knew of only three surfaces of minimal curvature in three-dimensional space that did not intersect themselves, technically known as embedded minimal surfaces. These were the infinite plane, which has zero curvature in all directions at each point, the helicoid, a sort of smoothed spiral staircase, and the catenoid, a sort of hourglass, all known in the nineteenth century. The first two surfaces were topologically related to a spherical surface with a circular opening cut in it, and the last to a spherical surface with two openings. In the 1980s a Brazilian graduate student named Celso Costa devised a set of equations that he could prove represented an infinite minimal surface, but could not determine whether it intersected itself. It was left to two American mathematicians, David Hoffman and Bill Meeks, and a graduate student in computer science, James Hoffman, at the University of Massachusetts to determine that it was in fact a non-intersecting and thus minimal surface. To truly understand the nature of these surfaces, these researchers relied heavily of new computer graphics programs. Not only were they ably to devise a proof Costa's surface did not intersect itself, but they were also able to demonstrate the existence of an infinite number of such embedded surfaces.
The work of topologists seldom attracts attention from the general public. This was not so in the case of catastrophe theory, developed by French mathematician René Thom (1923- ) and presented to the public in a book entitled Structural Stability and Morphogenesis, first published in French in 1972. Thom's theory deals with the type of system frequently encountered in the sciences in which the state of a system is described by two kinds of variables, called observables and parameters. The parameters may be quantities that could be externally controlled, or that would vary slowly from one instance to the other, while the observables are measurable properties of the system. For a given set of values of the parameters, the relatively stable states would correspond to minimums, or spots of greatest positive curvature in a manifold. In Thom's terminology, a catastrophe occurs when a stable condition becomes unstable when the parameters change by a small amount. Thom was able to argue that, with very limited exceptions, for systems with six or fewer parameters, there were only seven types of catastrophe possible—that is, seven different types of manifold that would allow catastrophe's to occur.
Thom's book met an initially highly positive response from reviewers. A reviewer for the (London) Times compared Thom's book to Sir Isaac Newton's Mathematical Principles of Natural Philosophy. In 1974, at the International Congress of Mathematicians held in Vancouver Canada, English mathematician Christopher Zeeman gave a major lecture on the topic, suggesting that catastrophe theory would find its main applications in the behavioral and social sciences. When the English translation of Thom's book was published in 1975, enthusiastic reviews appeared in a number of scientific periodicals, as well as in the New York Times and the widely read Newsweek.
There were negative responses to catastrophe theory as well. One mathematics professor at Rutgers University spent two years giving seminars at different meetings, claiming not that catastrophe theory was mistaken, but that the clams made for it in areas far from mathematics would eventually not be born out. Understanding was not helped by the fact that the term "catastrophe" has a less dramatic meaning in French than the same word in English, which is synonymous with calamity. Enthusiasm for catastrophe theory has died down somewhat, while the basic mathematical soundness of Thom's ideas has not been questioned.
The desire for a consistent theory of change has been a recurrent theme in philosophy. In his Physics, the Greek philosopher Aristotle (384-322 BC) attempted to provide a uniform explanation for change in the physical world, in living things and in societies, and to do so without mathematics. Thom's work, to some interpreters, represents a new, and mathematically informed, attempt towards such a synthesis.
DONALD R. FRANCESCHETTI
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