Krystyna Maria Trybulec Kuperberg

Quick Info

Born
17 July 1944
Tarnów, Poland

Summary
Krystyna Kuperberg is a Polish mathematician who works in America. She is known for her work in dynamical systems, geometry and topology.

Biography

Krystyna Kuperberg's name before she married was Krystyna Maria Trybulec. Her parents were Jan W Trybulec and Barbara H Kurlus. They were both trained pharmacists and owned a chemists shop in Tarnów where they dispensed medicines. The town of Tarnów, in South eastern Poland, had been administered by Austria until it was returned to Poland after the end of World War II, when Krystyna was still a very young child. She grew up in a family with an older brother Andrzej Trybulec who was interested in philosophy and mathematics.

Krystyna was brought up in Tarnów until she was 15 years old, when the family moved to Gdańsk on the northern coast of Poland. After attending high school for three years in Gdańsk she entered the University of Warsaw in 1962, where her brother Andrzej was studying. He had set out on a philosophy course but had changed to study mathematics. Krystyna decided in high school that she would study mathematics at university. The first course she studied was given by Andrzej Mostowski, then later she attended topology lectures given by Karol Borsuk and found the subject a fascinating one. She was not the only member of her family to love Borsuk's lectures, for her brother also found them far the best of all the mathematics lectures. Borsuk's lectures held another special significance for Krystyna, since a fellow student at these lectures was Wlodzimierz Kuperberg, whom Krystyna married.

After graduating, Krystyna Kuperberg undertook research at the University of Warsaw under Borsuk's supervision and, in 1966, she was awarded her Master's degree. Her husband went on to complete his Ph.D. in mathematics but Krystyna stopped her studies with the Master's Degree. Her first child Greg Kuperberg was born in 1967. Two years later, in 1969, the Kuperberg family left Poland and went to live in Sweden. Krystyna's second child, Anna, was born in 1969 not long after the family moved to Sweden.

In 1972 the Kuperberg family moved again, this time to the United States and Krystyna took up officially the graduate work she had begun in Warsaw under Borsuk's supervision. It was not that she had given up mathematics in the intervening years, merely that she had not been registered as a student or held a post. She had, however, continued research collaborating with her husband. The two published several papers on topology. In 1971 Krystyna and Wlodzimierz published the joint paper On weakly zero-dimensional mappings. In this paper they answered a question posed by A Lelek in 1967 when he asked if a Cantorian manifolds could be mapped by a weakly zero-dimensional mapping onto a space of lower dimension. They proved that indeed it was possible.

In 1972, the year Krystyna register as a research student at Rice University in Houston, Texas, she published An isomorphism theorem of the Hurewicz-type in Borsuk's theory of shape in Fundamenta Mathematicae. At Rice University her doctoral studied were supervised by W H Jaco and she was awarded her Ph.D. in 1974. She attended a topology conference at the University of North Carolina, Charlotte in the same year and presented a paper Two Vietoris-type isomorphism theorems in Borsuk's theory of shape, concerning the Vietoris-Čech homology and Borsuk's fundamental groups which was published in the conference proceedings in the following year. Also in 1974 both Krystyna and her husband Wlodzimierz Kuperberg were appointed to permanent positions on the Faculty at Auburn University in Auburn, Alabama.

Krystyna Kuperberg has remained at Auburn University since she was appointed in 1974. She was promoted to Full Professor at Auburn in 1984 and then awarded an Alumni Professorship there in 1994. While on the Faculty at Auburn she has held a number of visiting positions: Oklahoma State University (1982-83); the Courant Institute of Mathematical Sciences (1987); the Mathematical Sciences Research Institute at Berkeley (1994-95); and the University of Paris at Orsay (summer 1995).

In 1975 she published A note on the Hurewicz isomorphism theorem in Borsuk's theory of shape which improved on the results of her paper from 1972 published in Fundamenta Mathematicae which we mentioned above. Her interests moved towards applying topological ideas to the theory of dynamical systems. In 1981 she published A rest point free dynamical system on $\mathbb{R}^{3}$ with uniformly bounded trajectories. This paper, on which she collaborated with Coke Reed, provided a counter-example to a question posed by Ulam in 1935 and entered by him into the Scottish Book.

Continuing her collaboration with Coke Reed, Kuperberg published another paper on dynamical systems in 1989 when A dynamical system on $\mathbb{R}^{3}$ with uniformly bounded trajectories and no compact trajectories appeared in the Proceedings of the American Mathematical Society. In 1988 Kuperberg solved a problem relating to question of Knaster as to whether every homogeneous space is bihomogeneous. Kuratowski found a counter-example in 1922 but in 1930 T Dantzig had asked whether Knaster's question was true for continua. It was this question to which Kuperberg found a counter-example, announcing it in A homogeneous nonbihomogeneous continuum in 1988, and giving full details in On the bihomogeneity problem of Knaster published in the Transactions of the American Mathematical Society.

Kuperberg's most celebrated result, however, was discovered in 1993 and published in 1994 in A smooth counterexample to the Seifert conjecture in the Annals of Mathematics. John E Fornaess, reviewing the paper, explains its context:-
One of the basic concepts in dynamical systems is that of a fixed point or a periodic orbit. This was already observed by H Poincaré (in 1890), who discussed the existence of periodic orbits for the three-body problem in celestial mechanics. The idea is that once you have a periodic orbit of some complicated system, you can start analyzing the system near the periodic orbit and this gives you an initial handle on the description of the system. One question therefore is whether dynamical systems tend to have periodic orbits. One such case is for the three-sphere. The Seifert conjecture (1950), is that all vector fields on the three-sphere have at least one closed orbit. ...

The paper is an important contribution to the theory of dynamical systems, and it solves in a simple but elegant way the long-standing Seifert conjecture.
Having written papers with her husband, Kuperberg went on to write papers with her son Greg who had also become a mathematician. In 1996 they published Generalized counterexamples to the Seifert conjecture which improved on Kuperberg's results from two years earlier. They gave examples in this later paper which are real-analytic rather than only smooth. They also solve the Seifert conjecture by giving counter-examples in dimension three or more.

Kuperberg has already received several awards for her outstanding work and it is certain that she will receive further awards. Perhaps the most prestigious was in 1995 when she was awarded the Alfred Jurzykowski Award by the Kosciuszko Foundation. In the following year she received a Research Excellence Award from the College of Sciences and Mathematics of Auburn University. Also in 1996 she was elected to the Council of the American Mathematical Society having served on several Committees of that Society.

References (show)

1. B Cipra, (Vector) field of dreams, What's happening in the Mathematical Sciences 2 (1994), 47-51.
2. I N Stewart, Hairy balls in higher dimensions, New Scientist (13 November 1993), 18.