Michael H. Freedman has been recognized by the American Mathematical Society, the International Congress of Mathematicians, the United States Government, and the MacArthur Foundation for his research breakthroughs in topology, a branch of mathematics that deals with the invariant properties of geometric objects rather than their sizes and shapes. Freedman's work has been fundamental in making progress with some of the most difficult problems in four-dimensional geometry and topology. He is perhaps best known for his proof of the four-dimensional Poincaré conjecture, a problem dating from 1904.
Michael Hartley Freedman was born in Los Angeles on April 21, 1951, to Benedict Freedman and Nancy Mars Freedman. Freedman began his post-secondary education with a year at the Berkeley campus of the University of California in 1968. He then transferred to Princeton University, where he received his Ph.D. under William Browder four years later. While in college, he pursued his hobby of rock climbing, scaling the northeast ridge of Mount Williamson alone in 1970. A decade later, he won the Great Western boulder climbing championship.
Freedman joined the Department of Mathematics at the University of California, Berkeley as a lecturer in 1972. In 1974, he spent a year at the Institute for Advanced Study (IAS) in Princeton. Then he returned to California, this time to the University of California, San Diego (UCSD) campus, where he quickly progressed through the ranks of assistant professor, associate professor, and full professor. In 1985, Freedman was appointed by UCSD as the first professor to hold the newly endowed Charles Lee Powell chair of mathematics.
Throughout the 20th century, mathematicians have made progress in understanding geometric objects in terms of associated algebraic operations. In particular, topologists have tried to use algebra to classify manifolds (multidimensional surfaces). Visualizing surfaces in more than three-dimensional space is difficult. Four dimensions are somewhat intuitive if one considers the fourth dimension to be time. In his Fortune magazine description of Freedman's work, Gene Bylinsky suggested thinking of an eight-dimensional sphere as a ball with attached information about its age, color, temperature, weight, and bounciness.
In 1904, French mathematician Jules Henri Poincaré designed a system to classify manifolds. He imagined a loop of string wrapped around a surface and determined how far the loop could be shrunk. On a sphere, the loop could be shrunken to a single point. On a torus (doughnut-like shape), a loop encircling the hole cannot shrink smaller than the circumference of the hole. Thus, a sphere and a torus belong to different classifications.
Three-dimensional manifolds are especially difficult to classify because they can be stretched and folded in many different ways. Poincaré devised a series of tests that he believed could be used to identify any three-dimensional manifold, no matter how distorted, that was topologically equivalent to a sphere. The statement of this problem was refined over the years, but not until 1960 did Stephen Smale give the first proof of the Poincaré conjecture for all dimensions greater than four. Smale and other topologists followed algebraic guidelines in cutting the manifold apart and sewing it back together as a sphere, a technique known as surgery. However, manifolds of dimension three or four do not have as much "room" for maneuvering. Thus, the necessary surgery is much more difficult, and the four-dimensional Poincaré conjecture remained unsolved for another two decades.
Finally, after seven years of work, Freedman solved the surgery problem for simply connected four-dimensional manifolds in 1982. His paper "The Topology of Four-Dimensional Manifolds" gives a complete classification of all simply connected, four-dimensional manifolds in terms of two quantities. In the course of proving this theorem, Freedman exhibited several new four-dimensional manifolds,including the first examples of such manifolds that do not support a coordinate system for calculus. These results, along with nearly 50 papers on the structure and classification of three- and four-dimensional manifolds, resolved many fundamental issues in these physically significant dimensions.
John Milnor, a mathematician of considerable stature, wrote in Notices of the American Mathematical Society that Freedman's classification theorems for important classes of four-dimensional topological manifolds "are simple to state and use, and are in marked contrast to the extreme complications that are now known to occur in the study of differentiable and piecewise linear 4-manifolds."
The four years following his proof of the famous conjecture were eventful for Freedman. In 1983, he married Leslie Blair Howland, with whom he would raise three children. In 1984, he received a five-year MacArthur Foundation Fellowship to provide financial support while he continued his research. That same year he was elected to the National Academy of Sciences, and the following year to the American Academy of Arts and Sciences. In 1986 Freedman received the Fields Medal, the highest honor in mathematics.
The American Mathematical Society awarded the 1986 Oswald Veblen Prize in Geometry to Freedman for his work in four-dimensional topology. In his response to this award, Freedman discussed the importance of interchange among the various branches and applications of mathematics. That statement, which was printed in the Notices of the American Mathematical Society, included his assertion that "Mathematics is not so much a collection of different subjects as a way of thinking. As such, it may be applied to any branch of knowledge." In particular, he praised the movement among mathematicians to voice their opinions on such topics as education, energy, economics, defense, and world peace. He noted that "Experience inside mathematics shows that it isn't necessary to be an old hand in an area to make a contribution. Outside mathematics the situation is less clear, but I can't help feeling that there, too, it is a mistake to leave important issues entirely to the experts."
Freedman pushed the limits not only in terms of his expectations of mathematicians in society, but also in his technical research. Speaking to a joint meeting of the American Mathematical Society and the Mathematical Association of America in early 1997, he described the direction of his current work on the subject of computability. Computer scientists are interested in being able to identify which problems are "hard" in the sense of not being solvable with an efficient computer algorithm. The classic question, known as the Traveling Salesman Problem, asks for the most efficient itinerary for a person to visit each of a certain number of sites. Analogous problems have practical applications in areas such as cryptography and computer data security. According to Barry Cipra's article in Science magazine, Freedman thinks key to deciding whether a problem is "hard" may be to look at the limiting case of such a problem as the number of choices approaches infinity. Cipra quotes Freedman as saying, "This is always an attractive situation for the pure mathematician, when there's a very clear, well-defined problem blocking understanding. It's kind of like waving a red flag at a bull!"
Although Freedman has announced the broad direction of his assault on the two-decade-old Traveling Salesman Problem, the mathematics community has yet to see a detailed description of his tactic. The considerable respect with which Freedman is regarded by fellow mathematicians generates optimism that his approach may generate at least some progress on this elusive topic.
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