Two leading University of California researchers, one a physicist, the other a mathematician, will be joining the Cornell faculty in the coming months. The hirings are regarded as a major coup for the university, since both are seminal figures in their fields. J.C. Séamus Davis, professor of physics at UC-Berkeley (Read Davis' profile), is on the leading edge of the development of scanning tunneling microscopy as applied to superconductivity and quantum computing semiconductors. William Thurston, professor of mathematics at UC-Davis, is a world-renowned topologist, with a keen interest in the application of computing to mathematics.
Cornell's mathematics department is in a state of excitement over the fact that William Paul Thurston, now at the University of California-Davis, will join the faculty in the fall of 2003.
"It's like hiring one of the great Nobel Prize winners," said John Hubbard, professor of mathematics, who for the past 15 years has been writing a book about Thurston's work. There is no Nobel Prize for mathematics, but in 1982, at age 37, Thurston won the Fields Medal, which mathematicians regard as the equivalent.
He will have a joint appointment in the Department of Mathematics and the Faculty of Computing and Information Science.
Thurston's primary work is in topology, which deals with properties of geometric objects that remain unchanged when the object is bent or stretched. For instance, an inner tube or a balloon is topologically the same whether it is inflated or not. Particularly, he works with "manifolds," a generalization of surfaces. Balloons, inner tubes and your skin are good approximations of two-dimensional manifolds. A two-dimensional manifold can be created by taking a flat surface and bending it around until the edges can be glued together. The three-dimensional analog, called a 3-manifold, can be constructed by mathematically gluing together the limits of a three-dimensional space.
Thurston received the Fields medal for showing that a large class of such manifolds can be blown up to one particular "best shape" that would be, geometrically, a "perfect shape." It will be "perfect'' in the sense that just as on a sphere, every point looks like every other point. A football, in contrast, is not perfect: the ends are more curved than the midsection. "Perfection'' requires constant curvature.
The class of perfect objects Thurston dealt with are "hyperbolic" 3-manifolds. If you were a two-dimensional being living on the surface of a sphere, you would eventually discover that the space around you has positive curvature. Now imagine gluing together a lot of equilateral triangles. Six to a vertex makes a flat surface; five to a vertex makes an icosahedron, which is approximately spherical; seven to a vertex is an approximation to a hyperbolic plane, which has negative curvature. Hyperbolic 3-manifolds are made out of pieces of three-dimensional hyperbolic space glued together.
"Before Thurston came around, people thought hyperbolic manifolds were very unusual," Hubbard explains. "What he showed is that a very large class of hyperbolic manifolds have this natural perfect shape."
Thurston also has made important contributions in the study of foliations. A foliation of a manifold is a way of filling it with manifolds of one dimension lower -- for instance, filling a three-dimensional manifold with surfaces, which may wind around and accumulate in complicated ways. Some manifolds have an "obstruction," which makes it impossible to foliate them completely. The surface of a sphere, for example, can't be sliced entirely into circles because there will always be two single points -- the poles -- left over. Mathematicians say that a sphere has a "Euler characteristic" of two. In another breakthrough, Thurston proved that the Euler characteristic is the only obstacle to foliation, so any manifold whose Euler characteristic is zero can be foliated.
Thurston has co-written papers with members of the Cornell computer science faculty and lectured here recently on the application of computing to certain problems with manifolds. "He wants to run algorithms on real machines," said Robert Constable, dean of computing and information science. "It was only a small part of it, but I think our computing and information science faculty was the icing on the cake in getting him to come here."
The son of a homemaker and an aeronautical engineer, Thurston attended New College in Sarasota, Fla., and after graduation in 1967 studied under the noted mathematician Morris Hirsch at the University of California-Berkeley, earning his doctorate in 1972. He spent two years at the Institute for Advanced Study in Princeton, N.J., then taught for two years at the Massachusetts Institute of Technology before joining the faculty of Princeton University.
In 1991 he returned to UC-Berkeley as a faculty member, and in 1993 became director of the Mathematical Sciences Research Institute in Berkeley. His career has been noteworthy for efforts to improve the teaching of mathematics in high schools and for bringing more women and minorities into mathematics. In 1996 he moved to UC-Davis when his wife enrolled as a veterinary student there. Now, Hubbard says, Thurston has decided to move east and has been drawn to Cornell by the opportunity to work with a strong group of mathematicians in his field.
Thurston held an Alfred P Sloan Foundation Fellowship from 1974 to 1975. In 1976 he was awarded the Oswald Veblen Geometry Prize of the American Mathematical Society for his work on foliations. In 1979 he became the second person to receive the Alan T. Waterman Award. He is a member of the American Academy of Arts and Sciences and the National Academy of Sciences.
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