Emmy Noether Lectures

1993 Lecturer: Linda G. Keen

Hyperbolic Geometry and Spaces of Riemann Surfaces

Linda Goldway Keen was born in New York City. She attended the Bronx High School of Science, where she was first taken with the elegance of mathematics in her geometry class. After receiving her BS degree from the City College of New York, she studied at the Courant Institute of Mathematical Sciences and received her PhD in 1964. She wrote her thesis on Riemann surfaces under the direction of Lipman Bers.
After a year at the Institute for Advanced Study, Keen took a position at Hunter College and at the Graduate Center of the City University of New York. When, in 1968, the Bronx campus of Hunter became the independent Lehman College, Keen moved to Lehman and has been there ever since. She was made full professor in 1974. She has held visiting professorships at the University of California at Berkeley, Columbia University, Boston University, Princeton University, and the Massachusetts Institute of Technology, as well as at various mathematical institutes in Europe and South America.
Keen served as President of the Association for Women in Mathematics during 1985-1986 and as Vice-President of the American Mathematical Society during 1992-1995. In 1975, she presented an AMS Invited Address at the Joint Mathematics Meetings in Washington, D.C., and in 1989 she presented an MAA Joint Invited Address at the Joint Summer Meetings in Boulder, Colorado. She is an associate editor for the Journal of Geometric Analysis and a coordinating editor for the Proceedings of the AMS.
In her Noether Lecture, Keen focused on the interplay between the analytic and geometric aspects of classifying Riemann surfaces. She originally tackled this problem in her thesis and subsequent early work. In the early 1960s, Bers and Ahlfors showed that the space of conformal structures on a given Riemann surface can be modeled on a Banach space with a real analytic structure. Keen defined the set of parameters for this space in terms of the hyperbolic structure of a given surface determined by the conformal structure. In the mid-1980s, she returned to this problem, this time in collaboration with Caroline Series. By this time, Bers had proved that the space of conformal structures on Riemann surfaces admits a complex analytic structure, and Maskit had defined an embedding of that space into complex n-dimensional space for appropriate n. Using powerful techniques developed by Thurston that involve hyperbolic three-manifolds, Keen and Series gave a geometric interpretation to Maskit’s parameters.
Keen has also collaborated with Paul Blanchard, Robert Devaney, and Lisa Goldberg in the area of dynamical systems. She finds working with other mathematicians more exciting and less frustrating than working on her own. She says, “I am basically a social person and enjoy people.”
Keen’s father was an English teacher and she says, “not only was mathematics fascinating, but it seemed as far away from English as I could get.” Her father, though, was always encouraging. “I feel very lucky. First my father, then my thesis advisor, and finally my husband and children have been extremely supportive.”