Krystyna Kuperberg is a researcher and educator best known for disproving the famous Seifert conjecture in topology. Her counterexample, first announced in the mid-1990s, was termed a "small miracle" of geometry by Ian Stewart. It was quickly generalized and should prove central to the continued development of dynamic systems theory, by way of the vector fields used to study physical and statistical phenomena.
Kuperberg was born Krystyna M. Trybulec in Tarnow, a city in southern Poland, on July 17, 1944. Her parents, Jan W. and Barbara H. (Kurlus) Trybulec, were both trained pharmacists. Her brother, Ardrzej, also became a mathematician. After receiving a master's degree from Warsaw University in 1966, Kuperberg had to wait until settling in the United States to earn her Ph.D. This was awarded by Rice University in 1974. Upon graduating she accepted her first post at Auburn University. Kuperberg remains a member of the faculty at Auburn, and has been a full professor there since 1984. She has also held visiting positions at Oklahoma State University, the Courant Institute of Mathematical Sciences in New York, the Mathematical Sciences Research Institute at Berkeley, and l'Universite de Paris-Sud, Centre d'Orsay.
In 1974, the first counterexample to the famous Seifert conjecture was found by P. A. Schweitzer. His "plug" was devised to cancel out any circular orbit, but it broke down to two minimal sets. Kuperberg, who had begun publishing papers in 1971, was already interested in dynamical systems. She resolved a conjecture about fixed points in 1981 with Coke Reed, and built upon the methods used in this work to find a new kind of counterexample with only one minimal set. What she eventually found served to disprove the Siefert conjecture for all three-dimensional manifolds.
The Seifert conjecture is a higher-dimensional extension of the "hairy billiard ball" theorem for the two-dimensional surface of a sphere. The idea that you cannot comb down all the hairs on a fully hairy ball without getting a cowlick is really a geometric statement about a dynamical system. The one-dimensional version, the circle or "1-sphere," is "combable," allowing for a smooth vector field. The 2-sphere is combable because it contains at least one "bald spot" consisting of a fixed point around which the trajectories can flow. The fact that there is always some place on Earth where the wind is not blowing is a real-world example of this "bald spot."
More complex surfaces proved more difficult to analyze. In the case of a torus-shaped vector field or "hairy donut," for instance, whether a trajectory is fixed or not depends upon how it advances along the circumference of the torus as it flows. This explains why it was impossible to prove a simple conjecture about one of the three-dimensional shapes for more than 40 years. In 1950, Herbert Seifert had proposed that in the three-dimensional case any smooth vector field will have at least one "closed" or periodic orbit. It was already known that 3-spheres did not have any "bald spots," but it seemed reasonable enough to think that they would have at least one closed orbit.
Kuperberg disproved this conjecture in 1993 by constructing a smooth vector field with no closed orbits, and her construction applies not only to 3-spheres, but to all three-dimensional manifolds. To do this she used a Wilson plug, a kind of topological tool, to break up any closed orbits that might be present. This plug is a three-dimensional shape that traps the trajectories of one or more formerly closed orbits inside itself. The trick is to apply the plug without creating any new closed orbits. To accomplish this, Kuperberg modified the plug so that it "eats its own tail" like a snake. Thus, the trajectories that enter get trapped in an infinite spiral and no new closed orbits can be formed. In addition to disproving the Siefert conjecture, Kuperberg's construction produces a "minimal set" that may be of an entirely new kind, according to John Mather, a dynamical systems theorist at Princeton. Since minimal sets are basic components of dynamical systems, Kuperberg's plug may help mathematicians better understand the range of things that can happen in these systems.
Since her success in disproving the Siefert conjecture, Kuperberg has been especially in demand as a speaker at events devoted to topology or dynamical systems, and at honorary symposia worldwide. She delivered the MSRI-Evans lecture at Berkeley in 1994, and addressed the American Mathematical Society and Mathematical Association of America meetings in 1995 and 1996.
Kuperberg's husband and frequent collaborator, Wlodzimierz, received his Ph.D. in mathematics from Warsaw University. He is also a professor at Auburn University. Krystyna and Wlodzimierz were married in Poland and lived there until 1969. Their son, Greg, born in Poland in 1967, is also a mathematician; he received a Ph.D. in mathematics from the University of California at Berkeley. Their daughter, Anna, born two years later in Sweden, holds a M.F.A. from the San Francisco Art Institute.
In addition to awards from Auburn University for her research and professorship, and National Science Foundation grant support, Kuperberg also won the Alfred Jurzykowski Foundation Award in 1995 from the Kosciuszko Foundation in New York. In 1996, Kuperberg was elected to the American Mathematical Society council as a member at large for a three-year term. She also currently edits the Electronic Research Announcement of the American Mathematical Society.
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